22. Electromagnetic Casimir Energies as Path Integrals¶
PDF pages 1005–1106
22.1.1 Planar Interface: TE polarization¶
Chapter 22 Electromagnetic Casimir Energies as Path Integrals 22.1 Scalar Representation of Electromagnetism The main deficiency in the worldline method in the previous chapter for computing Casimir and Casimir– Polder energies is that it applies to an artificial field—a massless, scalar field coupled to a background ‘‘potential’’ is not the same thing as the electromagnetic field. The next step in improving the method is to develop worldline path integrals for scalar representations of elecromagnetism. In propagation problems, scalar representations often make for good approximations to the vector propagation in the paraxial ap- proximation, when the polarization does not vary much as the field propagates (as in the propagation of a well-collimated laser beam). In Casimir-related calculations, we might not expect the scalar approximation to hold, as fields propagating in all directions are involved. However, there are some situations in which a scalar treatment is effective: namely when the geometry of space has sufficient symmetry that the two possible polarizations decouple, in which case the two polarizations can act as independent scalar fields. A trivial example is in free space. Less trivial examples include the fields at a planar, dielectric interface, which we treat in more detail below, and a spherical interface (if we take the center of the sphere to define a vector multipole basis). 22.1.1 Planar Interface: TE polarization Consider the modes associated with a planar interface between vacuum and a (nondispersive) magnetodi- electric material (Section 8.9.1), or a more complicated interface with planar symmetry. The case of TE polarization is diagrammed below. The key observation here is that while the magnetic-field direction varies between the incident, reflected, and transmitted components, the electric-field direction is the same among the components.
Chapter 22. Electromagnetic Casimir Energies as Path Integrals e, m z = 0 y x z qi qr qt E0i (+) H0i (+) E0r (+) H0r (+) E0t (+) H0t (+) In particular, we may write the electric field as E = Eyˆy. (22.1) We will then proceed to work with the classical field, where the spatial profile of the classical field carries over to the quantized case. The electromagnetic Hamiltonian (in noncanonical coordinates) is the total field energy (as we will justify in Section 22.2.3) HEM = 1 Z d3r h
i = 1 Z d3r h ϵE2 + µH2i , (22.2) where we are assuming a linear magnetodielectric medium with possibly inhomogeneous permittivity ϵ(r) and permeability µ(r). We will continue by treating the field as if there is a single mode of wave vector k and frequency \omega = ck. In this case we can identify \nabla ≡ik,
(22.3) In view of the normal-mode decomposition of the electromagnetic field (Section 8.4), there is an implied sum over all such modes, in which case these identifications are justified in the general case. Then the Maxwell equation
(22.4) becomes B = 1
(22.5) and so H2 = 1 µ2 B2 =
k2 µ2\omega2 E 2 y = −
(22.6) where we have used the fact that k and E are orthogonal. Then we can write the Hamiltonian (22.2) in terms of only one component Ey of the field as HTE = 1 Z d3r ϵE 2 y −
. (22.7) We would then like to define a scalar potential that satisfies
(22.8) so that up to an arbitrary constant (corresponding to a choice of ‘‘zero’’ time), we should define
Z t dt′ Ey(r, t′). (22.9) (TE scalar potential)
22.1.2 Planar Interface: TM polarization¶
22.1 Scalar Representation of Electromagnetism In this case, the Hamiltonian finally becomes HTE = 1 Z d3r
. (22.10) (scalar TE Hamiltonian) This is equivalent to the Hamiltonian (22.2) in the sense of producing the same decomposition in normal modes, restricted to TE-polarized modes. Note that while we have effectively changed a curl to a gradient in the magnetic-field term, but the relevant boundary condition is still the same. In the vector case, the component of H parallel to the interface is continuous through the interface. In the scalar case, the normal derivative \partial z\phi is continuous across the interface, which is the same requirement as continuity the normal component of \nabla \phi. These are equivalent constraints on the magnetic-field vectors in each case. Note also that in restricting the electric-field representation to Ey, we have effectively restricted our analysis to a particular plane of incidence. However, our final Hamiltonian (22.10) does not depend on this choice, and thus functions as a sum over all planes of incidence.
Then the Hamilton equation ˙\pi\phi = −\deltaHTE/\delta\phi yields ϵ\partial 2
µ\nabla \phi = 0 (22.11) (TE scalar wave equation) as the appropriate wave equation for \phi. 22.1.2 Planar Interface: TM polarization For TM polarization, we have a similar situation, but now H acts as the scalar field, while E has a vector character. e, m z = 0 y x z qi qr qt E0i (+) H0i (+) H0r (+) E0r (+) E0t (+) H0t (+) Again, we will emphasize this by writing H = Hyˆy. (22.12) Then the Maxwell equation
(22.13) becomes E = −1
(22.14) and so E2 =
k2 ϵ2\omega2 H 2 y = −
(22.15)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals where we have used the fact that k and E are orthogonal. Then the Hamiltonian (22.2) becomes HTM = 1 Z d3r −1
y . (22.16) Then given the condition
(22.17) we define the alternate scalar potential
Z t dt′ Hy(r, t′), (22.18) (scalar TM field) so that the Hamiltonian becomes HTM = 1 Z d3r
ϵ (\nabla ϕ)2 . (22.19) (scalar TM Hamiltonian) This is again equivalent to the Hamiltonian (22.2) in the sense of producing the same mode decomposition, restricted to TM modes. Notice that this scalar Hamiltonian has the same form as the TE Hamiltonian (22.10), except that ϵ and µ are switched, while the scalar field now represents H instead of E. This symmetry is also apparent in the two Maxwell equations we used, if we write them in the form
(22.20) Here we see that under the combined transformation E −\rightarrow H, H −\rightarrow −E, ϵ −\rightarrow µ, and µ −\rightarrow ϵ, these two equations transform into each other. In analogy with the TE case, the (noncanonical) TM Hamiltonian (22.19) has the scalar momentum field \piϕ = −µ\partial tϕ conjugate to ϕ. Then the Hamilton equation ˙\piϕ = −\deltaHTM/\deltaϕ yields µ\partial 2
ϵ \nabla ϕ = 0 (22.21) (TM scalar wave equation) as the appropriate wave equation for ϕ. 22.2 Interaction Energy In computing Casimir–Polder potentials, we are considering the interactions of point dielectric particles with other material bodies. And generally for Casimir potentials, we are considering the variation in the electromagnetic field energy as a material configuration varies. There are some subtleties here, and so we will expend a fair amount of effort in carefully considering electromagnetic energies. For example: the electromagnetic-field energy in the presence of a magnetodielectric material can be written HEM = 1 Z d3r ϵE2 + µH2 . (22.22) If we consider an atom to be a localized, weak perturbation (\deltaϵ, \deltaµ) on the medium, a naïve approach would be to make the replacements ϵ −\rightarrow ϵ + \deltaϵ and µ −\rightarrow µ + \deltaµ in the Hamiltonian here, in which case we would obtain interaction-potential terms of the form \deltaϵE2 + \deltaµH2. If we interpret the fields as their unperturbed counterparts, then the first one has the wrong sign for an electric-dipole interaction, while the second one is (accidentally) correct. A more natural choice for the magnetic-field term, as we will see, is to write the energy as B2/µ, and if we expanded this, we would again obtain the wrong sign. So let’s be careful and sort all this out.
22.2.1 Maxwell Equations in Magnetodielectric Materials¶
22.2 Interaction Energy 22.2.1 Maxwell Equations in Magnetodielectric Materials We will begin with the fundamentals: a review the construction of the Maxwell equations in magnetodielectric media. Although we are proceeding classically, the same considerations will apply to the quantum-mechanical case. We will start with the vacuum Maxwell equations for the electric and magnetic fields E and B, respectively, including sources,
ϵ0
(22.23) (Maxwell’s equations with sources) where the charge density \rho and the current density j satisfy the continuity equation
(22.24) (continuity equation) Now we characterize the response of media via the polarization field P (electric-dipole-moment density) and the magnetization field M (magnetic-dipole-moment density). We can then introduce the fields D and H via the constitutive relations
H = 1 µ0 B −M. (22.25) (constitutive relations) We are already at the crucial point. The fields D and H are introduced for convenience, since they will simplify the representation of the medium response in Maxwell’s equations. The original fields E and B are the fundamental fields.1 We can then introduce the electric susceptibility \chi and the magnetic susceptibility \chim for a medium response,
(22.26) (linear medium response) where the susceptibilities are constant for a linear medium, but can also represent nonlinear-medium re- sponse. Note the asymmetry of the prevailing convention here:2 The polarization is induced by the electric field E, but by convention the magnetization is ‘‘induced’’ by the introduced field H—rather than what
Eqs. (22.25) become
(22.27) and thus defining permittivity ϵ and permeability µ for the medium by
(22.28) (permittivity, permeability) the constitutive relations become D = ϵE B = µH. (22.29) (linear constitutive relations) 1See John David Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), Section 5.8, especially p. 193. 2See David J. Griffiths, Introduction to Electrodynamics, 4th ed. (Prentice Hall, 2013), Section 6.4.1, p. 284.
Chapter 22. Electromagnetic Casimir Energies as Path Integrals Again, notice the asymmetry of the convention: it appears that the electric and magnetic cases are parallel by comparing ϵ and µ, and then by E vs. H and D vs. B. This is, in fact, basically the symmetry that we mentioned before for the Maxwell equations. in Section 22.1.2. However, keep in mind that this is a formal symmetry, induced by the definitions in the constitutive relations; the fundamental similarity is between E vs. B. Now let us apply the constitutive relations (22.25) to the Maxwell equations (22.23), which are still valid in the presence of magnetodielectric media. We will keep the polarization and magnetization fields explicit this time, rather than appeal to ϵ and µ as we did above to highlight an apparent symmetry in the fields. We will start with the first Maxwell equation,
ϵ0 , (22.30) where by the total charge label we are emphasizing that the charge density includes any migration of charges due to the polarization response of the medium. Using the first of Eqs. (22.26), we have
(22.31) We can interpret the last term in terms of a bound charge density
(22.32) (bound charge density) associated with the charge migration due to the dielectric polarization. Then defining the free charge density as the part of the total charge not associated with the medium response,
(22.33) (free charge density) we have the new Maxwell equation
(22.34) (first Maxwell equation) in which the medium response has shifted from the charge density to the D field. Similarly, we can take the fourth Maxwell equation in (22.23)
(22.35) where again we are using the total current label to emphasize that any medium response here is contained in the source term, which is now the current density. Using Eqs. (22.26) to eliminate both B and E, we have
(22.36) We can again interpret the last two terms as current densities due to the response of media to the fields. The first is a current density associated with charges moving due to a changing polarization,
(22.37) (polarization current density) Note that this current density is due to motion of bound charges, and as it should, it obeys a continuity equation:
(22.38) The last term in Eq. (22.36) represents a bound current density due to the magnetization of the medium:
(22.39) (magnetization current density)
22.2.2 Interaction Energies for Magnetodielectric Materials: Static Fields¶
22.2 Interaction Energy Once again, defining the free current density as the current density not associated with the medium re- sponse, jfree := jtotal −jP −jM, (22.40) (free current density) Eq. (22.36) becomes
(22.41) (fourth Maxwell equation) Now using Eqs. (22.34) and (22.41) in place of the corresponding equations in (22.23), we have the usual Maxwell equations for media:
(22.42) (Maxwell’s equations in media) Here, we have dropped the ‘‘free’’ labels, but it is important to remember that the explicit sources here no longer include sources associated with medium response. At the risk of belaboring the point, E and B are the original fields, and are the only fields appearing in the two homogeneous Maxwell equations. The two introduced fields D and H serve to reorganize the sources, so we only have to explicitly consider free sources. Note that the symmetry E \leftarrow \rightarrow −H simultaneously with B \leftarrow \rightarrow D (and ϵ \leftarrow \rightarrow µ, when written explicitly) of the Maxwell equations is apparent here, but only in the absence of sources. 22.2.2 Interaction Energies for Magnetodielectric Materials: Static Fields Having developed the Maxwell equations, we will now apply these in dealing with the energies form materials in static fields, which will be sufficient for many purposes (i.e., as long as we are ignoring dispersion), and which will serve as relatively simple studies for developing our intuition of electromagnetic energies and materials. 22.2.2.1 Static-Electric-Field Energies We begin with the free-space electromagnetic Hamiltonian [Eq. (8.34)] HEM = 1 Z d3r ϵ0E2 + 1 µ0 B2 (22.43) in terms of the fields, which we will take to be the total field energy. Our goal will be to modify this to encompass the field energy in the presence of magnetodielectric media. First, we will focus on the electric-field energy, HE = ϵ0 Z d3r E2, (22.44) and in going to the static, source-free case, since \nabla \times E = 0, we can introduce the scalar potential \phi such that
(22.45) Then the energy becomes HE = ϵ0 Z
Z
(22.46) after integration by parts. Then using the first Maxwell equation (22.23) in the form of the Poisson equation \nabla 2\phi = −\rho/ϵ0, the energy becomes HE = 1 Z
(22.47) (energy of static electric field)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals in terms of the potential and charge density. It is tempting at this point to try translating this result to a dielectric by separating the free and
this does not give the correct result, as it only computes the energy associated with assembling the total (free and bound) charge, but not the energy associated with polarizing the medium. Thus, we instead begin by considering the effect \deltaHE on the energy due to a small change \delta\rho in the charge density. Starting with Eq. (22.44), the small variation \deltaE in the electric field due to the change in the charge density
Z
(22.48) Then using Eq. (22.45) and integrating by parts,
Z
Z
(22.49) and once again using the Poisson equation, we find
Z
(22.50) Note that this differs from what we would have gotten from varying \rho in Eq. (22.47) by a factor of two, because the change in charge modifies both \rho and \phi. The interpretation here is that by varying \rho against a fixed field \phi, we are counting the energy due to bringing in charge elements from infinity to assemble \rho(r). In bringing in each of the charge elements, we must do work against the electromagnetic forces against the already-assembled charges, and the total work is the energy of the final configuration. Now using \nabla \cdot D = \rho, we have
Z
Z
Z
(22.51) Then if we assume a linear dielectric, D = ϵE, (22.52) with ϵ independent of E (but may depend on r or even be generalized to a tensor), then
Z
(22.53) and upon integration of the field from 0 to E, we find the electric-field energy in the presence of a dielectric:3 HE = 1 Z d3r ϵE2 = 1 Z d3r E \cdot D. (static-field energy for linear dielectric) (22.54) The important point here is that this result depends on the assumption of a linear dielectric; more generally, the relation can be far more complicated. 22.2.2.2 Static-Magnetic-Field Energies The magnetic case proceeds in analogy to the electric case. We begin with the magnetic part of the electro- magnetic Hamiltonian [Eq. (8.34)], HM = 1 Z d3r 1 µ0 B2. (22.55) 3John David Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), Section 4.7, especially p. 165; Julius Adams Stratton, Electromagnetic Theory (McGraw-Hill, 1941), Sections 2.7-2.8, pp. 104-111.
22.2 Interaction Energy Since \nabla \cdot B = 0, we can introduce the vector potential A such that
(22.56) with the further condition that \nabla \cdot A = 0, since the longitudinal part of A can’t affect B anyway. Then HM = 1 Z d3r 1 µ0
Z d3r 1 µ0
(22.57) where to integrate by parts we used the identity
(22.58) Using the last Maxwell equation (22.42) in the static limit and with no magnetic media,
(22.59) we then have the source-coupling form of the energy HM = 1 Z d3r A \cdot j (22.60) (energy of static magnetic field) in terms of the vector potential and the source current density. Proceeding along the same lines as before, the change in the energy due to a small change in the magnetic field is from Eq. (22.55)
µ0 Z
= 1 µ0 Z
= 1 µ0 Z
= Z
(22.61) Note that we have ended up with a varying field against a fixed current. The interpretation here is as follows: in order to have a static magnetic field in the presence of a source current, the magnetic field must be brought up from zero. In doing so, the induced electromotive forces attempt to modify the flow of charge in the source currents. In order to maintain fixed currents, work must be done against the induced electromotive forces. The total work done is then the total energy of the configuration. Note that we can alternately hold A fixed while varying j. However, this corresponds to a fixed magnetic field due to the sources, assembling the current density by bringing in current-density elements from infinity. This is not a natural interpretation in the absence of magnetic charge.
Z
Z
Z
(22.62)
magnetic material (e.g., not a ferromagnetic material), we have B = µH, (22.63) with µ independent of H, so then
µ Z
(22.64)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals and upon integration of the field from 0 to B, we find the magnetic-field energy in the presence of a magnetic material:4 HM = 1 Z d3r 1 µB2 = 1 Z d3r B \cdot H. (static-field energy for linear magnetic material) (22.65) To reiterate the main point, this result depends on the assumption of a linear magnetic material (in particular, without hysteresis, in which case the energy could depend on the entire history of the magnetic field). 22.2.2.3 Variation of Dielectric Materials Now a more useful question is: suppose we change the configuration of a dielectric medium by modifying ϵ(r). What is the difference in energy between the configurations?5 In answering this question, we should be careful to specify, what happens to the sources? In this section we will explicitly assume fixed sources while modifying the dielectric. And we are still working only with static fields and linear dielectrics. Suppose we have an energy associated with a ‘‘reference’’ dielectric configuration ϵ1(r), H(1) E = 1 Z d3r E1 \cdot D1, (22.66) and a second energy associated with a modified dielectric ϵ2(r), H(2) E = 1 Z d3r E2 \cdot D2, (22.67) again with the same sources in both cases. We then define the ‘‘interaction potential’’ as the difference of these energies,
E −H(1) E , (22.68) and our goal is to find expressions for V . The concept of an interaction potential is meaningful here in the sense of thinking of the second configuration as introducing a new object into the ‘‘background’’ defined by the reference configuration. Then V measures the energy associated with introducing the new object. First, we can write out the potential as VE = 1 Z d3r
= 1 Z d3r
+ 1 Z d3r E2 + E1 \cdot D2 −D1 . (22.69)
(22.70) and so the last integral is proportional to Z
D2 −D1 = − Z
D2 −D1 = − Z d3r \phi
= 0, (22.71) because we have assumed fixed sources. Thus, the interaction energy is VE = 1 Z d3r
= −1 Z d3r (ϵ2 −ϵ1)E2 \cdot E1. (energy difference between dielectric configurations) (22.72) 4John David Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), Section 5.16, p. 212; Julius Adams Stratton, Elec- tromagnetic Theory (McGraw-Hill, 1941), Section 2.10, p. 112. 5John David Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), Section 4.7, especially p. 167; Julius Adams Stratton, Electromagnetic Theory (McGraw-Hill, 1941), Section 2.10, p. 112; L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960), Section 11, p. 52.
22.2 Interaction Energy Note that the integrals here in principle extend over all space, but for a bounded object, the integration need only extend over the object. The second form highlights the following observation: increasing the dielectric, and in particular introducing a dielectric object into a vacuum background always decreases the electric-field energy, provided the sources are fixed.
susceptibility of the object, and using P = ϵ0\chiE2, the interaction energy becomes VE = −1 Z d3r P(r) \cdot E1(r). (22.73) This is the usual interaction with a polarizable medium, with a factor of 1/2 since the polarization is induced by the field. In particular, for a point dipole d at r0,
(22.74) so that the interaction becomes VE = −1 2d \cdot E1(r0). (22.75) This is the usual expression for the interaction energy of an induced point dipole in a background field E1. 22.2.2.4 Variation of Magnetic Materials The analogous problem of computing the change in the (static) magnetic-field energy when changing the (linear) magnetic material proceeds along similar lines to the electric case.6 Again, we will assume fixed sources—this time, meaning a fixed current density j(r). As before, suppose we have an energy associated with a reference magnetic-material configuration µ1(r), H(1) M = 1 Z d3r B1 \cdot H1, (22.76) and a second energy associated with a modified configuration µ2(r), H(2) M = 1 Z d3r B2 \cdot H2, (22.77) with the same source in both cases. The magnetic interaction potential is now the difference of these energies,
M −H(1) M . (22.78) Writing out the potential explicitly, VM = 1 Z d3r
= 1 Z d3r
+ 1 Z d3r B2 + B1 \cdot H2 −H1 . (22.79)
introduce a scalar potential ϕ such that
(22.80) and so the last integral is proportional to Z
B2 + B1 = − Z
B2 + B1 = 0, (22.81) 6John David Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), Section 5.16, especially p. 214; Julius Adams Stratton, Electromagnetic Theory (McGraw-Hill, 1941), Section 2.17, p. 126.
Chapter 22. Electromagnetic Casimir Energies as Path Integrals because \nabla \cdot B = 0. Thus, the interaction energy is VM = 1 Z d3r
= 1 Z d3r (µ2 −µ1)H2 \cdot H1 = 1 Z d3r 1 µ1 −1 µ2 B2 \cdot B1. (energy difference between magnetic-material configurations) (22.82) Again, the integrals here in principle extend over all space, but for a bounded object introduced in the case of µ2, the integration need only extend over the object. The last two forms highlight the following observation: increasing the magnetic permeability, and in particular introducing a paramagnetic object (µ2 > µ0) into a vacuum background material always increases the magnetic energy, provided the sources are fixed. This is opposite to the dielectric case. Note that everything up to the first of the above three expressions for VM is the same as the dielectric case with E −\rightarrow B and D −\rightarrow H. However, it is the asymmetry of the D = ϵE and B = µH that causes the energy change to move in the opposite direction.
as VM = 1 Z d3r µ2 µ0 −1 H2 \cdot B1. (22.83)
VM = 1 Z d3r M(r) \cdot B1(r). (22.84) This is the usual interaction energy of a magnetized body, but with the opposite sign compared to what we would normally expect, and with a factor of 1/2 that reflects the induced nature of the magnetization. As
energy of (1/2)m \cdot B1(r0), which again has the opposite sign to the usual dipole energy −m \cdot B. 22.2.2.5 Interpretation of Minus Signs The relative minus sign in the electric vs. magnetic interaction energies that we have developed in the last two sections highlights an important difference between the two cases, and also the importance of being precise about what ‘‘the energy’’ means. To illustrate this, let’s return to the dielectric case, but modify the conditions of the sources: rather than fixing the sources, we will instead consider fixing the potentials on some bounding surfaces,7 e.g., via conducting electrodes held at fixed voltage by a power supply. In this case, charge is free to migrate in order to maintain the fixed potentials. To compute the energy here, we will begin by considering a small variation in the energy (22.47) due to small changes in the potential and current:
Z d3r
. (22.85) Recall that we previously derived a similar expression, Eq. (22.50)
Z
(22.86) in the case where the dielectric was fixed. This implies that if the dielectric is held fixed, then the two terms in (22.85) are equal (but not in general equal if the dielectric is changed). To proceed, we will break the change in the dielectric at fixed potential into two steps: first, discon- nect the electrodes from the power supply and change the dielectric at fixed charge; second, reconnect the 7See John David Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), Section 4.7, especially pp. 168-9.
22.2 Interaction Energy electrodes, and allow charge to flow and restore the original potentials (at least on the electrodes). In step 1, \delta\rho = 0, so the energy change is \deltaH(1) E = 1 Z
(22.87) where \delta\phi1 is the potential change induced by the small dielectric change. We have already computed the energy change VE for a change in dielectric at fixed charge in Eqs. (22.72). In the second step, the batteries must undo the effect of the first-step change \delta\phi1, or written otherwise, \delta\phi2 = −\delta\phi1. Thus, the energy change in this step is \deltaH(2) E = 1 Z d3r
= − Z
E , (22.88) where we used the observation from above that for a fixed dielectric, the two terms have the same contribu- tion. Thus, the energy change from the second step is opposite to but double the change in the first, so the net energy change is −VE, or the same as the fixed-charge potential (22.72) except for a minus sign, so that in this case, introducing a dielectric object increases the total energy. As an example, these observations predict that in introducing a dielectric slap between the plates of a charged capacitor, the dielectric should be sucked into the capacitor if the electrodes are disconnected (maintained at fixed charge), and repelled from the capacitor if the plates are connected to a power supply. In the latter case, the power supply is also doing work on the dielectric, which yields the change in sign.8 With the above result, we can now more intuitively interpret the difference in minus sign due to introducing electric vs. magnetic objects, Eqs. (22.72) vs. (22.82).9 As we saw from the energy variation in Eqs. (22.61),
Z
(22.89) the magnetic-field energy (22.65) includes work done by moving charges (in the source current) against electromotive forces as the magnetic field is brought up from zero. This is a closer analogy to the dielectric energy at fixed voltage, which is dominated by the work done by moving charges against the potential. To see this more explicitly, suppose we calculate the work of the current against the electromotive forces. Using
Z
Z d3r E \cdot j (22.90) if the current is fixed. Integrating this over the time over which the potential is ramped to the steady-state value, the change in energy associated with work done against the electromotive forces is ∆HM = Z d3r A \cdot j, (22.91) where the A here is the steady-state potential. Note that this is twice the value of the magnetic-field energy (22.60), which led to the magnetic-material energy (22.65). Thus, if we subtract away this contribution to the energy, we are left with the same interaction energy, but with a negative sign, which is in closer analogy to the dielectric energy with fixed charges. Thus, the usual magnetic energy density of the form −M \cdot B (or equivalently, the dipole energy −m\cdotB) counts just the energy associated with (permanent) magnetic dipoles being placed in a magnetic field, after the dipoles have been created and made permanent elsewhere. The energy we have derived here, (1/2)M \cdot B, has the opposite sign (and a factor of 1/2, due to the linearity of the magnetic medium), because it additionally counts the energy associated with creating the magnetic dipoles. 8See David J. Griffiths, Introduction to Electrodynamics, 4th ed. (Prentice Hall, 2013), Section 4.4.4, pp. 202-4. 9The interpretations here are straight out of John David Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), Section 5.16, especially pp. 214-5.
Chapter 22. Electromagnetic Casimir Energies as Path Integrals 22.2.2.6 Small Material Perturbations One awkward feature of the interaction energies Eqs. (22.72) and (22.82) for variations in the electric and magnetic materials is that the expressions refer to products of fields in the presence of each of two different configurations. However, there is a useful limit where this complication goes away: that of a small change in the material. Returning to the dielectric interaction energy (22.72), suppose we set the background dielectric notation to ϵ1 = ϵ, and the perturbed dielectric ϵ2 = ϵ + \deltaϵ, with \deltaϵ = ϵ0\chiobj, where \chiobj represents the susceptibility of a weakly polarizable object. Then the interaction potential becomes VE = −1 Z
(22.92) To lowest order in \deltaϵ, we can write this solely in terms of the unperturbed field: VE = −1 Z
(22.93) If we take the small perturbation to be an atom modeled by a point dipole at r0 with (static) polarizability
(22.94) then the energy becomes VE = −1 2\alpha0E2(r0; ϵ), (22.95) where ϵ(r0) is the background dielectric (e.g., vacuum or a fluid). This is the usual interaction energy for a linear, induced dipole. It may seem objectionable that we expanded to lowest order in \deltaϵ, and then set it to a delta function, which diverges as a ‘‘function.’’ However, it is important to remember that the intent here is to examine the linear response of the atom by expanding to lowest order in the atomic response \alpha0. The extent of the atom, in the point-dipole regime, is smaller than any length scale in the field. Really, we should keep the atomic polarization as a localized but finite distribution, tempered by the small quantity \alpha0, and then at the end of the day, take the limit as the atom becomes arbitrarily small. Introducing the delta function is a shortcut for this well-defined procedure. Note also that in computing the atomic response, we have looked at the effect of a localized perturbation to ϵ(r) at r0, to linear order in the perturbation, dropped the perturbation, and replaced it by \alpha0. In the language of functional differentiation, we have thus shown that
\deltaHE \deltaϵ (22.96) is the interaction energy for a point-dipole atom at r0, with static fields and fixed sources. The same considerations apply to the magnetic-field case. Returning to Eq. (22.82), to lowest order in a perturbation \deltaµ on a background µ, VM = 1 Z
Z d3r \deltaµ µ2 B2(µ). (22.97) Again, we take the small perturbation to be an atom modeled by a point (magnetic) dipole at r0 with static,
\beta0µ0\delta(r −r0)H, we have
0 \delta(r −r0). (22.98) Note that this differs slightly in form from the electric case (22.94),
(22.99) 10The convention for the magnetizability induced by H is used, e.g., by John David Jackson, Classical Electrodynamics, 3rd
\beta0 dimensions of µ−1 times a volume, in analogy with \alpha0 having dimensions of ϵ0 times a volume.
22.2 Interaction Energy¶
becomes VM = µ 2
2 \beta0H2(r0; µ), (22.100)
the usual interaction for an induced dipole with a magnetic field, −m \cdot B/2, except without the minus sign that we would normally expect due to the reasons we covered in the previous section. Thus, we have shown that for the energy of an atom in a static magnetic field,
\deltaHM \deltaµ (22.101) in terms of the functional derivative. 22.2.2.7 Application to Casimir–Polder Potentials Closely related to the above electrostatic energy shifts is the Casimir–Polder effect, which is of course what we’re after in all this background material. Note that the Casimir–Polder effect is not a static-field phenomenon. However, recall that if the atom is far away from the surface compared to any of its transition wavelengths, the near-dc wavelengths are the only important ones (i.e., we can ignore dispersion), and we can use the dc properties of the atom. Also, the (effectively dc) sources are quantum-vacuum fluctuations in the magnetodielectric material, and will act essentially as fixed sources. In considering the Casimir–Polder force, we will consider some magnetodielectric body (or configuration of multiple bodies), described by ϵ(r) and µ(r). The Casimir–Polder energy is defined as the change in the energy when we introduce an atom (precisely, by bringing it in from infinity), which we can model as a point particle with (static) polarizability and magnetizability \alpha0 and \beta0, respectively. Thus, beginning with the electromagnetic Hamiltonian HEM = 1 Z d3r ϵE2 + 1 µB2 , (22.102) the idea is to consider small perturbations
(22.103) and consider the first-order expansion of the difference in vacuum expectation values (in the quantum case) VCP = D
E − D HEM(ϵ, µ) E = Z d3r′
\deltaϵ
\deltaµ \deltaµ(r′) , (22.104) in terms of the (partial) functional derivatives of the energy. If we take the perturbations to be due to an atom located at r, where the dimensions of that atom are small compared to other length scales in the calculation, we can formally replace the (finite but localized) perturbations by delta functions:
(22.105) This procedure gives a result for the energy that is lowest-order in terms of the atom’s size. Then the Casimir–Polder energy becomes
\deltaϵ
\deltaµ , (Casimir–Polder potential, no dispersion) (22.106) where the functional derivatives are to be evaluated at the atomic position (r). 11The expressions (22.98) and (22.99) are consistent with, for example, S. Y. Buhmann, H. Safari, Ho Trung Dung, and D.-G. Welsch, ‘‘Two-atom van der Waals interaction between polarizable/magnetizable atoms near magnetoelectric bodies,’’ Optics and Spectroscopy 103, 374 (2007) (doi: 10.1134/S0030400X07090068), Eqs. (6) and (12).
Chapter 22. Electromagnetic Casimir Energies as Path Integrals However, recall that the Casimir effect is generally by considering atomic interactions with vacuum fields, to leading order via dipole interactions of the form −d\cdotE and −m\cdotB. Our static treatment in the past few sections seems to be consistent with the former but opposite to the latter, as we see in Eq. (22.100). Thus, it seems like we have a reversed magnetic-field contribution, and it is tempting to take the Casimir–Polder energy to be
\deltaϵ −\beta0µ 2
\deltaµ (wrong!) (22.107) So how do we reconcile the Casimir–Polder expression (22.106) with the static-field results? The answer is that the sign in the magnetic-field case is already built into the result (22.106). Specifically, this equation gives the energy change for introducing a particle without direct reference to the fields. Thus, the static results give us intuition as to what will happen to the field energy as we make small changes to the magnetodielectric material. Specifically, these results suggest that when introducing a purely dielectric vs. purely magnetic atom into the same background configuration, the sign of the energy change should be different. (The magnitude is not necessarily the same, as each of the two possible ‘‘atoms’’ couples to different fields.) We will confirm this intuition in Section 22.11, where we will consider all combinations of purely electric and magnetic particles and planar surfaces. The result: the potential is attractive if both are dielectric or both are magnetic; the potential is repelling whenever the two have opposite characters. 22.2.2.8 Functional Derivatives and Expectation Values: A Second Look The results in the previous section are somewhat counterintuitive at first glance. To see this, let’s set up a heuristic (but wrong!) calculation of the Casimir–Polder potential. Suppose for simplicity that we have an atom located outside a dielectric body. A neutral atom interacting with an electric field VE = −1 2\alpha0E2 (22.108) if we ignore dispersion and model the atom as an electric dipole. This comes from the energy −d \cdot E, with d = \alpha0E and the factor of 1/2 arises for a dipole induced by the field. For the magnetic-dipole case we can analogously write VM = −µ 2 2µ2 \beta0B2. (22.109) We have seen these energies in the previous section. Quantum-mechanically, this holds as well. Even though the field is in the vacuum state, the field fluctuations are captured by the field variance, and we simply interpret the squared fields as expectation values VCP = −1 2\alpha0
E2 −µ 2 2µ2 \beta0
B2 , (22.110) representing the mean vacuum-induced energy induced by the magnetodielectric body. Everything we have done is fine so far. What is now tempting is to look at the Hamiltonian (22.102), and functionally differentiate to obtain \deltaHEM \deltaϵ = E2 2 , \deltaHEM \deltaµ = −B2 2µ2 (wrong!). (22.111) Now we could solve these for the squared fields, substitute into Eq. (22.110), and arrive at
\deltaHEM \deltaϵ
\deltaHEM \deltaµ
(wrong!). (22.112) However, this is wrong, since there is an overall minus sign on the dielectric term compared to the correct answer (22.106). The problem, to reiterate, is that the fields themselves also depend on ϵ and µ, so the variation should also vary the fields. However, their variation is not obvious, and must be established
22.2.3 Hamiltonian Structure of Electromagnetism and Linear Materials¶
22.2 Interaction Energy from the more careful arguments above, which precisely account for the variation of the fields with the magnetodielectric configuration. To take this a bit further: recall Eq. (22.93): VE = −1 Z
(22.113) This is the interaction energy for a small change in the permittivity, to lowest order. The point is, this has the opposite sign, compared to naïvely differentiating an energy involving ϵE2 with respect to ϵ while holding E fixed. Evidently, this means that to lowest order, the field variation is
1 −\deltaϵ ϵ E(ϵ), (22.114) and this variation in ϵE2 conspires with the obvious variation of ϵ to yield the overall minus sign in Eq. (22.113). However, note that D stays constant under the same variation, to linear order:
1 −\deltaϵ ϵ
(22.115) Thus, we can make a naïve version of the argument work out, but only if we rewrite the electromagnetic energy (22.102) in terms of D and B, HEM = 1 Z d3r D2 ϵ + B2 µ , (22.116) and then differentiate while holding the fields here fixed. This procedure leads to something more like the previous result (22.106). 22.2.3 Hamiltonian Structure of Electromagnetism and Linear Materials Why have we been dealing with static fields when Casimir–Polder potentials arise fundamentally from vacuum-field fluctuations? Here we will show that the results we have derived for vacuum potentials are still valid if we relax the assumption of static fields, and in doing so we will review some of the symplectic structure of the electromagnetic fields with media. To see how all this comes out of the Hamiltonian structure of the field, we can write the Lagrangian for the electromagnetic field coupled to matter with (linear) permittivity ϵ and (linear) permeability µ as LEM = 1 Z d3r
, (22.117) where as usual the fields are related by
(22.118)
Lagrangian becomes
Z d3r ϵ(\partial tA)2 −1
(electromagnetic Lagrangian with linear material) (22.119) in terms of the vector potential, which plays the role of the generalized-coordinate field. We can check that this Lagrangian is sensible by writing out the Euler–Lagrange equation, \deltaL
\deltaL
(22.120)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals which implies \nabla \times 1
−ϵ\partial 2 t A = 0, (22.121) yielding the Maxwell equation
(22.122) assuming a time-independent (and hence nondispersive) permittivity. Now to derive the Hamiltonian, we will make the usual transformation. That is, the momentum field is
(22.123) in which case we obtain the perturbed Hamiltonian
Z
= 1 Z d3r
= 1 Z d3r 1 ϵ \Pi2 + 1
(22.124) in terms of canonical coordinate fields. In terms of the electromagnetic fields, we have HEM = 1 Z d3r
, (22.125) (electromagnetic Hamiltonian) which we can identify as the (time-invariant) total energy of the electromagnetic field. 22.2.3.1 Material Perturbations Then to find the effect on the electrodynamic energy due to a change in material properties ϵ and µ (or equivalently, introducing an extra magnetodielectric object), we proceed as in the static case. Referring to the electromagnetic energy as represented by the Hamiltonian (22.125), we should write down our first electromagnetic energy for the reference magnetodielectric-material configuration, characterized by ϵ1(r) and µ1(r), as H(1) EM = 1 Z d3r
, (22.126) and a second energy associated with a modified configuration characterized by ϵ2(r) and µ2(r), as H(2) EM = 1 Z d3r
, (22.127) Again, we wish to hold the sources fixed between the two cases. The (dynamical) electromagnetic interaction potential is now the difference of these energies,
EM −H(1) EM. (22.128) Writing out the potential explicitly, VEM = 1 Z d3r
−1 Z d3r
= 1 Z d3r
+ 1 Z d3r
. (22.129)
22.2 Interaction Energy Then separating out cross-terms as before, VEM = 1 Z d3r
+ 1 Z d3r
+ 1 Z d3r hE2 + E1 \cdot D2 −D1 + B2 + B1 \cdot H2 −H1 i . (22.130) The first term here has the form of the static-electric interaction energy (22.72), while the second term has the form of the static-magnetic interaction energy (22.82). The last integral contains two terms that we showed vanished separately, under the assumption of static fields. We can’t make the same assumption, but we still wish to show that the last integral vanishes, and we can do this as follows. Noting that we can write the electric sum field in terms of potentials in the most general way as
(22.131) the first term in the integral has the form IED = 1 Z d3r E2 + E1 \cdot D2 −D1 = −1 Z d3r \nabla \phi \cdot D2 −D1 −1 Z d3r \partial tA \cdot D2 −D1 . (22.132) The first integral vanishes after integrating by parts, using \nabla \cdot D = \rho, and assuming fixed sources. Thus, we are left with IED = −1 Z d3r \partial tA \cdot D2 −D1 . (22.133) The second part of the last integral in (22.130) has the form IBH = 1 Z d3r B2 + B1 \cdot H2 −H1 = 1 Z d3r
\cdot H2 −H1 = 1 Z
H2 −H1 = 1 Z
D2 −D1 , (22.134) where we wrote the magnetic sum field in terms of the same vector potential as the electric sum field,
\nabla \times H = \partial tD + j, where the current density drops out because we assume it to be the same in both configurations. Now the critical point is to assume that a decomposition into normal modes exists; this is true because we can still assume monochromatic solutions of frequency \omega (e.g., right in the Maxwell equations), which separates the time and space variables of the problem, just as in the vacuum case. In this case, we can identify \partial t ≡−i\omega, and
Z d3r A \cdot D2 −D1 = −IBH. (22.135) Thus, the last integral in Eq. (22.130) vanishes for each mode, and we are left with the energy VEM = 1 Z d3r
+ 1 Z d3r
, (electromagnetic interaction energy between material configurations) (22.136) which is the same as the electrostatic and magnetostatic results added together. Thus, the intuition from the static case carries over to full electrodynamics, concerning the opposite signs for a dielectric vs. magnetic particle interacting with the same magnetodielectric body.
22.3.1 Gaussian Field Integral¶
Chapter 22. Electromagnetic Casimir Energies as Path Integrals 22.3 Development of the Path Integrals As usual in the development of path integrals, we will need to start with the Lagrangian. In scalar electro- magnetism, we have two Lagrangians, one for each of the two (uncoupled) polarizations. For TE polarization, we have the Hamiltonian (22.10), which arises from the Lagrangian LTE = 1 Z d3r
. (22.137) (scalar TE Lagrangian) For the TM polarization, we have from the Hamiltonian (22.19) the Lagrangian LTM = 1 Z d3r µ(\partial tϕ)2 −1 ϵ (\nabla ϕ)2 . (22.138) (scalar TM Lagrangian) Note in this latter case that this is a Lagrangian that is related to the Hamiltonian solely in terms of the field ϕ. In terms of the original electromagnetic Hamiltonian, recall that the kinetic and potential terms swapped roles in switching to the field ϕ. 22.3.1 Gaussian Field Integral Now we will proceed as in the case of the scalar field coupled to a potential, and we will first do the TE field. The Wick-rotated partition function is Z = Z D\phi exp i ¯h Z
, (22.139) where the temperature ‘‘length’’ is ˜\beta = ¯hc\beta, which becomes ZTE = Z D\phi exp −1 2¯hc Z d˜\beta Z d3r
= Z D\phi exp −ϵ0c 2¯h Z
−ϵ(r) ϵ0 \partial 2
µ(r)\nabla
∝ s det −ϵr\partial 2
µr \nabla −1 (22.140) with the Lagrangian (22.137). Here we have introduced the relative permittivity and permeability
ϵ0 ,
µ0 , (22.141) (relative permittivity/permeability) and we have used ϵ0µ0c2 = 1. Then log ZTE = −1 2 log det −ϵr\partial 2
µr \nabla = −1 2Tr log −ϵr\partial 2
µr \nabla , (trace-log form of partition function, TE) (22.142) up to an additive constant that disappears in renormalization. The TM case is the same as the TE case, but with ϵr \leftarrow \rightarrow µr.
22.3.3 Rescaling the Green Operator: Coupling to the Potential¶
22.3 Development of the Path Integrals 22.3.2 Imaginary-Time Green Operator As an alternate derivation of Eq. (22.142), for the TE polarization, we can start with the wave equation (22.11), ϵ\partial 2
µ\nabla
(22.143) which defines a Green function via µ0 ϵ\partial 2
µ\nabla
(22.144) where we have introduced an (arbitrary) overall factor of µ0 to define the Green function. This in turn implies the Green operator GTE = ϵr c2 \partial 2
µr \nabla −1 . (22.145) The imaginary-time Green operator is
−ϵr\partial 2
µr \nabla −1 (imaginary-time Green operator) (22.146) The short-cut partition-function rule is log ZTE = 1 2Tr log ˜GTE, (Green-operator form of partition function) (22.147) which is consistent with the result (22.142). The TM scalar wave equation (22.21) is the same as the TE wave function but with ϵ \leftarrow \rightarrow µ, so (with the appropriate normalization) the expression for log ZTM follows in the same way, but with ϵr \leftarrow \rightarrow µr in the end.12 22.3.3 Rescaling the Green Operator: Coupling to the Potential Now consider again the TE Green operator (22.146), in particular noting the possibility of space dependence of ϵr(r) and µr(r). When we change this partition function into a worldline path integral in the following section, recall that the Green-operator derivatives will become kinetic-energy terms. The factor µr(r) that accompanies \nabla 2 is particularly problematic, because for the worldline particle, the permeability will act as a space-dependent mass, effectively curving space-time. This drastically complicates the path integral. The factor ϵr(r) also introduces a space-dependent mass for the time direction of the worldline particle. While this technically also induces a nontrivial space-time, it turns out that this will be much easier to deal with. To deal with this curvature in advance, we can work with rescaled Green operators. Since we will be computing energies from the log partition function (22.147), it is important to recall that only differences in energies are physically meaningful (or even finite, in Casimir calculation). This means that we can rescale ˜G by an arbitrary constant factor, and as long as we are computing energy differences between different configurations (or ratios of partition functions), the scaling factor will cancel out. Similarly, we can also rescale ˜G by an arbitrary function. In this case, the function splits off in the determinant, and the functional determinants cancel in the energy difference (partition-function ratio). What is somewhat less obvious is that we can rescale ˜G by functions of ϵr and µr. These are not necessarily the same between the two different configurations, and so this rescaling requires more careful interpretation. Suppose we are considering the Casimir effect between two bodies. A sensible comparison would contrast two different configurations of the same bodies, but translated or rotated slightly (or even possibly deformed). An infinitesimal change would be the first step in the calculation of the Casimir force, for example. As long as there is the same amount 12Similar expressions for scalar Green operators were used by Julian Schwinger, ‘‘Casimir energy for dielectrics,’’ Proceedings of the National Academy of Sciences 89, 4091 (1992) (doi: 10.1073/pnas.89.9.4091).
Chapter 22. Electromagnetic Casimir Energies as Path Integrals of stuff in the two configurations, the functional determinants of ϵr and µr are equivalent in both cases, because they are effectively infinite products of these functions at every position in space, but the order in the products is not important. In particular, we would like to define ‘‘flat-space,’’ imaginary-time Green operators by rescaling via ¯G−1
TE \sqrtµr ¯G−1
TM \sqrtϵr. (22.148) (flat-space Green operators) Writing these out explicitly, ¯G−1
µr
¯G−1
ϵr
(22.149) We still have functions of position on the gradient term, but that term is overall ‘‘neutral’’ with respect to position-dependent functions.13 Using the operator-commutation relation
(22.150) we can then write g\partial x
x −g′2 g2 + \partial x, g′ g = \partial 2 x −2g′2 g2 + g′′ g (22.151) Putting g = \sqrt h, so that g′ = h′/2 \sqrt
\sqrt
\sqrt h \partial x h\partial x \sqrt
x + h′′ 2h −3h′2 4h2 . (22.152)
x log h = h′′/h −h′2/h2, we can alternately write \sqrt h \partial x h\partial x \sqrt
x + 1 2\partial 2 x log h −1 \partial x log h
x log \sqrt h − \partial x log \sqrt h 2. (22.153) Thus, we may rewrite the Green operators (22.149) as ¯G−1
¯G−1
(22.154) (flat-space Green operators) where we have introduced the material-induced ‘‘potentials’’ (which have dimensions of inverse area)
i
i . (22.155) (matter-induced potentials) We will now proceed by using these Green operators, which have no space-dependent masses on the \nabla 2 terms (but replaced by effective potentials), in the trace-log expression (22.147). 13Similar rescalings in path integrals were noted by S. Pasquali, F. Nitti, and A. C. Maggs, ‘‘Numerical methods for fluctuation- driven interactions between dielectrics,’’ Physical Review E 77, 016705 (2008) (doi: 10.1103/PhysRevE.77.016705).
22.3.4 Worldline Form of the Path Integral¶
22.3 Development of the Path Integrals 22.3.4 Worldline Form of the Path Integral We can then formally change to an integral representation of the logarithm, log Z = 1 Z \infty dT T Tr exp −T ˜G , (22.156) with the understanding that this expression is divergent unless renormalized. Also, we are dropping the polarization label, since this procedure will work for either polarization, given the appropriate choice of Green operator (22.154). Defining the momentum operators
(22.157) the partition density with (22.146) becomes log Z = 1 Z \infty dT T Tr ( exp " −T
ϵrµr p 2 ˜\beta 2¯h2 + p2 2¯h2 + V !#) , (22.158) where V is the potential (22.155) appropriate to the polarization. Notice that we have inserted a factor of 1/2 in the inverse Green operator here. We can do this because we are free to rescale T in Eq. (22.156) by an arbitrary factor. The trace of the exponential factor here has the form of a partition function Zeff := Tr e−T Heff , (22.159) (effective partition function, scalar EM) where T plays the role of the temperature parameter \beta, and
p 2 ˜\beta 2¯h2 + p2 2¯h2 + V (r) (effective Hamiltonian, scalar EM) (22.160) is the effective particle Hamiltonian for the scalar electromagnetic field coupled to the medium. The total (log) partition function is then log Z = 1 Z \infty dT T Zeff, (log partition function, in terms of effective Z) (22.161) in terms of the effective partition function. Note that there is no ordering ambiguity in this effective Hamiltonian. 22.3.4.1 Recap: Normalization of the Particle Path Integral At this point, our task will be to write the effective partition function (22.159) as a Monte-Carlo-type path integral. To do this, we will recall the corresponding path integral for a standard particle in one dimension of
(22.160). Reviewing our results before in Section 20.3.1, the partition function for the standard particle is Z = Z ˜Dx \delta[x(¯h\beta) −x0] exp " −1 ¯h Z ¯h\beta d˜\beta m
# (22.162) where the path-integration measure is defined as ˜Dx := m
N/2 N Y j=0 dxj. (22.163)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals Taking the kinetic-energy part of the Lagrangian to define the probability measure, we can write the partition function as the ensemble average Z = Z dx0 **
" −1 ¯h Z ¯h\beta d˜\beta V (x)
, (22.164) where the average is taken with respect to paths of the form
r ¯h m W(˜\beta), (22.165) where W(t) is a standard Wiener path. The delta function selects only the paths (of zero measure) that return to x0 at ‘‘time’’ ¯h\beta, so it is convenient to switch to more appropriate paths. We will thus use the normalization
\delta[W(T)] F[W(t)]
= \sqrt 2\piT
F[BT (t)]
, (22.166) where BT (t) is a Brownian bridge [i.e., equivalent to a standard Wiener path, but conditioned on the endpoint
Z = r m
Z dx0 ** exp " −1 ¯h Z ¯h\beta d˜\beta V (x)
p
, (22.167) without the delta function but with Brownian ‘‘loops’’ BT (t) in imaginary time. 22.3.4.2 Normalization of the Worldline Path Integral Transforming the partition function (22.159) into the form of Eq. (22.162) is a fairly straightforward tran- scription, with ¯h\beta −\rightarrow T and setting m = ¯h2 and V −\rightarrow ¯hV in the exponential. We also lose the distinction between \beta and ˜\beta, which also absorbs the 1/¯h in the exponential. The result of all this is Zeff = Z
" − Z T d\tau
- V (x) !# , (22.168) and we have the integration measures
¯h
N/2 N Y j=0 dx ˜\betaj N Y j=1 h ϵr(xj)µr(xj) i−1/2 , ˜Dx := ¯h
(D−1)N/2 N Y j=0 dD−1x. (22.169) Note that we are writing the expression for D dimensions (D −1 space dimensions and 1 imaginary-time
a variable allows us to track the origin of various factors, as well as to examine other D values as test cases (and even use noninteger values for dimensional regularization). Note also that ˜\beta here ranges from 0 to ¯hc\beta, including an extra factor of c compared to the standard particle case to measure the time-like coordinate in spatial units. The result here bears some explanation. Note that we had no ordering issues to contend with, as p ˜\beta commutes with ϵ(x). The main difference with the standard result (22.162) is the dependence of the p ˜\beta term in the Hamiltonian on ϵr(r) and µr(r). In the derivation of (22.162), recall that we had to perform integrations over momenta. Each one of these integrations generated a factor of \sqrtm. Because of the spatial dependence of ϵr(r) and µr(r), this is a bit more complicated. The idea is to then think of fixing the x coordinates in the path integral, while performing the integrals over the x ˜\beta coordinates. Then for these integrals, ϵr(x) and µr(x) appear as different (fixed) masses for each of the x ˜\beta integrals, and thus they contribute the extra factors in the measure (22.169), and they still appear in the time-like kinetic-energy
22.3 Development of the Path Integrals term in (22.168). The path integral then proceeds just as in the standard-particle case for the space-like dimensions. Turning this path integral into ensemble-average form, Zeff = Z dD−1x0 Z ¯hc\beta d˜\beta **
" − Z T d\tau V (x)
x(\tau) , (22.170) where the (four-dimensional) paths satisfy
p
(22.171) where \alpha here refers to the spatial dimensions, and x ˜\beta is the time-like coordinate of the path. Note that the choice of stochastic calculus is not important here, as the noise is additive. Carrying out the derivative and accounting for the \delta functions, Zeff = ¯hc\beta
Z dD−1x0 **
++ x(\tau) . (22.172) (effective Z, scalar EM) The path average here is defined as
T Z T
(22.173) (path average) for a function g of the path, where the time coordinate \tau of the path runs from 0 to T . The normalization factors came from the prefactor on the right-hand side of Eq. (22.166), which was the probability density for
(2\piT )−1/2, since the space-like part of the paths (22.171) are standard Wiener paths. The generalization in the x ˜\beta direction is additionally weighted by the root-mean-square average of the spatial factor in Eq. (22.166), resulting in the extra path-average factor. We also carried out the ˜\beta integral, which simply gives an extra factor of ¯hc\beta. In this case, the path in the x ˜\beta direction is irrelevant, and so we have
(22.174) (stochastic path, scalar EM) as the (D −1)-dimensional path, without any variable velocities, and we have switched to Brownian bridges to enforce the path condition on the paths that replaced the delta functions. Note that while BT (\tau) is a
BT (\tau) is a D-dimensional vector Brownian bridge, or D independent bridges bundled together as a single vector. Thus, the relevant paths are just Brownian bridges displaced to x0. Together with the expression (22.161) for the (field) log partition function, Eq. (22.172) gives the log partition function as log Z = ¯hc\beta
Z \infty dT
Z dD−1x0 **
++ x(\tau) , (22.175) or using the energy expression
(22.176)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals we have the energy EEM =\langle HEM\rangle : EEM = − ¯hc
Z \infty dT
Z dD−1x0 **
++ x(\tau) . (Casimir energy, scalar EM) (22.177) This is the (unrenormalized) ground-state Casimir energy in scalar electromagnetism in D space-time di- mensions, valid for either TE or TM polarization if we choose the potential V appropriately according to
for an energy in D = 4. Since T has the dimensions of squared length, this expression will have dimensions of energy-length and energy-area in D = 3 and D = 2, respectively. 22.3.4.3 Alternate Derivation of the Normalized Path Integral: Space-Dependent Mass As an alternative to working with the rescaled Green operators (22.148), we can also work directly with the ‘‘raw’’ TE Green operator (22.146) and its TM counterpart, repeated here: ˜GTE = −ϵr\partial 2
µr \nabla −1 ˜GTM = −µr\partial 2
ϵr \nabla −1 . (22.178) As we mentioned at the beginning of Section (22.3.3), the factor of µr(r) in the TE operator and the corresponding factor of ϵr(r) in the TM operator are problematic, as they represent space-dependent masses, which require more careful path-integration techniques. Fortunately, we have already done the hard work for this in Section 20.4.4.1. Recall that the one-dimensional Hamiltonian (20.370)
(22.179) which has the same form as the space components of the Green operators (22.178), leads to a partition- function path integral of the form (20.375):
r m
Z dx0 ** exp " −1 ¯h Z ¯h\beta d˜\beta g(x) V (x) −¯h2(2gg′′ −3g′2) 8mg2
p
. (22.180) This followed after converting to the proper curved-space momentum operators and developing the prop- agator path integral, rescaling time to remove the space dependence from the path-integral measure, and reorganizing the ordering potentials, which lead to the g-derivative potential terms here. Recall that due to the temporal scaling, the stochastic paths here are scaled Brownian bridges (i.e., flat-space paths). Now, noting that the effective ordering potential can be rewritten as
8g2 = 1 \partial x log p g(x) 2 −\partial 2 x log p g(x) , (22.181)
Then we can proceed by making the same identifications as we did before in Section 22.3.4.2. The spatial dimensions here give the same contributions to the worldline path integral as before, but the potentials VTE(r) and VTM(r) of Eqs. (22.155) come from the ordering potential [instead of from the rescaling step in Eqs. (22.148)]. Then the normalization of the imaginary-time dimension also goes through as before, but with one difference. It would seem from Eqs. (22.178) that the local path variance is scaled by ϵr[x(t)] or µr[x(t)] for the TE and TM cases, respectively. [See, e.g., Eqs. (22.171).] However, because we rescaled time by g[¯x(t)] in Eq. (20.365), the local path variance ends up being scaled by ϵr[x(t)] µr[x(t)], as before. The end result is the same path integral (22.172), but with more technical complication in the derivation.
22.3.5 Worldline Path Integral for Casimir-Polder Potentials¶
22.3 Development of the Path Integrals 22.3.5 Worldline Path Integral for Casimir–Polder Potentials Now if we are interested in the interaction of an atom with a magnetodielectric body (or bodies), we can adapt the classical result Eq. (22.106) to the present case:,
ϵ0 \deltaEEM \deltaϵr
\deltaEEM \deltaµr , (22.182) (Casimir–Polder potential) where we are ignoring dispersion (of the atom or the bodies), \alpha0 is the (static) polarizability of the atom, and \beta0 is the (static) magnetizability. Note that this expression is invariant under the duality transformation
To evaluate the functional derivatives here, we will first need the partial (functional) derivatives cor- responding to the first factor in Eq. (22.177), \delta \deltaϵr Z
\delta \deltaµr Z
(22.183) These derivatives require some explanation. Here we are regarding EEM[ϵr(r), µr(r)] as a functional of the relative permittivity and permeability, which are themselves functions over all space. The integration over all space, as in the Hamiltonian (22.125), reduces these functions to a scalar, and thus define the scalar nature of the functional. But when we vary a path integral of the form \delta Z
Z dD−1x0
, (22.184) we must identify the derivatives via the inner products of the form
\deltaE \deltaϵr , \deltaϵr
- \deltaE \deltaµr , \deltaµr
. (22.185) The integration over x0 is an obvious candidate for the inner product, but in the path integral, it is not so simple, because ϵr and µr are now functions of the coordinates x(\tau) along the entire path. Thus, the path average (or, equivalently, an integral along the path time \tau) is also necessary to define a scalar result, and thus a functional, which is the reduction to a scalar by integration over all path averages, where the paths are rigid translations of a single path. The ensemble average over paths is not necessary here to define the inner product, as it implements the integration over the intermediate coordinates x1, . . . , xN−1, whose dependence on x0 has already been removed by considering the rigid translation of paths. Thus, Eq. (22.184) becomes \delta Z
Z dD−1x0
−µr \deltaϵr
, (22.186) from which the derivatives (22.183) follow. A check on the the choice of inner product for the functional
14Stefan Yoshi Buhmann and Stefan Scheel, ‘‘Macroscopic Quantum Electrodynamics and Duality,’’ Physical Review Letters 102, 140404 (2009) (doi: 10.1103/PhysRevLett.102.140404); Hassan Safari, Dirk-Gunnar Welsch, Stefan Yoshi Buhmann, and Stefan Scheel, ‘‘van der Waals potentials of paramagnetic atoms,’’ Physical Review A 78, 062901 (2008) (doi: 10.1103/Phys- RevA.78.062901).
Chapter 22. Electromagnetic Casimir Energies as Path Integrals derivative \deltaE/\deltaϵr at r0 out of the inner product (22.185). This leads us to evaluate an integral of the form Z dD−1x0 D µr(x) \delta(x −r0) E
T Z dD−1x0 Z T
= 1 N Z dD−1x0 N−1 X j=0 µr(xj) \delta(xj −r0) = 1 N Z dD−1x0 N−1 X j=0 µr[x0 + (xj −x0)] \delta[x0 + (xj −x0) −r0] = 1 N N−1 X j=0 µr(r0)
(22.187) where we switched to the discrete representation of the path average and used the fact that (xj −x0) is constant for any rigid translation of a particular path. This procedure also leads to the derivatives (22.183). We will also need functional derivatives corresponding to the exponent (i.e., the potential) of the path integral. Writing the TM potential from Eqs. (22.155) in the form VTM = 1
i = 1 \nabla log ϵr 2 −1 4\nabla 2 log ϵr = 3 \nabla ϵr 2 8ϵ 2r −\nabla 2ϵr 4ϵr , (22.188) we can write the partial derivatives for the TM potential as \partial \partial ϵr VTM = −3 \nabla ϵr 2 4ϵ 3r
4ϵ 2r = −1 \nabla ϵr 2 ϵ 3r
r ! = 1 4ϵr \nabla 2 log ϵr −2 \nabla log ϵr 2 \partial
4ϵ 2r = −3 4\nabla ϵ −1 r = 3 4ϵr \nabla log ϵr \partial
VTM = −1 4ϵr , (22.189) where we used \partial xh−1 = −h′/h2, \partial 2
with similar expressions for the corresponding µr derivatives of the TE potential. Then, recalling that for a functional of the form F[x, xt, xtt, . . . ; t] = Z t2 t1 f(t, x, xt, xtt) dt, (22.190) where xt ≡\partial x/\partial t, the functional derivative is [see Eq. (8.12)] \deltaF
\partial x −d dt \partial f \partial xt + d2 dt2 \partial f \partial xtt . (22.191) Adapting the functional derivative to the present case, we will need the following derivatives of the partial
22.3 Development of the Path Integrals derivatives (22.189), ∂ ∂ϵr VTM ¶
4ϵr \nabla 2 log ϵr −2 \nabla log ϵr 2 P(r)
\partial
VTM
4ϵr \nabla 2 log ϵr − \nabla log ϵr 2 P(r) −3 4ϵr
\nabla 2 \partial
VTM
4ϵr \nabla 2 log ϵr − \nabla log ϵr 2 P(r) + 1 2ϵr
4ϵr \nabla 2P(r), (22.192) where P(r) is an arbitrary function that represents the rest of the integrand. Combining these terms gives \partial \partial ϵr VTM
\partial
VTM
\partial
VTM P(r) = −1 4ϵr \nabla 2 log ϵr P(r) −1 4ϵr
4ϵr \nabla 2P(r) = −1 4ϵr h \nabla 2 log ϵr
P(r). (22.193) Finally, assembling all the parts from Eqs. (22.183) and (22.193) the functional derivatives of Eq. (22.177) become \deltaE (TE) EM \deltaϵr = ¯hc
Z \infty dT
**
++ x(\tau) \deltaE (TE) EM \deltaµr = ¯hc
Z \infty dT
\times **
2µr \nabla 2 log µr
++ x(\tau) , (22.194) with analogous expressions for the TM case. Putting these functional derivatives into Eq. (22.182) leads to the somewhat cumbersome expression V (TE) CP
¯hc
Z \infty dT
ϵ0
−\beta0µ0T 2µr \nabla 2 log µr
++ x(\tau) . (unrenormalized Casimir–Polder potential, TE polarization) (22.195) For TM polarization, the expression is essentially the same, but with ϵr and µr reversed, and correspondingly with \alpha0/ϵ0 and \beta0µ0 reversed. Thus, the resulting TM expression is V (TM) CP
¯hc
Z \infty dT
ϵ0
−\alpha0T 2ϵ0ϵr \nabla 2 log ϵr
++ x(\tau) . (unrenormalized Casimir–Polder potential, TM polarization) (22.196)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals expression simplifies drastically: V (TE) CP
¯hc\alpha0
Z \infty dT
**
x(\tau) ++ x(\tau) (unrenormalized, nonmagnetic Casimir–Polder potential, TE) (22.197)
region of constant ϵr (i.e., if the atom is in vacuum outside a dielectric body), then the TM expression also simplifes, though not quite as much: V (TM) CP
¯hc\alpha0
Z \infty dT
**
2ϵr \nabla 2
x(\tau) = ¯hc\alpha0
Z \infty dT
x(\tau) −T 2ϵr \nabla 2
++ x(\tau) . (unrenormalized, nonmagnetic Casimir–Polder potential, TM) (22.198) Recall that the magnetic path-integral potential is defined in (22.155) as
i . (22.199) (matter-induced potentials) Also, note that in all these expressions, the paths (22.174) are defined after the functional derivatives such that
(22.200) (stochastic path, scalar EM) That is, the atomic location r in the interaction potential is the beginning and end-point of the paths (Brownian bridges); the integral over this terminus point was lost in the functional differentiation. Note that, had we derived analogous expressions to (22.197) and (22.198) for a magnetic atom interacting with a magnetic surface, we would obtain the same results, but with the following changes: \alpha0 is replaced by \beta0µ2, ϵr is replaced by µr, ϵ0 is replaced by µ0, TE and TM polarizations are interchanged, and there is an extra overall minus sign for both polarizations. That is, the geometry dependence would be the same, but the total force would change sign. 22.3.5.1 Interaction with a Point Particle: Path-Integral View (TE Polarization) Thus far, we have relied on a rather formal expression (22.182) for the Casimir–Polder potential in terms of functional derivatives of the energy in path-integral form. To gain a more intuitive view of this interaction, we can also exhibit this interaction directly in the path integral. To do this, let’s return to the path integral
E (TE) EM = − ¯hc
Z \infty dT
Z dD−1x0 **
x(\tau) ++ x(\tau) . (22.201) Now let’s consider the interaction of a point-dipole atom with a dielectric body (or bodies), which amounts to separating out the atomic contribution from the permittivity as
ϵ0 \deltaD−1(x −r), (22.202) where the atom (of static polarizability \alpha0) is at position r. Making this replacement in the path integral to obtain the Casimir–Polder potential, V (TE) CP
¯hc
Z \infty dT
Z dD−1x0
E−1/2 x(\tau) ++ x(\tau) = − ¯hc
Z \infty dT
Z dD−1x0
ϵ0T ℘[x(\tau); r] −1/2 x(\tau) ++ x(\tau) , (22.203)
22.3 Development of the Path Integrals where we have defined the point-occupation time in D −1 dimensions
Z t dt′ \deltaD−1[y(t′) −r] (22.204) for y(t) at r, assuming that the process y runs from 0 to t. Note that in one dimension, this reduces to the local time of y, whereas in higher dimensions this statistic has the typical value of zero unless r is the starting point of the process.
V (TE) CP
¯hc
Z \infty dT
Z dD−1x0 **
x(\tau) −
x(\tau) ++ x(\tau) . (22.205) The first term disappears when we renormalize against the atom and ϵr body being separated by an arbitrarily large distance. The integral renormalized thusly is V (TE) CP
¯hc
Z \infty dT
Z dD−1x0 **
x(\tau) ++ x(\tau) . (22.206) Due to the presence of ℘, only paths x(\tau) that intersect r will actually contribute to the path average due to the presence of ℘in the integrand. However, any path that passes through r is equivalent to a path that begins at r, and the path itself defines an equivalence class of paths that are cyclic permutations of the path in time, starting at different points along the path. For the purposes of the path average over the permittivity, we will replace all such paths by the equivalent path at r, which is generated with the same probability as each possible path by virtue of being a path. Thus, we replace the contribution of the path
V (TE) CP
¯hc
Z \infty dT
Z dD−1x0 **
xr(\tau) ++ xr(\tau) , (22.207) and by changing the ensemble average to reference xr instead of x, we have effectively dropped the null contribution of any path that doesn’t pass through r. Then the only part of the integrand that depends on x0 is the ℘factor, since this refers to the original path x(\tau). By shifting the path, we can carry out this integral: Z
Z
(22.208) The last equality here follows from integrating the definition (22.204) over all r. [This is essentially the same argument we used in Eq. (22.187), but here in continuous notation.] Thus, we finally have V (TE) CP
¯hc\alpha0
Z \infty dT
**
xr(\tau) ++ xr(\tau) . (22.209) This is equivalent to the result (22.197) that we derived from functional differentiation. Note that in this approach, we have perturbed the dielectric permittivity to introduce an atom without regard for what fields, potentials, or sources to hold fixed, as we discussed at length in Section 22.2. Here, we have little choice: the fields and sources (the sources being effective sources due to vacuum fluctuations) are buried in the path integral (22.201), and so we assume that the whatever needs to be fixed is so. This is justified from our procedure in Section 22.3.5, which was based on the more careful analysis of electromagnetism.
Chapter 22. Electromagnetic Casimir Energies as Path Integrals 22.3.5.2 Interaction with a Point Particle: Path-Integral View (TM Polarization) In the TM case, we can return to Eq. (22.177) for a nonmagnetic interaction, E (TM) EM = − ¯hc
Z \infty dT
Z dD−1x0 **
++ x(\tau) , (22.210) where we now must also deal with the potential defined in (22.155):
\nabla log ϵr 2 −1 4\nabla 2 log ϵr. (22.211) Again letting
ϵ0 \deltaD−1(x −r), (22.212) we will expand to lowest order in the atomic polarizability. Note that we will simply expand in spite of the delta function, which only makes sense when the delta function is regularized by a localized function of finite height. This expansion is more involved than in the TE case, because of the nonlinear functions of ϵr in the potential that appear inside the path average. Expanding the potential first, using log(a + b) + b/a + O(b2), the potential with the atom separated out is VTM(x) −\rightarrow 1
ϵ0 \nabla 1 ϵr \deltaD−1(x −r) 2 −1
ϵ0 \nabla 2 1 ϵr \deltaD−1(x −r) . (22.213) Expanding to lowest order in \alpha0, VTM(x) −\rightarrow 1 \nabla log ϵr 2 −1
4ϵ0 h
ϵr \deltaD−1(x −r), (22.214) gives the original potential plus an atomic interaction term. Thus, the replacement (22.212) in the path integral (22.210) gives E (TM) EM = − ¯hc
Z \infty dT
Z dD−1x0 \times **
ϵ0 \deltaD−1(x −r) −1/2 x(\tau) e −T D
r \deltaD−1(x−r) E x(\tau) ++ x(\tau) . (22.215) Now expanding to first order in \alpha0, and dropping the zeroth-order component (which disappears in renor- malization against the configuration sans atom), we obtain E (TM) EM = − ¯hc
Z \infty dT
Z dD−1x0 \times ** − \alpha0
\deltaD−1(x −r)
x(\tau)
x(\tau)
− \alpha0T
x(\tau) h
ϵr \deltaD−1(x −r)
x(\tau)
++ x(\tau) . (22.216) Now we can integrate by parts twice on the second term, and note that the path average of the delta function takes care of the x0-integration as in the last section. The result is E (TM) EM = ¯hc\alpha0
Z \infty dT
\times **
2ϵr h
++ x(\tau) , (22.217)
22.4.1 TE Polarization: Strong Coupling¶
22.4 Casimir–Polder Potential Near a Dielectric Half-Space
and also to Eq. (22.198) if the gradient of ϵr vanishes at the atomic location. 22.4 Casimir–Polder Potential Near a Dielectric Half-Space To illustrate the Casimir path integrals we have developed so far, we will evaluate some examples, in particular for the Casimir–Polder path integrals (22.197) and (22.198) for an atom near a (planar) dielectric half-space. 22.4.1 TE Polarization: Strong Coupling Let’s begin with a test-drive of the TE path integral (22.197) in a simple regime: a perfectly conducting, planar surface: V (TE) CP
¯hc\alpha0
Z \infty dT
**
x(\tau) ++ x(\tau) . (22.218) Recall that we must renormalize this by subtracting the same path integral with the atom and surface separated by an arbitrarily large distance. In this case it suffices to subtract the same integral with ϵr = 1: V (TE) CP
¯hc\alpha0
Z \infty dT
**
x(\tau) −1 ++ x(\tau) . (22.219) For a perfect conductor, the susceptibility of the surface is very large. If any part of the path touches the surface, that part will dominate the path average, causing it to diverge, and thus the inverted average
x(\tau) will vanish for any path that touches the surface. With the renormalization term, the quantity in the ensemble average is 0 for paths not touching the surface, and −1 for paths touching the surface. With the path average, the ensemble average is −1 times the probability that the paths touch the surface. The relevant path statistic is the boundary-crossing probability, which for a standard Brownian bridge is [Eq. (17.380)]
(22.220) where d is the distance of the boundary from the initial and final point of the bridge. The paths here, defined in Eq. (22.200), go to time T , so they are effectively ‘‘larger’’ by a factor \sqrt T . We can adapt the statistic by setting d = z/ \sqrt T , where z is the distance to the surface. Note that this applies to only one of the spatial dimensions; the others are irrelevant to the problem except for normalization. Then the path integral becomes V (TE) CP
¯hc\alpha0
Z \infty dT T 1+D/2 Pcross z \sqrt T = − ¯hc\alpha0
Z \infty dT T 1+D/2 Pcross \sqrt 2T = − ¯hc\alpha0
Z \infty dT
(22.221) where we rescaled T in the second step to remove the z dependence from the crossing probability and to introduce a factor of 2, and then we inserted the explicit crossing probability. The result is V (TE) CP
(Casimir–Polder potential, TE strong coupling) (22.222) Setting D = 4 we obtain V (TE) CP
¯hc\alpha0
32\pi2ϵ0z4 1 , (Casimir–Polder potential, TE strong coupling) (22.223)
22.4.2 TE Polarization: Weak Coupling¶
Chapter 22. Electromagnetic Casimir Energies as Path Integrals which is the standard result for the TE part of the total potential [see Eq. (14.214) and the following
V (TE) CP
16\piϵ0z2 , (22.224) which is not a standard result, but easily verified by mode-summation (Problem 22.9). 22.4.2 TE Polarization: Weak Coupling Now, in the limit of a rarefied dielectric surface, we can write
(22.225) and calculate to first order in the susceptibility \chi. In this limit, the TE path integral (22.197) becomes V (TE) CP
3¯hc\alpha0
Z \infty dT
x(\tau) , (22.226) where we have already dropped the \chi-independent (zeroth-order) term in renormalization. For a dielectric half-space, we have
(22.227) where the dielectric occupies z < 0 and we will compute the potential for z > 0. The path integral then becomes V (TE) CP
Z \infty dT
x(\tau) . (22.228) The relevant statistic here is the sojourn time, measuring the time a process y(t) spends beyond a barrier at d [Eq. (17.499):
Z t dt′ \Theta[y(t′) −d]. (22.229) Thus, the potential is V (TE) CP
Z \infty dT
Ts[x(\tau); z]
x(\tau) , (22.230)
view of the martingale nature of x(\tau). We will again want to transform the sojourn time of x(\tau) into the sojourn time of a standard Brownian bridge, again with \tau = tT :
Z T
= T Z 1 dt \Theta[x(tT ) −z] = T Z 1 dt \Theta[ \sqrt T B(t) −z] = T Z 1 dt \Theta B(t) − z \sqrt T = T Ts B(t); z \sqrt T . (22.231)
22.4.3 TE Polarization: General Coupling¶
22.4 Casimir–Polder Potential Near a Dielectric Half-Space We then have the potential V (TE) CP
Z \infty dT
** Ts B(t); z \sqrt T
B(t) = −
Z \infty dT
** Ts B(t); \sqrt 2T
B(t) , (22.232) where we have scaled away z in the integral by letting T −\rightarrow 2z2T . The mean sojourn time for a standard Brownian bridge is given by [Eq. (17.588)] DD Ts[B(t); d] EE = e−2d2 − r\pi 2 d erfc h\sqrt 2 d i , (22.233) and so V (TE) CP
Z \infty dT
e−1/T − r \pi T erfc 1 \sqrt T . (22.234) Carrying out the integral, the result is V (TE) CP
(Casimir–Polder potential, TE weak coupling) (22.235) Putting D = 4, we find the result V (TE) CP
32\pi2ϵ0z4 \chi . (Casimir–Polder potential, TE weak coupling) (22.236) This is the standard result for the TE part of the total potential in this regime [see Eq. (14.214) and the
V (TE) CP
64\piϵ0z2 , (22.237) as can be verified by mode-summation (Problem 22.8). 22.4.3 TE Polarization: General Coupling Using basically the same methods, in the TE case at least, we can derive the potential for an atom interacting with a dielectric half-space of arbitrary (dispersionless) susceptibility \chi. Returning to the TE path integral (22.197) after renormalization, V (TE) CP
¯hc\alpha0
Z \infty dT
**
x(\tau) −1 ++ x(\tau) , (22.238) In this case, we can write the path average for the permittivity as
B(\tau); z \sqrt T , (22.239) in terms of the sojourn time for the path, which represents the fraction of the path inside the dielectric, which is a distance z from the atom, and we have rescaled the paths into standard Brownian bridges. Then
22.4.4 TM Polarization: Strong Coupling¶
Chapter 22. Electromagnetic Casimir Energies as Path Integrals the potential in terms of the sojourn time is V (TE) CP
¯hc\alpha0
Z \infty dT
B(\tau); z \sqrt T −3/2 −1 ++ B(\tau) = − ¯hc\alpha0
Z \infty dT
** 1 −
B(\tau); \sqrt 2T
B(\tau) , (22.240) To compute the path average here, we will write DD
= Z 1
= Z 1
"r 4(1 −x) \pixT
1 −2 T e−1/T erfc r x (1 −x)T # , (22.241) where we are integrating against the probability density for the standard Brownian bridge from Eq. (17.584),
h 1 −e−2d2i
r 8d2(1 −x) \pix e−2d2/(1−x) + (1 −4d2) e−2d2 erfc r 2d2x 1 −x ! , (22.242)
\sqrt 2T . Then computing the T integral for D = 4, Z \infty dT T 3 DD
= Z 1
Z \infty dT T 3 "r 4(1 −x) \pixT
1 −2 T e−1/T erfc r x (1 −x)T # = Z 1
3(1 −\sqrtx)2 2\sqrtx (22.243) which for \alpha = −3/2 becomes Z \infty dT T 3 DD
\chi
\chi3/2 . (22.244)
V (TE) CP
32\pi2ϵ0z4
6 + 1 \chi −
2\chi
2\chi3/2 ! . (Casimir–Polder potential, TE) (22.245) The parenthetic quantity is \etaTE from the Green-tensor treatment in the full electromagnetic case [Eq. (14.214)], and the rest of the result is the strong-coupling result from full electromagnetism. This factor is \chi/40 and 1/6 for small and large \chi, respectively, demonstrating the consistency with the weak-coupling calculation (22.236) and the strong-coupling version (22.223). 22.4.4 TM Polarization: Strong Coupling To begin, a disclaimer: in this section we will do something wrong, for illustrative purposes. The idea is to now switch to the TM path integral, which is substantially more complicated due to the presence of a divergent potential. It’s important to be careful with this potential in order to avoid trouble.
22.4 Casimir–Polder Potential Near a Dielectric Half-Space For the TM path integral, evaluated near a planar dielectric interface, we return to Eq. (22.198), V (TM) CP
¯hc\alpha0
Z \infty dT
x(\tau) −T 2ϵr \nabla 2
++ x(\tau) . (22.246)
V (TM) CP
¯hc\alpha0
Z \infty dT
x(\tau) −T 2 \nabla 2
++ x(\tau) . (22.247) There are two main, extra complications here that we didn’t have in the TE case: the potential and the derivative. The derivative will be easy to handle, but let’s first take a closer look at the potential. From Eq. (22.155) the potential is
i . (22.248) If the dielectric has susceptibility \chi, then we can write the permittivity as
(22.249) for a vacuum–dielectric interface, if the dielectric occupies z > 0 (we choose this convention to keep the signs simple). Then putting this into the potential, we have a rather divergent mess:
log p
2 \delta2(z) − log p
\delta′(z) . (22.250) The first term is divergent in the path average of any nonvanishing path (with unit probability). This is not terribly problematic, because it means that any path touching the surface, the (unrenormalized) contribution of that path vanishes, which is precisely what should happen in the strong-coupling limit. However, this shouldn’t happen for arbitrary \chi, which is where we are at the moment, so this term is problematic. The second term in the potential is also problematic: the derivative of the delta function picks out the derivative of the path in the path average, which is divergent (either positive or negative) [see the local-time derivative in Eq. (17.660) and the following discussion]. Here, if the overall term swings large and positive, the result is the same as for the second term. If it swings large and negative, then problems can happen, as now the exponential factor can become arbitrarily large. But let’s ignore these problems for the moment and try to work with the potential in the strong- coupling limit. To do this, we will assume that, for sufficiently large \chi, that the second term in the potential dominates the first, because asymptotically log2 \chi ≫log \chi. This only really makes sense if we regularize the delta functions, replacing them with corresponding narrow and sharply (but finitely) peaked functions. To proceed, we need to evaluate (renormalized) path integrals of the form
Z \infty dT
**
++ x(\tau) . (22.251) The first term in the path average vanishes whenever the path touches the surface, which is a distance z away. Thus, we are back to the crossing probability as in Section 22.4.1. Using the Brownian-bridge crossing probability of e−2d2, and then setting d = z/ \sqrt T as before, we have
Z \infty dT T 1+D/2 Pcross z \sqrt T
Z \infty dT T 1+D/2 Pcross \sqrt 2T
Z \infty dT
(22.252)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals Then writing Eq. (22.247) in terms of this integral as V (TM) CP
¯hc\alpha0
I(D, 3/2; z) −1 2\partial 2 z I(D −2, 1/2; z) = − ¯hc\alpha0
z z−(D−2) \Gamma[(D −2)/2] = − ¯hc\alpha0
. (22.253) Simplifying the last factor, we find V (TM) CP
. (Casimir–Polder potential, TM strong coupling) (22.254) Note that we were a bit slippery here with the derivative. The derivative nominally operates on the atomic position, but this is equivalent to letting it operate on the atom-surface distance. Equivalently, we could also fix the atom and let the derivative operator work on the surface position. In the case D = 4, we have V (TM) CP
5¯hc\alpha0
3¯hc\alpha0 32\pi2ϵ0z4 5 (wrong!), (22.255) which is the standard result for the TM part of the total potential [see Eq. (14.214) and the following discussion], except for an overall sign, as this potential should be negative (attractive). Note that in 1D
V (TM) CP
¯hc\alpha0 16\piϵ0z2 (wrong!), (22.256) which should be identical to the TE result (22.224): in 1D, all waves are normally incident, so the distinction between TE and TM waves is irrelevant. However, we are again off by an overall sign. What happened? The flaw is in dropping the log \chi term in the potential compared to the log2 \chi term, which is invalid even in the limit \chi −\rightarrow \infty. It is the balance of the two terms in the potential that is critical in obtaining a sensible answer (though we will see in the next section that one of these terms can be dropped for small \chi). We will see this minus sign showing up later in Section 22.8.6.1 when we treat the TM potential properly in the limit of large \chi. 22.4.4.1 Fluctuations and the Interface Potential Now that we see how the strong-coupling limit works here (or doesn’t work, as it were), we can wax a bit philosophical. It was the first (\delta2) term in the potential (22.250) that produced the strong-coupling limit (albeit with the wrong sign), and evidently the second (\delta′) term is the one that produces deviations from it. We will see in the next calculation that for small \chi, the first (\delta2) term is ignorable, and the second (\delta′) term yields the correct result. In intermediate regimes, it is the interplay of these two terms that conspire to give the correct result. As we have seen, it is even true in the strong-coupling regime that this interplay must hold, since the two terms must conspire to produce an overall minus sign, compared to what we would expect for only the (\delta2) term. In terms of this interplay, one can intuitively think of a path touching an interface with some inter- mediate \chi. The second term will always attempt to make the potential large and positive, trying to bring about the strong-coupling result. The first term can swing either way as the path crosses the surface, so most of the time the strong-coupling result will indeed happen. Recall that in the unrenormalized path integral, the strong-coupling result is simply a null integrand. However, it may happen that as the path crosses the surface, the total potential will swing large and negative. In this case, rather than having a null contribu- tion, the integrand can be huge. The relative rarity of these events tempers their large contribution, and the
22.4.5 TM Polarization: Weak Coupling¶
22.4 Casimir–Polder Potential Near a Dielectric Half-Space resulting average, while incorporating large fluctuations, converges to the correct intermediate result. Note that the T integral rescales the paths, so even for a particular path, this outer integral already averages over all these possibilities as different parts of the path are in contact with the interface. 22.4.5 TM Polarization: Weak Coupling In the case of weak coupling, we can linearizing and then regularize the path integral (although technically the regularization should come before the linearization). In this case, we again have the potential
i . (22.257) with permittivity
(22.258) so that
log p
2 \delta2(z) − log p
\delta′(z) , (22.259) but now in the limit of small \chi. Expanding to first order in \chi, we can discard the first term and obtain
4 \partial 2 z \Theta(z). (22.260) Note that by rewriting the derivative of the delta function in terms of the Heaviside function, we are already implementing our regularization strategy for the divergent potential (which is the only way in which an expansion like this could possibly make sense). Now in the path integral (22.247), V (TM) CP
¯hc\alpha0
Z \infty dT
x(\tau) −T 2 \nabla 2
++ x(\tau) , (22.261) we can insert the potential and permittivity, and expand to first order in \chi. The result is V (TM) CP
Z \infty dT
−3 2 + T 2 \partial 2 z −T 2 8 \partial 4 z
++ x(\tau) (22.262) after renormalization against vacuum. Identifying the path average as the sojourn time for x(\tau) starting a distance z away from the surface, or equivalently a standard Brownian bridge sojourning a distance z/ \sqrt T from the surface [see Eqs. (22.231) and the associated discussion], we have V (TM) CP
Z \infty dT
**3 2 −T 2 \partial 2 z + T 2 8 \partial 4 z Ts B(t); z \sqrt T
B(t) = −
Z \infty dT
**3 2 −T \partial 2 z + T 2 2 \partial 4 z Ts B(t); z \sqrt 2T
B(t) . (22.263) Here, we have integrals to evaluate of the form
z Z \infty dT
** Ts B(t); z \sqrt 2T
B(t) . (22.264) We have already computed the integral here, back in the solution to Eq. (22.234), so
z \Gamma(D/2)
(22.265)
22.5.1 TE Reflection Coefficient: Vacuum-Dielectric Interface¶
Chapter 22. Electromagnetic Casimir Energies as Path Integrals
regularization, as we will see. For example, if the differentiation is performed naïvely here, by setting D = 0 too soon, for example, a spurious null result would obtain. The correct (dimensionally regularized) procedure is to do the differentiation before fixing D, and then taking the limit D −\rightarrow 0. Now writing Eq. (22.262) in terms of this integral as V (TM) CP
3 2I(D, 0; z) −I(D −2, 2; z) + 1 2I(D −4, 4; z) = −
3\Gamma(D/2)
4(D −3)\Gamma(D −4) . (22.266) Simplifying the last factor, the result is V (TM) CP
[4D(D −1) −5]\Gamma(D/2) 4(D + 1) . (Casimir–Polder potential, TM weak coupling) (22.267) For D = 4, the last factor is 43/20, and we have V (TM) CP
32\pi2ϵ0z4 43 . (Casimir–Polder potential, TM weak coupling) (22.268) This is the standard result in the small-\chi regime for the TM part of the total potential [see Eq. (14.214) and the following discussion]. For D = 2, the last factor is 1/4, and so V (TM) CP
64\piϵ0z2 . (22.269) This is equivalent to the TE result (22.237), as it should be in 1D electromagnetism, where the distinction between the polarizations is lost. Evidently, in obtaining results that are not quite correct, we have played fast and loose with the divergent potential. Although it should in principle be possible to regularize the potential, in the limit where a sharp interface is a good idealization, the behavior of the potential path average is a violent one, swinging between zero and a large positive value. What we are seeing here is that in a perturbative treatment that misses this behavior, the predictions aren’t correct. 22.5 Fresnel Reflection Coefficients In applying the method that follows, we will repeatedly encounter expressions involving the Fresnel reflection coefficients at a dielectric interface, but in a somewhat disguised form. Thus, we will have a short digression and develop some expressions that will greatly simplify algebra and interpretation in later calculations. 22.5.1 TE Reflection Coefficient: Vacuum–Dielectric Interface The Fresnel reflection coefficient for TE polarization is given by15
, (22.270) where the incident angle is \theta1 from the medium of index n1 into the medium of index n2, and we have Snell’s law,
(22.271) which determines the angle \theta2 in the transmitting medium in terms of \theta1. 15Daniel A. Steck, Classical and Modern Optics (2006), Chapter 9, available online at http://steck.us/teaching/. Note that rTE ≡rS.
22.5 Fresnel Reflection Coefficients 22.5.1.1 Vacuum-Side Reflection (TE)
\chi is the susceptibility of the medium, to obtain
= cos \theta1 − q
q
p
p
p
p
. (22.272) Now suppose introduce the shorthand notation
(22.273) Then the reflection coefficient becomes
\sqrt
\sqrt
. (TE reflection coefficient, vacuum side) (22.274) Note that we are explicitly indicating the dependence on the medium parameter \chi and the angle parameter \lambda. 22.5.1.2 Dielectric-Side Reflection (TE) If a wave is incident from the dielectric side, the reflection coefficient takes on a somewhat different form.
=
p 1 −sin2 \theta2
p 1 −sin2 \theta2 =
q
q
=
p
p
. (22.275) Again identifying \lambda with cos2 \theta1, we can then write r′
p
p
p
p
, (TE reflection coefficient, dielectric side) (22.276) where the prime indicates that the sense of the reflection is from the dielectric side, compared to the vacuum- side coefficient in Eq. (22.274). Note that for the same incidence angles on each side, these two expressions should be equivalent up to a minus sign. However, the forms here are different because \lambda refers to different angles in each case.
22.5.2 TM Reflection Coefficient: Vacuum-Dielectric Interface¶
Chapter 22. Electromagnetic Casimir Energies as Path Integrals 22.5.2 TM Reflection Coefficient: Vacuum–Dielectric Interface The reflection coefficient for TM polarization takes only a slightly different form: 16
. (22.277) Again, we will work out what this looks like if we replace cos2 \theta1 with \lambda, for a wave incident on either side. 22.5.2.1 Vacuum-Side Reflection (TM) For incidence from the vacuum side we again put we can put n1 = 1 and n 2
= p
p
= p
p
= p
p
. (22.278) With \lambda = cos2 \theta1, we can write
\sqrt
\sqrt
, (TM reflection coefficient, vacuum side) (22.279) for a the reflection coefficient of a TM vacuum-incident wave. 22.5.2.2 Dielectric-Side Reflection (TM) From the dielectric side, we take n2 = 1 and n 2
= q
q
= q
q
= q
q
= p
p
= p
p
. (22.280) 16Daniel A. Steck, Classical and Modern Optics (2006), Chapter 9, available online at http://steck.us/teaching/. Note that rTM ≡rP in the notation there. Note also that the convention here is that rTM = +1 if the reflected wave is in phase with the incident wave, while the opposite-sign convention is common. In particular, this means that rTE = rTM at normal incidence onto the same interface.
22.6.1 Laplace and Mellin Transforms¶
22.6 Laplace–Mellin Method for Evaluating Path Integrals Thus, we can write r′¶
p
\sqrt \lambda p
\sqrt \lambda , (TM reflection coefficient, vacuum side) (22.281) where again the prime denotes incidence from the dielectric side. 22.6 Laplace–Mellin Method for Evaluating Path Integrals In the solutions we have gotten so far, we have required probability densities for path statistics to evaluate the path integrals. In the TM case, the path integral is complicated by the presence of multiple statistics (path average of the potential and a function of the sojourn time), so in these cases we would need the joint density of path statistics, which are hard to come by in closed form. In the strong-coupling TM limit, we were able to reduce the problem to something manageable, so that we didn’t need a complicated statistic. In the weak-coupling TM limit we were also able to simplify the path integral, but not without running into problems. All this motivates a more general method for evaluating path integrals that does not rely on specific probability densities for path statistics. We will begin by introducing some useful transforms and general integral relations. 22.6.1 Laplace and Mellin Transforms For a function f(t), recall that we can define the Laplace transform by
Z \infty dt e−stf(t). (22.282) (Laplace transform) We can also define the Mellin transform via
Z \infty
Z \infty dt t1−z f(t). (22.283) (Mellin transform) These transforms obviously are only defined whenever the integrals converge. However, when they are sensible, the Mellin transform is invertible in the same sense as the Laplace transform, although it is somewhat more obscure. Suppose that you have a Laplace transform of a function, but you want the Mellin transform. Obviously you could invert the Laplace transform and then stick the result into the Mellin integral. But what if the original function is not readily available? Here we will show that you can obtain the Mellin transform directly from the Laplace transform by computing a Mellin transform of the Laplace transform itself :17
\Gamma(1 −z)M L [f] (1 −z). (Laplace–Mellin conversion formula) (22.284) In more direct notation, we can also write this as
\Gamma(1 −z) Z \infty ds s−zF(s). (22.285) (Laplace–Mellin conversion formula) Here,
Z \infty dt tz−1e−t = M e−t (z) (22.286) 17J. S. Lew, ‘‘On Some Relations between the Laplace and Mellin transforms,’’ IBM Journal of Research and Development 19, 582 (1975) (doi: 10.1147/rd.196.0582).
22.6.2 Laplace Transform and Inverse Moments¶
Chapter 22. Electromagnetic Casimir Energies as Path Integrals is the usual gamma function. To show this, we begin by writing out the Mellin transform of the Laplace transform F(s) explicitly: M L [f]
F
Z \infty
Z \infty ds sz−1 Z \infty dt e−stf(t). (22.287) Then changing z to 1 −z, M L [f]
Z \infty ds s−z Z \infty dt e−stf(t). (22.288) Changing the order of integration gives M L [f]
Z \infty dt f(t) Z \infty ds s−z e−st, (22.289) and scaling s −\rightarrow s/t, M L [f]
Z \infty dt tz−1f(t) Z \infty ds s−z e−s. (22.290) Using
Z \infty ds s−ze−s, (22.291) and recognizing the remaining t integral as F(z), we have M L [f]
(22.292) which is the result we wanted. Note that we relied on a coordinate transformation that is ill-defined at the limits of integration. The integrations should really be taken between finite limits a and b, where 0 < a < b, and then the limits a −\rightarrow 0 and b −\rightarrow \inftyshould be taken at the end. 22.6.2 Laplace Transform and Inverse Moments Recall that the Laplace transform of a probability density f(x) DD e−sxEE = Z \infty
(22.293) acts as a moment-generating function, where the moments arise via differentiation: DD xnEE
s DD e−sxEE s=0 . (22.294) It is also possible (and useful) to use the moment-generating function for shifted, inverse moments,18 \Gamma(\alpha) Z \infty ds s\alpha−1e−s\betaDD e−sxEE =
, (shifted-inverse moment formula) (22.295) which holds for any \alpha and \beta where the integrals make sense (\alpha is not necessarily an integer).
\Gamma(\alpha)M L [f] (\alpha). (22.296) 18For this and other related formulae, see Edward B. Rockower, ‘‘Integral Identities for Random Variables,’’ The American Statistician 42, 68 (1988) (doi: 10.1080/00031305.1988.10475526).
22.6.4 Application to Worldline Path Integrals¶
22.6 Laplace–Mellin Method for Evaluating Path Integrals Writing out the Mellin-transform integrals from the definition (22.283), Z \infty
\Gamma(\alpha) Z \infty ds s\alpha−1L f. (22.297) Then writing out the Laplace transform as an expectation value, Z \infty
\Gamma(\alpha) Z \infty ds s\alpha−1DD e−sxEE . (22.298) Finally, changing variables via x −\rightarrow x + \beta and identifying the left-hand side as an expectation value gives the desired result. 22.6.3 Feynman–Kac Formula: Review The third ingredient we will need in this method is the Feynman–Kac formula (Section 17.11). Recall that the Feynman–Kac formula states that the distribution f(x, t) of solutions to stochastic differential equations can be written both as the solution of a diffusion-type PDE and as an ensemble average over stochastic trajectories. In particular, we will be interested in the steady-state solution f(x) ≡f(x, t −\rightarrow \infty) for time-independent damping and driving functions, in which case the Feynman–Kac formula for exponentially stopped paths with stopping rate \lambda [from Eqs. (17.438) and (17.440)] is
Z \infty dt ** g[x + W(t)] exp −\lambdat −s Z t dt′ V [x + W(t′)]
, (Feynman–Kac formula) (22.299) where the function f(x) also satisfies the ODE
f(x) −2g(x). (22.300) (PDE for Feynman–Kac formula) By choosing the functions V (x) and g(x) appropriately, we can derive various statistics about Wiener paths and Brownian bridges (see Section 17.12.1 and following sections). The basic idea is to obtain an analytic solution to the ODE (22.300). Then observing that the right-hand side of Eq. (22.299) has the form of a double Laplace transform in the variables \lambda and s, the idea is to invert the Laplace transforms, thereby obtaining the desired probability density for a statistic (the density here is hidden in the ensemble average).
statistics for Brownian bridges. Many variations on this theme are possible.19 22.6.4 Application to Worldline Path Integrals The point of all this is to apply these pieces to evaluating worldline path integrals, schematically of the form I = Z \infty dT
**
++ BT (\tau) , (22.301) where we have shifted away the center x0 of the paths, and \gamma(x) and V(x) are some scalar functions of position. Then the general strategy is: 1. Design a related path integral of the form (22.299), where V must reflect both s\gamma (where s is an auxiliary parameter) and V. Choose g(x) to pin the path W(t) to 0 at T . 19A. N. Borodin and I. A. Ibragimov, Limit Theorems for Functionals of Random Walks (Proceedings of the Steklov Institute of Mathematics, vol. 195) (American Mathematical Society, 1995).
22.7.1 Feynman-Kac Formula¶
Chapter 22. Electromagnetic Casimir Energies as Path Integrals 2. Solve the ODE (22.300) to obtain a solution to the path integral (22.299). 3. Use the Mellin-transform formula (22.284) to change the \lambda Laplace transform into the T integral.
BT (\tau) factor. The details vary with the details of the path integrals, so we will move on to solving some path integrals. 22.7 TE Casimir–Polder Path Integral: Dielectric Interface As a first example, we will apply this method to the TE path integral for the Casimir–Polder potential in Eq. (22.238), V (TE) CP
¯hc\alpha0
Z \infty dT
**
x(\tau) −1 ++ x(\tau) , (22.302) for a vacuum–dielectric interface. 22.7.1 Feynman–Kac Formula Since the path integral involves the path average of ϵr in one dimension, the relevant statistic is the sojourn time at distance d:
Z t dt′ \Theta[y(t′) −d]. (22.303) We have already set up the Feynman–Kac formula to obtain the relevant path integral with exponential stopping. The form that we desire is Eq. (17.563)
Z \infty dt \sqrt t e−\lambdat
exp (−sTs[Bt; d])
= r\pi \lambda " 1 −e−2 \sqrt 2\lambda d \sqrt
\sqrt \lambda \sqrt
\sqrt \lambda !# = r\pi \lambda h
\sqrt 2\lambda di , (sojourn-time solution to Feynman–Kac formula) (22.304) where we are using the TE reflection coefficient (22.274):
\sqrt
\sqrt
. (22.305) Again, Eq. (22.304) is equivalently a formula for the iterated Laplace transform of the sojourn time for Brownian bridges pinned to 0 at time t. As a complete illustration of the general method for solving worldline path integrals, we will briefly
select only Brownian bridges. Further, we will only write down the path integral at x = 0, which is the only point we will need:
Z \infty dt ** eikW (t) exp −\lambdat −s Z t dt′ \Theta[W(t′) −d]
= Z \infty dt e−\lambdat ** eikW (t) exp (−sTs[W(t); d]) ++ . (22.306) From Eq. (22.300), we can obtain an expression for f(0) by solving the ODE
f(x) −2eikx. (22.307)
22.7.2 Mellin Transform¶
22.7 TE Casimir–Polder Path Integral: Dielectric Interface The equation is easy to solve piecewise, and the result is [Eq. (17.556)]
Ae− p
2eikx
(x > d) Be \sqrt
2eikx
(x < d). (22.308) Requiring continuity of the solution and its derivative yield the coefficients [Eq. (17.557)] A = \sqrt 2 s \sqrt 2\lambda −ik e p
\sqrt
\sqrt
, B = − \sqrt 2 s p
e− \sqrt
\sqrt
\sqrt
, (22.309) completing the Feynman–Kac-formula solution. Now to complete the pinning of the solution W(t). Integrating the solution (22.306) over k introduces a delta function of the path W at the end-time t: 2\pi Z \infty −\infty
Z \infty dt e−\lambdat ** \delta[W(t)] exp (−sTs[W(t); d]) ++ . (22.310) Then using the path-integral relation [Eq. (17.561)]
\delta[W(t)] F[W(t′)]
= \sqrt 2\pit
F[Bt(t′)]
, (22.311)
\sqrt 2\pi Z \infty −\infty
Z \infty dt \sqrt t e−\lambdat ** exp (−sTs[Bt; d]) ++ , (22.312) which is the integral form we desire in Eq. (22.302). What remains is to evaluate the integral of the solution f(0) to the diffusion ODE. If we assume d > 0, corresponding to the case of a Casimir–Polder potential for an atom outside the dielectric, then Eq. (22.308) becomes
(22.313) and thus, with B as in Eq. (22.309), we obtain the solution [see Eq. (17.563)] \sqrt 2\pi Z \infty −\infty
\sqrt 2\pi Z \infty −\infty dk B +
= r\pi \lambda " 1 −e−2 \sqrt 2\lambda d \sqrt
\sqrt \lambda \sqrt
\sqrt \lambda !# . (22.314) This is the right-hand side of our desired result (22.302). 22.7.2 Mellin Transform Now we can use the Mellin-Laplace-transform conversion formula (22.284) to compute the integral
Z \infty dT
e−sTs[BT ;d] −1
, (22.315) where BT (t) is a Brownian bridge pinned to 0 at time T . First, we rewrite the integral in Eq. (22.304) as
Z \infty dT
exp {−sTs[BT ]}
= r\pi \lambda h
\sqrt 2\lambda di . (22.316)
22.7.3 Inverse Moments¶
Chapter 22. Electromagnetic Casimir Energies as Path Integrals
\Gamma
Z \infty
i = \sqrt\pi \Gamma
Z \infty
\sqrt 2\lambda d. (22.317) Note that we can also regard this Laplace–Mellin transformation as being an example of the integral formula Z \infty
. (22.318) This transformation formally takes care of the first Laplace transform (in \lambda). Note that the path integral (22.315) that we computed here has the form of a potential-coupled scalar field, where the potential has the form of a step function in one direction. However, this isn’t quite the integral that we want for the dielectric interface. The reason is that the path integral (22.302) contains the path average of ϵr. If we take the the path terminus r to be the origin and the interface to be at distance z, then, considering 1D paths, the path average is
T = T −1
(22.319) in terms of the sojourn time Ts[BT ; d] of Brownian bridges BT . Thus, we need the potential in our path integral here to involve this functional. Note that we can adapt Eq. (22.316) for this purpose by letting s −\rightarrow s\chi and then \lambda −\rightarrow \lambda + s, so that
Z \infty dT
e−s(T +\chiTs[BT ])
= r \pi
h
p
. (22.320) This allows us to compute the alternate integral, analogous to (22.315),
Z \infty dT
e−s(T +\chiTs[BT ;z]) −e−sT
, (22.321) using the same procedure as for (22.317):
\Gamma
Z \infty
i = \sqrt\pi \Gamma
Z \infty
\sqrt
p
(22.322) The motivation for this form of the integral isn’t completely clear at this point, but the main idea is that it contains the relevant functional for the problem. 22.7.3 Inverse Moments Now going back to the original path integral (22.302), we see that for a dielectric interface we need to evaluate a path integral of the form
Z \infty dT
**
++ , (22.323) where \chi \ge 0 is the dielectric susceptibility, z is the distance from the vacuum–dielectric interface to the atom (which is on the vacuum side).
22.7.4 Result: 3D Electromagnetism¶
22.7 TE Casimir–Polder Path Integral: Dielectric Interface The slightly more useful form of Eq. (22.323) is
Z \infty dT
**
++ , (22.324)
\Gamma(\alpha) Z \infty ds s\alpha−1DD e−sxEE = x\alpha
, (22.325) and comparing this to Eq. (22.321), we see that
\Gamma(\alpha) Z \infty
(22.326) Putting in the result (22.322) for ˜ID(z),
\sqrt\pi \Gamma(\alpha)\Gamma
Z \infty ds s\alpha−1 Z \infty
\sqrt
p
(22.327) Now we just need to simplify this. Scaling out the distance dependence by letting \lambda −\rightarrow \lambda/8z2 and s −\rightarrow s/8z2,
\sqrt\pi
zD Z \infty ds s\alpha−1 Z \infty
\sqrt
(22.328) where we have used the property of the reflection coefficient (22.274) that it is invariant if both parameters are scaled in the same way. Then letting \lambda −\rightarrow \lambda −s,
\sqrt\pi
zD Z \infty ds s\alpha−1 Z \infty s
\sqrt \lambda \sqrt \lambda
(22.329) and then letting \lambda −\rightarrow \lambdas,
\sqrt\pi
zD Z \infty ds sD/2−1 Z \infty
\sqrt \lambdas \sqrt \lambda
= \sqrt\pi
zD Z \infty
\sqrt \lambda Z \infty ds sD/2−1 e− \sqrt \lambdas, (22.330)
zD Z \infty
(22.331) This is as far as we can go without being more explicit about \alpha and D. 22.7.4 Result: 3D Electromagnetism Now comparing the original path integral (22.302) with the integral (22.323), we see that for the vacuum– dielectric interface we may write V (TE) CP
¯hc\alpha0
ID,3/2(z). (22.332)
22.7.5 Result: 1D Electromagnetism¶
Chapter 22. Electromagnetic Casimir Energies as Path Integrals With the result (22.331), this becomes V (TE) CP
\Gamma(D)¯hc\alpha0
D/2 −1 ϵ0zD Z \infty
(22.333) Now using the duplication formula for the gamma function20
(22.334) and setting z = D/2, we have the useful result
(22.335)
V (TE) CP
¯hc\alpha0
Z \infty
(TE Casimir–Polder potential) (22.336) where again
\sqrt
\sqrt
. (22.337) For D = 4, we have V (TE) CP
3¯hc\alpha0 128\pi2ϵ0z4 Z \infty
\sqrt \lambda
\sqrt \lambda ! , (22.338) and evaluating the remaining integral, we find V (TE) CP
32\pi2ϵ0z4
6 + 1 \chi −
2\chi
2\chi3/2 ! . (TE Casimir–Polder potential) (22.339) This matches the earlier result (22.245) that we obtained using the explicit probability density for the sojourn time. 22.7.5 Result: 1D Electromagnetism
handle the removable singularity here via dimensional regularization, letting D −\rightarrow 2+ only after we remove the divergences. We begin by putting in D = 2 wherever it is not objectionable to do so: V (TE) CP
32\piϵ0z2 Z \infty
(22.340) Integrating by parts, we have V (TE) CP
(D −2)¯hc\alpha0 D/2 −1 32\piϵ0z2 h
i\infty − (D −2)¯hc\alpha0 D/2 −1 32\piϵ0z2 Z \infty
h
i . (22.341) 20Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions (Dover, 1965), p. 256, Eq. (6.1.18).
22.8 Evaluation of the TM Interface Potential Since D > 2, the boundary terms vanish at both the lower limit (due to the \lambda −1 factor) and upper limits (since the integrand, except for the D-dependent factor, decays asymptotically more quickly than \lambda−3/2, and we are assuming D is close to 2). Also, canceling the (D −2) terms, V (TE) CP
¯hc\alpha0 16\pi\Gammaϵ0z2 Z \infty
h
i . (22.342) Now the expression is regular at D = 2, so we will take D −\rightarrow 2+: V (TE) CP
16\piϵ0z2 Z \infty
h
i . (22.343) The remaining integration is easy, and so V (TE) CP
¯hc\alpha0
16\piϵ0z2
,
(22.344)
and that the potential is overall negative because of the sign of the coefficient. Also the reflection coefficinet here approaches −1 as \chi −\rightarrow \infty, so this result is consistent with our earlier strong-coupling calculation
(22.269). To see that this result is consistent with the earlier calculations of Section 22.4.3, note that the analogue
Z \infty dT T 2 DD
= Z 1
Z \infty dT T 2 "r 4(1 −x) \pixT
1 −2 T e−1/T erfc r x (1 −x)T # = Z 1
1 \sqrtx −1 =
(22.345)
V (TE) CP
16\piϵ0z2 Z \infty dT T 2 ** 1 −
B(t); \sqrt 2T
B(t)
16\piϵ0z2
. (22.346) Thus, we obtain the same result with that method. 22.8 Evaluation of the TM Interface Potential Recall from Sections 22.4.4 and 22.4.5 that the TM potential in the case of a dielectric,
i , (22.347) becomes singular and difficult to work with for a dielectric interface. Here, we will demonstrate how to work with this potential for an arbitrary dielectric susceptibility. To keep things simple, we will work with
22.8.1 Regularized TM Potential¶
Chapter 22. Electromagnetic Casimir Energies as Path Integrals a one-dimensional geometry and work with a simple model path integral that involves only this potential (and hence will have no obvious physical interpretation). Going to one dimension, the potential (22.347) becomes
\partial z log p ϵr(z) 2 −\partial 2 z log p ϵr(z) . (22.348) We want to consider the simple case of a planar interface, where
(22.349)
In this case the potential becomes
log p
2 \delta2(z −d) − log p
\delta′(z −d) , (22.350) which is, again, a rather singular potential to be stuck with. 22.8.1 Regularized TM Potential As a regularized form of this potential, let us take the logarithm of the permittivity to ramp linearly over a transition length a, log p
(z < d) (z −d) \Xi a (d < z < d + a) \Xi (z > d + a), (22.351) where we have introduced the coupling parameter \Xi := log p
(dielectric coupling parameter for TM potential) (22.352) It is not yet obvious, but the explicit form of the function tanh \Xi = \chi
(22.353) (path-integral coupling) will be useful in what follows, as this is what will appear in the resulting path integrals. In this case, the regularized potential (22.348) becomes
2a h
i , (regularized TM dielectric potential) (22.354) where 1A(x) is the indicator function for the set A (i.e., the function is unity if x \in A and zero otherwise). Note that this potential ‘‘converges’’ to the singular potential (22.350) as a −\rightarrow 0. (The convergence here is in the sense of any delta-function limit: the limit should only be taken after appropriate integrations have taken place that do away with the singularities.) Thus, a path integral over Brownian bridges Bt(t′) involving this potential requires consideration of the regularized interface functional M[Bt; d, \Xi, a] := Z t dt′ VTM Bt(t′) = \Xi 2a Z t dt′ \Xi a 1[d,d+a] h Bt(t′) i
h Bt(t′) −(d + a) i −\delta h Bt(t′) −d i = \Xi 2a \Xi a Ts Bt; [d, d + a] + ℓ Bt; d + a −ℓ Bt; d . (regularized interface functional) (22.355)
22.8.3 Feynman-Kac Formula¶
22.8 Evaluation of the TM Interface Potential In the last expression here, we have replaced the integral of the three terms with the sojourn time Ts of Bt in the interval [d, d + a] Ts[y; A] := Z t dt′ 1A[y(t′)] (22.356) (this generalizes the sojourn time across a boundary, as in Section 17.12), and the local time of the process Bt at d and d + a (Section 17.13): ℓ[y; a] := Z t dt′ \delta y(t′) −a . (22.357) The regularized interface functional is, of course, interesting as a route to evaluating path integrals involving the interface functional, which we will define via N[Bt; d, \Xi] := lim a\rightarrow 0 M[Bt; d, \Xi, a] = \Xi Z t dt′ \Xi \delta2h Bt(t′) −d i −\delta′h Bt(t′) −d i . (22.358) (interface functional) Again, any calculations for N will need to involve M, only then taking the defined limit when possible without causing problems. 22.8.2 Model Path Integral Now having defined the interface functional, we will evaluate the model path integral
Z \infty dt \sqrt t e−\lambdat
exp −N[Bt; d, \Xi] Bt (22.359) to demonstrate the TM potential and gain some intuition. Of course, to do this, we will need to consider the regularized path integral
Z \infty dt \sqrt t e−\lambdat
exp −sM[Bt; d, \Xi, a] Bt , (22.360) and then take the limit a −\rightarrow 0. We have also introduced the parameter s to highlight the delicate interaction of the \delta2 and \delta′ parts of the interface functional; as it turns out, that the a −\rightarrow 0 limit will only lead to sensible results if s = 1. 22.8.3 Feynman–Kac Formula 22.8.3.1 Setup To work out the path integral (22.360), we start with the ODE part of the FK formula (22.300),
f(x) −2g(x), (22.361) choosing the potential
(22.362) where
2a,
2a2 . (22.363)
For x < d or x > d + a, the ODE is
(22.364)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals with general solutions
\sqrt
\sqrt
eikx
(22.365) for coefficients \alpha\pm to be determined in each region. On the other hand, for d < x < d + a, the ODE is
(22.366)
p
p
eikx
(22.367) for coefficients \alpha\pm to be determined. Picking the bounded solutions in each domain, we have the solution
Ae− \sqrt
eikx
(x > d + a) Be− p
p
eikx
(d < x < d + a) De \sqrt
eikx
(x < d), (22.368) for undetermined constants A, B, C, and D. 22.8.3.2 Boundary Matching
Ae− \sqrt
p
p
- eik(d+a)
Be− p
p
eikd
\sqrt
eikd
(22.369)
− \sqrt 2\lambdaAe− \sqrt
−2\sigma Ae− \sqrt
= − p
p
- p
p
ikeik(d+a)
− p
p
- p
p
- ikeik(d+a)
\sqrt 2\lambdaDe \sqrt
ikeikd
De \sqrt
eikd
, (22.370)
Since we are only in interested in f(0), we only need solution of these four equations for the coefficient D. As it turns out, this expression is cumbersome, but it is somewhat less so after integration over k: 2\pi Z \infty −\infty dk D = − \sqrt 2\lambda
\sqrt
sinh h a p
i e−2 \sqrt 2\lambda d
h a p
i −2 p
h a p
i . (22.371)
22.8 Evaluation of the TM Interface Potential Then note that¶
(22.372) and since we want to calculate
\sqrt 2\pi Z \infty −\infty
Z \infty dt \sqrt t e−\lambdat
exp −sM[Bt; d, \Xi, a] , (22.373) from the FK formula (22.299), we can use the integral formula \sqrt 2\pi Z \infty −\infty dk e−ikc
r\pi \lambda e− \sqrt 2\lambda|c| (22.374) to find
r\pi \lambda 1 −
\sqrt
sinh h a p
i e−2 \sqrt 2\lambda d
h a p
i −2 p
h a p
i , (22.375) which is our basic result for the regularized model integral (22.360). 22.8.3.3 Overall Scaling of the Potential Now we turn to the dependence of this result on the parameter s. Recall that we have used the notation (22.363)
2a,
2a2 (22.376) to render the expression a bit more compact. In the case of s = 1,
2a2 = 0, (22.377) and so
r\pi \lambda 1 + \sqrt
h a p
i e−2 \sqrt 2\lambda d \lambda sinh h a p
i + p
h a p
i . (22.378) This is important in analyzing the subsequent limit a −\rightarrow 0, where the 2\sigma2 −\sigma′ terms in the numerator and denominator would dominate the other terms, leading to the \Xi-independent result
r\pi \lambda 1 −e−2 \sqrt 2\lambda d , (22.379) This result is characteristic of a ‘‘strong-coupling’’ result. For example this is equivalent to the path-integral
the right balance between local and occupation times to produce a nontrivial limit. This also shows that VTM (alternately, N) is not a potential in the usual sense (for example, doubling it does not double the ‘‘energy’’); rather its function is to enforce a boundary condition at the interface, which has an arbitrarily large ‘‘cost’’ in terms of the energy. 22.8.3.4 Sharp-Interface Limit Then in the limit a −\rightarrow 0, a p
p
(22.380)
22.8.4 Weak-Coupling Expansion¶
Chapter 22. Electromagnetic Casimir Energies as Path Integrals
r\pi \lambda 1 + tanh(\Xi) e−2 \sqrt 2\lambda d , (model TM path-integral result) (22.381) which is a nontrivial result. As \Xi −\rightarrow 0, this reduces to L(\lambda, s; d, \Xi, a) −\rightarrow r\pi \lambda, (22.382) which is reasonable, as it is the same as the d −\rightarrow \inftydecoupling limit. For small \Xi, the result is
r\pi \lambda 1 + \Xi e−2 \sqrt 2\lambda d . (22.383) On the other hand, as \Xi −\rightarrow \infty, this reduces to
r\pi \lambda 1 + e−2 \sqrt 2\lambda d , (22.384) which is also a reasonable result, in the sense of avoiding pathologies and depending sensibly on d. 22.8.4 Weak-Coupling Expansion As a check for the small-\Xi limit, we can keep only the first-order terms in \Xi in the potential (22.358), N[Bt; d, \Xi] = lim a\rightarrow 0 M[Bt; d, \Xi, a] = lim a\rightarrow 0 \Xi 2a Z t dt′ \delta h Bt(t′) −(d + a) i −\delta h Bt(t′) −d i = −\Xi Z t dt′ \delta′h Bt(t′) −d i = −\partial 2 d \Xi Z t dt′ \Theta h Bt(t′) −d i . (22.385) In this regime, the path integral (22.359) becomes
Z \infty dt \sqrt t e−\lambdat
exp −N[Bt; d, \Xi] Bt = Z \infty dt e−\lambdat
1 −N[Bt; d, \Xi]
Bt = Z \infty dt \sqrt t e−\lambdat
d \Xi Z t dt′ \Theta h Bt(t′) −d i Bt = Z \infty dt \sqrt t e−\lambdat
d \Xi DD Ts[Bt; d] EE , (22.386) where we have written this now as an integral over the first moment of the sojourn time. We can use the expression (17.588), generalized to pinning at time t, DD Ts[Bt, d] EE = t
e−2d2/t − r \pid2 2t erfc "r 2d2 t #! , (22.387) and then evaluate the integral, with the result
r\pi \lambda 1 + \Xi 2 \partial 2 d e−2 \sqrt 2\lambda d 4\lambda = r\pi \lambda 1 + \Xi e−2 \sqrt 2\lambda d . (22.388)
22.8.5 Transfer-Layer Formalism¶
22.8 Evaluation of the TM Interface Potential This agrees with the earlier small-\Xi calculation (22.383). Note that there are no issues with obtaining the wrong sign in the expanded integral here. 22.8.5 Transfer-Layer Formalism
derivation of Section 22.8.3 to take this limit as early as possible, and to thereby do it ‘‘once and for all.’’
function. Then for x < d or x > d + a we are considering the homogeneous ODE
(22.389) and thus referring back to the function f(x) defined in Eq. (22.368), we should instead work with the solution
Ae− \sqrt 2\lambda x (x > d + a) Be− p
p
(d < x < d + a) De \sqrt 2\lambda x (x < d). (22.390)
as
p
p
Be− p
p
(22.391) and we can rewrite the boundary conditions (22.370) for f ′(x) as
p
p
- p
p
− p
p
p
p
(22.392)
solving these four equations for f(d −0+) and f ′(d −0+), while eliminating B and C, then substituting in
2a,
2a2 , (22.393) then setting s = 1 and taking the limit a −\rightarrow 0, we find the remarkably simple result that the interface functional defines a discontinuity in the solution f(x) according to
(TM boundary condition in FK solution) (22.394) These boundary conditions replace the usual continuity conditions at the interface. For example, suppose we apply this boundary condition to
Ae− \sqrt 2\lambda x (x > d) Be− \sqrt
\sqrt 2\lambda x (0 < x < d) De \sqrt 2\lambda x (x < d). (22.395)
22.8.6 Mellin Transform¶
Chapter 22. Electromagnetic Casimir Energies as Path Integrals
Ae− \sqrt
\sqrt
\sqrt 2\lambda d − \sqrt 2\lambdaAe− \sqrt
\sqrt 2\lambdaBe− \sqrt
\sqrt 2\lambdaCe \sqrt 2\lambda d, (22.396) and the boundary conditions at x = 0 read
\sqrt
\sqrt
\sqrt 2\lambdaC. (22.397) Solving for D gives D = \sqrt 2\lambda 1 + tanh(\Xi) e−2 \sqrt 2\lambda d , (22.398) and since
\sqrt
\sqrt
r\pi \lambda 1 + tanh(\Xi) e−2 \sqrt 2\lambda d , (22.399) we have recovered the result (22.381), but now by directly applying the interface boundary conditions (22.394).
with the solution function
Ae− \sqrt
eikx
(x > d) Be \sqrt
eikx
(x < d), (22.400) which is basically the solution (22.368), but with no ‘‘middle’’ region d < x < d + a. The analysis of the thin boundary layer still applies, since the inhomogeneous part of the solution is ignorable in that region.
Ae− \sqrt
eikd
Be \sqrt
eikd
e−\Xi − \sqrt 2\lambdaAe− \sqrt
ikeikd
\sqrt 2\lambdaBe \sqrt
ikeikd
e\Xi, (22.401) and solving for B gives B = \sqrt 2\lambda −ike\Xi e\Xi −1 eikd− \sqrt 2\lambda d
\sqrt
. (22.402) Then using
\sqrt 2\pi Z \infty −\infty
\sqrt 2\pi Z \infty −\infty dk
, (22.403) we again recover the result (22.381), but with considerably less effort. 22.8.6 Mellin Transform To continue an example with the TM interface potential, we will compute the path integral
Z \infty dT
e−N[BT ;z,\Xi] −1
BT . (22.404)
22.8 Evaluation of the TM Interface Potential This has the form of the TM path integral (22.247), but simplified to only the exponential factor, as a ‘‘warm up’’ to doing the full path integral. Here we will follow the same method as in Section 22.7.2. Beginning with Eq. (22.381),
r\pi \lambda 1 + tanh(\Xi) e−2 \sqrt 2\lambda z , (22.405) we then use Eq. (22.284) in the form
\Gamma
Z \infty
i =
\Gamma
Z \infty
\sqrt 2\lambda z, (22.406) or evaluating the last integral, we have
23D/2−1\Gamma
zD . (22.407) Using Eq. (22.335) in the form
(22.408) we have
2D/2 zD
2D/2 zD \chi
, (22.409) (model TM path integral) as the result for our model TM path integral, where we have used Eq. (22.353) to connect the result back to the susceptibility \chi. 22.8.6.1 Comparison to Strong-Coupling It is useful to compare this result with the strong-coupling limit of physical path integrals. For example, we can take the TE strong-coupling result (22.222), and remove the prefactor in the expression (22.219) to obtain the analogous result I (TE) D
\chi\rightarrow \infty Z \infty dT
**
x(\tau) −1 ++ x(\tau)
2D/2zD . (22.410) Note that this is a generic result, occurring whenever there is a strong effect on a path touching the dielectric interface. For example, the sojourn-time path integral (22.315),
Z \infty dT
e−sTs[BT ;z] −1
, (22.411) has the form of a Casimir–Polder path integral for a massless scalar field coupled to a potential step, where s governs the ‘‘strength’’ of the potential. The result (22.317) for this path integral was
\sqrt\pi \Gamma
Z \infty
\sqrt 2\lambda z \sqrt
\sqrt \lambda \sqrt
\sqrt \lambda ! . (22.412) In the limit s −\rightarrow \infty, this becomes
\sqrt\pi \Gamma
Z \infty
\sqrt 2\lambda z = −
23D/2−1\Gamma
zD , (22.413)
22.8.7 Generalization for Open Bridges¶
Chapter 22. Electromagnetic Casimir Energies as Path Integrals which after applying Eq. (22.408) is equivalent to Eq. (22.410). Thus, we will compare the TM model path integral (22.409) to the strong-coupling result (22.410). The factor (−tanh \Xi) appearing in (22.409) is an ‘‘efficiency factor’’ multiplying the expected strong-coupling limit, and note that the sign is opposite to what one might expect for an efficiency. This is because while most of the potential is positive, there is a negative component, which when exponentiated can spike to large, positive values, thus yielding a positive result even after subtracting away the z −\rightarrow \inftypart (normally, a negative potential keeps the exponential less than one, and when subtracting one, the result is strictly negative). In the limit \Xi −\rightarrow \infty(equivalently, \chi −\rightarrow \infty), this integral is equivalent to the strong-coupling result, except for the overall minus sign. This explains the sign discrepancy that we found in Section 22.4.4 when we tried to cheat to obtain the strong-coupling result by considering only the \delta2 part of the potential. There, we obtained the ‘‘usual’’ strong-coupling result, which is wrong by an overall sign compared to the ‘‘correct’’ result we are getting here. 22.8.7 Generalization for Open Bridges A useful generalization of the path integral in Eqs. (22.359) and (22.381) is to change the path from ‘‘loop’’ Brownian bridges Bt(t′) to bridges Bt(0\rightarrow c)(t′) pinned at 0 and some other point c. In particular, our goal will be to compute the interface path integral
e−N[Bt(0\rightarrow c);d,\Xi]
Bt(0\rightarrow c) , (22.414) (interface path integral) which is useful in numerically evaluating path integrals with the interface potential. 22.8.7.1 Feynman–Kac Formula To start, we will derive an expression for the path integral
Z \infty dt e−\lambdat e−c2/2t \sqrt t
exp −N[Bt(0\rightarrow c); d, \Xi] Bt(0\rightarrow c) , (22.415)
and the interface functional N here is defined as N[Bt; d, \Xi, c] := lim a\rightarrow 0 \Xi 2a Z t dt′ \Xi a 1[d,d+a] h Bt(0\rightarrow c)(t′) i
h Bt(0\rightarrow c)(t′) −(d + a) i −\delta h Bt(0\rightarrow c)(t′) −d i = lim a\rightarrow 0 \Xi 2a \Xi a Ts Bt(0\rightarrow c); [d, d + a] + ℓ Bt(0\rightarrow c); d + a −ℓ Bt(0\rightarrow c); d
(22.416) in analogy to the path case in Eqs. (22.355) and (22.358). The path integral here generalizes the integral (22.359) for path bridges. In solving this problem, we will employ the transfer-layer formalism in the limit a −\rightarrow 0, embodied by
function (22.400)
Ae− \sqrt
(x > d) Be \sqrt
(x < d), (22.417)
22.8 Evaluation of the TM Interface Potential¶
Ae− \sqrt
Be \sqrt
e−\Xi − \sqrt 2\lambdaAe− \sqrt 2\lambda d + ikeik(d−c)
\sqrt 2\lambdaBe \sqrt 2\lambda d + ikeik(d−c)
e\Xi, (22.418) and solving for B gives B = \sqrt 2\lambda −ike\Xi e\Xi −1 eik(d−c)− \sqrt 2\lambda d
\sqrt
. (22.419) Then using
\sqrt 2\pi Z \infty −\infty
\sqrt 2\pi Z \infty −\infty dk e−ikc
, (22.420) we obtain
r\pi \lambda
e− \sqrt
1 −e−\Xi sgn(d −c) + e−\Xi 1 + e−2\Xi e− \sqrt
! . (22.421) Throughout, we have assumed d > 0, so we can also write this as
r\pi \lambda
e− \sqrt
1 −e−\Xi sgn(d −c) + e−\Xi 1 + e−2\Xi e− \sqrt
! . (22.422) The absolute value we introduced here is important in maintaining the choice of bounded solution when we generalize to d < 0. An equivalent result also arises if we make the simultaneous replacement d −\rightarrow −d, c −\rightarrow −c, and \Xi −\rightarrow −\Xi, given the reflection symmetry of the problem (which reverses the ‘‘direction of crossing’’ through the boundary, hence the change in \Xi),
r\pi \lambda
e− \sqrt
1 −e\Xi −sgn(d −c) + e\Xi 1 + e2\Xi e− \sqrt
! , (22.423) so in general, we have
r\pi \lambda
e− \sqrt
1 −e−sgn(d)\Xi sgn(d) sgn(d −c) + e−sgn(d)\Xi 1 + e−sgn(d)2\Xi e− \sqrt
! = r\pi \lambda
e− \sqrt
sgn(d −c) esgn(d)\Xi/2 + sgn(d) e−sgn(d)\Xi/2 cosh(\Xi) e− \sqrt
! , (22.424) which is now valid for any c and d (positive or negative) 22.8.7.2 Inversion of the Laplace Transform Now to compute the path integral (22.414), we must invert the Laplace transform in Eq. (22.415). Using the inverse Laplace transform L −1 r\pi \lambda e− \sqrt
\sqrt
(22.425) we have L −1h L(\lambda; d, \Xi, c) i
\sqrt t +
sinh(\Xi/2) sgn(d −c) esgn(d)\Xi/2 + sgn(d) e−sgn(d)\Xi/2 cosh(\Xi) !
\sqrt t . (22.426)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals Equating this with the inverse transform of Eq. (22.415), L −1h L(\lambda; d, \Xi, c) i
\sqrt t
exp −N[Bt(0\rightarrow c); d, \Xi] Bt(0\rightarrow c) , (22.427) we then find
sgn(d −c) esgn(d)\Xi/2 + sgn(d) e−sgn(d)\Xi/2 cosh(\Xi) e c2−(|d|+|c−d|)2 /2t (interface path integral) (22.428) as our desired result for the path integral defined in Eq. (22.414). Note that the exponential factor in the second term is unity if d is between 0 and c (i.e., if the path endpoints straddle the boundary), and is the
and d). The \Xi-dependent factor reduces to sech (\Xi)−1 again if d is between 0 and c (i.e., a boundary-crossing
is restricted to the range (−1, 1), and so the functional (22.428) is strictly positive, as it should be, from the definition (22.414). The expression for N(d, \Xi, t, c) has the interpretation of representing probabilities for reflection and transmission at the interface. We can see this by letting d −\rightarrow 0, so that the crossing probability reduces to unity, being careful to maintain the sign of d. In this case, we may write
tN (\Xi) (c > 0) 1 + rN (\Xi) (c < 0)
1 −rN (\Xi) (c > 0) tN (\Xi) (c < 0), (interface path integral, starting at interface) (22.429) where the effective reflection and transmission coefficients are
\chi
(reflection/transmission coefficients for interface path integral) (22.430) The interpretation here is as follows. Suppose we follow a (Wiener) stochastic path until it just bumps into the interface. The probability density for the path position x after a time t later is then given by the Gaussian measure exp[−(x−d)2/2t]/ \sqrt 2\pit, multiplied by the appropriate factor in Eq. (22.429) for each side of the interface. An enhancement of probability on one side compared to the other is exactly what we expect from a reflection/transmission process at an interface. However, since sech2x + tanh2 x = 1, it is not the case that rN + tN = 1; in fact, rN + tN \ge 1, with equality only achieved for \chi = 0 or in the limit \chi −\rightarrow \infty. Rather, the coefficients satisfy the ‘‘probability-conservation relation’’ r 2 N + t 2 N = 1. Thus, each ‘‘collision’’ of the path with the interface does not conserve the path amplitude. Of course, we do not necessarily require this, since the path integral must only reproduce some correct value on average. However, it is interesting to note that for a closed Brownian bridge, which must have an equal number of crossings in either direction, and thus transmission factors contribute only in powers of t 2 N . On the other hand, evidently any power of rN may arise. In the limit where \chi −\rightarrow \infty, we have rN −\rightarrow 1 and tN −\rightarrow 0, indicating perfect reflection (and amplitude conservation on the left-hand side of the interface). Note that for finite \chi, in evaluating the TM potential, there are also losses on the right-hand side of the interface associated with the \langle \chi\rangle −\alpha factor in the Casimir path integral that compensate for the extra probability generated on each reflection.
22.8.8 Assembling Closed Bridges¶
22.8 Evaluation of the TM Interface Potential 22.8.8 Assembling Closed Bridges The path integral (22.414), which takes on the value given in Eq. (22.428), is an average of an exponential functional over a Brownian bridge pinned from 0 to c in time t. Thus, if we split the Brownian bridge into two temporal ‘‘segments’’ (0, t/2) and (t/2, t), then note that the exponential factors, and by construction we should be able to recover the original result by summing over all possible values of the Brownian bridge at time t/2, with the appropriate weight. Specifically, we are saying that by construction N(d, \Xi, t, c) satisfies
Z \infty −\infty dc N(d, \Xi, t/2, c) N(d −c, \Xi, t/2, −c) fG(c; 0, t/4), (22.431) where
\sqrt
(22.432) is the normal (Gaussian) probability density. Recall that from our analysis of finite-bridge generation
points B0, . . . BN, where B0 = BN = 0, by using the forward recurrence
Bn = zn s N −n
N −n N −n + 1 Bn−1. (22.433)
variance t/4 for the density of c follows from the distribution of B1 for the case N = 2. As a numerical example, the plot below shows the integrand of Eq. (22.431) for three values of \Xi, with
Xo=o0.001 Xo=o1 Xo=o1000 c -5 -4 -3 -2 -1 integrand Note the discontinuity at c = d in the integrand, where the source point of the Brownian bridge just touches
well-approximated by the Gaussian factor, with increasingly large deviations above and below the Gaussian as \Xi increases. The impact of this observation is as follows. Suppose we evaluate the integral in a Monte- Carlo fashion, by choosing random values of c according to the Gaussian factor fG(c; 0, t/4), and averaging the values taken on by the rest of the integrand for each chosen c. This would be fairly simple at small \Xi, but at large \Xi, this introduces relatively large fluctuations in the integrand value, and thus slowed convergence. Of course, the slow convergence gets dramatically worse if we put in many intermediate integrals, developing a path integral. For example, the generalization of (22.431) for two intermediate points is
Z dc1 dc2 N(d, \Xi, t/3, c1) N(d−c1, \Xi, t/3, c2−c1) N(d−c2, \Xi, t/3, −c2) f1(c1)f2(c2), (22.434)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals where from Eqs. (22.433), we have
(22.435) since the cn corresponds to a particular value for Bn, and in this case N = 3. We are also here using the usual notations ∆t := t N ,
N −n N −n + 1. (22.436) Note the recursive nature of the definition here, such that successive integrations have the form of convolu- tions. For the general case of N −1 intermediate points, this generalizes to
Z dc1 \cdot \cdot \cdot dcN−1 N(d −cN−1, \Xi, ∆t, cN −cN−1) N−1 Y j=1 N(d −cj−1, \Xi, ∆t, cj −cj−1) fj(cj), (path integral for TM potential) (22.437) where c0 = cN = 0. Again, whenever \Xi is large and there is significant ‘‘overlap’’ of the integral with the TM boundary at d, the N factor will have large variation. For large N, there will be many such factors, leading to large fluctuations of the path-integral samples (with c-paths), and thus poor convergence, becoming increasingly worse for large N. Of course, this is an artifact of choosing the product of all fj(cj) as the path-integral measure, when really we should absorb the fluctuations into the path measure to the greatest extent we can. In particular, we should define the ‘‘tempered’’ path-step distributions
(22.438) where we have introduced the normalization factors \etaj, defined by \eta−1 j := Z \infty −\infty dcj N(d −cj−1, \Xi, ∆t, cj −cj−1) fj(cj) = Z \infty −\infty dcj N(d −cj−1, \Xi, ∆t, cj −cj−1) fG(cj; \xijcj−1, \xij∆t) = Z \infty −\infty dx N(d −cj−1, \Xi, ∆t, x) fG[x; (\xij −1)cj−1, \xij∆t]. (22.439) This has the analytic solution \eta−1 j
cosh \Xi + sgn(d −cj−1)sinh2(\Xi/2) cosh \Xi erf
d −\xijcj−1 p 2\xij∆t ! + tanh \Xi " sgn(d −cj−1) −erf
p 2\xij∆t !# e2(\xij−1)d(d−cj−1)/∆t
N −j N −j + 1, (22.440) at least for cj−1̸ = d, since this factor has a discontinuity at cj−1 = d. The precise value cj−1 = d doesn’t really matter for integration, but it is sensible to define the function at this point to have the mean of the values for cj−1 = d + 0\pm. Note again that all these are defined recursively in terms of the previous point cj−1 of the path. Thus, we have the tempered form of the path integral (22.437):
Z \infty −\infty dc1 \cdot \cdot \cdot dcN−1 N(d −cN−1, \Xi, ∆t, cN −cN−1) N−1 Y j=1 \eta−1 j ¯fj(cj). (22.441) Then taking the product of the ¯fj(cj) as the path measure, the tempered path integral becomes
** N(d −¯cN−1, \Xi, ∆t, ¯cN −¯cN−1) N−1 Y j=1 \eta−1 j ++ ¯c(t′) (22.442)
22.9.1 Feynman-Kac Formula¶
22.9 TM Casimir–Polder Path Integral: Dielectric Interface in Monte-Carlo form, which behaves much better than the Gaussian path integral (as the \etaj factors are essentially smoothed versions of the N distributions), but at the expense of more complicated paths. Note in this last expression that we have switched the path to the ¯c notation to emphasize that this path is generated by the ¯f path measure. 22.9 TM Casimir–Polder Path Integral: Dielectric Interface To apply the method for handling the interface functional in the previous section, we will evaluate the TM path integral (22.247) V (TM) CP
¯hc\alpha0
Z \infty dT
x(\tau) −T 2 \nabla 2
++ x(\tau) (22.443) for a planar, vacuum–dielectric interface. The derivation here will parallel the TE treatment in Section 22.7, but with additions from the TM-potential treatment in Section 22.8. 22.9.1 Feynman–Kac Formula We will begin by deriving an expression for the path integral
Z \infty dt \sqrt t e−\lambdat
exp −N[Bt; d, \Xi] −sTs[Bt; d] Bt , (22.444)
sojourn time, and the functional N is as defined in Eq. (22.358). This generalizes the corresponding TE expression (22.304) by introducing the interface functional at the boundary. In working out this integral, we will employ the transfer-layer formalism of Section 22.8.5 (in the sharp-interface limit a −\rightarrow 0), embodied by the boundary conditions (22.394). We will thus generalize the
function (22.400),
Ae− p
eikx
(x > d) Be \sqrt
eikx
(x < d), (22.445) which modifies the denominator in the x > d region to account for the sojourn time. Again, we are taking
the setup for the sojourn-time calculation [cf. Eq. (17.556)], except that we will apply the modified boundary conditions (22.394) instead of simple continuity of the function and the derivative.
Ae− p
eikd
Be \sqrt
eikd
e−\Xi − p
p
ikeikd
\sqrt 2\lambdaBe \sqrt
ikeikd
e\Xi, (22.446) and solving for B gives B = e− \sqrt
\sqrt
\sqrt −
\sqrt
\sqrt
\sqrt
\lambda . (22.447) Then using
\sqrt 2\pi Z \infty −\infty
\sqrt 2\pi Z \infty −\infty dk
, (22.448)
22.9.3 Inverse Moments¶
Chapter 22. Electromagnetic Casimir Energies as Path Integrals we obtain
r\pi \lambda " 1 −e−2 \sqrt 2\lambda d \sqrt
\lambda \sqrt
\lambda !# = r\pi \lambda h 1 + e−2 \sqrt
i , (22.449) where we are using the shorthand
\lambda − \sqrt
e2\Xi\sqrt
\sqrt
, (22.450) which will later become the usual TM reflection coefficient. Notice the similarity here of Eq. (22.449) to Eq. (22.304), to which it reduces as \Xi −\rightarrow 0. 22.9.2 Mellin Transform Now we want to use the Mellin-Laplace-transform conversion formula (22.284) to compute the path integral
Z \infty dT
e−N[BT ;z,\Xi]−s(T +\chiTs[BT ;z]) −e−sT
, (22.451) in analogy to the TE integral (22.321). To do this, we rewrite (22.449) by replacing t with T , let s −\rightarrow s\chi and then let \lambda −\rightarrow \lambda + s:
Z \infty dT \sqrt T e−\lambdaT
e−s(T +\chiTs[BT ])
= r \pi
h 1 + e−2 \sqrt
i , (22.452) as in Eq. (22.320). Then we can transform this result using Eq. (22.284),
\Gamma
Z \infty
i = \sqrt\pi \Gamma
Z \infty
\sqrt
p
(22.453) as in Eq. (22.322). Notice that our procedure here parallels exactly the TE calculation, except for the reflection coefficient, which has includes two factors of e2\Xi compared to rTE(\lambda + s; s\chi). Recall that \Xi is determined completely by \chi [Eq. (22.352)], so we won’t need to note the dependence of the result on \Xi. (As far as the derivation was concerned, they were independent parameters, but we are now identifying them as being related.) 22.9.3 Inverse Moments In converting the result (22.453) into the worldline path-integral result, the calculation here continues to closely parallel that of Section 22.7.3. We thus require the TM version of the TE integral (22.323), which is
Z \infty dT
** e−N[BT ;z,\Xi]
++ . (22.454) To do this, we apply the inverse-moment formula (22.295) to Eq. (22.453). Again, the functional form of the integral here is essentially the same as what leads to the result (22.331), except for the (constant) factors of
zD Z \infty
= −
zD Z \infty
(22.455)
22.9.4 Result: 3D Electromagnetism¶
22.9 TM Casimir–Polder Path Integral: Dielectric Interface where we used¶
\sqrt
\sqrt
(22.456) with rTM(\lambda; \chi) being the usual Fresnel coefficient for TM polarization, as defined in Eq. (22.279). Again, to proceed, we will need to choose specific values of \alpha and D, and to obtain Casimir–Polder energies, we will need to consider combinations of these integrals with different values of \alpha and D. 22.9.4 Result: 3D Electromagnetism Comparing the TM path integral (22.443) for the Casimir–Polder potential V (TM) CP
¯hc\alpha0
Z \infty dT
x(\tau) −T 2 \nabla 2
++ x(\tau) (22.457) to the integral definition (22.454), we see that we can write the (renormalized) TM potential as V (TM) CP
¯hc\alpha0
ID,3/2(z) −1 2\partial 2 z ID−2,1/2(z) (22.458) for an atom at a distance z from a planar dielectric interface. Thus, to begin, we will need
\Gamma(D)
Z \infty
Z \infty
(22.459) where we used the duplication formula in the form (22.335),
(22.460) which we can rewrite as \Gamma(D)
\sqrt\pi . (22.461) Similarly, we will need
\Gamma(D −2)
Z \infty
(22.462) to obtain the differentiated form \partial 2
\Gamma(D)
Z \infty
Z \infty
(22.463) Thus, we have the combination ID,3/2(z) −1 2\partial 2
Z \infty
(22.464)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals and hence the potential from (22.458): V (TM) CP
¯hc\alpha0
Z \infty
(TM Casimir–Polder potential) (22.465) Recall that from from Eq. (22.279),
\sqrt
\sqrt
. (22.466) This is the analogue of the TE potential (22.336).
V (TM) CP
3¯hc\alpha0 128\pi2ϵ0z4 Z \infty
\sqrt
\sqrt
! . (22.467) This integral is equivalent to the one that comes out of the Green-tensor analysis of the atom–wall problem
\sqrt
integral]. Adapting the solution there, we have V (TM) CP
32\pi2ϵ0z4 7
2\chi
2\chi3/2
sinh−1p
. (TM Casimir–Polder potential) (22.468) The bracketed quantity has asymptotes 43\chi/120 for small \chi and 5/6 for large \chi, and acts as an ‘‘efficiency’’ \etaTM for the TM energy compared to the full-electromagnetism, perfect-conductor result [see Eq. (14.214)]. The small-\chi result agrees with our earlier calculation (22.268). The large-\chi result also agrees with our naïve strong-coupling result (22.255), except of course for the overall sign. 22.9.4.1 Digression: Derivative-Free Path Integrals Although we have shown that we can evaluate the path integral (22.457) without problem, from a numerical point of view the presence of derivatives is a nuisance, as they tend to amplify numerical and statistical fluctations. Thus, we will spend a bit of time deriving alternative path integrals for the TM polarization without any such derivatives. One obvious approach is to start, as we have already noted, that the derivative with respect to the atomic position is a derivative with respect to the coordinate of the interface, up to a minus sign. But varying the distance to the surface is has a similar effect to varying the total path time T , because it varies the portion of the path that ‘‘contacts’’ the interface. Because a Brownian bridge running from 0 to T has an extent that scales as \sqrt T , we expect the path average in the integrand of the T integral to depend on d and T only via the combination d/ \sqrt T . This is explicitly the case, for example, for the sojourn time that appears explicitly in the TE path integral as well as the TM case [see, e.g., Eq. (22.231)]. Thus, consider a derivative with respect to the atom–surface distance d of such an integrand, which we can write schematically as \partial df d \sqrt T = \sqrt T f ′ d \sqrt T . (22.469) The analogous T derivative reads \partial T f d \sqrt T = −d 23/2 f ′ d \sqrt T . (22.470) Comparing these two expressions, we may then identify \partial d ≡−2T d \partial T (22.471)
22.9 TM Casimir–Polder Path Integral: Dielectric Interface whenever operating on such an integrand. Then in general, we have an integrand with an operator of the form
2T d \partial T
T d3 (d2\partial T )
T 3/2 d3 (d2\partial T )
d \partial T T 3/2 d3 (d2\partial T ) = 4
(22.472) where note that we were careful to ensure the variables to the right only appeared in the combination d/ \sqrt T before changing the derivative variable. Now we can integrate by parts twice under the T integral to eliminate the derivatives:
d2
d2
d2 T −D/2. (22.473) Under this substitution, the path integral (22.457) becomes V (TM) CP
¯hc\alpha0
Z \infty dT
x(\tau) −T (D −1)(D −2) d2
++ x(\tau) . (TM path integral, derivative-free) (22.474) Note, however, that while we have eliminated the derivatives, we have had to explicitly introduce the atom– surface distance d. While this is fine for the single-planar-interface calculation, it does not obviously gener- alize to more general geometries. A better path integral without an explicit distance results if we start from Eq. (22.455),
\Gamma(D)
Z \infty
= − \Gamma(D)
Z \infty
(22.475) which we can rewrite as
\Gamma(D)
Z \infty
= − \Gamma(D)
Z \infty
- \Gamma(D)
Z \infty
= 1 8\partial 2 z ID−2,1/2(z) −1 2ID,3/2(z), (22.476) after comparison with Eqs. (22.459) and (22.463). Rearranging, we have 2\partial 2
(22.477)
22.9.5 Result: 1D Electromagnetism¶
Chapter 22. Electromagnetic Casimir Energies as Path Integrals Then we can eliminate the derivative term in Eq. (22.458), which becomes V (TM) CP
¯hc\alpha0
. (22.478) Evidently, we may rewrite Eq. (22.457) in derivative-free form as V (TM) CP
¯hc\alpha0
Z \infty dT
x(\tau)
++ x(\tau) . (TM path integral, derivative- and distance-free) (22.479) Tracing the algebra for the ID,\alpha(z) back to Eq. (22.327), we can see that the replacements for the derivatives \partial 2 z (the factor of 4(D −2) on the second term, and the inverted minus sign on the first term) ultimately came from the derivative changing the power of (\lambda + s) in the integrand, due to the exponential factor that involved z. This type of factor is generic in Casimir–Polder calculations, so we expect this result to hold even in the general case of multiple objects [for two surfaces, for example, see the TE two-plane Casimir–Polder calculation, Eq. (22.619)]. 22.9.5 Result: 1D Electromagnetism
which we will again remove via dimensional regularization. The procedure is the same as before. First
V (TM) CP
32\piϵ0z2 Z \infty
(22.480) Integrating by parts gives V (TM) CP
(D −2)¯hc\alpha0 D/2 −1 32\piϵ0z2 h
i\infty − (D −2)¯hc\alpha0 D/2 −1 32\piϵ0z2 Z \infty
h
i . (22.481) Again, since D > 2, the boundary terms vanish at both the lower limit (due to the \lambda −1 factor) and upper
V (TM) CP
16\piϵ0z2 Z \infty
h
i . (22.482) Now we can evaluate the integral. The result evaluated at the upper limit vanishes. Therefore, we find V (TM) CP
¯hc\alpha0
16\piϵ0z2
,
(22.483) where we used
(22.484) which follows from Eqs. (22.274) and (22.279). This agrees with the TE result (22.344), as it should, since in 1D electromagnetism (where all waves are effectively at normal incidence), there should be no distinction between the two polarizations.
22.9.6 Casimir-Polder Potential: Full Electromagnetism¶
22.9 TM Casimir–Polder Path Integral: Dielectric Interface 22.9.6 Casimir–Polder Potential: Full Electromagnetism Now that we have worked out the TM Casimir–Polder path integral, after doing the same for the TE case quite a while back in Section 22.7.4, we will wrap up this discussion by briefly summarizing the combined results of both sections to obtain the Casimir–Polder potential in full electromagnetism, especially in the mainly relevant case of 3D electromagnetism. First, combining Eqs. (22.336) and Eqs. (22.465), we see that we can write the total potential as
¯hc\alpha0
Z \infty
i , (Casimir–Polder potential) (22.485) where the reflection coefficients are defined in Eqs. (22.274) and (22.279). We see here that the contributions add in the form of combined reflection coefficients, with an extra factor of (2\lambda −1) in the TM case. Recall that, in terms of a mode sum, this is a geometric factor: for a dielectric particle interacting with the electric field, the TE field behaves as a scalar (the electric field is always parallel to the planar interface) while it behaves as a vector in the TM case, so that the geometric factor is related to the dot product of the incident and reflected wave vectors for vacuum-incident modes.
(22.468) as potentials V (TE) CP
V (TM) CP
(TE and TM Casimir–Polder potentials) (22.486) where we have defined ‘‘efficiencies’’ relative to the strong-coupling (\chi −\rightarrow \infty) case of [cf. Eq. (14.214)]
6 + 1 \chi −
2\chi
2\chi3/2
2\chi
2\chi3/2
sinh−1p
. (TE and TM Casimir–Polder efficiencies) (22.487) We have already discussed the asymptotic behaviors of these efficiencies in Sections 22.7.4 and 22.9.4. In the full-electromagnetism case, we can thus most compactly combine the individual efficiencies
(22.488) in which case the full potential is
(TE + TM Casimir–Polder potential) (22.489) Written out explicitly, the full efficiency is [cf. Eq. (14.212)]
2\chi
2\chi3/2
sinh−1p
. (TE + TM Casimir–Polder efficiency) (22.490) This efficiency is asymptotically 23\chi/60 for small \chi, and for large \chi, \eta(\chi) −\rightarrow 1 (which of course was the point of the definition).
Chapter 22. Electromagnetic Casimir Energies as Path Integrals 22.10 Casimir–Polder Potentials Within Media A relatively straightforward generalization of the above results is to compute the Casimir–Polder potential for an atom near a dielectric interface, but with the atom within the dielectric. This is a model for an atom in a gas near an interface, or an atom in a liquid near a bubble. 22.10.1 TE Polarization 22.10.1.1 Feynman–Kac Formula For TE polarization, we return to the path integral (22.197): V (TE) CP
¯hc\alpha0
Z \infty dT
**
x(\tau) ++ x(\tau) . (22.491) The calculation parallels the calculation with the atom in vacuum of Section 22.7. The first difference is that in the Feynman–Kac solution (22.308), we should choose the solution for x > d to put the atom inside the dielectric. Then instead of Eq. (22.314), we have
\sqrt 2\pi Z \infty −\infty dk f(0) = \sqrt 2\pi Z \infty −\infty dk A +
= r \pi
" 1 + e2 p
\sqrt
\sqrt \lambda \sqrt
\sqrt \lambda !# = r \pi
h 1 −e2 p
i , (22.492) where we are using the reflection coefficient (22.274) incident from the vacuum side, even though we are considering a particle on the dielectric side (although the sign of the reflection coefficient differs, as we expect for the dielectric side). The reason for this will become more clear shortly. Note that the result here can be obtained from the previous result (22.314) by formally interchanging \lambda and \lambda + s, and changing the sign of d. 22.10.1.2 Mellin Transform Proceeding in analogy to Section 22.7.2, we can use the replacements s −\rightarrow s\chi and \lambda −\rightarrow \lambda + s to write
Z \infty dT
e−s(T +\chiTs[BT ])
= r \pi
h 1 −e2 p
i . (22.493) We will use this to compute the integral
Z \infty dT
e−s(T +\chiTs[BT ;z]) −e−s(T +\chiTs[BT ;z\rightarrow \infty])
, (22.494) which is the counterpart to Eq. (22.321). Note that here, we are renormalizing by subtracting the limit of large distance z = −d > 0 from the interface. In the previous case, this was equivalent to setting \chi = 0, but this time the ‘‘background’’ case is an atom in a uniform dielectric. Using the same procedure as for
22.10 Casimir–Polder Potentials Within Media (22.317), we can write¶
\Gamma
Z \infty
i = − \sqrt\pi \Gamma
Z \infty d\lambda
p
p
(22.495) Again, this is the same as the atom-in-vacuum case so far, except for the factors of (1 + \chi). 22.10.1.3 Inverse Moments Now for the dielectric-interface path integral, the integral that we need to compute is of the form
Z \infty dT
**
++ , (22.496) in analogy to Eq. (22.323). Note that the renormalization here is different, as appropriate for a particle on the dielectric side of the interface. This cuts off the divergence at T , where the paths become arbitrarily small, and so Ts[BT ; z] = T on the dielectric side. Rearranging this expression to be more compatible with Eq. (22.495),
Z \infty dT
**
++ . (22.497) Then using (22.326),
\Gamma(\alpha) Z \infty
(22.498) with Eq. (22.495) this becomes
\sqrt\pi \Gamma(\alpha)\Gamma
Z \infty ds s\alpha−1 Z \infty d\lambda
p
p
(22.499) as in Eq. (22.327). Simplifying as before by letting \lambda −\rightarrow \lambda/8z2 and s −\rightarrow s/8z2, then letting \lambda −\rightarrow \lambda −s, and then letting \lambda −\rightarrow \lambdas,
\sqrt\pi
zD Z \infty
Z \infty ds sD/2−1 e− p
(22.500) which compares to Eq. (22.330). Then letting \lambda −\rightarrow \lambda −\chi,
\sqrt\pi
zD \times Z \infty
\sqrt \lambda Z \infty ds sD/2−1 e− \sqrt \lambdas, (22.501) In the next step, we will let \lambda −\rightarrow (1 + \chi)\lambda, and we note that
p
p
p
p
= −r′
(22.502) where we used Eqs. (22.274) and (22.276), and we see that we are switching via the natural transformations of the problem from the vacuum-side to the dielectric-side reflection coefficient. Thus with \lambda −\rightarrow (1 + \chi)\lambda in Eq. (22.501), we have
zD Z \infty
\sqrt \lambda Z \infty ds sD/2−1 e− p
(22.503)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals The integral over s now has the value 2\Gamma(D)[\lambda(1 + \chi)]−D/2, so
Z \infty
as in Eq. (22.331). Note, in fact, that the only differences here compared to Eq. (22.331) are the replacement of rTE with r′ TE and the factor of (1 + \chi)−\alpha. Now we will proceed by being more specific about \alpha and D. 22.10.1.4 Result: 3D Electromagnetism Now since the Casimir–Polder path integral (22.302) contains the integral (22.496) after renormalization, we have V (TE) CP
¯hc\alpha0
ID,3/2(z), (22.505) just as in Eq. (22.332). With the result (22.331), this becomes V (TE) CP
\Gamma(D)¯hc\alpha0
D/2 −1
Z \infty
(22.506) With Eq. (22.335), this result simplifies to V (TE) CP
¯hc\alpha0
Z \infty
(TE Casimir–Polder potential, dielectric side) (22.507) where again [Eq. (22.276)] r′
p
p
p
p
. (22.508) Notice again that this potential has the same form as the vacuum-side potential, except for the factor (1 + \chi)−3/2, and the dielectric-side reflection coefficient r′ TE appears in place of the vacuum-side version rTE.21 For D = 4, we have V (TE) CP
3¯hc\alpha0
Z \infty
p
p
p
p
! , (22.509) and evaluating the remaining integral, we find V (TE) CP
3¯hc\alpha0 32\pi2ϵ0z4 5 6 + 1 \chi −
2\chi
2\chi3/2
(TE Casimir–Polder potential, dielectric side) (22.510) The \chi-dependent factor here is strictly positive. Thus the force here is repulsive, unlike the vacuum-side potential (22.339). For small \chi, the \chi-dependent factor is \chi/40, which is the same as the vacuum-side case (except for the overall minus sign). However, as \chi −\rightarrow \infty, the potential here decays to 0 as (5/6−\pi/4)\chi−3/2, which is quite different to the vacuum-side case where the \chi-dependent part levels off at 1/6. 21See Fei Zhou and Larry Spruch, ‘‘van der Waals and retardation (Casimir) interactions of an electron or an atom with multilayered walls,’’ Physical Review A 52, 297 (1995), Eq. (4.55) (doi: 10.1103/PhysRevA.52.297).
22.10 Casimir–Polder Potentials Within Media 22.10.1.5 Result: 1D Electromagnetism
result V (TE) CP
¯hc\alpha0 16\piϵ0z2 r′
¯hc\alpha0 16\piϵ0z2
(22.511) Again, this has the same form as the vacuum-side expression (22.344), except for the overall sign and the
22.10.2 TM Polarization For the TM case of the Casimir–Polder potential for an atom on the dielectric side of the interface, we return to the path integral (22.198), V (TM) CP
¯hc\alpha0
Z \infty dT
**
2ϵr \nabla 2
x(\tau) , (22.512) which is more general than the path integral (22.247) that we used in the vacuum-side calculation. 22.10.2.1 Feynman–Kac Formula The derivation parallels that of the vacuum-side case in Section 22.9. The first difference is that we need the solution to the A coefficient from Eqs. (22.446): A = e p
\sqrt
\sqrt −
\sqrt
\sqrt
e−2\Xi\sqrt
\sqrt \lambda . (22.513) Then from the form of the solution f(x) in Eq. (22.445), we need the function
\sqrt 2\pi Z \infty −\infty dk f(0) = \sqrt 2\pi Z \infty −\infty dk
= r \pi
" 1 + e2 p
\sqrt
\lambda \sqrt
\lambda !# = r \pi
h 1 −e2 p
i , (22.514) where L(\lambda, s; d, \Xi) is defined by Eq. (22.444). This result is in analogy to (22.449), but d < 0 here. We are also using the notation (22.450)
\lambda − \sqrt
e2\Xi\sqrt
\sqrt
. (22.515) 22.10.2.2 Mellin Transform The next step is to compute the path integral
Z \infty dT
e−N[BT ;z,\Xi]−s(T +\chiTs[BT ;z]) −e−s(T +\chiTs[BT ;z\rightarrow \infty])
, (22.516)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals which parallels (22.451), but with the more appropriate renormalization for this case. The result, as in Eq. (22.453), is
\Gamma
Z \infty
i = − \sqrt\pi \Gamma
Z \infty d\lambda
p
e−2 p
(22.517) where we are now taking z = −d. 22.10.2.3 Inverse Moments Next, we compute the integral (22.454),
Z \infty dT
** e−N[BT ;z,\Xi]
++ , (22.518) but modified here for a distant-interface renormalization. To do this, we take the inverse-moment formula (22.295), in the form of Eq. (22.326),
\Gamma(\alpha) Z \infty
(22.519) and apply it to Eq. (22.517):
\sqrt\pi \Gamma(\alpha)\Gamma
Z \infty ds s\alpha−1 Z \infty d\lambda
p
e−2 p
(22.520) Now absorbing a factor of \sqrt 8z2 into \lambda and s, letting \lambda −\rightarrow \lambda −s, and then \lambda −\rightarrow \lambdas,
\sqrt\pi
zD Z \infty
Z \infty ds sD/2−1 e− p
(22.521) Using Eq. (22.456) to eliminate r in favor of the TM reflection coefficient,
\sqrt\pi
zD Z \infty
Z \infty ds sD/2−1 e− p
(22.522) and then letting \lambda −\rightarrow \lambda −\chi,
\sqrt\pi
zD \times Z \infty
\sqrt \lambda
Z \infty ds sD/2−1 e− \sqrt s\lambda. (22.523)
zD Z \infty
\sqrt \lambda Z \infty ds sD/2−1 e− p
(22.524) where we switched to the dielectric-side reflection coefficient using
\sqrt \lambda − p
\sqrt
p
−r′
(22.525)
22.10 Casimir–Polder Potentials Within Media which follows from Eqs. (22.279) and (22.281). The integral over s again has the value 2\Gamma(D)[\lambda(1 + \chi)]−D/2, so
Z \infty
(22.526) This is the analogue of Eq. (22.504) in the TE dielectric-side case, or Eq. (22.455) in the TM vacuum-side case. 22.10.2.4 Result: 3D Electromagnetism Now to evaluate the path integral (22.512) for the Casimir–Polder potential, we see from the definition (22.518) that we can write the potential as V (TM) CP
¯hc\alpha0
ID,3/2(z) −1 2ϵr \partial 2 z ID−2,1/2(z) (22.527) for an atom at a distance z from a planar dielectric interface, on the dielectric side. Notice the factor of
the vacuum-side expression (22.458). Then we can take the special case of Eq. (22.526):
\Gamma(D)
Z \infty
Z \infty
(22.528) where we used Eq. (22.461) to transform the gamma functions. Similarly, \partial 2
\Gamma(D)
Z \infty
Z \infty
(22.529) Thus, we have the combination ID,3/2(z) −1 2ϵr \partial 2
Z \infty
(22.530) in analogy to Eq. (22.464). Then from Eq. (22.527), we have the potential V (TM) CP
¯hc\alpha0
Z \infty
(TM Casimir–Polder potential, dielectric side) (22.531) where from Eq. (22.281), r′
p
\sqrt \lambda p
\sqrt \lambda . (22.532) This is the analogue of the TM vacuum-side potential (22.465), having the same form except for the factor (1 + \chi)−3/2 and the appropriately modified reflection coefficient (the same changes that occurred in the TE case). For D = 4, this becomes V (TM) CP
3¯hc\alpha0
Z \infty
p
\sqrt \lambda p
\sqrt \lambda ! . (22.533)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals Evaluating the integral, we have V (TM) CP
3¯hc\alpha0 32\pi2ϵ0z4 5
+
tanh−1
(TM Casimir–Polder potential, dielectric side) (22.534) For small \chi, the \chi-dependent factor is 43\chi/120, as in the vacuum-side case (except for the overall minus sign). For large \chi, this again drops to zero as (5/6)\chi−3/2, unlike the vacuum-side case, which levels off to 5/6. 22.10.2.5 Casimir–Polder Efficiencies In analogy to the vacuum-side analysis of Section 22.9.6, we will combine and summarize the TM and TE results of our calculations above for the dielectric-side Casimir–Polder potentials, and rewrite our results in terms of efficiencies relative to the strong-coupling result. First, the combined Casimir–Polder potential on the dielectric side of the interface is, from Eqs. (22.507) and (22.531), V (TE) CP
¯hc\alpha0
Z \infty
r′
i . (TE + TM Casimir–Polder potential, dielectric side) (22.535) as in the vacuum-side result (22.485), the total involves the combination of reflection coefficients, with a ge- ometric factor on the TM coefficient. The dielectric-side Fresnel coefficients here are defined in Eqs. (22.276) and (22.281).
and (22.534) as potentials V (TE) CP
3¯hc\alpha0
TE(\chi), V (TM) CP
3¯hc\alpha0
TM(\chi). (TE and TM Casimir–Polder potentials, dielectric side) (22.536) Note that we are comparing to the vacuum-side, strong coupling limit, but without the minus sign, so that the efficiencies are positive (and, as it turns out, they will never achieve unity, even when combined). We are using primes to notate the efficiencies here, compared to the vacuum-side efficiencies (22.487), to denote that these efficiencies quantify the dielectric-side potentials. Explicitly, the efficiencies are \eta′
5 6 + 1 \chi −
2\chi
2\chi3/2 tan−1 \sqrtx \eta′
5
+
tanh−1
. (TE and TM Casimir–Polder efficiencies, dielectric side) (22.537) as can be read off directly from Eqs. (22.510) and (22.534). We have already discussed the asymptotic behaviors of these efficiencies in Sections 22.10.1.4 and 22.10.2.4, and it is worth reiterating that \eta′ TE > 0 and \eta′ TM > 0 for all \chi \ge 0. In the full-electromagnetism case, we can then combine the individual efficiencies
TM(\chi), (full Casimir–Polder efficiency, dielectric side) (22.538)
22.10 Casimir–Polder Potentials Within Media in which case the full potential is
3¯hc\alpha0
(TE + TM Casimir–Polder potential, dielectric side.) (22.539) Since \eta′(\chi) here is fairly cumbersome, we won’t bother to write it out here. The efficiency is asymptotically 23\chi/60 for small \chi, just as in the vacuum-side case, and for large \chi, \eta′(\chi) scales as (5/3 −\pi/4)\chi−3/2. Plotting the ‘‘inside’’ (dielectric-side) efficiency \eta′(\chi) from Eq. (22.538) along with the ‘‘outside’’ (vacuum-side) efficiency \eta(\chi) from Eq. (22.490), we see the matching at small \chi, but a dramatic difference in scaling at large \chi, reflecting a sort of ‘‘screening’’ of the interface by the dielectric. outside inside c 10-6 10-4 10-2 ho(c) 10-7 10-1 10-2 10-3 10-4 10-5 10-6 We are also displaying the asymptotes for the curves for small and large \chi. Plotting the individual TE and TM components for the ‘‘inside’’ (dielectric-side) efficiencies [Eq. (22.537)] and the ‘‘outside’’ (vacuum-side) efficiencies [Eq. (22.487)], we can see the relative contributions of each po- larization. TE outside TM outside TE + TM outside TE inside TM inside TE + TM inside c 10-6 10-4 10-2 hoo(c) 10-7 10-1 10-2 10-3 10-4 10-5 10-6 The TE polarization contributes relatively little compared to the TM polarization, on either side of the interface.
Chapter 22. Electromagnetic Casimir Energies as Path Integrals 22.11 Electric vs. Magnetic Casimir–Polder Interactions Recall that the general forms (22.195) and (22.196) for the Casimir–Polder potential encapsulate both electric and magnetic interactions of particles with surfaces. So far, we have focused on the special case of a dielectric particle (atom) of polarizability \alpha0, interacting with a dielectric interface, where the dielectric function is constant in the neighborhood of the particle. This led to the path-integral expressions (22.197) and (22.198) for the TE and TM energies, respectively. For convencience we reproduce these here: V (TE) CP
¯hc\alpha0
Z \infty dT
**
x(\tau) ++ x(\tau) V (TM) CP
¯hc\alpha0
Z \infty dT
x(\tau) −T 2ϵr \nabla 2
++ x(\tau) . (22.540) In this section we will briefly review the analogous forms for other combinations of electric and magnetic interactions, and compare them to this one. The point of this exercise is to gain some more intuition for the general path integrals and for magnetic interactions in particular. Of course, the dielectric–dielectric interaction is the most important, since typically atoms and materials with large magnetic responses have even larger dielectric responses: in atoms, magnetic dipoles are comparable to electric quadrupoles, and in materials, magnetic materials are typically metallic conductors. 22.11.1 Dielectric Particle, Magnetic Surface First, let’s consider a dielectric particle (atom) interacting with a purely magnetic surface. We can do this
V (TE) CP
¯hc\alpha0
Z \infty dT
**
++ x(\tau) V (TM) CP
¯hc\alpha0
Z \infty dT
**
x(\tau) −T
x(\tau) ++ x(\tau) . (unrenormalized Casimir–Polder potential, magnetic body) (22.541) We have also set the derivatives of µr to zero at the particle location, as appropriate, for example, for a particle in vacuum outside a purely magnetic body. Recall that the TE potential here is given by
i , (22.542) as in Eq. (22.155). 22.11.1.1 Weak-Coupling Limit These path integrals can be evaluated using the same techniques as in the dielectric-particle-dielectric- surface integrals. As an example, we will consider the weak-coupling limit of a dielectric particle at a planar vacuum–magnetic interface. That is, if
(22.543) then we will expand to lowest order in \chim. Doing this in the path integrals (22.541), we find V (TE) CP
(TM) CP
¯hc\alpha0
Z \infty dT
−3 2 + T 4 \nabla 2
x(\tau) . (22.544)
To evaluate the two integral terms, we can read off the dielectric result from Eqs. (22.226) and (22.235) as Z \infty dT
x(\tau) =
(22.545)
22.11 Electric vs. Magnetic Casimir–Polder Interactions where the particle-surface distance is z. Using this in the path integrals (22.544), V (TE) CP
(TM) CP
¯hc\alpha0
−3
4\nabla 2
(D −1)2D/2zD−2 =
− 3\Gamma(D/2)
=
(2D −1)\Gamma(D/2)
(22.546) For D = 4, the result is V (TE) CP
(TM) CP
3¯hc\alpha0 32\pi2ϵ0z4 7\chim . (22.547) This gives a total electromagnetic potential of (7\chim/60) times the strong-coupling result, but with an overall positive result, so the potential in this case is repulsive.22 22.11.2 Magnetic Particle, Magnetic Surface For a purely magnetic atom interacting with a purely magnetic body, we can take \alpha0 = 0 and ϵr = 1 in Eqs. (22.195) and (22.196), with the resulting path integrals V (TE) CP
¯hc\beta0µ0
Z \infty dT
x(\tau) −T 2µr \nabla 2
++ x(\tau) V (TM) CP
¯hc\beta0µ0
Z \infty dT
**
x(\tau) ++ x(\tau) , (unrenormalized Casimir–Polder potential, magnetic particle/magnetic body) (22.548) where we have set any gradients of µr at this particle location to zero. Note that these are the same as the dielectric–dielectric path integrals (22.540), with the following changes: \alpha0 −\rightarrow \beta0µ0, ϵ0 −\rightarrow µ0, ϵ −\rightarrow µ, and the TE and TM integrals have interchanged. Thus, all the calculations for the electric-dipole atom interacting with a dielectric surface, in Sections 22.4–22.10 apply here with the same changes. In particular, for a magnetic response equivalent to the electric response, the total (TM+TE) Casimir–Polder potential in this case is the same as for the dielectric–dielectric case. 23 22.11.3 Magnetic Particle, Dielectric Surface
and (22.196), to obtain V (TE) CP
¯hc\beta0µ0
Z \infty dT
**
x(\tau) −T
x(\tau) ++ x(\tau) V (TM) CP
¯hc\beta0µ0
Z \infty dT
**
++ x(\tau) . (unrenormalized Casimir–Polder potential, magnetic particle/dielectric body) (22.549) 22cf. S. Y. Buhmann, H. T. Dung, T. Kampf, and D.-G. Welsch, ‘‘Casimir-Polder interaction of atoms with magnetodielectric bodies,’’ European Physical Journal D 35, 15 (2005), Eq. (99) (doi: 10.1140/epjd/e2005-00044-6). 23See Timothy H. Boyer, ‘‘Van der Waals forces and zero-point energy for dielectric and permeable materials,’’ Physical Review A 9, 2078 (1974) (doi: 10.1103/PhysRevA.9.2078). See also Stefan Yoshi Buhmann, Dispersion Forces I: Macroscopic Quantum Electrodynamics and Ground-State Casimir, Casimir–Polder and van der Waals Forces (Springer, 2012). Compare
Chapter 22. Electromagnetic Casimir Energies as Path Integrals where we have set any gradients of ϵr at this particle location to zero. Note that these have the form of the path integrals for a dielectric particle and a magnetic body as in Eqs. (22.541), but with the TE and TM polarizations interchanged, and the electric and magnetic matter functions (and parameters) interchanged. Thus, the weak-coupling result (22.11.1.1) also applies if we replace \chim by \chi and \alpha0 with \beta0µ0, giving a total Casimir–Polder potential from (22.547) of
(TE) CP (r) + V (TM) CP
32\pi2z4 7\chi . (22.550) That is, for positive \chim and \beta0, this configuration also produces a repulsive potential.24 We can confirm this intuition by considering a mode summation based on the mode diagrams in Sections 22.1.1 and 22.1.2. Recall that a dielectric particle interacting with the TE modes has path integrals given in Eqs. (22.540), and the overall negative sign of the (renormalized) potential is set by the Fresnel reflection coefficients, which are negative for a source outside the dielectric. Further, the vector nature of the electric field in the TM case is represented by the \nabla 2 term in the TM path integral. By contrast, for a magnetic particle interacting with the same modes, note from the TE diagram in Section 22.1.1 that now the incident and reflected magnetic fields have an opposite orientation compared to the electric fields, and they also have a vector character, since the field vectors are not parallel unless the mode is normally incident. Thus, the \nabla 2 appears in the TE path integral in Eqs. (22.549) here. The overall minus sign is buried in the path integral, but is reflected in the positive, small-\chim result (22.550). From the TM diagram in Section 22.1.2, we see that the magnetic field takes on a scalar character, and there is another relative minus sign in the incident vs. reflected fields, compared to the electric-field case. Hence, the \nabla 2 does not appear in the TM path integral in Eqs. (22.549) here. However, the TM potential does appear, as it maintains the proper TM boundary conditions across the interface (thus implementing the proper reflection coefficient). 22.12 Casimir Potential As an example of the Casimir interaction between two macroscopic bodies, we will evaluate the path integrals (22.177) for the two polarizations E (TE) EM = − ¯hc
Z \infty dT
Z dD−1x0 **
++ x(\tau) E (TM) EM = − ¯hc
Z \infty dT
Z dD−1x0 **
++ x(\tau) . (Casimir energy, scalar EM) (22.551) For convenience, we will reproduce the matter potentials (22.155) for the two polarizations here:
i
i . (22.552) (matter-induced potentials) 24For the equivalence of the dielectric atom–magnetic surface with the magnetic atom–dielectric surface, see Stefan Yoshi Buhmann, Dispersion Forces I: Macroscopic Quantum Electrodynamics and Ground-State Casimir, Casimir–Polder and van
the factors of ϵ0 and µ0 work out in the prefactors. For more about the repulsive nature of a magnetic atom and a conducting surface, see H. Haakh, F. Intravaia, C. Henkel, S. Spagnolo, R. Passante, B. Power, and F. Sols, ‘‘Temperature dependence of the magnetic Casimir-Polder interaction,’’ Physical Review A 80, 062905 (2009) (doi: 10.1103/PhysRevA.80.062905). The repulsive force between a dielectric body and a permeable body is discussed in Timothy H. Boyer, ‘‘Van der Waals forces and zero-point energy for dielectric and permeable materials,’’ Physical Review A 9, 2078 (1974) (doi: 10.1103/PhysRevA.9.2078). See also the online talk at http://cnls.lanl.gov/casimir/PresentationsSF/henkel-sfe.pdf.
22.12 Casimir Potential Of course, the techniques we have used so far for Casimir–Polder potentials will apply here. The main difference is that here we will need to integrate over the source-point location x0 of the Brownian bridges. To simplify the calculation, we will take the case of two identical dielectric interfaces (i.e., a vacuum gap between two dielectric half-spaces) of susceptibility \chi, separated by a distance d. Thus, we will set
22.12.1 TE Path Integral For the TE-polarized component, we will then need to evaluate the path integral E (TE) EM = − ¯hc
Z \infty dT
Z dD−1x0 **
x(\tau) ++ x(\tau) , (Casimir energy, scalar EM, TE polarization) (22.553) where we have taken µr = 1. To evaluate this integral, we will follow the procedure of Section 22.7. 22.12.1.1 Feynman–Kac Formula We will begin by considering the path integral
Z \infty dt \sqrt t e−\lambdat
exp −sTds h Bt(t′); d, d0 i Bt , (22.554)
the ‘‘double-sojourn time’’ Tds[Bt; d, d0] := Z t dt′ 1(−\infty,d0]∪[d0+d,\infty) h Bt(t′) i , (22.555) which is the combined time that Bt(t′) spends sojourning in either (−\infty, d0] or [d0 + d, \infty). Equivalently, the indicator function here counts the amount of time the bridge spends outside the ‘‘gap’’ [d, d + d0] (this interval may or may not include the point 0). Starting with the Feynman–Kac ODE (22.300), we can obtain an expression for f(0), which will yield the path integral (22.554) by solving
f(x) −2eikx, (22.556)
(22.557)
Z \infty dt e−\lambdat ** eikW (t) exp − Z t dt′ V W(t′)
, (22.558) which we will later integrate over k to obtain the desired path integral. Then for x < d0 or x > d0 + d, the ODE is
(22.559) with general solutions
p
p
eikx
(22.560)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals for coefficients \alpha\pm to be determined in each region. On the other hand, for d0 < x < d0 + d, the ODE is the same, but with s = 0. Picking the bounded solutions in each domain, we have the solution
Ae− p
eikx
(x > d0 + d) Be− \sqrt
\sqrt
eikx
(d0 < x < d0 + d) De p
eikx
(x < d0), (22.561)
the conditions Ae− p
eik(d0+d)
\sqrt
\sqrt
Be− \sqrt
\sqrt
eikd0
p
eikd0
(22.562) respectively, and continuity of the derivatives gives the conditions − p
p
\sqrt 2\lambdaBe− \sqrt
\sqrt 2\lambdaCe \sqrt
− \sqrt 2\lambdaBe− \sqrt
\sqrt 2\lambdaCe \sqrt
p
p
ikeikd0
(22.563) The solution for the cofficients are fairly complicated, but they simplify somewhat after integration over k: 2\pi Z \infty −\infty dk A = p
s sinh h d \sqrt 2\lambda i e2 p
h d \sqrt 2\lambda i + 2 p
h d \sqrt 2\lambda i 2\pi Z \infty −\infty dk B = \sqrt 2\lambda −se \sqrt
p
e− \sqrt 2\lambda d
h d \sqrt 2\lambda i + 2 p
h d \sqrt 2\lambda i 2\pi Z \infty −\infty dk C = \sqrt 2\lambda −se− \sqrt
p
e− \sqrt 2\lambda d
h d \sqrt 2\lambda i + 2 p
h d \sqrt 2\lambda i 2\pi Z \infty −\infty dk D = p
s sinh h d \sqrt 2\lambda i e−2 p
h d \sqrt 2\lambda i + 2 p
h d \sqrt 2\lambda i . (22.564) Then since
A +
(0 > d0 + d)
(d0 < 0 < d0 + d) D +
(0 < d0), (22.565) and we want to calculate
\sqrt 2\pi Z \infty −\infty
Z \infty dt \sqrt t e−\lambdat
exp −s1(−\infty,d0]∪[d0+d,\infty) h Bt(t′) i Bt , (22.566)
22.12 Casimir Potential and so¶
r \pi
1 + s sinh h d \sqrt 2\lambda i e2 p
h d \sqrt 2\lambda i + 2 p
h d \sqrt 2\lambda i (d0 + d < 0) r\pi \lambda 1 − s cosh h (2d0 + d) \sqrt 2\lambda i −
p
e− \sqrt 2\lambda d
h d \sqrt 2\lambda i + 2 p
h d \sqrt 2\lambda i (d0 < 0 < d0 + d) r \pi
1 + s sinh h d \sqrt 2\lambda i e−2 p
h d \sqrt 2\lambda i + 2 p
h d \sqrt 2\lambda i (d0 > 0). (22.567) Eliminating the hyperbolic functions in favor of exponentials, we have
r \pi
1 + s 1 −e−2 \sqrt 2\lambda a e2 p
\Lambda+ −\Lambda−e−2 \sqrt 2\lambda d (d0 + d < 0) r\pi \lambda 1 − s 1 + e−2 \sqrt
e2 \sqrt 2\lambda d0 −2\Lambda−e−2 \sqrt 2\lambda d \Lambda+ −\Lambda−e−2 \sqrt 2\lambda d (d0 < 0 < d0 + d) r \pi
1 + s 1 −e−2 \sqrt 2\lambda d e−2 p
\Lambda+ −\Lambda−e−2 \sqrt 2\lambda d (d0 > 0), (22.568) where the symbols
p
= \sqrt
\sqrt \lambda 2 (22.569) encapsulate much of the dependence on \lambda and s. Recalling the TE reflection coefficient (22.274)
\sqrt
\sqrt
, (22.570) we can see that r 2
\Lambda+ , (22.571) and s \Lambda+ = \sqrt
\sqrt \lambda \sqrt
\sqrt \lambda \Lambda+
(22.572) Then we can rewrite Eq. (22.568) as
r \pi
1 − rTE(\lambda; s) 1 −e−2 \sqrt 2\lambda d e2 p
1 −r 2 TE(\lambda; s) e−2 \sqrt 2\lambda d (d0 + d < 0) r\pi \lambda 1 + rTE(\lambda; s) 1 + e−2 \sqrt
e2 \sqrt
TE(\lambda; s) e−2 \sqrt 2\lambda d 1 −r 2 TE(\lambda; s) e−2 \sqrt 2\lambda d (d0 < 0 < d0 + d) r \pi
1 − rTE(\lambda; s) 1 −e−2 \sqrt 2\lambda d e−2 p
1 −r 2 TE(\lambda; s) e−2 \sqrt 2\lambda d (d0 > 0), (22.573) now in terms of Fresnel coefficients.
Chapter 22. Electromagnetic Casimir Energies as Path Integrals 22.12.1.2 Mellin Transform Our next goal here is to compute the integral
Z \infty dT
, (22.574) where the second term is a renormalization against separated plates, which in each region means considering the case where the interfaces are displaced far away from the point of consideration (i.e., the source point of the Brownian bridges). We will proceed in analogy to Section 22.7.2. First, we will make the replacements s −\rightarrow s\chi and \lambda −\rightarrow \lambda + s, which from Eq. (22.566) means we have now computed
Z \infty dT \sqrt T e−\lambdaT
e−s{T +\chiTds[BT ;d,d0]}
BT . (22.575) Note that we must compare this to the reference case
Z \infty dT \sqrt T e−\lambdaT
BT , (22.576) which again corresponds to taking the limit as the interfaces move far away from the source point (the origin) in L(\lambda + s, s\chi; d, d0). In each region this corresponds to the limit where all of the exponential factors that depend on d or d0 vanish. This allows us to drop the leading terms in Eq. (22.568):
r \pi
= − r \pi
rTE(\lambda; s) 1 −e−2 \sqrt 2\lambda d e2 p
1 −r 2 TE(\lambda; s) e−2 \sqrt 2\lambda d (d0 + d < 0) r\pi \lambda rTE(\lambda; s) 1 + e−2 \sqrt
e2 \sqrt
TE(\lambda; s) e−2 \sqrt 2\lambda d 1 −r 2 TE(\lambda; s) e−2 \sqrt 2\lambda d (d0 < 0 < d0 + d) − r \pi
rTE(\lambda; s) 1 −e−2 \sqrt 2\lambda d e−2 p
1 −r 2 TE(\lambda; s) e−2 \sqrt 2\lambda d (d0 > 0). (22.577) Then, as in Eq. (22.321), we can compute the integral (22.574) as
\Gamma
Z \infty
(22.578) which for now we will leave in terms of the cumbersome expression (22.577). 22.12.1.3 Inverse Moments In analogy to Eq. (22.324), we will now compute the integral
Z \infty dT
**
\alpha ++ , (22.579) using the transformation (22.326):
\Gamma(\alpha) Z \infty ds s\alpha−1 ˜ID−2\alpha(d, d0). (22.580)
22.12 Casimir Potential With Eq. (22.578), this becomes¶
\Gamma(\alpha)\Gamma
Z \infty ds s\alpha−1 Z \infty
(22.581) We then carry out the other transformations in Section 22.7.3. First letting \lambda −\rightarrow \lambda −s and then \lambda −\rightarrow \lambdas,
\Gamma(\alpha)\Gamma
Z \infty ds sD/2−1 Z \infty
. (22.582) Then letting s −\rightarrow s/8,
\sqrt\pi
Z \infty ds sD/2−1 Z \infty
\times hp
i . (22.583) The last factor is hp
i = −
1 −e− \sqrt s\lambda d e p
1 −r 2
\sqrt s\lambda d (d0 + d < 0) \sqrt \lambda
1 + e− \sqrt
e \sqrt
\sqrt s\lambda d 1 −r 2
\sqrt s\lambda d (d0 < 0 < d0 + d) −
1 −e− \sqrt s\lambda d e− p
1 −r 2
\sqrt s\lambda d (d0 > 0), (22.584) by making the replacements in Eq. (22.577). 22.12.1.4 Spatial Integration and Renormalization The next step is to compute the integral of the function ID,\alpha(d, d0) in Eq. (22.583) over all d0, which corresponds to the component of the x0 integral normal to the interfaces. All the d0-dependent components are in the factor in Eq. (22.584), so we will focus on this last part. We will call the integrals over the respective regions of the three piecewise parts II, III, and IIII, in the order the components are listed in Eq. (22.584). Beginning with the third section, IIII = −
1 −e− \sqrt s\lambda d
h 1 −r 2
\sqrt s\lambda d i Z \infty dd0 e− p
= −
1 −e− \sqrt s\lambda d
h 1 −r 2
\sqrt s\lambda d i. (22.585) By symmetry, the first section is equivalent: II = IIII. (22.586)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals Finally, in the middle region we have III =
\sqrt \lambda h 1 −r 2
\sqrt s\lambda d i Z 0 −d dd0 1 + e− \sqrt
e \sqrt
\sqrt s\lambda d =
\sqrt \lambda h 1 −r 2
\sqrt s\lambda d i Z 0 −d dd0 e \sqrt
\sqrt
\sqrt s\lambda d =
\sqrt \lambda h 1 −r 2
\sqrt s\lambda d i \sqrt s\lambda 1 −e− \sqrt s\lambda d
\sqrt s\lambda d =
h 1 −r 2
\sqrt s\lambda d i
1 −e− \sqrt s\lambda d + d \sqrt \lambda
\sqrt s\lambda d . (22.587) The sum over all three regions is therefore
h 1 −r 2
\sqrt s\lambda d i
1 −e− \sqrt s\lambda d + d \sqrt \lambda
\sqrt s\lambda d =
\sqrt \lambda h 1 −r 2
\sqrt s\lambda d i r \lambda s 1 \lambda −
1 −e− \sqrt s\lambda d
\sqrt s\lambda d . (22.588) Anticipating the final result, note that we normally define the Casimir energy (22.553) to be zero when the separation between objects becomes large. Note, though, that Eq. (22.588), which encapsulates all the spatial dependence of the energy (22.553), does not have this property, since it does not vanish as d −\rightarrow \infty. The renormalization is only necessary for the first term, and has the form 1 −e− \sqrt s\lambda d 1 −r 2 TE e− \sqrt
r 2 TE −1 e− \sqrt s\lambda d 1 −r 2 TE e− \sqrt s\lambda d , (22.589) Thus, after renormalization, Eq. (22.588) becomes (II + III + IIII)renorm =
\sqrt \lambda h 1 −r 2
\sqrt s\lambda d i r \lambda s 1 \lambda −
r 2
e− \sqrt
\sqrt s\lambda d . (22.590) We can simplify this further by using \sqrt \lambda[r 2
1 \lambda −
=
\chi \sqrt
= "\sqrt
\sqrt
− \sqrt
\sqrt
\chi \sqrt¶
= " −4 p
−\chi # \chi \sqrt
=
(22.591) where we used the explicit form (22.570) for the reflection coefficient. Thus, we have
2r 2
\sqrt s\lambda d \sqrt \lambda h 1 −r 2
\sqrt s\lambda d i p
- d (22.592)
22.12 Casimir Potential as the simplified form of Eq. (22.590). Then we can write out the original result that we were trying to calculate Z \infty −\infty
\sqrt\pi
Z \infty ds sD/2−1 Z \infty
\times 2r 2
\sqrt s\lambda d \sqrt \lambda h 1 −r 2
\sqrt s\lambda d i d + p
(22.593) from Eq. (22.583), again after renormalization. 22.12.1.5 Casimir Energy Now we will assemble all the parts to obtain the Casimir energy. We started with the path-integral expression (22.553), which we can write as E (TE) EM = − ¯hc
Z dD−2x0 Z dd0 ID,1/2(d, d0), (22.594) in terms of the integral (22.579). Note that we have split the original x0 integral into the d0 integration normal to the interfaces, and the rest of the D −2 integrals, where the integrand is constant. Thus, we may define the (divergent) (D −2)-dimensional ‘‘area’’ A := Z dD−2x0, (22.595) and then write the Casimir energy density as E (TE) EM A = − ¯hc
Z \infty −\infty dd0 ID,1/2(d, d0). (22.596)
E (TE) EM A = − ¯hc
Z \infty ds sD/2−1 Z \infty
\sqrt \lambda " r 2
\sqrt s\lambda d 1 −r 2
\sqrt s\lambda d # d + p
. (22.597) which has already been renormalized (shifted) such that this energy vanishes as d −\rightarrow \infty. Then using
\sqrt
\sqrt
= − rTE p
(22.598) and \partial \lambda h r 2
\sqrt s\lambda di
\sqrt
r s \lambda r 2
\sqrt s\lambda d = − r s \lambda r 2
e− \sqrt s\lambda d
d + p
! , (22.599) Eq. (22.597) becomes E (TE) EM A = 2¯hc
Z \infty
Z \infty
r 2
\sqrt s\lambda d 1 −r 2
\sqrt s\lambda d = − 2¯hc
Z \infty
Z \infty
h 1 −r 2
\sqrt s\lambda di . (22.600)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals Integrating by parts, we then have E (TE) EM A = (D −2)¯hc
Z \infty
Z \infty
h 1 −r 2
\sqrt s\lambda di . (TE dielectric Casimir energy) (22.601) Note that we can rescale s −\rightarrow s/d2 to separate out the distance dependence as E (TE) EM A = (D −2)¯hc
Z \infty
Z \infty
h 1 −r 2
\sqrt s\lambdai . (TE dielectric Casimir energy) (22.602) In the general, dispersive case, however, the reflection coefficient also depends on s, and this rescaling can’t be done (indicating a crossover between different power-law potentials). Often these results are differentiated with respect to d to give a Casimir force (pressure).
E (TE) EM A = ¯hc 128\pi2 Z \infty ds \sqrts Z \infty d\lambda log h 1 −r 2
\sqrt s\lambda di = ¯hc 128\pi2d3 Z \infty ds \sqrts Z \infty d\lambda log h 1 −r 2
\sqrt s\lambdai . (TE dielectric Casimir energy, 3D) (22.603)
E (TE) EM A = ¯hc 128\pi2d3 Z \infty ds \sqrts Z \infty d\lambda log h 1 −e− \sqrt s\lambdai = ¯hc 128\pi2d3 −4\pi4 , (22.604) and thus we have the strong-coupling result25 E (TE) EM A
1440d3 . (TE dielectric Casimir energy, 3D strong coupling) (22.605)
E (TE) EM A = − ¯hc\chi2 2048\pi2d3 Z \infty ds \sqrts Z \infty d\lambda \lambda2 e− \sqrt s\lambda = − ¯hc\chi2 2048\pi2d3 8 , (22.606) and so the weak-coupling result is26 E (TE) EM A = − ¯hc\chi2 1280\pi2d3 . (TE dielectric Casimir energy, 3D weak coupling) (22.607) Note that this goes as \chi2 to lowest order, with one factor of \chi to leading order for each of the two surfaces in the interaction. The O(\chi) terms—corresponding to the one-body energies—were lost in the renormalization when we subtracted away the d −\rightarrow \inftylimit. 25See, e.g., Julian Schwinger, ‘‘Casimir energy for dielectrics,’’ Proceedings of the National Academy of Sciences 89, 4091 (1992) (doi: 10.1073/pnas.89.9.4091). 26Julian Schwinger, op. cit.
22.12 Casimir Potential 22.12.1.6 Small-\chi Limit The above calculation was fairly cumbersome, so to get a more intuitive look, we will consider the dilute- medium limit. In the limit of small \chi, the path integral (22.553) becomes E (TE) EM = − ¯hc
Z \infty dT
Z dD−1x0 ** 1 −1
8\langle \chi\rangle 2 x(\tau) ++ x(\tau) , (22.608) where we have expanded to second order in \chi. Again, this is because we are interested in an interaction energy between two bodies. This is perhaps more clear if we label the susceptibilities of the two bodies as \chi1 and \chi2; then the term we seek is of the form \chi1\chi2, while the O(\chi1) and O(\chi2) terms are one-body energies that drop out in the renormalization. Thus, dropping the (obvious) background and one-body contributions, we have the weak-coupling result E (TE) EM = − 3¯hc
Z \infty dT
Z dD−1x0 ** \langle \chi\rangle 2 x(\tau) ++ x(\tau) . (Casimir energy, scalar EM, TE polarization, small \chi) (22.609) The factor of 3/8 here is similar to the analogous factor of 3/4 that we obtained in the Casimir–Polder potential: the first factor of 1/2 from the functional derivative in Eq. (22.182), and the remaining 3/2 from making the small-\chi expansion as in Eq. (22.226). In fact, in evaluating the path integral, it is best to emphasize the two-body interaction by letting \chi = \chi1 + \chi2, and again drop the one-body contributions of the form \chi 2 1 and \chi 2 2 to obtain E (TE) EM = − 3¯hc
Z \infty dT
Z dD−1x0 **
++ x(\tau) . (Casimir energy, scalar EM, TE polarization, small \chi) (22.610) Note that since there are two terms in \chi2 corresponding to the two-body interaction, we obtain the factor of 2 that brings this path integral in line with the analogous Casimir–Polder path integral (22.226). Now this path integral is linear in both \chi1 and \chi2 separately. Thus, we can take a Green-function approach, letting
Z \chi2 dD−1r′ \delta(r −r′), (22.611) where we assume \chi2 models a dielectric body that is uniform over some region in space, and the integral extends over this region. Then the path integral (22.610) becomes E (TE) EM
Z \chi2 dD−1r′ Z \infty dT
Z dD−1x0 **
++ x(\tau) . (22.612) This has precisely the same form as the Casimir–Polder path integral (22.206), once expanded to lowest order in \chi, where \chi refers to \chi1, and \chi2 is equivalent to \alpha0/ϵ0. Thus, the delta function is removed in the same way as in the Casimir–Polder calculation, and we have E (TE) EM A
Z \infty d dz Z \infty dT
**
++ x(\tau) , (22.613) where we are taking \chi1 to correspond to the region z < 0, and \chi2 to z > d for the two-plane interaction, and we have already divided through by the divergent transverse-integral factor to give the energy per unit area.
(22.236) to write Z \infty dT T 3
x(\tau) = \chi 40z4 . (22.614)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals Thus, Eq. (22.613) becomes E (TE) EM A
1280\pi2 Z \infty d dz
1280\pi2d3 . (22.615)
22.12.1.7 Casimir–Polder Potential: Two Parallel Planes As a by-product of the Casimir-energy analysis above, we can also immediately obtain the Casimir–Polder potential for an atom in the vicinity of the same two dielectric planes. For example, to obtain the potential between two dielectric planes, we can start with Eqs. (22.583) and (22.584) in the middle region to write the integral
\sqrt\pi
Z \infty ds sD/2−1 Z \infty
\sqrt \lambda \times
1 + e− \sqrt
e \sqrt
\sqrt s\lambda d 1 −r 2
\sqrt s\lambda d . (22.616) Then using Eq. (22.332), V (TE) CP
¯hc\alpha0
ID,3/2(z), (22.617) we can write-the Casimir–Polder potential as V (TE) CP
¯hc\alpha0
Z \infty ds sD/2−1 Z \infty
\sqrt \lambda \times
e \sqrt
\sqrt
- 2r 2
\sqrt s\lambda d 1 −r 2
\sqrt s\lambda d . (22.618) Note here that d > 0 and d0 < 0. Thus, it is more natural to set z = −d0 > 0 as the atomic distance to the left-hand interface, with d the interface separation, and 0 < z < d: V (TE) CP
¯hc\alpha0
\times Z \infty ds sD/2−1 Z \infty
\sqrt \lambda
e− \sqrt
\sqrt s\lambda(d−z) 1 −r 2
\sqrt s\lambda d . (22.619) Note that we have dropped the r 2 TE term in the numerator of the last factor, which is independent of the atomic position z, and only contributes to the Casimir energy between the interfaces. Also, notice that in the limit d −\rightarrow \infty, only the first term in the numerator of the last factor stays, and we can do the integral over s; the result is just the one-interface energy (22.336). In general the integrals here are difficult to carry out, but they simplify in the case of perfectly
V (TE) CP
¯hc\alpha0
Z \infty
\sqrt \lambda Z \infty ds sD/2−1 e− \sqrt
\sqrt s\lambda(d−z) 1 −e− \sqrt s\lambda d , (22.620) where we have changed the order of integration. Carrying out the s integration, V (TE) CP
¯hc\alpha0
Z \infty
\sqrt \lambda 2\Gamma(D)
, (22.621)
22.12 Casimir Potential where \zeta(z, a) is the generalized zeta function. Cleaning this up, we have the result V (TE) CP
Z \infty
i , (22.622) where we used Eq. (22.335) to simplify the gamma functions. Carrying out the remaining integral, V (TE) CP
h
i . (TE Casimir–Polder potential between dielectric planes) (22.623) Now for the 3D case, we take D = 4: V (TE) CP
¯hc\alpha0 64\pi2ϵ0d4 h
i . (22.624) The generalized zeta functionis related to the polygamma function \psi(n)(z) by
(22.625) or in the present case,
(22.626) Thus, V (TE) CP
¯hc\alpha0 384\pi2ϵ0d4 h
i . (22.627) Finally, using the formula27
z cot \piz, (22.628) or specifically,
sin4 \piz , (22.629) we have V (TE) CP
192ϵ0d4
. (TE Casimir–Polder potential between dielectric planes, 3D) (22.630) Note that for small z/d, the last factor reduces to 3(d/\piz)4, and then this result reduces to the one-interface, strong-coupling result (22.223). Of course, we can also include the z-independent ‘‘Lamb-shift’’ component (the final term scaling as r 2 TE) from (22.618), V (TE) L
¯hc\alpha0
Z \infty ds sD/2−1 Z \infty
\sqrt \lambda 2r 2
\sqrt s\lambda d 1 −r 2
\sqrt s\lambda d . (22.631) In the perfect-conductor limit, we have V (TE) L
¯hc\alpha0
Z \infty
\sqrt \lambda Z \infty ds sD/2−1 e− \sqrt s\lambda d 1 −e− \sqrt s\lambda d . (22.632) The s integral gives the factor 2d−D\lambda−D/2\Gamma[D]\zeta[D], V (TE) L
Z \infty
(22.633) 27Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions (Dover, 1965), p. 260, Eq. (6.4.7).
Chapter 22. Electromagnetic Casimir Energies as Path Integrals and the \lambda integral gives a factor 4/D(D −2): V (TE) L
(22.634) For D = 4, we have V (TE) L
2880ϵ0d4 , (22.635) and thus Eq. (22.630) becomes V (TE) CP
192ϵ0d4 1
, (TE Casimir–Polder potential between dielectric planes, 3D) (22.636) including the z-independent potential shift between the plates. 22.12.2 TM Path Integral For the TM polarization, we will return to Eq. (22.551), which is E (TM) EM = − ¯hc
Z \infty dT
Z dD−1x0 **
++ x(\tau) (Casimir energy, scalar EM) (22.637)
2 −\partial 2 z log \sqrtϵr i , (22.638) (matter-induced potentials) if we assume the only variation in the dielectric occurs along the z-direction. The calculation here is analogous to the Casimir–Polder calculation of Section 22.9. 22.12.2.1 Small-\chi Expansion We will start with a weak-coupling expansion analogous to the TE calculation in Section 22.12.1.6 and the TM Casimir–Polder calculation of Section 22.4.5. Again, we will expand the energy to second order in \chi, with ϵr = 1 + \chi. Starting with the potential,
2 −2\partial 2
z \chi2i
(22.639) the energy (22.551) becomes E (TM) EM = − ¯hc
Z \infty dT
Z dD−1x0
1 −1
8\langle \chi\rangle 2 x(\tau) \times 1 + T h (\partial z\chi)2
z
\chi2 x(\tau) i + T 2 h \partial 2
= − ¯hc
Z \infty dT
Z dD−1x0 \times ** 1 −1
8\langle \chi\rangle 2
h (\partial z\chi)2
z
\chi2 x(\tau) i + T 2 h \partial 2
i2 −T
++ . (22.640)
22.12 Casimir Potential To obtain the lowest-order interaction between the surfaces, we will again set \chi = \chi1 + \chi2, discarding all terms that are not of the form \langle \chi1\rangle \langle \chi2\rangle . We will also discard the
(\partial z\chi)2 and
\chi2 terms, since these will vanish provided the two bodies do not overlap. First tossing out these latter terms and the first-order terms in \chi, E (TM) EM = − ¯hc
Z \infty dT
Z dD−1x0 ** 8\langle \chi\rangle 2
h \partial 2
i2 −T
++ , (22.641) and now labeling the individual bodies, we have E (TM) EM = − ¯hc
Z \infty dT
Z dD−1x0 **
- T 2 h \partial 2
ih \partial 2
i −T
++ . (Casimir energy, scalar EM, TM polarization, small \chi) (22.642) Note that the first term is simply the TE energy, which we have already computed. For convenience we will repeat here the calculation following Eq. (22.610), where we compute the interaction of a small element of the \chi2 body with the \chi1 body, and then integrate over the profile of the \chi2 body. Thus we have the integral
z1 Z \infty dT
Z dz0 **
++ x(\tau) = \chi2 Z \infty d
z Z \infty dT
**
++ x(\tau) , (22.643) where z1 is the location of the boundary of the \chi1 region. Now following the small-\chi calculation of the TM atom–wall potential in Section 22.4.5,
Z \infty d
z Z \infty dT
**
++ x(\tau)
Z \infty d
z Z \infty dT
** Ts B(t); z \sqrt T
x(\tau)
Z \infty d
z z−D Z \infty dT
** Ts B(t); \sqrt T
x(\tau)
\Gamma(D) \chi1\chi2 Z \infty d
Z \infty dT
** Ts B(t); \sqrt T
x(\tau)
Z \infty dT
** Ts B(t); \sqrt T
x(\tau)
Z \infty dT
** Ts B(t); \sqrt 2T
x(\tau)
Z \infty dT
** Ts B(t); \sqrt 2T
x(\tau)
(22.644) For D = 4 and \alpha = 0, this is
120d3 , (22.645)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals so that the first term in the Casimir energy (22.646) leads to E (TE) EM A = − 3¯hc
Z \infty dT
Z dD−1x0 **
++ = − 3¯hc
1280\pi2d3 , (22.646) in agreement with the prior calculation (22.615). The last two terms in Eq. (22.642) then have the form E3,4 = − ¯hc
Z \infty dT
Z dD−1x0 ** −T
++ = − ¯hc
−1 8I(D −2, 2; d) = − ¯hc
−1
2D/2(D −1)\Gamma(D −2)dD−1 , (22.647) and the second term in Eq. (22.642) has the form E2 = − ¯hc
Z \infty dT
Z dD−1x0 ** T 2 h \partial 2
ih \partial 2
= − ¯hc
1 16I(D −4, 4; d) = − ¯hc
1
2D/2−1(D −3)\Gamma(D −4)dD−1 , (22.648) where we integrated by parts in the second step, and we have implicitly carried out dimensional regularization in the last step. Assembling all the pieces, E (TM) EM = −
3 \Gamma(D/2)\Gamma(D −1)
\Gamma[(D −4)/2]\Gamma(D −1) 2D/2−1(D −3)\Gamma(D −4) −1 \Gamma[(D −2)/2]\Gamma(D −1) 2D/2(D −1)\Gamma(D −2) = −
(22.649) The quantity in parenthesis is 43/480 for D = 4, so the small-\chi Casimir energy is E (TM) EM
3840\pi2d3 (Casimir energy, scalar EM, TM polarization, small \chi) (22.650) Note that this effect is substantially stronger than the TE contribution in Eq. (22.615). Combining the two polarizations, we obtain EEM = E (TE) EM + E (TM) EM
1920\pi2d3 (Casimir energy, small \chi) (22.651) for the total Casimir potential in the weak-coupling limit. 22.12.2.2 Feynman–Kac Formula To proceed with the calculation for arbitrary dielectric strength, will begin in analogy to Section 22.12.1.1 by considering the path integral
Z \infty dt \sqrt t e−\lambdat
exp −sTds h Bt(t′); d, d0 i Bt , (22.652)
22.13 Exercises as in Eq. (22.652), but also involving the TM potential as in ???????, Here, we are again using the double- sojourn time, defined in Eq. (22.555) by Tds[Bt; d, d0] := Z t dt′ 1(−\infty,d0]∪[d0+d,\infty) h Bt(t′) i . (22.653) 22.13 Exercises Problem 22.1 Beginning with the electric-field energy (22.47) HE = 1 Z
(22.654) separate the free and bound charge densities, \rho = \rhofree + \rhoP, and use
(22.655) to introduce the dielectric polarization. Show that the result you obtain is not the correct energy for a linear dielectric. Problem 22.2 Consider a scalar field that obeys the wave equation
t \phi. (22.656) Write down the plane-wave solutions and dispersion relation for this wave equation. Problem 22.3 Consider again the scalar wave equation (22.656). We will identify the ‘‘electromagnetic fields’’
(22.657) Now start with the Lagrangian L = ϵ0 Z d3r ϵ ϵ0
, (22.658) and work out the Hamiltonian, both in canonical coordinates and in terms of the fields. Problem 22.4 Quantize the wave equation in Problem 22.2, by decomposing the Hamiltonian from Problem 22.3 into normal modes. In doing this, you should develop the conjugate normal-mode operators
r ¯h 2\omegakϵ0 [fk(r) ak + H.c.]
ϵ0 r ¯h\omegakϵ0 [fk(r) ak −H.c.] , (22.659) with second-quantized field operators
X k \phik,
X k \pik, (22.660)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals and where the mode functions fk(r) satisfy the orthonormality condition Z d3r ϵ(r) ϵ0 fk(r)f ∗
(22.661) as is analogous to the full electromagnetic case. Problem 22.5 For the quantized scalar field coupled to a dielectric, as in Problem 22.4, show that the vacuum energy is given by the expression
X k Z d3r ¯h\omegak ϵ(r) ϵ0 |fk(r)|2 . (22.662) Problem 22.6 A dielectric film (slab) of thickness d surrounded by vacuum is described in ordinary electromagnetism by the field reflection coefficient28 r (TM,TE) film
1 −r 2
(22.663) for the two polarizations in terms of the appropriate Fresnel coefficients for the incident reflection, where the round-trip phase \phi in the film is given by
q
(22.664) and k is the optical wave number and n is the refractive index. The other parameters here are the refractive index n(\omega) of the film and the film thickness d. Show that, in the limit of a very thin film, and at normal incidence, that the reflection coefficient is given by rfilm = ikd\chi (22.665) Problem 22.7 Using the reflection coefficient from Problem 22.6, use a mode summation for the 1D version of the scalar field in Problem 22.4 and the formula V (TE) CP = −1
(22.666) in terms of the vacuum expectation value to compute the (TE component of the) Casimir–Polder potential for an atom a distance z from a very thin dielectric thin film of thickness d. Your result should be V (TE) CP
32\piϵ0z3 . (22.667) 28Daniel A. Steck, Classical and Modern Optics, available online at http://steck.us/teaching.
22.13 Exercises Problem 22.8 In analogy with Problem 22.7, use a mode-summation approach to compute the (TE component of the) Casimir–Polder potential in 1D electromagnetism for an atom a distance z away from a dielectric half space of susceptibility \chi, in the limit \chi ≪1. Your result should be V (TE) CP
64\piϵ0z2 . (22.668) Problem 22.9 In analogy with Problem 22.7, use a mode-summation approach to compute the (TE component of the) Casimir–Polder potential in 1D electromagnetism for an atom a distance z away from a planar, perfect conductor. Problem 22.10 Use the weak-coupling path integral [Eq. (22.226)] V (TE) CP
3¯hc\alpha0
Z \infty dT
x(\tau) (22.669) to compute the (TE component of the) Casimir–Polder potential for an atom a distance z from a very thin dielectric thin film of thickness d and susceptibility \chi. Show that your results are consistent with Problem 22.7 in one dimension. Problem 22.11 Use the TE-polarization path integral [Eq. (22.197)] V (TE) CP
¯hc\alpha0
Z \infty dT
**
x(\tau) ++ x(\tau) (22.670) to compute the TE component of the Casimir–Polder potential for an atom at a distance z from a thin dielectric plane of the form
(22.671) for the case D = 4. Show that your results are consistent in the appropriate limit with the results of Problem 22.10. Problem 22.12 Using the path integral (22.553) for the TE component of the Casimir energy E (TE) EM = − ¯hc
Z \infty dT
Z dD−1x0 **
x(\tau) ++ x(\tau) , (22.672) for a pair of thin dielectric planar membranes separated by distance d:
(22.673) Begin by defining the ‘‘double-local time’’ ℓdl[Bt; d, d0] := Z t dt′ \delta Bt(t′) −d0
Bt(t′) −d −d0 , (22.674)
Chapter 22. Electromagnetic Casimir Energies as Path Integrals and then use the Feynman–Kac formula to compute the path integral
Z \infty dt \sqrt t e−\lambdat
exp −sℓdl h Bt(t′); d, d0 i Bt , (22.675) which can be transformed into the Casimir energy (22.672), following the procedure of Section 22.12.1. Analyze the energy in the limits of large and small d, noting any pathologies that you find. Problem 22.13 Recalculate the path integral
Z \infty dt \sqrt t e−\lambdat
exp −N[Bt; d, \Xi] Bt , (22.676) where N[Bt; d, \Xi] := lim a−\rightarrow 0 M[Bt; d, \Xi, a] = \Xi Z t dt′ \Xi\delta2h Bt(t′) −d i −\delta′h Bt(t′) −d i M[Bt; d, \Xi, a] := Z t dt′ VTM Bt(t′) = \Xi 2a Z t dt′ \Xi a 1[d,d+a] h Bt(t′) i
h Bt(t′) −(d + a) i −\delta h Bt(t′) −d i = \Xi 2a \Xi a Ts Bt; [d, d + a] + ℓ Bt; d + a −ℓ Bt; d , (22.677)
Problem 22.14 Give integral expressions for the Casimir–Polder potential (both polarizations) in 2D electromagnetism for an atom on the vacuum side of a planar, dielectric interface. Obtain explicit expressions for the strong- and weak-coupling limits. Problem 22.15 Evaluate the derivative-free TM path integral (22.479) V (TM) CP
¯hc\alpha0
Z \infty dT
x(\tau)
++ x(\tau) (22.678) for an atom near one planar, dielectric interface in the limit of small \chi. Do this by expanding the path integral to first order in \chi right at the beginning. Problem 22.16 The goal of this problem is to review the techniques for analytically summing the worldline path integral for a perfectly conducting plane, for the TE-polarization field. (a) Starting with the Green-function form of the Feynman–Kac diffusion equation [Eq. (22.300),
f(x) −2g(x), (22.679)
22.13 Exercises¶
the image method to satisfy the Dirichlet boundary condition. (c) Use the result to compute the Casimir–Polder potential, assuming TE polarization, following the general method for computing the potential at a dielectric interface from Section 22.7. (d) Repeat the calculation in steps (a)–(c), but using the full 3D Green function solution (i.e., keep the ‘‘ignorable’’ transverse dimensions explicitly).