21. Path-Integral Calculation of Casimir Energies¶
PDF pages 965–1004
21.1.1 Quantum Scalar Field¶
Chapter 21 Path-Integral Calculation of Casimir Energies 21.1 Scalar Theory So far, in Chapters 13 and 14, we have developed two methods for computing atom–surface interactions: explicit mode summation with dipole-approximation atom–field coupling (with the assumption of perfectly conducting boundaries), and the more general Green-tensor formalism, which can handle dielectrics. Both of these approaches are difficult when the geometry of the surface is not highly symmetric, and it is useful to develop methods amenable to numerical computation in such cases. Here we will develop the path-integral, or world-line method for computing more general Casimir forces (including forces between macroscopic bodies, not just the Casimir–Polder effect). The original for- mulation of this method1 is for the scalar field, which is unphysical, but a simple way to introduce the method. The method amounts to a Monte–Carlo solution of a path integral, and is thus quite different from the approaches we have covered up to now. In evaluating the path integrals, we will make extensive use of the results on stochastic processes that we have developed in Chapter 17. 21.1.1 Quantum Scalar Field We will take the scalar field \phi(r) to be defined by the Lagrangian L = ¯h2 Z ddr
¯h2 \phi2 (21.1) in d spatial dimensions, which includes coupling to a mass m (for the analogous mass coupling in the electromagnetic field, see Problem 8.16). Note that we are in mks units, where \phi2 has dimensions of inverse volume, and ¯h/mc is the Compton length. The Lagrangian thus has overall dimensions of square energy. As a model for interactions with material bodies, we can consider a space-dependent mass,
(21.2) where m0 is the ‘‘asymptotic’’ value of the mass function, representing the mass of the field, and the deviation \deltam(r) models the effects of the material bodies. Then as
(21.3) 1Holger Gies, Kurt Langfeld, and Laurent Moyaerts, ‘‘Casimir effect on the worldline,’’ Journal of High Energy Physics 06, 018 (2003) (doi:
For an overview of preceding work on world lines, see Christian Schubert, ‘‘An Introduction to the Worldline Technique for Quantum Field Theory Calculations,’’ Acta Physica Polonica B 27, 3965 (1996), arXiv.org preprint (arXiv: hep-th/9610108v2); C. Schubert, ‘‘Perturbative quantum field theory in the string-inspired formalism,’’ Physics Reports 355, 73 (2001) (doi: 10.1016/S0370-1573(01)00013-8), arXiv.org preprint (arXiv:
21.1.2 Partition Function¶
Chapter 21. Path-Integral Calculation of Casimir Energies we can define a background ‘‘potential’’ function V (r) such that
h
i c4, (21.4) which also has dimensions of squared energy. In this case, we can write the Lagrangian as L = 1 Z d3r h
, (model scalar-field Lagrangian) (21.5) Again, the potential will ultimately represent the macroscopic bodies of the Casimir effect, but there is no simple correspondence to the usual ϵ(r) of the electromagnetic field. The conjugate momentum to \phi is
(21.6) Thus, the Hamiltonian is
= 1 Z d3r \pi2
. (21.7) Hamilton’s equations give the first-order-in-time evolution in terms of the two coupled fields \phi and \pi,
(21.8) while after decoupling these equations, we find the inhomogeneous wave equation \nabla 2 −1 c2 \partial 2 t
m 2 0 c2 ¯h2 + ¯h2c2 V (r) \phi (21.9) (scalar-field wave equation) for uncoupled but second-order-in-time evolution. In quantum mechanics, we simply regard the conjugate \pi(r) and \phi(r) fields to be operators. That is, envision a harmonic oscillator at each r representing the amplitude of the field there. Then the operator \phi(r) acts as the operator x ∝(a + a\dagger) for that particular oscillator (but note that this x refers to the temporal part of the field, giving the local, time-dependent amplitude). Correspondingly, \pi(r) acts as the local conjugate (quadrature) operator p ∝(a −a\dagger) at that point in space. (More on this in Section 21.1.4.2.) Normally we will take the background mass m0 to be zero for a massless scalar field, but we can note here that we can simply absorb any nonzero mass into an overall constant offset in V (r). Henceforth, we will not write an explicit mass m0, although the treatment here can cover massive scalar fields as well. 21.1.2 Partition Function We ultimately want the energy associated with the various configurations of V (r), which is specified by the Hamiltonian. A general way to get at the energy is to consider the partition function, Z := Tr e−\betaH . (21.10) where \beta := 1/kBT for temperature T and Boltzmann constant kB. Expressing this in terms of the basis |n\rangle of energy eigenstates (of energy En), Z = X n
X n e−\betaEn. (21.11)
21.1 Scalar Theory Recall that according to the Boltzmann distribution in statistical mechanics, each term here represents the relative probability of occupying state |n\rangle , and the partition function is the sum of these relative probabilities, and thus gives the proper probability normalization. That is, the probability of occupying state |n\rangle is e−\betaEn/Z. Then the energy expectation value is
X n e−\betaEn Z En = −1
X n
(21.12) or
(21.13) (energy in terms of partition function) Thus, we will seek to compute log Z, and thereby compute the Casimir energy. Generally speaking, the Casimir energy is the energy associated with the vacuum (i.e., the effect is due not to real photons, but rather only to virtual photons). In the zero-temperature limit, the vacuum is occupied with unit probability, which we can see from the partition function, where the ground-state-energy term dominates the other ones as \beta −\rightarrow \infty: lim
(21.14) Thus, differentiating the partition function directly yields the ground-state energy. Of course, we can also derive temperature-dependent corrections to the ground-state energy with this method. 21.1.2.1 Free Energy When we consider finite-temperature effects (e.g., in Section 21.1.6), rather than considering the energy E, we will need to compute the Helmholtz free energy F := E −TS, (21.15) (Helmholtz free energy) where T is the temperature and S is the entropy. To see why, recall the first law of thermodynamics, dE = dQ −dW, (21.16) which gives the small change in internal energy E in terms of the a small heat dQ transferred to the system and the work dW done by the system (commonly written P dV in terms of pressure and volume). Since the entropy change is dS = dQ/T, we can rewrite the first law as −dW = dE −T dS = dF, (21.17) where we assume the differential dF to be at constant temperature. Thus, at constant temperature, we can interpret the free energy F as the relevant Casimir potential, as its change represents the work involved in rearranging the material bodies. (We use −dW here as the work done on the system by rearranging the bodies, rather than the work dW done by the system.) If we regard the work as being due to moving one of the bodies through a small displacement dr against a force F, we have
(21.18) and thus
(21.19) (force and free energy) That is, the thermodynamic force on the body (at constant temperature) is given by the gradient of the free energy, taken with respect to a coordinate that represents the position of the body. To put F in terms of the partition function, we can begin with
\partial T , (21.20)
21.1.3 Path Integral of the Quantum Field¶
Chapter 21. Path-Integral Calculation of Casimir Energies where the temperature derivative is taken with no work (e.g., constant volume). Then
F T , (21.21)
F = −kBT log Z = −1 \beta log Z. (free energy in terms of partition function) (21.22) Note that in the zero-temperature limit, we can see from Eq. (21.14) that lim
(21.23) and thus for zero temperature, both F and E give the ground-state energy E0. That E and F coincide at
21.1.2.2 Partition-Function Renormalization As we have seen, the Casimir energy is divergent, and must be renormalized by subtracting away the energy without coupling between the bodies. We will revisit this issue later when discussing particular examples, but for now we note that we will need a subtraction of the form
Z0 , (21.24) (renormalized Casimir energy)
the field. The main point here is that overall factors in the partition function (such as normalization factors) will cancel in renormalization, and so we are free to drop such factors along the way without changing the final result. Of course, a similar renormalization applies to the free energy in the case of nonzero temperature. 21.1.3 Path Integral of the Quantum Field Now it is convenient to introduce a basis of eigenstates of the field operator \phi, which must sum to the identity: Z
(21.25) The labels \phi(r) for these states correspond to something like the set of all classical fields of a localized excitation, of all possible amplitudes. We can then write the partition-function trace in (21.10) in this basis: Z = Z
(21.26) Note that in the notation here, a volume integration is implied, Z d\phi ≡ Z ddr d\phi(r), (21.27)
if we split \beta into N pieces ∆\beta := \beta/N, we can insert identities in between each of the split exponential factors: Z = Z
Z
Z
e−∆\betaH Z
= Z
N−1 Y j=1 Z d\phij
(21.28)
21.1 Scalar Theory This is the same procedure that we used for the quantum-mechanical path integral in Section 20.1.1, but generalized to the quantum-field case. Then if we take
(21.29) in analogy to the endpoint identification (20.128), we can rewrite the partition function as Z = Z N Y j=1
(21.30) Introducing the shorthand for the path-integral volume element
N Y j=1
N Y j=0
(21.31) (field path-integration element) we then have Z = Z D\phi N Y j=1
(21.32) (field path integral) where the path integral is periodic due to the identification |\phiN\rangle ≡|\phi0\rangle . 21.1.3.1 Momentum Projectors and Imaginary Time The path integral (21.32) that we have written down so far can be interpreted as something like the propa- gation of the field state through many small increments ∆\beta in imaginary time. This observation motivates the next step in the calculation. An exponential evolution operator generates first-order evolution in time, but the scalar-field evolution, as we have seen, is second-order in time, or first-order if we consider coupling to the conjugate \pi field. This is also a sensible step since the Hamiltonian, and thus the evolution operator,
−∆\beta Z d3r \pi2 ¯h2 exp −∆\beta Z d3r h
(21.33) Notice that we have split the exponential into the conjugate parts; we are assuming the limit of vanishingly small ∆\beta, so we can ignore the higher-order corrections here. Concentrating on a single factor in Eq. (21.32), we can insert the identity in terms of the basis of momentum-field eigenstates |\pi\rangle for each such factor:
= Z
= Z
−∆\beta 2¯h2 Z d3r \pi2 exp −∆\beta Z d3r h
= Z
−∆\beta 2¯h2 Z d3r \pi 2 j exp −∆\beta Z d3r h
j−1 i . (21.34) Note that the \phij−1 in the potential term could be just as written in terms of any combination of \phij and \phij−1. The inner products correspond to inner products of conjugate-variable eigenstates, e.g., \langle x|p\rangle = eipx/¯h/ \sqrt 2\pi¯h, so
∝ Z d\pij exp Z d3r i
2¯h2 \pi2 j exp −∆\beta Z d3r h
j−1 i . (21.35)
21.1.4 Reduction of the Hilbert Space¶
Chapter 21. Path-Integral Calculation of Casimir Energies Focusing on the momentum-dependent factor, we can carry out the momentum integral explicitly. First we complete the square, Z d\pij exp Z d3r i
2ϵ0 \pi2 j = Z d\pij exp " −∆\beta 2¯h2 Z d3r \pij −i ¯h
2# exp −1 Z
∆\beta , (21.36) again keeping only terms to order ∆\beta in splitting the exponential. We can evaluate the Gaussian integral in analogy to
exp −\alphax2 dx = Z \infty −\infty exp −\alphax2 dx = r\pi \alpha (Re[\alpha] > 0), (21.37) since shifting the contour in the imaginary direction in the complex plane does not affect the integral’s value (due to analyticity of the Gaussian). The \pij integral thus gives an overall factor that is independent of the fields, so we can drop it from the partition function. We can return the remaining quadratic factor, merging it into the remaining exponential in Eq. (21.35):
−∆\beta Z d3r
∆\beta2
j−1 . (21.38) Now the partition function (21.32) contains the product of many such factors. Taking the limit ∆\beta −\rightarrow d\beta, and taking ˜\beta := ¯hc\beta to be the ‘‘Boltzmann length,’’ Z = Z D\phi exp −1 2¯hc Z d˜\beta d3r
. (field path integral, \beta-integral form) (21.39) Here, we have defined the derivative corresponding to the (j −1) field:
∆˜\beta . (21.40) There is no other dependence on \phi in the kinetic term, so there is no concern with ordering or a ‘‘space’’- dependent mass. Note that the partition function now has the form Z = Z D\phi exp i ¯h Z
, (21.41) where L is the Lagrangian (21.5). This has the form of the (coherent) path-integral propagator, but with the imaginary-time replacement t \rightarrow −i˜\beta/c = −i¯h\beta, which converts the unitary-evolution integral into a diffusive path integral. This is the Wick rotation of the path integral, referring to the effective change from a Minkowski to Euclidian metric via the change in minus sign of the temporal term in the quadratic form of the Lagrangian. 21.1.4 Reduction of the Hilbert Space Because we are dealing with a linear system (i.e., a quadratic Hamiltonian), we can simplify the above path integral greatly. This is fortunate, as a path integral with field integration variables is not necessarily an easy thing to handle. 21.1.4.1 Evaluation of the Gaussian Functional Integral First, note that we have an exponentiated quadratic form in the partition function, Z = Z D\phi exp −1 2¯hc Z
−¯h2c2\partial 2
, (21.42)
21.1 Scalar Theory where we have integrated by parts in the derivative terms. We thus have a (functional) Gaussian integral, which we can always do. To do this, we note that the functional integral is the extension of the finite- dimensional Gaussian integral, Z
−1 2z\alpha S−1
(21.43) Here, S\alpha\beta =\langle z\alphaz\beta\rangle , where the expectation value is taken with respect to the normalized Gaussian function, and we can always assume the matrix in such a quadratic form to be symmetric. Thus, it is diagonalized by an orthogonal transformation, and making this transformation, the integral is of the form Z
" −1 X \alpha z 2 \alpha \sigma 2 \alpha #
N Y \alpha=1
(21.44) where the \sigma 2 \alpha are the eigenvalues of S\alpha\beta, and we have carried out the N independent Gaussian integrals. The eigenvalue product then just becomes the determinant. Applying this argument to the functional case, we drop the constant factors and thus write Z = r det c2−¯h2\partial 2
- 2V −1 , (21.45) or log Z = −1 2 log det c2−¯h2\partial 2
- 2V . (21.46) Also noting that for any symmetric matrix A, log det A = Tr log A, (21.47) (which is clear in the diagonal case, where the log of the eigenvalue product is the sum of the eigenvalue logarithms, and follows in the general case by an orthogonal transformation), we can finally write log Z = −1 2Tr log c2−¯h2\partial 2
- 2V . (trace-log form of partition function) (21.48) Thus we have resolved the functional integral into an evaluation of the trace (or equivalently, determinant) of an operator. Recall that in dropping overall factors from Z, this expression is only meaningful in the context of renormalization. Note that it is common in field theory to use units where ¯h = c = 1, and not to bother with the factor of 2 in the potential term, where we would find log Z = −1 2Tr log −\partial 2
. (21.49) We would obtain this same form by removing a factor ¯h2c2 from the log, and absorbing the factor 2/¯h2c2 into the potential (whereupon it would gain the dimensions of inverse squared length). However, we will keep the units as written in (21.48) to better connect to the relativistic particle, and the utility of the factor of 2 will become apparent soon. Now, observe that the operator in Eq. (21.48) is just the (scaled) operator for the wave equation (21.9), \partial 2 t
¯h2c2 V (r)
(21.50) but under the same thermal-time replacement t −\rightarrow i˜\beta/c = i¯h\beta: −\partial 2
¯h2c2 V (r)
(21.51)
Chapter 21. Path-Integral Calculation of Casimir Energies The (classical) Green function G(r, t; r′, t′) for the scalar field is defined to be the solution of the wave equation corresponding to a perturbation localized to the spacetime point (r′, t′): ¯h2\partial 2
(scalar-field Green function) (21.52) Note that G can be defined up to an arbitrary factor, and we have made one particular choice here. We can think of G(r, t; r′, t′) as a matrix element of an operator G, expressed in a basis of space-time states |r, t\rangle , so that the defining equation for the Green operator is ¯h2\partial 2
G = 1, (21.53) or G = ¯h2\partial 2
(21.54) Thus, in terms of the Green function, the partition function (21.48) can be written log Z = 1 2Tr log ˜G, (Green-function form of partition function) (21.55) where
(21.56) is the imaginary-time Green operator. 21.1.4.2 Digression: Second Quantization and Mode Summation In computing Casimir energies, at zero temperature we are just summing the zero-point energies of all the field modes, or rather the differences in the zero-point energies when comparing different boundary configurations. Here we will connect our expressions thus far to the mode-summation picture to gain some intuition into what we are doing. To begin, though, we will be more precise about the quantum field modes by explicitly quantizing the scalar field. The time and space dependence of the scalar wave equation (21.9) are separable if we take
, (21.57) which gives the Helmholtz-like equation
¯h2c2 V (r)
j c2 fj(r), (21.58) where we use j as a mode index, the eigenvalue of the jth mode is \omega 2 j /c2, and we label the mode functions by fj, normalized so that Z
(21.59) This is the normal-mode decomposition of the scalar field. We can quantize the normal modes by changing the time dependence e−i\omegat of the mode to an annihilation operator, so that we obtain the quantized normal- mode operators
p 2¯h\omegaj [fj(r) aj + H.c.] , (21.60) (quantum normal-mode fields) where we have chosen the overall constant to make the Hamiltonian come out nicely below. The quantized conjugate normal-mode operators are given by (21.6) as
r ¯h\omegaj [fj(r) aj −H.c.] , (quantum normal-mode momentum fields) (21.61)
21.1 Scalar Theory with second-quantized field operators
X j \phij,
X j \pij. (21.62) (quantized field operators) Then noting that the eigenvalue equation (21.58) implies \phij
¯h2c2 V (r)
j c2 \phi 2 j , (21.63) we can put the mode operators (21.62) into the Hamiltonian (21.7) to obtain H = 1 Z ddr \pi2
\phi = 1 X j Z ddr " \pi 2 j
¯h2c2 V 2(r) \phij # = 1 X j Z ddr " \pi 2 j
j \phi 2 j # = 1 X j Z ddr ¯h\omegaj h a\dagger
j i |fj(r)|2 = 1 X j ¯h\omegaj h 2a\dagger jaj + 1 i , (21.64) or finally, H = X j ¯h\omegaj a\dagger jaj + 1 , (21.65) which is a sum of harmonic oscillators for each normal mode. In deriving this, we used the fact that only terms like a\dagger jaj and aja\dagger j could contribute to the Hamiltonian, simplifying the algebra, and that [a, a\dagger] = 1 for a harmonic oscillator. In particular, at zero temperature, the field is in the vacuum state, and the field energy is
X j ¯h\omegaj 2 , (mode summation for vacuum energy) (21.66) which is the sum of all zero-point energies of the (classical) normal modes. (Again, this quantity is obviously divergent, and must be regularized by computing the difference in this quantity between two configurations.) Alternately, we can leave the spatial integration in (21.64), to write the Hamiltonian as H = X j Z ddr |fj(r)|2 ¯h\omegaj a\dagger jaj + 1 , (21.67) where the integrand represents an energy density of the field. We can define an energy density at zero temperature as the vacuum expectation value of the integrand, or
X j ¯h\omegaj |fj(r)|2 , (mode summation for vacuum energy density) (21.68) which is again a sum over zero-point energies, weighted by the local value of the corresponding (squared) normalized mode function. The energy density here can provide an intuitive way to think about Casimir– Polder effects, where an atom ‘‘samples’’ the local energy density of the field, which has some spatial dependence due to the combination of all the mode functions.
Chapter 21. Path-Integral Calculation of Casimir Energies 21.1.4.3 Digression Part II: Mode Summation of the Functional Determinant Recall from Eq. (21.45) or (21.55) that we have a partition function that is basically the determinant of the Green operator: Z = p det ˜G. (21.69) In terms of the eigenvalues \lambdaj of ˜G, this becomes Z = s Y j \lambdaj, (21.70) and then the logarithm changes this into an eigenvalue sum: log Z = 1 X j log \lambdaj. (21.71) In the zero-temperature limit, the zero-point energy is proportional to log Z, as we see from Eq. (21.14). However, here we have a sum of logarithms of eigenvalues, while the energy mode sum (21.66) is a sum of (square roots of) eigenvalues, without the same logarithm. It is not so obvious that we are still doing the same mode summation, so we will work out the determinant directly in terms of its eigenvalues to recover the former mode sum. The key here, of course, is that the eigenvalues \omegaj of the wave operator are not the same as the eigenvalues of the Green operator. We will work with the inverse Green operator in imaginary time, ˜G−1 ∝−\partial 2
¯h2c2 V, (21.72) which has the form of the wave operator in Eq. (21.58), but with the extra, imaginary-time dimension ˜\beta. This is a free, scalar wave in this dimension (in the sense of not coupling to the potential V ), but is bounded in extent from 0 to ¯h\beta, with periodic boundary conditions, as we will justify in more detail in Section 21.1.6. Thus, denoting the eigenvalues of ˜G by \lambdan,j, we can write (up to a constant factor) c2 \lambdan,j = 2\pin ¯h\beta 2 + \omega 2 j , (21.73) where the first term on the right-hand side gives the squared frequency that respects the periodic boundary conditions (where n \in Z), and the second term comes from Eq. (21.58). Then using these eigenvalues in Eq. (21.70), we have Z = \infty Y n=−\infty Y j "2\pin ¯h\beta 2 + \omega 2 j #−1/2 , (21.74) where we are dropping overall factors of c. Noting that the function is even in n, we can combine the negative and positive n in the product to find Z = Y j \omegaj \infty Y n=1 "2\pin ¯h\beta 2 + \omega 2 j #−1 . (21.75) In working with the eigenvalues of the Green operator (21.72) with the extra dimension, we expect there to be many more eigenvalues than we had from the wave operator, an expectation that is of course confirmed by the extra mode index n here. Therefore, this quantity should be even more divergent than the mode sum we seek. We will be careful to normalize against the free extra dimension, or in other words we will normalize the partition function by the partition function of the inverse Green operator −\partial 2 ˜\beta to mitigate the
21.1 Scalar Theory effects of the extra degree of freedom here. Thus, we will work with Z Z(−\partial 2
Y j \omegaj \infty Y n=1 2\pin ¯h\beta 2 2\pin ¯h\beta 2 + \omega 2 j = Y j " \omegaj \infty Y n=1 1 +
2\pin 2#−1 , (21.76) noting that we exclude the zero-eigenvalue mode that would otherwise cause problems in the sum. Using the product identity sinh z = z \infty Y j=1 " 1 + z \pij 2# , (21.77) we find Z Z(−\partial 2
Y j 2 ¯h\beta sinh
−1 . (21.78)
Z Z(−\partial 2
Y j ¯h\beta
. (21.79) The bracketed quantity here is the standard form of the partition function for a quantum harmonic oscillator of frequency \omegaj, and the overall factors of ¯h\beta correspond to overall energy offsets that are discarded in renormalization. Dropping factors of 2/¯h\beta and computing the log of the partition function, we find log Z Z(−\partial 2
X j log sinh
. (21.80) Using \partial x log sinh x = coth x, we can then differentiate the log of the partition function to find the energy:
Z Z(−\partial 2
X j ¯h\omegaj coth
. (mode-summation energy, thermal state) (21.81) In the zero-temperature limit, we use limx\rightarrow \inftycoth x = 1 to obtain E0 = X j ¯h\omegaj 2 , (21.82) which is the same mode sum that we obtained in Eq. (21.66). Eq. (21.81) then gives the generalization of this mode-sum energy for finite temperatures. 21.1.4.4 Integral Representation of the Logarithm Now to continue with the path-integral construction, we implement a transformation2 that allows us to transform the partition function into another path integral. We can write the integral identity ˜G = Z \infty dT exp −T ˜G (21.83) 2This transformation was used, e.g., by Julian Schwinger, ‘‘On Gauge Invariance and Vacuum Polarization,’’ Physical Review 82, 664 (1951) (doi: 10.1103/PhysRev.82.664). Thus, T is often called the Schwinger proper time or the Fock–Schwinger proper time, after the earlier work of V. Fock, Proper time in classical and quantum mechanics Physikalische Zeitschrift der Sowjetunion 12, 404 (1937), reprinted in Selected Works: V. A. Fock, L. D. Faddev, L. A. Khalfin, and I. V. Komarov, Eds. (Chapman & Hall, 2004), p. 421 (doi: 10.1201/9780203643204.ch10b).
Chapter 21. Path-Integral Calculation of Casimir Energies for the Green operator, and then consider the variation of the inverse, \delta( ˜G−1), using this integral form:
Z \infty dT exp −T ˜G
Z \infty dT T exp −T ˜G . (21.84) It is worth emphasizing here that we are regarding ˜G−1 as the ‘‘independent variable’’ here of the variation.
log ˜G = Z \infty dT T exp −T ˜G . (21.85) The partition function (21.55) then becomes log Z = 1 Z \infty dT T Tr exp −T ˜G . (21.86) Technically, this integral is divergent at T = 0, but this divergence is cured if we consider any difference between two such integrals, as we can see: lim ϵ−\rightarrow 0 Z \infty ϵ dT T e−AT − Z \infty ϵ dT T e−BT = −log A B . (21.87) Here, note that the subtraction removes the singularity at T = 0, and the result follows from applying Eq. (21.85). In terms of the partition function (21.86), this means it is now time to explicitly renormalize it, as we discussed in Section 21.1.2.2: log Z Z0 = 1 Z \infty dT T Tr exp −T ˜G −exp −T ˜G0 . (21.88) Here, Z0 and ˜G0 are respectively the partition and Green functions corresponding to a background potential V0. Writing out the Green function explicitly, log Z Z0 = 1 Z \infty dT T Tr exp h −T c2−¯h2\partial 2
- 2V i −exp h −T c2−¯h2\partial 2
- 2V0 i . (renormalized partition function, four-dimensional form) (21.89) The choice of V0 depends on the context of the problem, but could correspond to the scalar-field mass or
moving all objects to arbitrarily large separation.) To simplify this expression for log Z somewhat before proceeding, we can rescale T −\rightarrow 2T /¯h2c2, with the result log Z = 1 Z \infty dT T Tr exp −T −1 2\partial 2 ˜\beta −1
V ¯h2c2 . (partition function, four-dimensional form, rescaled Green operator) (21.90) To keep the expressions simple, we are suppressing the renormalization term, with the understanding that the resulting expressions only make sense as a difference between comparable material configurations. We can now carry out the ˜\beta integral as follows. First, we note that we can express the trace above in terms of the space-time basis |r, ˜\beta\rangle , e.g., we can write log Z = 1 Z d3r d˜\beta Z \infty dT T
exp −T −1 2\partial 2 ˜\beta −1
(21.91)
21.1 Scalar Theory where ˜V is shorthand for V /¯h2c2. If we focus on the ˜\beta-dependent factor, we have a Gaussian integral, Z ¯hc\beta
T 2 \partial 2 ˜\beta
Z ¯hc\beta d˜\beta Z
T 2 \partial 2 ˜\beta
= Z ¯hc\beta d˜\beta Z d\xi exp −T 2¯h2 \xi2
= 2\pi¯h Z d\xi exp −T 2¯h2 \xi2 Z ¯hc\beta d˜\beta
2\pi¯h s 2\pi¯h2 T = ¯hc\beta \sqrt 2\piT , (21.92) where we have introduced the conjugate ‘‘frequency’’ \xij to ˜\betaj, and we have again used the one-dimensional, normalized plane-wave state \langle x|p\rangle = eipx/¯h/ \sqrt 2\pi¯h. Then we have the simplified expression
\sqrt 8\pi Z \infty dT T 3/2 Tr exp −T −1
, (unrenormalized partition function, three-dimensional form) (21.93) or using Eq. (21.24) to obtain the energy, we have E = −¯hc \sqrt 8\pi Z \infty dT T 3/2 Tr exp −T −1
, (unrenormalized scalar-field Casimir energy, three-dimensional form) (21.94) where now the trace is only over the spatial states. Again, this expression for the energy must be renormalized by comparison between two sensible configurations to obtain a finite and physically relevant result. 21.1.4.5 Particle Path Integral Now the idea is to repeat the path-integration construction of Section 21.1.3, but with ordinary coordinate states. This, of course, follows closely the procedure in Section 20.1. First, we continue the trace in terms of the spatial basis |r\rangle : E = −¯hc \sqrt 8\pi Z d3r Z \infty dT
exp −T −1
|r\rangle . (21.95) Dividing the ‘‘proper-time’’ variable T into N small bits, and inserting space-time identities, we have Z d3r \langle r| exp −T −1
Z d3r \langle r| N−1 Y j=1 Z d3rj
= Z N Y j=1 d3rj \langle rj| exp −∆T −1
|rj−1\rangle = Z
Dr N Y j=1 \langle rj| exp −∆T −1
|rj−1\rangle , (21.96) where we are again using the notation r0 ≡rN ≡r, (21.97) and Dr := N Y j=1 ddrj = N Y j=0 ddrj \deltad(rN −r0). (21.98) (path-integration element)
Chapter 21. Path-Integral Calculation of Casimir Energies Now in each matrix element, we can split the exponential by dropping a negligible, O(∆T 2) term, and then insert a momentum-basis identity: \langle rj| exp −∆T −1
= Z ddpj \langle rj|pj\rangle \langle pj|e∆T \nabla 2/2e−∆T ˜V (rj−1)|rj−1\rangle = Z ddpj\langle rj|pj\rangle \langle pj|e−∆T p 2 j /2¯h2e−∆T ˜V (rj−1)|rj−1\rangle = (2\pi¯h)d Z ddpj eipj\cdot(rj−rj−1)/¯he−∆T p 2 j /2¯h2e−∆T ˜V (rj−1). (21.99) Here, we have introduced the conjugate momentum pj to rj, and we have also introduced the number d of
(recalling that imaginary parts of the integration variable can be shifted away), \langle rj| exp −1 2∆T
|rj−1\rangle = (2\pi¯h)d Z ddpj eipj\cdot(rj−rj−1)/¯h−∆T p 2 j /2¯h2e−∆T ˜V (rj−1) = (2\pi¯h)d Z ddpj e−(∆T /2¯h2)[pj−i(rj−rj−1)¯h/∆T ]2e−(rj−rj−1)2/2∆T e−∆T ˜V (rj−1) = (2\pi¯h)d Z ddpj e−(∆T /2¯h2)p 2 j e−(rj−rj−1)2/2∆T −∆T ˜V (rj−1) =
( −∆T " rj −rj−1 ∆T 2 + ˜V (rj−1) #) , (21.100) where we have ignored O(∆T 2) terms. Then assembling all the factors in Eq. (21.96), Z ddr \langle r| exp −T −1
|r\rangle =
Z
Dr N Y j=1 exp ( −∆T " rj −rj−1 ∆T 2 + ˜V (rj−1) #) =
Z
Dr exp " − Z T d\tau
- ˜V (r) # . (21.101) As in the field path integral, the exponentiated integral is a shorthand for the product of many close-to-unity exponential factors. The choice here of rj−1 (instead of rj, or some combination) is of course not unique, and as we know from Section 20.4, the time-slicing choice here corresponds to a particular choice of stochastic calculus (here, It¯o calculus). This turns out to be of no consequence for this problem, because there is no curvature or space-dependent mass—all the explicit space dependence here is in the potential. It is, however, a more important distinction in the electromagnetic case. Then, for the sake of completeness, we can write out the Casimir energy (21.95) as E = −¯hc \sqrt 8\pi Z \infty dT
Z
Dr(\tau) exp " − Z T d\tau
- V (r) ¯h2c2 # , (scalar-field Casimir energy (unrenormalized)) (21.102)
expression is only sensible after renormalization by subtracting the analogous background energy in terms of V0. Notice that the T and \tau (proper-time) variables that we have introduced in fact have dimensions of squared length.
21.1 Scalar Theory 21.1.4.6 Monte-Carlo Integration and Stochastic “Loops” Now we convert this integral into a Monte-Carlo average, suitable for evaluation on a computer. We motivate the basic idea as follows (as in Section 20.3.1): suppose f(x) is a normalized, nonnegative (probability) distribution; then we may rewrite an integral involving f(x) as an expectation value, Z
g(x)
f(x), (21.103) where the double-bracket expectation value is an average over the distribution f(x). Computationally, this allows us to throw random deviates xn, chosen from the distribution f(x), and simply average the values g(xn) to obtain an estimate for the integral. In the integral (21.102), we would like to choose the factor exp " − Z T
(21.104) as the probability distribution. To normalize it, we note the equivalence Z d3r \langle r| exp T 2 \nabla 2 ¶
Z
Dr exp " − Z T
, (21.105) which follows from Eq. (21.101) by removing the potential. Thus, we can work out the normalization of this factor, following the procedure of Eq. (21.92): \langle r| exp T 2 \nabla 2 ¶
Z ddp \langle r| exp T 2 \nabla 2
= Z ddp exp −T 2¯h2 p2
= (2\pi¯h)d Z ddp exp −T 2¯h2 p2 = (2\pi¯h)d 2\pi¯h2 T d/2 =
(21.106) Note that we have not computed the remaining integral with respect to r, which would cause this result to diverge; this could be interpretated as something like the energy density at any point may be finite, but the total background energy diverges, as we expect. Recalling that this is the same integration variable as r0 = rN in the path integral, we can then identify the normalization Z D′r exp " − Z T
=¶
T 2 \nabla 2
(21.107) and again, we are only integrating over all the intermediate path coordinates: D′r = N−1 Y j=1 ddrj. (21.108) Thus, to convert to a Monte-Carlo average, we should replace this part of the integral with an expectation value over this distribution, and tack on the factor on the right-hand side of Eq. (21.106) to compensate for the normalization of the probability distribution. Thus, Eq. (21.102) becomes E = − ¯hc
Z \infty dT
Z ddx0 ** exp " − ¯h2c2 Z T
x(\tau) , (scalar-field Casimir energy, Monte-Carlo form) (21.109)
21.1.5 Analytic Evaluation of Scalar Casimir Energies¶
Chapter 21. Path-Integral Calculation of Casimir Energies
4; recall that we normalized the timelike dimension separately, but in contributes to the overall normalization in the same way as the spacelike dimensions). We have also changed the remaining integration over r to x0. Here, the expectation value is taken over functions x(\tau) chosen according to the (unnormalized) Gaussian probability density exp " − Z T
, # (21.110)
where points in 3N-dimensional Cartesian space are chosen according to the Gaussian probability density for the increments N Y j=1 exp " −(xj −xj−1) 2∆T 2# , (21.111) again where x0 = xN = r is the beginning and termination point of the ‘‘path’’ here, which is also the spatial integration variable in (21.102). Now let’s take a closer look at the interpretations of Eqs. (21.102) and (21.109). To simplify the
integral (Nd dimensions, in the limit of large N), where the integrand has Gaussian velocity-weighting factors of the form exp (xj −xj−1)2 2∆T . (21.112) Thus, successive positions (separated by ∆T in ‘‘time’’) in the discrete form of the parameterized many- dimensional coordinate x(\tau) must be close together—of the order of \sqrt ∆T or less—otherwise the contribution of that particular point x(\tau) to the integral is negligible. As such, only a very small fraction of all possible x(\tau)—essentially, those that correspond to continuous paths in the large-N limit—can possible contribute to the integral. In Eq. (21.109), the idea is to make the integral much easier to evaluate. We focus only on those paths that whose amplitudes are not destroyed by the factor (21.112), by using precisely these factors to determine which paths to choose (at random). The coordinates rj and rj−1 [equivalently, r(\tau) and r(\tau −∆T )] are separated by a random distance of variance ∆T (in each direction). Because the first and last points in this stochastic path are identical, and this is a stochastic ‘‘loop.’’ The path is a continuous random walk, where the components of x(\tau) have the form
\sqrt T B \tau T , (21.113) with B(t) a vector Brownian bridge, having scalar Brownian bridges Bj(t) as its d components. Recall from Section 17.7 that Brownian bridges are the continuous limits of Gaussian random walks that return to
Carlo calculation, being parameterized by the ‘‘proper time’’ T . In any case, for the evaluation of the Casimir energy in Eq. (21.109), we must still weight each path according to the Gaussian potential factor for both the configuration potential V (r) and the background potential V0(r). Rather than discuss these abstractly, we will illustrate the calculation of Casimir potentials in this formalism with a couple of simple examples. 21.1.5 Analytic Evaluation of Scalar Casimir Energies 21.1.5.1 Strong-Coupling Limit: Atom–Plane Interaction To compute the Casimir–Polder interaction of an atom with a conducting plane, we will first note that this theory does not incorporate dispersion, so we will stick to a perfectly conducting plane, and we will not expect to get the crossover from z−3 to z−4 behavior, as this requires dispersion of the atom (frequency-dependent polarizability). Also, to compute the interaction with the perfectly conducting plane, we will consider the
21.1 Scalar Theory¶
the body). For concreteness, we may write
(21.114) where the coupling parameter \chi −\rightarrow \infty, \Theta(x) is the Heaviside function, the boundary of the plane is given by z = d, and the atom is situated at the origin so that d is also the atom–mirror separation. d Recall [see Eq. (1.60)] that the dipole potential for an atom interacting via its induced dipole moment
this case, we do not have a steady field, so we should interpret the squared electric field as an expectation value in the vacuum [as we did in Section 13.6 for the atomic dipole moment in the near-field Casimir–Polder potential; see also the discussion leading up to Eq. (13.201)]: Vdipole = −1 2\alpha0
E2 . (21.115) Now we must connect this expression to the Casimir energy in Eq. (21.109). First, note that the vacuum energy density of the electromagnetic field at any particular point has the similar form
E2(r)
. (21.116) We can compare this expression to the energy density that we can deduce from Eq. (21.109), by noting that the spatial integral represents an energy integrated over the energy density, and also taking the renormal-
8\pi2 Z \infty dT T 3 ** 1 −exp " − Z T
x(\tau) . (21.117)
Combining the above three expressions, we can then write Vdipole = −¯hc\alpha0 16\pi2ϵ0 Z \infty dT T 3 ** 1 −exp " − Z T
x(\tau) (21.118) for the world-line form of the Casimir–Polder potential. In this form, the potential is completely general, specified in terms of an arbitrary potential V (r) and an atomic location r as the source point for the paths. Actually, in the above argument for the interpreting the spatial integrand of the total energy (21.109) as the energy density, we should be more careful: there are many other functions whose integral could add up to the right energy. To see this more directly, note that we can trace through the whole derivation again, starting with a Lagrangian density L (r) that samples the fields at only one point in space, L = Z d3r L (r), (21.119) and then carrying through the derivation with L instead of L. We can fix the Lagrangian density uniquely by starting with a partition-function ‘‘density’’ analogous to Eq. (21.10),
h e−\betaH (r) d3ri , (21.120) where H (r) is the Hamiltonian density, defined by H =: Z d3r H (r). (21.121)
Chapter 21. Path-Integral Calculation of Casimir Energies The conjugate fields are defined with respect to the full Hamiltonian H, as the fields must respect global boundary conditions. Note that we can decompose the total partition function into the partition density by writing out the trace, Z = X n
= X n \langle n| exp −\beta Z d3r H (r) |n\rangle = X n \langle n| Y r exp −\betaH (r) d3r |n\rangle = Y r X n \langle n| exp −\betaH (r) d3r |n\rangle , (21.122) where the product is over all points in space (as the integral is a sum over all points in space), and the last step follows by inserting identities of the form P n′ |n′\rangle \langle n′| in terms of energy eigenstates. Then we have log Z = Z d3r log Z(r), (21.123) where we have dropped an additive constant (log d3r) on the right-hand side. Because we log Z to compute the energy, we see that the energy density simply integrates to the total energy, in terms of the logarithms of the corresponding partition functions. Furthermore, Z(r) clearly only represents the field energy at a single point r, so it produces the proper energy density. Everything in the derivation then carries through with R d˜\beta d3r replaced by R d˜\beta up through Eq. (21.42). At this point, the functional determinants and traces refer only to ˜\beta, and no longer to r. Thus, the expression of the trace in Eq. (21.91) in terms of a Euclidean space-time integral is just a time (˜\beta) integral, and the result in deriving the path integral is just the spatial integrand of (21.109), as desired. That is to say, the spatial integral in the total energy (21.109) is exactly the spatial integral that appears in the Hamiltonian (21.7). Coming back to the planar-mirror case, this expression is extremely simple to interpret in the strong- coupling limit. If the path touches the surface, then the argument in the exponential of Eq. (21.117) diverges negatively, so the exponential vanishes, and the overall contribution of that path to the expectation value is unity. Otherwise, the contribution of a non-touching path simply vanishes (as a direct result of renormalization, since these paths instead contribute to the Lamb shift). Then the z-coordinate of the path is the only one relevant to the calculation, and we must only keep track if its maximum excursion sup[z(\tau)] takes it past z = d. We can write this as an average over Heaviside functions of the paths as Vdipole = −¯hc\alpha0 16\pi2ϵ0 Z \infty dT T 3
\Theta n sup[z(\tau)] −d o x(\tau)
16\pi2ϵ0 Z \infty dT T 3
\Theta n\sqrt T sup[B(t)] −d o B(t) , (21.124)
Thus, we see that the renormalization simply cuts off the lower (divergent) end of the integral. To evaluate this analytically, we will implement the cutoff separately for each path: Vdipole = −¯h\alpha0 16\pi2ϵ0 Z \infty d2/ sup2[B(t)] dT T 3
B(t)
32\pi2ϵ0
d2
B(t) = − ¯hc\alpha0 32\pi2ϵ0d4
{sup[B(t)]}4
B(t) . (21.125)
21.1 Scalar Theory In scaling out the path sizes, we see that the z−4 scaling of the Casimir–Polder force is built into the path scale and the T −3 dependence of the integral. The path statistics enter here simply to give an overall
well-defined probability density [Eq. (17.383)]:
(21.126) The fourth moment of this distribution is 1/2 [from Eq. (17.385)]. Putting in this value for the fourth moment, we find the scalar result Vdipole = − ¯hc\alpha0
3¯hc\alpha0
1 , (Casimir–Polder potential, perfectly conducting plane, scalar result) (21.127) The asymptotic expression for the electromagnetic field in the limit of large z was [see Eq. (13.60)] VCP = − ¯hc\alpha0
(Casimir–Polder potential, perfectly conducting plane, far-field electromagnetic result) (21.128) which is the same except for the factor of 1/6. Again, since we are ignoring dispersion, it is most sensible to compare to the far-field result, where only the dc atomic polarizability contributes. While the scalar result does not quite agree with the electromagnetic Casimir–Polder potential, the scalar result does reproduce the contribution of only the TE modes to the total potential; the factor of 1/6 here is the same factor that appears in Eq. (13.181). This is sensible, since TE modes at a planar surface behave as scalar waves, since the polarization is the same for the incident, reflected, and transmitted waves, and effectively the polarization drops out of the calculation. By contrast, the polarization is different for the same three components of TM waves, and the scalar result does not capture this extra complexity. 21.1.5.2 Strong-Coupling Limit: Atomic Interaction with Two Planes Next, we consider the slightly more complicated case of an atom interacting with two parallel, conducting planes. The atom is situated at z = 0, with the barriers at z = −a and z = L −a. L a We consider only the case of the atom between the two planes, as if the atom is not in between them, the potential is simply the one-plane potential due to the nearer surface. To evaluate the potential here, we need the probability that a stochastic path touches either surface. This is equivalent to the problem of calculating the escape probability of a Brownian bridge outside the interval defined by [−a, L−a]. We have already calculated this before, and the result for a standard Brownian bridge is [Eq. (17.415)] Pescape = e−2a2 + \infty X j=1 h e−2(jL−a)2 + e−2(jL+a)2 −2e−2j2L2i . (21.129) However, we have paths x(\tau), which by their definition are equivalent to Brownian bridges B2T (t) running in time from 0 to 2T , for which the escape probability is given by scaling all squared lengths by this ‘‘time interval’’:
\infty X j=1 h
. (21.130)
Chapter 21. Path-Integral Calculation of Casimir Energies This is precisely the path average in Eq. (21.118), since the path average is of a function that is unity whenever the bridge escapes (touches either plane), and 0 otherwise. Thus, Eq. (21.118) becomes Vdipole = −¯hc\alpha0 16\pi2ϵ0 Z \infty dT T 3
\infty X j=1
, (21.131) and using the integral result Z \infty dx e−a/x x3 = 1 a2 , (21.132) we finally have Vdipole = −¯hc\alpha0 64\pi2ϵ0 1 a4 + \infty X j=1 (jL −a)4 + (jL + a)4 − j4L4 (Casimir–Polder potential between two conducting planes) (21.133) Note that the first term here agrees with the single-plate result (21.127), while the other terms represent ‘‘reflected images’’ due to the other mirror; all the other terms vanish in the single-plane limit L −\rightarrow \infty. Note also that this result is invariant under the replacement a −\rightarrow L −a, as it must be. The a-independent term here technically doesn’t influence the atomic potential, and can be dropped in the renormalization, though we will keep it here for illustrative purposes. Also, note the similarity of the summation structure to the frequency and lifetime shifts of a Lorentz atom between two parallel, planar conductors (Problem 1.3). Finally, note that we can use the summation formula \infty X j=1
, (21.134)
formula follows by differentiating the series formula3
\infty X j=1 x
\infty X j=1 1 j − j + x (21.135) (which technically is invalid when x is a negative integer, but this won’t be a problem here). Thus, we have Vdipole = −¯hc\alpha0 64\pi2ϵ0 1 a4 − \pi4 45L4 + 6L4 h
i (Casimir–Polder potential between two conducting planes) (21.136) as an analytic form for the potential of an atom between two planes, within scalar theory. Again, the L−4 term only contributes an overall offset to the atomic potential, and can be dropped. 21.1.5.3 Strong-Coupling Limit: Plane–Plane Interaction As a final example we evaluate directly the Casimir interaction between two parallel planes separated by distance L, this time without any atom involved. L 3Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions (Dover, 1965), p. 259, Eq. (6.3.16).
21.1 Scalar Theory To do this, we return to Eq. (21.109) for the (unrenormalized) Casimir energy. As a reference, we wish to reference the absolute energy such that the Casimir energy vanishes as L −\rightarrow \infty. Thus, we will take V (x) to refer to the two parallel planes separated by L, and V0(x) to refer to the two parallel planes with arbitrarily large separation. As before, the expectation value of each exponential in Eq. (21.109) is equivalent to 1−Ptouch, where Ptouch is the probability of the path x(\tau) to touch the surface defined by the appropriate potential function V (x). But we have to be careful when computing the touch probabilities here. The touch probability for V (x) is straightforward: Ptouch refers to the probability to touch either plate. However, the touch probability for V0(x) is a bit trickier. Since we are counting the case of widely separated plates, we must count all the touch probabilities that are equivalent to the touching path in the V (x) case. Each path that touches V (x) should be compared with two background paths: one that is ‘‘near’’ each of the now widely separated plates. Thus, the path average with renormalization in Eq. (21.109) is h 1 −Ptouch[V (x)] i − h 1 −Ptouch[V0(x)] i
(21.137) That is, we only count paths that touch both planes; paths that touch only one plane (or neither plane) are dropped after renormalization. With the global minus sign in the Casimir energy, the overall energy is negative in this renormalization scheme (leading to an attractive two-body force). Now we will work out the relevant probabilities and compute the energy in two parts, representing the exterior and interior of the pair of planes. First, the exterior. Consider a point a exterior to the pair of planes. The probability for a path starting and ending at a to touch both planes is just the probability to touch the more distant plane, which is a distance L + a away. The probability for a standard Brownian bridge to cross a boundary a distance d away is [Eq. (17.380)]
(21.138) Again, we are effectively considering Brownian bridges over a time interval T , so the probability for the path x(\tau) to touch the more distant surface is given by scaling d2 down by T , and then letting d −\rightarrow L + a:
(21.139) The energy associated with the exterior then is just Eq. (21.109), integrated over the extent of each exterior region. These are equivalent, so we just count twice the result for a single region: (E −E0)exterior A = −¯hc 8\pi2 Z \infty dT T 3 Z \infty
= −¯hc 16\pi2 Z \infty da (L + a)4 = − ¯hc 32\pi2L3 2 . (21.140) We again used the integral (21.132) to evaluate the T integral. Also, we only integrated one dimension out of the full volume integral in Eq. (21.109), dividing by the cross-sectional area A in lieu of performing the (divergent) transverse integrals. Note that while this appears to be a contribution from the ‘‘exterior’’ of the two planes, this is really an artifact of the renormalization. In the exterior region, the contributions in Eqs. (21.137) for touching the near plane and for touching either plane exactly cancel. The leftover is the contribution for touching the far plane, which is a contribution from subtracting one of the one-body energies. This can be viewed as an interior contribution, but from when the planes are far apart. For the interior contribution, we need the probability for a path to touch both surfaces. However, it is easier to use the probability to touch either surface, since that is the escape probability that we used above,
Chapter 21. Path-Integral Calculation of Casimir Energies Eq. (21.129). Choosing an interior point a distance a away from the two surfaces, the escape probability for x(\tau) is precisely what we used before for the atom between two planes, Eq. (21.130). Thus, what we need is
−e−2a2/T − \infty X j=1 h
\infty X j=1 h
. (21.141) Here we have again used the single-barrier crossing probability for a Brownian bridge by rescaling the distance in Eq. (21.138) appropriately. Putting this probability in for the expectation value in Eq. (21.109) and integrating over the interior of the two planes, (E −E0)interior A = − ¯h 8\pi2c3 Z \infty dT T 3 Z L da e−2(L−a)2/T − \infty X j=1 h
. (21.142) Again carrying out the T integral first, (E −E0)interior A = −¯hc 32\pi2 Z L da (L −a)4 − \infty X j=1 (jL −a)4 + (jL + a)4 − j4L4 = ¯hc 32\pi2 Z L da \infty X j=2 (jL −a)4 + \infty X j=1 (jL + a)4 − j4L4 = ¯hc 32\pi2 3L3 \infty X j=2 (j −1)3 −1 j3 + \infty X j=1 1 3L3 1 j3 − (j + 1)3 − j4L3 = ¯hc 32\pi2L3 \infty X j=1 2 1 j3 − (j + 1)3 −2 j4 . (21.143) Here, note that the sums over 1/j3 and 1/(j + 1)3 are equivalent, except that the second sum is missing a term of unity. We have already found that the sum over 1/j4 is \pi4/90. Thus, (E −E0)interior A = ¯hc 32\pi2L3 2 3 −\pi4 . (21.144) Then we obtain the total energy by adding Eqs. (21.144) and (21.140). The common 2/3 term cancels out, leaving only the \pi4/45 term, with the result E −E0 A
1440L3 . (Casimir energy density of two conducting planes, scalar result) (21.145) Note that this energy appears as an offset in the atom–two-planes potential (21.136). This result is exactly half of the true Casimir energy for two conducting planes,4 E −E0 A
720L3 , (Casimir energy density of two conducting planes, electromagnetic result) (21.146) suggesting that the two polarizations of the electromagnetic field in this geometry act as two independent scalar fields. 4H. B. G. Casimir, ‘‘On the attraction between two perfectly conducting plates,’’ Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 51, 793 (1948).
21.1.6 World Lines at Finite Temperature¶
21.1 Scalar Theory 21.1.6 World Lines at Finite Temperature Thus far, we have focused on the zero-temperature limit of the scalar Casimir effect. However, it is not difficult to adapt this theory to the case of nonzero temperature.5 To do this, we return to the partition- function expression (21.91), and again focus on the ˜\beta-dependent factor, but convert it into a path integral
Z ¯h\beta
T \partial 2 ˜\beta
Z ¯h\beta
N−1 Y j=1 Z d˜\betaj h e∆T \partial 2
i e∆T \partial 2
= Z N Y j=1
= Z
D ˜\beta N Y j=1
(21.147) Here we must be somewhat careful with the integration limits. While the endpoint ˜\beta spans 0 to ¯hc\beta, the other ˜\betaj are unrestricted. Since they are merely intermediate coordinates we may define as we wish, and in particular we want to be able to introduce the usual momentum eigenstates (plane waves) for these coordinates. Again, in each matrix element, we can insert a momentum-basis identity, where \xij is conjugate to ˜\betaj, and then complete the square and carry out the Gaussian integral:
Z
= Z
= 2\pi¯h Z
j /¯h2 = 2\pi¯h Z
= \sqrt 4\pi∆T exp "
4∆T 2 # . (21.148) Again, we should be a bit careful here, as technically this procedure doesn’t work for the conjugate momentum to ˜\betaN, because it has a discrete spectrum, corresponding to a bounded ‘‘time.’’ We can fix this by introducing |\xiN−1\rangle again for this last matrix element, and obtain the equivalent result. Now putting this matrix element back into Eq. (21.147), Z ¯hc\beta
T \partial 2 ˜\beta
Z
D ˜\beta N Y j=1 exp "
4∆T 2 # =
Z
D ˜\beta exp " − Z T
(21.149) As we computed in Eq. (21.92) or in Eq. (21.106), the normalization factor for this path integral is Z ¯hc\beta¶
T \partial 2 ˜\beta
¯hc\beta \sqrt 4\piT . (21.150) Then we have shown that the path average in ˜\beta is normalized such that ¯hc\beta \sqrt 4\piT −1
Z
D ˜\beta exp " − Z T
= 1, (21.151) 5Klaus Klingmüller and Holger Gies, ‘‘Geothermal Casimir phenomena,’’ Journal of Physics A: Mathematical and Theoretical¶
Chapter 21. Path-Integral Calculation of Casimir Energies and so, in switching to a path average, we are making the replacement Z ¯hc\beta
T \partial 2 ˜\beta
¯hc\beta \sqrt 4\piT
. (21.152) After differentiating log Z with respect to \beta to get the energy, the factor of \beta here disappears, and we obtain the same expression as before, Eq. (21.109), but with a different interpretation: the path average is now with respect to four-dimensional paths, where the fourth dimension is the ˜\beta-direction (‘‘time’’ direction). Of course, the potential doesn’t couple to this dimension, so we will evaluate it separately from the three spatial dimensions. To evaluate the time-dimension path integral, first we should return to the ˜\beta integral, as for example in Eq. (21.150). Recall that ˜\beta entered as the ‘‘imaginary time’’ in the partition function (21.39), running from
reads
(21.153) Since the only dependence on ˜\beta is periodic with period ¯hc\beta, we may take ˜\beta itself to be a periodic coordinate, with period ¯hc\beta. This helps us to make sense of the path integral in Eq. (21.150), where recall that the intermediate steps in the path integral are defined on an unbounded (aperiodic) coordinate. Thus it is possible for a path beginning at some ˜\beta to wander out of [0, ¯h\beta] as \tau −\rightarrow T , but the next-to-last and last points must be close [O( \sqrt ∆T )] together. This can still work if the path reconnects to an ‘‘image’’ of ˜\beta at ˜\beta + n¯hc\beta, for some integer n. Ultimately, this is equivalent to having paths on a cylindrical manifold, where the ˜\beta direction is periodic, and the spatial dimensions are extended. Then paths can either reconnect directly to the same spot, as in the spatial dimensions, or reconnect by winding around the ˜\beta-direction. In the zero-temperature limit, the extent of the ˜\beta-direction becomes arbitrarily large, reducing space to an extended four-dimensional manifold. Now from what we know about Brownian bridges (Section 17.7), any bridge B(t) can be ‘‘deformed’’ to connect the point 0 to the point c by introducing a constant drift. The relative probability of the deformed bridge, or the weight of the deformed bridge compared to the standard bridge, is
\sqrt 2\pi
\sqrt
(21.154) However, we are considering paths that are pinned at time 2T rather than 1, so we really should consider
(21.155) Thus, if we normalize to the zero-temperature case, we can count all possible paths with winding num- ber n, and weight them by this factor, with c −\rightarrow n¯hc\beta. We will then include the many more possible paths in the nonzero-temperature case, but organized by winding number, with each set mapped onto the zero-temperature paths, and weighted explicitly. Then the path average is only with respect to the zero- temperature paths, and except for these probability weights, the ˜\beta part of the path integral goes away. Thus, the path-average expression (21.156) for the (unrenormalized) Casimir energy becomes E = −¯hc 32\pi2 Z \infty dT T 3
\infty X n=−\infty
! Z d3r * exp " − Z T
+ , (scalar-field Casimir energy, nonzero temperature) (21.156) where we have not written the renormalization term to keep the expression (relatively) simple, and the new factor is the sum over all winding numbers. Note that this reduces to the zero-temperature case when \beta −\rightarrow \infty, so that only the n = 0 term survives, and the sum is replaced by unity.¶
21.2.1 Action¶
21.2 Worldlines and the Relativistic Scalar Particle 21.1.6.1 Example: Temperature-Dependent Atom–Planar-Conductor Potential With temperature dependence, we can similarly adapt the atom–surface potential (21.118) to read Vdipole = −¯hc\alpha0 64\pi2ϵ0 Z \infty dT T 3
\infty X n=−\infty
!* 1 −exp " − Z T
+ x(\tau) . (21.157) We will evaluate this for a planar surface in the strong-coupling limit, where the atom–surface distance is d. Again, the path average here is the probability of a Brownian bridge pinned at time 2T to touch the surface, which is exp(−d2/T ), as we argued in Eq. (21.139). Thus, Vdipole = −¯hc\alpha0 64\pi2ϵ0 Z \infty dT T 3 \infty X n=−\infty¶
64\pi2ϵ0 Z \infty dT T 3 \infty X n=−\infty
= − ¯hc\alpha0 64\pi2ϵ0d4 \infty X n=−\infty
= − ¯hc\alpha0 64\pi2ϵ0d4 2\pid ¯hc\beta coth 2\pid ¯hc\beta , (21.158) where we have again used the integral formula (21.132). Writing out the explicit temperature, our result is Vdipole = − ¯hc\alpha0
2\pidkBT ¯hc coth 2\pidkBT ¯hc . (atom–surface potential, nonzero T, scalar result) (21.159)
use limx\rightarrow \inftycoth x = 1 to obtain Vdipole = −kBT\alpha0 (4\piϵ0)8d3 (atom–surface potential, high T, scalar result) (21.160) Interestingly, this is half of the electromagnetic result (14.344), rather than 1/6 as in the zero-temperature limit. 21.2 Worldlines and the Relativistic Scalar Particle To wrap up the worldline calculation of Casimir energies of scalar particles, we will review some of the general theory of classical and quantum scalar particles. This will clarify some aspects of the form of the worldline path integral, as well as the ‘‘worldline’’ nomenclature for the partition-function path integral. 21.2.1 Action In general relativity, the trajectories of point particles correspond to geodesic motion. In flat space, we will take this to mean that the proper time
Z d\tau (21.161) is extremized, where in one spatial dimension,
(21.162)
Chapter 21. Path-Integral Calculation of Casimir Energies Note that \tau is defined up to an arbitrary constant, unless the integral is taken between two events, in which case we interpret \tau as a proper-time interval. This is the same as the space-time interval s up to a factor of c. Thus,
Z p
Z dt r 1 −˙x2 c2 . (21.163)
case |ds2| enters under the square root to maintain a real integrand for ‘‘timelike’’ particle motion.) We will then take the action for the relativistic particle to be
Z dt r 1 −˙x2 c2 , (21.164) where \alpha is some constant (with units of energy) yet to be determined. In addition, we can add in a background potential V (x, t) by tacking it on as in the nonrelativistic action:6 S = Z dt " \alpha r 1 −˙x2 c2 −V (x, t) # . (21.165) We can then identify the bracketed quantity as the Lagrangian L(x, ˙x). Computing the conjugate momentum,
1 −˙x2 c2 −1/2 ˙x c2 . (21.166) For this to coincide with the usual relativistic momentum, we should choose \alpha = −mc2, and thus p = m ˙x q 1 −˙x2/c2 . (21.167) The Euler–Lagrange equation \partial L \partial x −d dt \partial L
(21.168) then simply yields
\partial x , (21.169) as we expect in the nonrelativistic limit. Turning to the Hamiltonian, we obtain
r 1 −˙x2 c2 + V (x, t) = mc2 r 1 −˙x2 c2 p2c2 + m2c4 + V (x, t). (21.170) Solving Eq. (21.167) for ˙x gives ˙x2 c2 = p2c2 p2c2 + m2c4 , (21.171) or 1 −˙x2 c2 = m2c4 p2c2 + m2c4 , (21.172) 6Herbert Goldstein, Charles Poole, and John Safko, Classical Mechanics, 3rd ed. (Addison Wesley, 2001), Section 7.9, p. 312.
21.2.2 Reparameterization Independence¶
21.2 Worldlines and the Relativistic Scalar Particle so that the Hamiltonian (21.170) becomes
p p2c2 + m2c4 + V (x, t), (21.173) (relativistic-particle Hamiltonian) which is the usual expression for the relativistic energy, with an external potential included. Then to summarize, we have the relativistic action and Lagrangian S[x] = Z dt L(x, ˙x)
r 1 −˙x2 c2 −V (x, t). (relativistic-particle Lagrangian and action) (21.174) Note that if T denotes the particle kinetic energy, then we can deduce T from the Hamiltonian by subtracting the rest mass mc2 and the potential: T = p p2c2 + m2c4 −mc2. (21.175) (relativistic-particle kinetic energy) Note that expanding to order p2 gives the usual nonrelativistic result T = p2/2m. This amounts to assuming the Hamiltonian is of the form
(21.176) (relativistic-particle Hamiltonian) which is a constant of the motion if V does not depend explicitly on time. This then amounts to a definition
Eq. (21.173) becomes E2 −p2c2 = mc2, (21.177) (mass-shell condition) which is called the mass-shell condition. This appears as a constraint on the momentum, so in the one-dimensional case, the momentum is not an independent variable. In the relativistic case, note that the Lagrangian (21.174) is not of the form T −V . However, if we define ˜T = −mc2 r 1 −˙x2 c2 = − m2c4 p p2c2 + m2c4 , (21.178) then L = ˜T −V , which is as close as we will get to the ‘‘standard’’ nonrelativistic form for the Lagrangian. 21.2.2 Reparameterization Independence Right now, the action (21.174) is parameterized in terms of the local-time coordinate t. We can also write it in terms of the proper-time parameter as S[x, t] = −mc2 Z d\tau "r dt d\tau 2 −1 c2 dx d\tau 2 −dt d\tau V (x, t) # . (21.179) In this form, we see that the action has a particular property. If we introduce a (bijective) function \lambda(\tau) that acts as a rescaled time (possibly rescaled in a nonlinear way), then using
d\lambda d\tau d\tau, (21.180) the action has the same form: S[x, t] = −mc2 Z d\lambda "r dt d\lambda 2 −1 c2 dx d\lambda 2 −dt d\lambda V (x, t) # . (21.181)
21.2.3 Quadratic Action¶
Chapter 21. Path-Integral Calculation of Casimir Energies The point is that the action is invariant under arbitrary rescaling of the time parameter, or that it is reparameterization-independent. Reparameterization independence is an important concept in relativity. The idea is that the action S should characterize the world line of a particle—the set of all space-time events occupied by the particle. Recall that space-time events are observer-independent, and thus manifestly independent of any coordinate system used to describe them. The world line itself is thus a geometric object, independent of a particular choice of coordinate system. Since the world line fully determines the action (and vice versa), the action should also be described only in terms of the world-line geometry, and it should not include any ‘‘artifacts’’ introduced by a coordinate system. Hence the importance of reparameterization independence. In this sense, reparameterization we have something like a gauge freedom: choosing different proper- time-like parameters are analogous to different gauge choices that lead to the same physical quantities (i.e., the world line). We will see the significance of this soon when we treat this gauge invariance explicitly in the action. 21.2.3 Quadratic Action The action (21.174) and Hamiltonian (21.173) are perfectly valid, and we have seen that the action has the advantage of being independent of the world-line parameterization. However, the obvious feature is the square root, which becomes awkward in some calculations, particularly when carrying the Hamiltonian over to quantum mechanics. Now, using a proper-time parameterization, we will seek to write down a variational principle that is more similar to the nonrelativistic case, hiding the awkward square root. We will begin by replacing Eq. (21.164) with the action principle
Z
Z dt2 −dx2/c2 d\tau 2
Z " dt d\tau 2 −1 c2 dx d\tau 2#
Z ˙t2 −˙x2 c2 d\tau (21.182) where we are explicitly maintaining the action integral in terms of the proper time \tau instead of the local time t, and \alpha is again an undetermined (and potentially a different) constant parameter, with dimensions of energy. Then, while we will refrain for the moment from introducing a background potential V (x), we will introduce an energy offset of mc2/2, with the form of a potential energy, whose purpose will become clear later: S = Z \alpha ˙t2 −˙x2 c2 −1 2mc2 d\tau. (21.183) The momentum conjugate to the spatial coordinate x is
c2 dx d\tau , (21.184) while the momentum conjugate to the local time is
d\tau . (21.185)
2 . (21.186) Then the conjugate momenta are px = mdx d\tau , pt = −mc2 dt d\tau . (21.187) The Euler–Lagrange equation for the action (21.183) then gives d2t
d2x
(21.188) (Euler–Lagrange equations)
21.2 Worldlines and the Relativistic Scalar Particle The two equations together imply straight world lines. The first equation in particular relates local time and proper time. In particular, it states that dt/d\tau = a for some constant a. Then inverting this and writing
p dt2 −dx2/c2, we find 1/a = p 1 −˙x2/c2, and thus
r 1 −˙x2 c2 . (21.189) This is the usual relation between local and proper time. However, this came from inserting the definition (21.162) of the proper time, not from the dynamical equation. The Hamiltonian in this case is H = px ˙x + pt ˙t −L = px ˙x + pt ˙t −m 2 ˙x2 + mc2 ˙t2 + mc2 2 , (21.190) or in canonical coordinates,
x 2m − p 2 t 2mc2 + mc2 2 . (21.191) Using the conjugate momenta (21.187) and (21.189), we can see that p 2 x 2m − p 2 t 2mc2 = −mc2 2 , (21.192) and thus
(21.193) In the absence of the external potential, the Hamiltonian always has the value H = 0, which is characteristic of the ‘‘extended’’ phase space (including the temporal degree of freedom and the proper time as the new time). This is the reason for introducing the −mc2/2 in the Lagrangian; otherwise the Hamiltonian could have some other (arbitrary) constant value. Then to summarize the action and Lagrangian here (after dropping the external potential), we have S[x] = Z
2m ˙x2 −1 2mc2 ˙t2 −1 2mc2, (relativistic-particle Lagrangian and action, proper-time parameterization) (21.194) with Hamiltonian
x 2m − p 2 t 2mc2 + mc2 2 . (Hamiltonian, proper-time parameterization) (21.195) For this action principle, the parameterization can be changed, but an alternate parameter \lambda will only work as a parameter for the Euler–Lagrange equations, provided \lambda = a\tau + b for some constants a and b. We can see this by writing out d2x
d\lambda dx
d\lambda d d\tau d\tau d\lambda dx
d\tau 2 , (21.196) where the final equality only holds if d\tau/d\lambda is constant. In this case \lambda is called an affine parameter.7 Note that had we introduced a background-potential term of the form R d\tau V (x) [or R dt V (x), as in the square-root action (21.174)], in the action (21.183), we would have arrived at the odd-looking result
is somewhat unnatural—remember that if the Hamiltonian contains no explicit reference to a time parameter, then the Hamiltonian is a constant of the motion. We will see shortly how to introduce a potential in a way consistent with the simpler result H = 0. 7The actions (21.174) and (21.194) impose different requirements on reparameterizations in the free-particle case; in particular the former allows any parameterization, while the latter admits only affine parameters. For more details, see Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler, Gravitation (W. H. Freeman, 1973), p. 322, Boxes 13.2 and 13.3.
Chapter 21. Path-Integral Calculation of Casimir Energies 21.2.3.1 Massless-Particle World Lines While the action (21.194) works fine for massive relativistic particles, it is problematic when it comes to massless particles. [Note that the first action (21.174) is similarly problematic for massless particles.] This is due both to the presence of the mass in the kinetic-energy terms, as well as the parameterization by the proper time (the change in proper time is always zero along a light cone). We can cure this by letting \tau −\rightarrow m\tau/c2 in the action integral. This amounts to parameterizing the paths by \lambda = c2\tau/m, which remains well-defined in the limit m −\rightarrow 0. In this case, the action becomes S[x] = Z d\lambda Lm(x, t, ˙x, ˙t; \lambda), (21.197) (reparameterized action) where we are defining the rescaled Lagrangian
2c2 ˙x2 −1 ˙t2 −m2c4 , (reparameterized Lagrangian) (21.198) and ˙x and ˙t now refer to derivatives with respect to \lambda. The rescaled version Hamiltonian (21.191) is then
xc2 −p 2 t 2 + m2c4 , (reparameterized Hamiltonian) (21.199) where px = ˙x/2c2 and pt = −˙t/2. Everything here is well-defined as m −\rightarrow 0. Note, though, that both Hamiltonian and Lagrangian here have dimensions of square energy. Note that from Eq. (21.193), we still have Hm = 0, and the Hamilton and Euler-Lagrange equations are equivalent to the ones before, but the Hamilton equations have a slightly different form, owing to different canonical momenta, px = 1 c2 dx d\tau , pt = dt d\tau . (21.200) which are equivalent to the former momenta (21.187) under the same rescaling. 21.2.3.2 Variable-Mass Potential With this rescaled Hamiltonian, we can introduce an external potential by regarding the mass as a space- dependent quantity. If we allow the mass to vary in space, separating constant and variable components via
(21.201) then we may assign the space-dependent part to a potential via
h
i c4, (21.202) which again has dimensions of squared energy. Then the Hamiltonian (21.199) becomes
xc2 −p 2 t 2 + m 2 0 c4 + V (x), (reparameterized Hamiltonian with variable-mass potential) (21.203) and the Lagrangian (21.198) becomes
2c2 ˙x2 −1 ˙t2 −m 2 0 c4 −V (x). (reparameterized Lagrangian) (21.204) Now both have a suitable potential that keeps a constant (null) value of the Hamiltonian. Note that in implementing the mass rescaling of the proper time to obtain these functions, in the case of a space-dependent mass, the reparameterization had to be done on a trajectory-dependent basis.
21.2.4 Constrained Action¶
21.2 Worldlines and the Relativistic Scalar Particle Then the new Hamilton equations are dpx
\partial x , dpt
(21.205) where the first equation gives the force law, which is slightly different from the earlier law (21.169), due to a difference in factor of dt/d\tau, and in the background mass m0 not entering the force law here. 21.2.4 Constrained Action Now, informed by our attempts above, let’s rederive a quadratic action for the relativistic particle, but now keeping both a sensible m −\rightarrow 0 limit, and making sure the action is (generally) independent under reparameterizations. Thus, let’s first start with a Lagrangian of the form (21.204),
2c2 ˙x2 −1 ˙t2 −m2c4 , (21.206) and corresponding action S[x, t] = 1 Z d\tau " c2 dx d\tau 2 − dt d\tau 2 −m2c4 # . (21.207) For simplicity, we are temporarily ignoring the external potential (i.e., lumping it into the mass m). Again, the problem that we saw is that this action is not generally reparameterization-independent: it is only
S[x, t] = 1 Z d\lambda ˙\lambda " ˙\lambda2 c2 dx d\lambda 2 −˙\lambda2 dt d\lambda 2 −m2c4 # = 1 Z d\lambda " ˙\lambda c2 dx d\lambda 2 −˙\lambda dt d\lambda 2 −1 ˙\lambda m2c4 # . (21.208) Note that this has the same form as before only if ˙\lambda = 1, which is a stronger requirement than \lambda being an affine parameter. More generally, to preserve the form of the action, we can introduce an ‘‘einbein’’ function e(\tau) in the action, so that it reads S[x, t, e] = 1 Z d\tau " ec2 dx d\tau 2 −1 e dt d\tau 2 −em2c4 # . (relativistic-particle action with einbein) (21.209) Now recalling that a time reparameterization is basically a gauge freedom, the ‘‘gauge transformation’’ here is that we are switching from \tau to \lambda(\tau) as the parameter, and to keep the form of the action, we must make the simultaneous parameter and einbein replacements
d\lambda d\tau −1 , (21.210) (gauge freedom) under which the form of the action is explicitly invariant. As far as the action (21.209) is concerned, now e appears as an extra variable, but note that the action is explicitly independent of ˙e, which signals the presence of a constraint (i.e., the gauge freedom). The Euler-Lagrange equation for e gives \deltaS
e2c2 + ˙t2 e2 −m2c4 = 0, (21.211) which reduces to the constraint equation ˙x2
(21.212)
Chapter 21. Path-Integral Calculation of Casimir Energies That is, on any given world line, e is determined by the coordinates along the world line (in particular, the velocities). Solving for e, e = p˙t2 −˙x2/c2 mc2 , (21.213) and putting this into the action (21.209) S[x, t, e] = −mc2 Z d\tau q ˙t2 −˙x2/c2. (21.214)
To get a bit more intuition for the constraint (21.212), first note that the conjugate momenta from the action (21.209) are px = ˙x ec2 , pt = − ˙t e. (21.215) Then we can write Eq. (21.212) as p 2 t −p 2 xc2 = m2c4, (21.216) (mass-shell condition)
usually identify p2c2 + m2c4 with the square of the relativistic energy E, as in Eq. (21.177), in which case we have the more familiar form E2 −p 2 xc2 = m2c4 (21.217) (mass-shell condition) for the mass-shell condition. In any case this acts as a constraint on pt, showing that it is not independent from px. 21.2.4.1 Gauge Fixing From this discussion, it seems that we may just as well ‘‘gauge fix’’ the einbein at e = 1 and ignore it in the development of the action, which is essentially what we did in the first quadratic action that we developed in Section 21.2.3. That is, what do we get by introducing the einbein e? The reason is the necessity of having the constraint (21.216), which comes from the gauge freedom of e. This is analogous to electromagnetism, in which it is fine to fix a particular gauge, but we must also make sure to implement Gauss’ law as a constraint—a constraint that arose from the gauge freedom of electromagnetic. To see that we don’t naturally get the constraint (21.216), from the gauge-fixed theory, consider the Euler–Lagrange equations corresponding to the action (21.209). First, the x equation gives d d\tau 1 e dx d\tau = 0, (21.218) and with the momentum px in (21.215), dpx
(21.219) Similarly, for the temporal momentum, dpt
(21.220) This implies that px and pt are constants of the motion, and thus that the combination p 2 t −p 2 xc2 from Eq. (21.216) is constant. However, it does not tell us what that constant is; that comes only from an explicit treatment of the gauge freedom. In the gauge-fixed version, this value must be supplied as extra information, in the form of an initial value for this constant of the motion. This can also come in the form of assuming a proper-time parameterization, as we did in Section 21.2.3.
21.2 Worldlines and the Relativistic Scalar Particle 21.2.4.2 Terminology Finally, before continuing, it’s worth noting why e is called an ‘‘einbein.’’ The reason is that under the reparameterization \tau −\rightarrow \lambda, the coordinates transform like scalars:
(21.221) However, from Eq. (21.210), the einbein transforms as
(21.222) where the twiddles indicate the \lambda parameterization. Then e transforms like a one-form (i.e., it transforms covariantly) in one dimension, hence the name ‘‘einbein.’’ 21.2.4.3 Variable-Mass Potential We can now introduce a variable-mass potential, as in the gauge-fixed treatment of Section 21.2.3.2. Thus again if
(21.223) and we associate the space-dependent part of the mass with a potential,
h
i c4, (21.224) then we obtain the action S[x, t, e] = Z
2ec2 dx d\tau 2 −1 2e dt d\tau 2 −em 2 0 c4 −eV (x). (relativistic-particle action with einbein and variable mass) (21.225) Note that in doing this, we still obey the mass-shell constraint (21.216), and we do not have any awkward results with a nonconstant Hamiltonian value as we saw in Section section:relativistic-ptte-quadratic-action. Now let’s work out the Euler–Lagrange equations. The equation for the local time is the same as in the constant-mass case, Eq. (21.220). The x equation is more interesting, however. Writing out the result
d d\tau 1 ec2 dx d\tau
dx . (21.226) Using the momentum px in Eqs. (21.215), we can write this out compactly in the form dpx
dx . (21.227) We can also write out the equation of motion in a more explicit, but less compact form, by writing out (but not gauge-fixing) the einbein. Using the value (21.213) for e in the form e = p
mc2 = dt d\tau p
mc2 , (21.228) which is still valid provided we interpret m as the space-dependent mass, we find d dt
m(dx/dt) p
! = − p
mc2 dV (x) dx . (21.229) Note that the left-hand side here is the same as in the analogous result for the square-root action with external potential, Eq. (21.169). However, the potential couples in slightly differently. There is a factor of mc2 on
21.2.6 Path Integral¶
Chapter 21. Path-Integral Calculation of Casimir Energies the right-hand side, which makes the dimensions come out correctly. Also, there is a Lorentz-contraction factor on the right-hand side, which amounts to the same factor dt/d\tau that appears in the square-root action (21.179). Not to mention, the mass here is space-dependent, so a better analogous expression may be to put the momentum on the left-hand side in terms of the mass offset m0: d dt
m0(dx/dt) p
! = − p
[m2(x)/m0]c2 dV (x) dx . (21.230) This puts all the spatial dependence of the force on the right-hand side. In principle, then, we can choose a mass function m(x) to emulate whatever mass and force law we like in the square-root-action equation of motion, Eq. (21.169). Note that in this form of the action, any external potential we tack on in a parameterization- independent way (i.e., that includes a factor of e) has the form of a space-dependent mass. Again, the nonconstant Hamiltonian value as we saw in Section 21.2.3 shows that introducing an external potential in a form that does not modulate the mass introduces problems. 21.2.5 Klein–Gordon Equation The Klein–Gordon equation is the wave equation corresponding to the relativistic particle. The motiva- tion for the Klein–Gordon equation starts8 with the relativistic energy E = p p2c2 + m2c4. (21.231) If we take this to be the Hamiltonian, apply the usual Schrödinger equation via i¯h\partial t ≡H with the momentum identification p ≡−i¯h\partial x (in one dimension), then we obtain
q −¯h2c2\partial 2
(21.232) However, the square root here is awkward, as its Taylor expansion implies the presence of derivatives at all orders, and thus an inconveniently nonlocal wave equation. The Klein–Gordon alternative is to try the square of the identification i¯h\partial t ≡H, which gives −¯h2\partial 2 t ≡ H2. The corresponding wave equation is \partial 2 x −1 c2 \partial 2 t
mc ¯h 2 \phi, (21.233) (Klein–Gordon equation) after a bit of rearrangement. This is the Klein–Gordon equation. The cost of getting rid of the square root is to introduce negative-energy solutions of the form E = − p p2c2 + m2c4. These have the interpretation of solutions propagating backwards in time (or equivalently, antiparticles). Again, we can add in a background potential V (x), \partial 2 x −1 c2 \partial 2 t
m0c ¯h 2
¯h2c2 V (x) \phi, (Klein–Gordon equation, with background potential) (21.234) by regarding the mass m(x) to be space-dependent. 21.2.6 Path Integral One approach to developing the path integral for the Klein–Gordon equation (21.233) is to start with the action (21.174) corresponding to the relativistic particle, and assume that the nonrelativistic form (20.24)
Z Dx exp i ¯h Z t t0 dt L(x, ˙x) = Z Dx exp i ¯h Z t t0 dt S[x] (21.235) 8James D. Bjorken and Sidney D. Drell, Relativistic Quantum Mechanics (McGraw–Hill, 1964), Chapter 1.
21.2 Worldlines and the Relativistic Scalar Particle for the propagator carries over. Unfortunately, the square root in the action here is again problematic, and leads to non-Gaussian integrals. It is possible to pursue this, and obtain an explicit expression for the short- time propagator in terms of a Bessel function.9 Whether one chooses to go this way or to simply object to the use of the exponentiated action for this propagator,10 clearly this is not the ‘‘world-line’’ path integral that we seek here. 21.2.6.1 Proper-Time Path Integral An alternate approach to the path-integral quantization of the Klein–Gordon equation is due to Feynman.11 Suppose that we introduce an auxiliary ‘‘energy’’ parameter E in in the Klein–Gordon equation (21.233), −¯h2c2 \partial 2 x + ¯h2 2 \partial 2 t
0 c4
(21.236)
of E, but introducing the extra parameter allows us to leave the effective Hamiltonian for the Klein–Gordon equation intact.) Then defining the ‘‘E-rotating’’ wave function by
(21.237) we can write the Klein–Gordon equation in terms of this variable as
−¯h2c2 \partial 2 x + ¯h2 2 \partial 2 t ϕ + m 2 0 c4 ϕ + V (x) ϕ. (21.238) This has the form of the Schrödinger equation with ‘‘time’’ T (which has dimensions of inverse-squared energy), which acts something like a ‘‘proper time’’ with respect to the other space-time coordinates (x, t). In the solution of this form of the equation, ϕ(x, t; T ) contains contributions from all values of E. Since the equation is linear in the wave function, we can write the solution as a superposition
2\pi Z \infty −\infty dE ˜ϕ(x, t; E) e−iET , (21.239)
solution of the original equation (21.233). We can obtain the component for any E by a simple Fourier projection:
Z \infty −\infty dT ϕ(x, t; T ) eiET . (21.240) The case E = 0 becomes the simple integral
Z \infty −\infty dT ϕ(x, t; T ), (21.241) which we can easily verify by
2\pi Z \infty −\infty dT Z \infty −\infty
Z \infty −\infty
(21.242) Thus, by solving the extended equation (21.238), we can in principle obtain the propagator for the Klein– Gordon equation (21.233). 9Ian H. Redmount and Wai–Mo Suen, ‘‘Path integration in relativistic quantum mechanics,’’ International Journal of Modern Physics A 8, 1629 (1993) (doi: 10.1142/S0217751X93000667), arXiv.org preprint (arXiv: gr-qc/9210019). 10for a rather critical account, see Arlen Anderson, ‘‘Use of exp(iS[x]) in the sum over histories,’’ Physical Review D 49, 4049 (1994) (doi: 10.1103/PhysRevD.49.4049). 11R. P. Feynman, ‘‘Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction,’’ Physical Review 80, 440 (1950) (doi: 10.1103/PhysRev.80.440); L. S. Schulman, Techniques and Applications of Path Integration (Wiley, 1981), Chapter 25.
Chapter 21. Path-Integral Calculation of Casimir Energies To do this, we will be careful and use the language of Green functions and resolvent operators from Chapter 15. First, defining the effective Hamiltonian for the Klein–Gordon equation as H = −¯h2c2 \partial 2 x + ¯h2 2 \partial 2 t + m 2 0 c4 + V (x), (21.243) then the extended Klein–Gordon equation (21.236) becomes
(21.244) This has the form of a time-independent Schrödinger equation, which defines a retarded, energy-domain Green operator (resolvent operator) via [see Eq. (15.7)]
(21.245) The tilde here emphasizes that the Green operator is in energy space, and the T subscript is a reminder that this is a Green operator for the extended Hamiltonian. Similarly, Eq. (21.238) is the time-domain version of the extended Schrödinger equation, which we may rewrite as
(21.246) This as well defines the retarded Green function (in proper time) via [Eq. (15.14)]
(21.247) These two Green functions are related via the integral [Eq. (15.8)] G+
i Z \infty
T (T , 0), (21.248) which for E = 0 becomes ˜G+
i Z \infty dT G+ T (T , 0). (21.249) Now the propagator for the Klein–Gordon equation should satisfy Eq. (21.234) with a delta-function term added. Specifically, upon rescaling, we can write this as −¯h2 2 \partial 2 t + ¯h2c2 \partial 2 x −m 2 0 c4 −V (x)
(defining relation for Klein–Gordon propagator) (21.250) in analogy with Eq. (21.247). Since the operator on the left-hand side is −H, by comparison with (21.245), we have
(21.251)
Z \infty
T (T , 0)|x0, t0\rangle . (21.252) The matrix element on the right-hand side is the propagator for the extended Klein–Gordon equation (21.238). Since this is formally a Schrödinger equation, we can write this in terms of the standard non- relativistic propagator. Adapted to this case from Eq. (20.24), the extended propagator for this solution reads
Z Dx(\tau) exp " i Z T d\tau ˙x2 2c2 − ˙t2 2 −m 2 0 c4 −V (x) # , (21.253)
21.2 Worldlines and the Relativistic Scalar Particle where the Dx refers to integration over (x, t). Notice that the integrand in the exponential is just the
Z \infty dT Z (x,t) (x0,t0) Dx(\tau) exp " i Z T d\tau ˙x2 2c2 − ˙t2 2 −m 2 0 c4 −V (x) # = Z \infty dT Z (x,t) (x0,t0) Dx(\tau) exp " i Z T d\tau Lm(x, t, ˙x, ˙t; \tau) # (world-line Klein–Gordon propagator) (21.254) as the ‘‘world-line’’ path integral form for the Klein–Gordon equation.12 Essentially, this is a path integral over world lines that travel from (x0, t0) to (x, t) in proper time T ; then, we integrate over all possible proper times T . In a more ‘‘string-inspired’’ approach, the world-line parameter T plays the role of a constraint, and the integral form here is effectively a particular choice of gauge.13 21.2.6.2 Imaginary Time Imagining that we are to do statistical mechanics for the relativistic particle, by analogy with the nonrela- tivistic case, we can consider the propagator in imaginary (local) time:
Z \infty dT Z (x,−i¯h\beta) (x0,0) Dx exp " i Z T d\tau
(\partial \taux)2 2c2
−m 2 0 c4 −V (x) !# . (21.255) Here, ˜\beta := ¯h\beta as usual. Further, going to imaginary proper time via the Wick rotation \tau −\rightarrow −i\tau and T −\rightarrow −iT ,
Z \infty dT Z (x,−i¯h\beta) (x0,0) Dx exp " − Z T d\tau
(\partial \taux)2 2c2
- m 2 0 c4 + V (x) !# . (21.256) Note, however, that we cannot directly associate this with the partition function for the relativistic particle simply by tracing over x, because this identification was based on the analogy of the nonrelativistic evolution operator with the partition function. In particular, in imaginary (local and proper) times t −\rightarrow −it and \tau −\rightarrow −i\tau, what appears is the Hamiltonian
xc2 + p 2 t 2 + m 2 0 c4
xc2 + p 2 t 2 + m2c4 (21.257) [c.f. Eq. (21.203)]. To evaluate this, we recall that this was a reparameterized form of
x 2m + p 2 t 2mc2 + mc2 2 , (21.258) where the proper-time parameter was scaled by a factor of m/c2, with corresponding changes to the momenta (which we must restore to find proper factors of the mass that are otherwise hidden). The partition function corresponding to the imaginary-time propagator is then the trace of exp[−\betaH(x, t, px, pt; \tau)], which might seem reasonable. However, using the conjugate momenta px = 1 c dx d\tau , pt = dt d\tau , (21.259) 12cf. C. Schubert, ‘‘Perturbative quantum field theory in the string-inspired formalism,’’ Physics Reports 355, 73 (2001) (doi: 10.1016/S0370-1573(01)00013-8), arXiv.org preprint (arXiv: hep-th/0101036v2), especially Eq. (3.10), where the same propagator is given after evident Wick rotation and change to imaginary times. 13Philip R. Johnson, ‘‘Relativistic Particle Trajectories from Worldline Path Integral Quantization,’’ Proceedings of the 2001 Particle Accelerator Conference, 1781 (2001) (doi: 10.1109/PAC.2001.987181).
Chapter 21. Path-Integral Calculation of Casimir Energies and using the conjugate momenta (21.187) and (21.189), we find p 2 x 2m + p 2 t 2mc2 = p2 m + mc2 2 , (21.260) with p = px the momentum, now referred to the local time. Thus,
m + mc2. (21.261) This means that the parameterized form was
(21.262) which is the square of the relativistic energy. This would lead to a partition function with terms of the form exp(−\betaE 2 n), which is not directly useful. Nevertheless, this form of the imaginary-time Green function enters the world-line path integral for the scalar-field partition function, albeit with a different weighting of the integral over T . 21.3 Exercises Problem 21.1 Recalling that the mean energy is given by the partition-function derivative (21.13)
(21.263) show that the second derivative \sigma 2 E =
= \partial 2 \beta log Z (21.264) gives the variance in the energy. What does this say about fluctuations in Casimir energies at zero temperature? Problem 21.2 The inhomogeneous Helmholtz equation
¯h2c2 V (r)
(21.265) where f(r) is an arbitrary source function and V (r) is an added background-potential function, has a Green function (resolvent) defined by [see Eq. (14.57), noting that we are ditching the ϵ0 but keeping the minus sign]
¯h2c2 V (r)
(21.266) (See also Problem 15.2). Show that the retarded Green function has the world-line representation14
Z \infty dT eik2T Z r′ r Dx(\tau) exp " i Z T d\tau ˙x2 4 − ¯h2c2 V (x) # , (21.267) or with the Wick rotations T −\rightarrow −iT and k −\rightarrow i\kappa to the ‘‘Euclidean’’ form,
Z \infty
Z r′ r Dx(\tau) exp " − Z T d\tau ˙x2 2 + ¯h2c2 V (x) # , (21.268) 14Marco Schäfer, Idrish Huet, and Holger Gies, ‘‘Energy-momentum tensors with worldline numerics,’’ International Journal of Modern Physics Conference Series 14, 511 (2012) (doi: 10.1142/S2010194512007647), arXiv.org preprint (arXiv: quant- ph/0605180v3).
21.3 Exercises after also rescaling T by a factor of 2. Both expressions should be interpreted in the renormalized
down a normalized (Monte-Carlo) form of the Euclidean path integral in terms of Brownian bridges. Problem 21.3 Show that the world-line form of the Casimir–Polder energy (before renormalization)
¯hc\alpha0
Z \infty dT T 3 ** exp " − ¯h2c2 Z T
x(\tau) (21.269) for paths in D spacetime dimensions beginning and ending at r, as is consistent with Eq. (21.118), can be obtained from the expression
¯h\alpha0 2\piϵ0c2 Z \infty ds s2 ˜G+(r, r; is), (21.270) where ˜G+(r, r; is) is the retarded Green function for the Helmholtz equation from Problem 21.2, and
Problem 21.4 (a) Starting with the worldline path integral (21.109) for the Casimir energy, and using the calculation for the strong-coupling Casimir energy between two planar mirrors in Section 21.1.5.3, show that the following path integral yields the Casimir force per unit area for this configuration: F A = − ¯hc
Z dx0 ** \Theta −min[x(\tau)]
d(x0) T D/2 min [x(\tau)] ++ x(\tau) . (21.271) This expression implicitly assumes two mirrors located at x = 0 and x = L, and gives the force on the x = L mirror. In the expression, d is the number of spatial dimensions, D = d + 1 is the spacetime
the smallest value of the path running time T such that the path touches the x = L mirror. Note that
(b) How does this expression generalize to arbitrary geometries?