Regularization versus renormalization: Why are Casimir energy differences so often finite?

One of the very first applications of the quantum field theoretic vacuum state was in the development of the notion of Casimir energy. Now field theoretic Casimir energies, considered individually, are always infinite. But differences in Casimir energies (at worst regularized, not renormalized) are quite often finite --- a fortunate circumstance which luckily made some of the early calculations, (for instance, for parallel plates and hollow spheres), tolerably tractable. We shall explore the extent to which this observation can be made systematic. For instance: What are necessary and sufficient conditions for Casimir energy differences to be finite (with regularization but without renormalization)? And, when the Casimir energy differences are not formally finite, can anything useful nevertheless be said by invoking renormalization? We shall see that it is the difference in the first few Seeley--DeWitt coefficients that is central to answering these questions. In particular, for any collection of conductors (be they perfect or imperfect) and/or dielectrics, as long as one merely moves them around without changing their shape or volume, then physically the Casimir energy difference (and so also the physically interesting Casimir forces) are guaranteed to be finite without invoking any renormalization.


Introduction
Quantum field theoretic Casimir energies (considered in isolation) are typically infinite, requiring both regularization and renormalization to extract mathematically sensible answers, this at the cost of sometimes obscuring the physics [1][2][3][4][5]. On the other hand Casimir energy differences are quite often finite, and have a much more direct physical interpretation [1,2]. Additional background and general developments may be found in references [6][7][8][9][10][11][12][13][14][15]. In this article, I shall first argue (mathematically) that there are a large number of interesting physical situations where the Casimir energy differences, (and so the Casimir energy forces), are automatically known to be finite, even before starting specific computations. Secondly, I shall argue (mathematically) that one can often develop physically interesting "reference models" such that the Casimir energy difference between the physical system and the "reference model" is known to be finite, even before starting specific computations. (I will not actually calculate any Casimir energies -knowing that the result you are after is finite is often more than half the battle.) I shall first start with a simple formal argument to get the discussion oriented, and then provide a more careful argument in terms of regularized (but not renormalized) Casimir energies.

Formal argument
The formal argument starts with the exact result that: Now let ω n and (ω * ) n be two infinite sequences of numbers then, again as an exact result: Then in terms of the heat kernel K(t) defined by we formally have: But, (now assuming that the ω 2 n and (ω * ) 2 n are in fact the eigenvalues of some secondorder linear differential operators), by the standard Seeley-DeWitt expansion we have both and Note d is the number of space dimensions. As will be discussed more fully below, the integer indexed a n have both bulk and boundary contributions, while the half-integer indexed a n+ 1 2 have only boundary contributions. Then for the difference in heat kernels we have: (2.9) Here the designation "UV finite" means that any remaining terms contributing to the "UV finite" piece are now guaranteed to not have any infinities coming from the t → 0 region of integration. That is, taking E Casimir = 1 2 n ω n , we have the formal result: All of the potentially UV-divergent terms are now concentrated in the d + 2 leading terms proportional to the ∆a i . The rest of the article will involve several refinements on this simple theme.
Generally, in d space dimensions, if we are comparing two physical systems for which the first d + 2 Seeley-DeWitt coefficients are equal, then the difference in Casimir energies will be finite.

Exact argument
Let us now regularize everything a little more carefully, to develop an exact rather than formal argument. (Initially we shall use the complementary error function [erfc(x) = 1 − erf(x)] as a particularly simple and mathematically transparent reguator, but will subsequently show that physically almost any smooth cutoff function will do.) We have the exact result that: This leads to the further exact result that: But, because all the relevant quantities are guaranteed finite, we can now exchange sum and integral to obtain the exact (no longer just formal) result: Then in terms of the heat kernel: Now apply the Seeley-DeWitt expansion: But then (now choosing N = d ) for the heat kernel term we have: Working with the integral term is a little trickier. In the integral we instead choose N = d + 1. Then, treating the logarithmic term separately, we have That the a (d+1)/2 term leads to logarithmic term in the Casimir energy (and effective action) is well-known. See for instance references [5,16,17]. Performing the remaining integrals: ∞ Now assembling all the pieces: (3.9) We now have the exact result: -5 - For our current purposes the specific values of the dimensionless coefficients k i are not important.

(4.5)
We can now safely take the limit as the cutoff is removed (Ω → ∞). We have:

Theorem 2 (Casimir energy differences)
If we compare two systems where the first d + 2 Seeley-DeWitt coefficients are equal, then: This is a very nice mathematical theorem, but how relevant is it to real world physics? Just how general is this phenomenon?

Unchanging Seeley-DeWitt coefficients
Perhaps unexpectedly, there are very many physically interesting situations where the (first few) Seeley-DeWitt coefficients are unchanging. The pre-eminent cases are these: • Parallel plates.
In both of these cases an infra-red regulator is needed, and some subtle thought is still required. Much more radically: • Take any collection of perfect conductors. Move them around relative to each other. (Without distorting their shapes and/or volumes.) • Then the change in Casimir energy is finite.
• Then the Casimir forces are finite.
(Subsequently, we shall show that similar comments can be made for both imperfect conductors and dielectrics.) To establish these results we note that for a region V with boundary ∂V we have the quite standard results that: Here the { , , } denote various species-dependent linear combinations of the relevant terms. For current purposes we do not need to know the specific values of any of the dimensionless coefficients. (There are also contributions to the a i from kinks and corners; but let's stay with smooth boundaries for now.) Above we have retained terms due to both intrinsic and extrinsic curvature, plus a scalar potential V (x). One could in principle obtain even more terms from background electromagnetic or gauge fields, but the terms retained above are sufficient for current purposes.

Parallel plates
Working with QED (so V = 0) in flat spacetime (Riemann tensor zero) with flat boundaries (extrinsic curvature zero): So for finite Casimir energy differences one just needs to keep volume and surface area fixed. For example: Apply periodic boundary conditions in d − 1 spatial directions, and apply conducting box boundary conditions in the remaining spatial direction.
-8 -Physically this means you put the Casimir plates inside a big box, of fixed size, with two faces parallel to the plates. Consider the situation where one varies the distance between the Casimir plates while keeping the size of the big box (the infra-red [IR] regulator) fixed. From the above, and with no further calculation required, we can at least deduce that the Casimir energy difference (and so the Casimir force between the plates) is finite.

Hollow spheres
We are now working with QED in flat spacetime with thin spherical boundaries. The idea is to understand as much as we can regarding Boyer's calculation [2], but without explicit computation. (We shall assume 3+1 dimensions.)

Step I (QED in flat spacetime)
Using only the fact that we are working with QED (V = 0) in flat spacetime (Riemann tensor zero): a 0 ∝ (volume); Since the extrinsic curvature is non-zero, K = 0, keeping control of the higher a i , the higher-order Seeley-DeWitt coefficients, is now a little trickier.

Step II (thin boundaries)
As long as the boundaries are thin, then K inside = −K outside , leading to cancellations in both a 1 and a 2 . Similarly the thin boundaries take up zero volume, so the total volume is held fixed. (The outermost boundary, the IR regulator, is always held fixed.) Then: ∆a 2 → 0.

5.2.3
Step III (rescaling -conformal invariance) As long as the inner boundaries for the two situations we are considering are simply rescaled versions of each other, then ∂V KK √ g 2 d 2 x is scale invariant, thus leading to a cancellation in a 3/2 . (The outermost boundary, the IR regulator, is always held fixed.) Then: Note we still have to deal with ∆a 1/2 .

Step IV (TE and TM modes)
In spherical symmetry, one can easily define TE and TM modes. Note that they have equal and opposite contributions to a 1/2 , again leading to a cancellation in a 1/2 . (The outermost boundary is always held fixed.) Then: This finally is enough to guarantee finiteness of the Casimir energy difference.

Step V (finiteness)
From the above we have ∆(Casimir Energy) = (finite). (5.1) This observation underlies the otherwise quite "miraculous cancellations" in Boyer's calculation of the Casimir energy of a hollow sphere [2]. Comparing two hollow spheres of radius a and b; and letting the IR regulator (which is the same for each sphere) move out to infinity: Boyer uses Riesz resummation, (the so-called "Riesz means"), which is justified only in hindsight. If you know the answer you want is finite, then any of the standard "regular" resummation techniques will do [18]. In contrast if you don't know beforehand that the answer you want is finite, then blindly calculating is asking for trouble.

Arbitrary arrangement of fixed-shape fixed-volume perfect conductors
Consider now any collection of fixed-shape fixed-volume perfect conductors in 3+1 dimensions. We are working with QED (V = 0) in flat spacetime (Riemann tensor zero). Then: a 0 ∝ (volume); a 1/2 ∝ (surface area); Fixed-shape fixed-volume implies fixed extrinsic curvature, so all the ∆a i ≡ 0. That is: • Take any collection of perfect conductors. Move them around relative to each other. (Without distorting their shapes and/or volumes.) • Then the change in Casimir energy, and the Casimir forces, are finite.
We shall subsequently see how to generalize this result to imperfect conductors and/or dielectrics.

Reference models
Consider now a non-zero potential (V = 0), in flat spacetime (Riemann tensor zero), with periodic boundary conditions (so that there is no boundary). We have: So for finiteness we "just" need to keep a 0 , a 1 , and a 2 fixed.

1+1 dimensions
In (1+1) dimensions let us define the spatial average Compare the two situations: eigenvalues ω 2 n .
Then: In fact in this situation the reference eigenvaluesω n can be written down explicitly as The ω n depend on V (x) and can be quite messy; the difference between the ω n and the reference problem ω n is however well behaved.
-13 -7 What if Casimir energy differences are not finite?
Now there are certainly (mathematical) situations where the ∆a i = 0 and the Casimir energy difference is not naively finite. This merely means one has to be more careful thinking about the physics. For instance: • Real metals and real dielectrics are transparent in the UV.
• The UV cutoff Ω is then merely a stand-in for all the complicated physics.
For real metals and real dielectrics the cutoff represents real physics. See for instance the discussion in references [20][21][22] and compare with the discussion in [23][24][25][26]. Note that the discussion regarding real metals and real dielectrics has often lead to some considerable disagreement regarding interpretation [27][28][29]. (My own view, as should be clear from the current article, is that Casimir energies are ultimately determined by looking at differences in zero-point energies, summed over all relevant modes.)

General class of cutoff functions
Let us write a general class of cutoff functions as Note f (0) = 1, while f (∞) = 0, and f (ω/Ω) is monotone decreasing.
To see just how general this class of cutoff functions is, we proceed by noting that So we see Substituting χ = 1/ξ 2 we obtain But this is just the Laplace transform of g(χ −1/2 )/χ, evaluated at the point s = ω 2 /Ω 2 . Consequently, as long as the inverse Laplace transform of f (s 1/2 ) exists, which is a relatively mild condition on the cutoff function f (s 1/2 ), then we can determine g(ξ) in terms of f (ω/Ω).
Indeed, there is a little-known algorithm due to Post [30], see also Bryan [31], and reference [32], that allows for inversion of Laplace transforms by taking arbitrarily high derivatives. Specifically, if G(s) is the Laplace transform of g(z) then This algorithm may not always be practical, since one needs arbitrarily high derivatives. Even if not always practical, it again settles an important issue of principle -knowledge of the cutoff f (ω/Ω) in principle allows one to reconstruct an equivalent weighting g(ξ).
The point is that almost any cutoff function f (ω/Ω) can be cast in this "weighted integral over erf-functions" form. (In particular we could rephrase all of the preceding discussion concerning erf-regularization in terms of this more general f -regularization, but when ∆a i = 0 nothing new is obtained. It is only when ∆a i = 0 that general f -regularization becomes at all interesting.)

f -regularized Casimir energy
Let us now consider a generic regularized sum of eigen-frequencies: n ω n f ω n Ω . becomes The integrals over g(ξ) can be absorbed into redefining the dimensionless constants k i in a f -dependent manner. That is:

Theorem 3 (Physical cutoff )
For a general cutoff f (ω/Ω) one has The [k(f )] i are dimensionless phenomenological parameters that depend on the detailed physics of the specific cutoff function f (ω/Ω). However k (d+1)/2 is cutoff independent.
The Ω dependence represents real physics. Live with it! The [k(f )] i are dimensionless phenomenological parameters that depend on the detailed physics of the specific cutoff function f (ω/Ω). However k (d+1)/2 is cutoff independent.
The Ω dependence represents real physics. Live with it! Part of the reason it was never worthwhile to keep explicit track of the k i is that, once the f -cutoff is introduced, the k i would in any case be replaced by the purely phenomenological and cutoff dependent [k(f )] i .
Furthermore, if the first d + 2 of the ∆a i are zero, then the cutoff dependence drops out of the calculation. That is, even for imperfect conductors and dielectrics, if one is comparing two situations where the conductors/dielectrics have merely been moved around, (without changing shape and/or volume), then the difference in Casimir energies (and so the Casimir forces) are guaranteed finite.

Forcing finiteness?
Can one force the Casimir energy difference to be finite?
By hook or by crook find a number of "simple" problems D i such that Then it is certainly safe to say Casimir energy of D i = (finite). (8.2) Of course this does not calculate the "finite piece" for you, but it gives you some confidence regarding what to aim for before you start calculating. More formally, if the D i are sufficiently simple one might apply analytic techniques (such as zeta functions [5,19] or the like) to argue that it might make sense to define: Casimir energy of D i + (finite). while analytically continued to be finite, is purely formal. It need not be a physical energy difference. In short, one should seek at all times to calculate Casimir energy differences between clearly defined and specified physical systems. This might, at a pinch, involve differences between linear combinations of physical systems, but to get a physically meaningful Casimir energy one must either enforce ∆ m i=1 a j D i = 0, or develop an explicit physical model for the cutoff f (ω/Ω).

Conclusions
In (d + 1) dimensions, iff the first d + 2 Seeley-DeWitt coefficients agree, ∆a 0 = ∆a 1/2 = . . . ∆a (d+1)/2 = 0, (9.1) then the difference in Casimir energies is guaranteed finite. This is an extremely useful thing to check before you start explicitly calculating. The erfc function, in the form erfc(ω/Ω), is a perhaps unexpectedly useful regulator erfc(0) = 1; erfc(∞) = 0. For real metals and real dielectrics, which become transparent in the UV, the cutoff is physical, and its influence on the Casimir energy is encoded in a small number of dimensionless parameters [k(f )] i and an overall cutoff scale Ω. Various generalizations of this argument, (such as counting differences in eigenstates, or calculating differences of sums of powers of eigenvalues), are also possible. Similar arguments, regarding differences in Seeley-DeWitt coefficients, can also be applied to the one-loop effective action [33]. Finally, I should emphasise that I have not renormalized anything anywhere in this article, the worst I have done is to temporarily regularize some infinite series, to allow some otherwise formal manipulations to be mathematically well-defined.