Regularization and Renormalization of the Vacuum Energy
Abstract and Keywords
This chapter uses the method of heat kernel expansion together with cutoff regularization to separate the divergent part of the vacuum energy. After the vacuum energy has been regularized, the procedure of renormalization is considered. The divergent contributions have a structure which allows their removal by a redefinition of the parameters in the ‘noninteracting theory’ (including the parameters of a classical background field if one is present). However, this procedure is not always possible. In the case of background fields (if these are singular or if one uses some limiting process which makes them singular) these questions are not completely settled, and they are briefly discussed. The cases of a single body and two separate bodies are considered in detail.
As previously discussed, the vacuum energy is a divergent quantity. In the vacuum, quantum field theory assigns half a quantum to each of the infinitely many degrees of freedom. These divergences are of an ultraviolet nature similar to that known from higher loop expansions. Their treatment, however, requires special approaches because of the presence of boundaries. Powerful methods are available for this. The most general one is the heat kernel expansion, which can be considered as the standard method and the natural language to represent these divergences. From a mathematical point of view, it is closely related to spectral geometry [see e.g. the book by Gilkey (1995)]. The heat kernel expansion is also related to zeta function regularization, which can be considered as the most elegant among the many different regularization schemes. In the present chapter, we use this method together with cutoff regularization to separate the divergent part of the vacuum energy.
After having regularized the vacuum energy, we consider the procedure of renormalization. We start with the case where some smooth background fields are present. To some extent this is not central to this book, but it is necessary for an understanding of the renormalization procedure. In a smooth background field, the renormalization procedure is the same as that known in quantum field theory. The divergent contributions have a structure which allows their removal by a redefinition of the parameters in the “noninteracting theory” (including the parameters of a classical background field if one is present). However, this procedure is not always possible. For the case of background fields (if these are singular or if one uses some limiting process which makes them singular), these questions are not completely settled, and we shall discuss them briefly in Section 4.3.
For the Casimir energy resulting from the boundary of a single body, geometric characteristics such as the volume, surface area, and curvature should be used for renormalization. If such characteristics are not available, the vacuum energy cannot be given a physical meaning except when the divergences are absent. The same also holds for other quantities such as the Casimir pressure and force. We shall discuss some examples below.
The most important case is the Casimir force between separate objects. Here the situation is completely different. In general, this force is always finite, as opposed to the interaction energy, which becomes finite when the contribution of the vacuum energy of free space is removed. This will be discussed in the last (p.56) section of this chapter.
4.1 Regularization schemes
Regularization is a method to change an infinite quantity into a finite one. A regularization parameter is introduced such that, in the appropriate limit, the original expression is restored. Of course, this procedure is not unique, and different schemes are possible, some of which will be discussed below. Beyond this formal definition, regularizations sometimes have a direct physical meaning. For instance, since ideal conductors do not exist in nature, one has in all real applications some natural frequency, usually of the order of the plasma frequency, beyond which the reflectivity rapidly decreases. However, this decrease might not provide a regularization for some systems. The most important regularization schemes are frequency cutoff, point splitting, and zeta function regularization.
In a frequency cutoff regularization, one introduces some cutoff function in the mode expansion which makes the corresponding sum/integral converge. Equation (3.60) defines the nonregularized (infinite) vacuum energy E 0. In this case we introduce the regularized vacuum energy E 0(δ), and in place of eqn (3.60) we get
This regularization was used in the original work by Casimir and also in Section 2.1. A modification of this scheme would be a sharp frequency cutoff in place of the exponential in eqn (4.1). In addition, any other sufficiently fast-decreasing function of ωJ (such as a sufficiently fast-decreasing frequency-dependent permittivity) or a momentum-dependent decreasing function (a momentum cutoff) can be used.
In a point splitting regularization, one starts from the representation (3.93) of the vacuum energy in terms of a Green’s function without carrying out the coincidence limit. Then
In a zeta function regularization, one temporarily changes the power of the frequency ωJ in the mode sum (3.60), leading to (p.57)
This converges for Re s > (d + 1)/2, where d is the dimensionality of the space. The factor μ2s, where μ has the dimension of a mass, is arbitrary. It is introduced in order to keep the dimension of E 0. It disappears on removing the regularization in the limit s → 0. This regularization is called zeta function regularization because the vacuum energy E 0(s) is given by
The zeta function ζP(s) is associated with an elliptic boundary value problem. In our case this problem is specified by eqn (3.39) with the operator −Δ + m 2 along with some boundary conditions, such as eqn (3.41) or (3.42). This zeta function can be viewed as a generalization of the Riemann zeta function
It must be mentioned that, owing to the analytic properties of ζp(s), the vacuum energy in this regularization is defined on the entire complex plane for the parameter s with the exception of the poles. In three-dimensional space, its sum representation (4.5) is valid for Re s > 2 only, but still serves as a starting point for analytic continuation.
4.2 The divergent part of the vacuum energy
The regularizations (4.1), (4.2), and (4.3) were introduced to have a finite representation of the vacuum energy. Here, we consider that part of this representation which becomes singular in the limit of removing the regularization.
4.2.1 The divergent part in the cutoff regularization
We consider the vacuum energy for a scalar field in a finite volume V enclosed by a surface S where boundary conditions, such as those discussed in Section 3.2, are (p.58) imposed. In this case the eigenvalues in eqn (3.39) can be labeled by an integer n = 1, 2, …. The frequencies also become labeled by the same integer n. Note that in some specific cases (e.g. the rectangular boxes considered in Chapter 8) it is convenient to use several integer indices. However, the values of these indices can be renumbered and represented by one index with integer values. For δ → 0, the divergent part of the vacuum energy (4.1) results from the asymptotic behavior of the eigenvalues for n → ∞,
From the pole of the Riemann zeta function (4.6) at z = 1, it follows that Ẽ 0(s) has simple poles at s = 2, 3/2, 1, 1/2 and double poles at s = 0, -1/2, -1, … because of the poles of the gamma function. From the residues at these poles, the divergent part of the vacuum energy is
The highest-order divergence is 1/δ4. This is proportional to the volume and corresponds to the contribution of empty space. The next-order divergence, 1/δ3, is proportional to the surface area. The weakest divergence is proportional to ln δ, which comes from the first double pole.
We note that these are the contributions which must be subtracted from the vacuum energy in order to get a finite expression when the regularization is removed. We postpone discussion of the justification and interpretation of the subtraction procedure. From eqn (4.12), we see that the first five terms in the asymptotic expansion (4.7) of the eigenvalues contribute to the divergent part of the vacuum energy. Thus, for an arbitrary surface S, direct numerical approaches to the calculation of the vacuum energy as a sum over the eigenvalues have not yet been successful.
4.2.2 The divergent part in the zeta function regularization and the heat kernel expansion
The powers of the frequency in eqn (4.3) can be identically represented as an integral,
Interchanging the order of the summation and integration, the vacuum energy in the zeta function regularization can be expressed as
In general, the heat kernel is the key object in the theory of heat conduction. It is important for the Casimir effect because its behavior for small t describes the divergences in the vacuum energy. An important feature of the heat kernel is that it has an asymptotic expansion for small t,
The heat kernel coefficients have a very long history. They were introduced independently several times and are known under different names such as the Minakshisundaram-Pleijel coefficients and the Seeley or Seeley-deWitt coefficients. The heat kernel coefficients have been very well investigated [recently, an excellent review was given by Vassilevich (2003)]. They are universal in the sense that they depend only on the geometric characteristics of the volume V and its enclosing surface S, such as the curvature and its derivatives, and on the type of the boundary conditions.
In the following, we consider a volume V with a background field U(r), which can be introduced as a position-dependent mass density by the substitution m2 → m2 + U(r) in the operator (3.6). We postpone considering a curvature such as in eqn (3.10) for later. Furthermore, we assume that the volume V is bounded by a surface S. The geometric properties of the surface can be expressed in terms of its second fundamental form at a point r,
(p.61) Here n is the outward-pointing normal vector to the surface at a point r (Gray 1997). The heat kernel coefficients are represented as a sum of two local integrals, one over the volume (bulk part) and the other over the surface (surface part),
Here, we use the same parametrization for S as in Section 3.6. The surface part is absent if the volume V has no boundary (for example, an interval with periodic conditions). We must warn readers that several different notations for the heat kernel coefficients are used in the literature. Sometimes the factor (4 < π)−3/2 is included in their definition. Sometimes the enumeration is done with integer numbers, i.e. a k/2 → ak. In the notation of eqns (4.18) and (4.21) and for Dirichlet boundary conditions (upper entry in the curly brackets) and Neumann boundary conditions (lower entry in the curly brackets), the first few coefficients read
Here, there is a summation over the repeated indices a, b = 1, 2, i.e.
When inserted into eqn (4.21), the coefficient b 0 leads to the volume of V, a 0 = V, and c 1/2 is proportional to the area S of the surface. It should be noted that the coefficients with half-integer numbers result only from the boundary.
Below, we shall consider a sphere without a background field. Here the second fundamental form is simply Lab = δab/R, such that Laa = 2/R and the coefficients become
Thus, in terms of the heat kernel expansion, complete information about the divergences of the vacuum energy is available. This is contained in the poles of (p.62) E 0(s). These poles follow from eqn (4.14) together with eqn (4.18), from the integration in the vicinity of t = 0. We divide the integration over t into t ∈ [0, 1] and t ∈ [1,∞). The integral over the second interval gives a regular expression. The contribution to E 0(s) from the interval [0, 1], denoted by Ẽ 0(s) can be calculated after a power series expansion of the exponent in eqn (4.14) has been performed. The result is
As we are interested in the pole at s = 0, we separate the pole part of Ẽ 0(s)
This is the part of the vacuum energy which diverges when the regularization is removed. It contains the coefficients up to and including a 2. Higher-order coefficients do not contribute.
For massive fields, the heat kernel expansion provides an expansion in inverse powers of m. This can be obtained by inserting the heat kernel expansion (4.18) into eqn (4.14) and performing the integrations in each term of the sum:
It must be stressed that this is an asymptotic expansion for E 0(s).
The terms of this expansion for 0 ≤ k ≤ 4 diverge when s → 0. These terms contain nonnegative powers of the mass. Expanding the terms with k < 4 in eqn (4.29) in powers of s around the point s = 0, we arrive at
We call this the divergent part of the vacuum energy in zeta function regularization, although it also contains some finite contributions. This definition makes sense only for a theory with a nonzero mass m or, equivalently, with a gap in the spectrum. For a massless field or a gapless spectrum, one must return to the divergent pole part, eqn (4.28), which is also meaningful for m = 0.
(p.63) For comparison, it is instructive to express the vacuum energy using cutoff regularization, eqn (4.1), in terms of the heat kernel coefficients. As can be seen from eqns (4.8) and (4.3), the Mellin transform Ẽ 0(s) (4.8) of the vacuum energy in cutoff regularization is related to the vacuum energy in zeta function regularization by means of
Substituting the series (4.26) into eqn (4.31) and using the inverse Mellin transform (4.9), we obtain the divergent part of the vacuum energy in cutoff regularization in terms of the heat kernel coefficients,
The coefficients ã1, ã2 are defined in eqn (4.27). We mention that a comparison of this formula with eqn (4.12) allows one to establish a connection between the heat kernel coefficients and the coefficients ci in the Weyl expansion of the eigenvalues (4.7).
Now we consider the representation (3.112) of the vacuum energy which follows from the effective action. Using eqn (3.111), we represent the effective vacuum energy in the form
Here “Tr” is understood as the sum of all of the diagonal matrix elements calculated with the functions
It is easily seen that the matrix of the Green’s function (3.87) is diagonal in the basis (4.34). Because of this, we obtain
Calculating Tr of eqn (4.35) using the orthonormality of the basis functions (4.34), we get
Note that the integration with respect to t in eqn (4.36) results in 2πδ(ω′ − ω) and the subsequent integration with respect to ω′ and t′ gives a total time T. Substituting eqn (4.36) into eqn (4.33) and using eqn (3.39), we arrive at
Again we are faced with an infinite expression. Its zeta function regularization is
Using the definition of the generalized zeta function (4.5) and calculating the integral (Gradshtein and Ryzhik 1994), we get
In this representation, a remarkable property of the vacuum energy defined by eqn (3.112) follows. This energy is not singular if the zeta-function-regularized vacuum energy has at most a simple pole in s = 0. Indeed, representing this energy as
We note that this expression does not contain a singularity for s → 0, and, as a result, E 0, eff(s) in eqn (4.38) is finite for s → 0 (provided E 0(s) has only a single pole at s = 0). It should be mentioned that sometimes, because of this property, the zeta function regularization resulting in eqn (4.43) has been interpreted as zeta function renormalization. This interpretation, however, has limited applicability since it does not provide a unique definition of the vacuum energy. This follows from the fact that the quantity contains terms depending on μ [see eqn (4.30)],
We conclude this section with a definition of the divergent part
Again, similarly to eqn (4.43), this expression is finite for s → 0, and the notation has been chosen in uniformity with eqns (4.30) and (4.32). As expected, the first term in eqn (4.46) coincides with that in eqn (4.44).
4.3 Renormalization of the vacuum energy
After we have obtained regularized expressions for the vacuum energy, it is necessary to remove the divergences, give an interpretation of this procedure, and address the key question about its uniqueness. Nonunique features are always present owing to the choice of the regularization scheme and parameters such as μ in eqn (4.3).
The simplest case of renormalization is that of a quantum field coupled to a smooth background field. We start with this case, where one can follow the well-known procedures from quantum field theory. Next, complications to this can be added in two ways, either by adding a boundary or by making the background field singular. We conclude this section with the easiest (from the renormalization point of view) case offorces between two separate bodies, which are always finite.
(p.66) 4.3.1 Smooth background fields
Here we consider the vacuum energy of a quantum field of mass m with a background of a smooth classical field of mass M in unbounded space-time. Some physical examples are the quantum fields for matter and radiation in a gravitational or electrodynamic background. However, there are no smooth background fields in Casimir problems, and we discuss this case only to illustrate the renormalization procedure. Thus, it is reasonable not to deal with electromagnetic or gravitational fields but instead to choose a technically simpler example of a classical scalar field ϕ(x) (the background field) and a quantum field φ(x) with the action
Here we have included a self-interaction term for the background field with some constant λ. In the action of the quantum field, the interaction term can be viewed as an additional position-dependent mass density. We assume the background field to be static, i.e. ϕ(x) → ϕ(r). Thus, when the quantum field is in the vacuum state, the system has a definite energy,
The eigenvalues in eqn (4.50), after the replacement are subject to the equation
These coefficients follow from the bulk part in eqn (4.22), where we have inserted (a 1/2 and a 3/2 are zero since we have no boundary). We drop the (p.67) contribution to eqn (4.30) from a 0 = V, which is infinitely large in unbounded space (see the discussion at the end of this subsection). As a result,
It should be noted that the divergent part repeats the structures present in the classical energy.
Representing the complete energy (4.48) in the form
Thus, the renormalized classical part of the energy is given by
The renormalized vacuum energy is then given by
The same approach can be used in other regularization schemes. For instance, we can consider the cutoff regularization. When using, instead of eqn (4.50), the vacuum energy in cutoff regularization (4.1), one needs to use the divergent part given in eqn (4.32). The heat kernel coefficients are the same as before [eqn (4.52)], and the only change in the above scheme will be slightly different formulas for the renormalized mass M ren and the self-interaction constant λren,
Finally, we consider this procedure for the vacuum energy (3.112),
In this case the renormalized parameters of the classical field are
We thus obtain a finite vacuum energy (4.57) which must be added to the classical energy (4.56). As explained above, the parameters of the classical energy have been renormalized. This is, however, not a problem, since the renormalized values must be determined independently anyway (usually experimentally). This is the general scheme of renormalization known in quantum field theory.
It must be mentioned that in this model, renormalization requires the self-interaction term in the classical part, and this is in agreement with the standard counting of the superficial degrees of divergence. Furthermore, we remark that there is no renormalization of the term containing the derivatives in the classical energy. This follows from the absence of a corresponding structure in the heat kernel coefficients and implies the nonrenormalization of the classical field Φ(r).
Also, note that a renormalization scheme such as that suggested by eqn (4.55) or (4.58) is not unique. This is due to the fact that with an infinite renormalization, we can always include a finite renormalization and still remove the singularities. This is similar to a change in the definition of the divergent part of the vacuum energy. Also, the parameter μ and the choice of the regularization lead to nonuniqueness.
A discussion of this nonuniqueness involves deeply the particular model considered. For instance, within the model given by eqn (4.47), it would be natural to look for a minimum of the complete energy E in eqn (4.48). It is clear that one may perform a redistribution of the energy between the two parts of eqn (4.48). This can be viewed as an additional finite renormalization. Consequently, here, the vacuum energy does not have an independent meaning.
Another method of proceeding is to impose a normalization condition on the renormalized vacuum energy such that it becomes uniquely defined after the regularization is removed (regardless of the regularization scheme). One of these conditions is the so-called no-tadpole condition introduced by Graham et (p.69) al. (2004). A second approach follows from a consideration of the mass of the quantum field together with the large-mass expansion (4.29), by demanding that
The motivation for this condition is that an infinitely heavy field should not have quantum fluctuations and hence should not produce a vacuum energy. This condition, as follows from the heat kernel expansion, is equivalent to the subtraction of all contributions involving the heat kernel coefficients a 0 through a 2 because these enter the large-mass expansion with nonnegative powers of the mass. In this manner, one can give the vacuum energy a unique meaning independent of a classical model. The definitions (4.30), (4.32), and (4.46) of the divergent part are given in such a way that the corresponding renormalized vacuum energy fulfills the normalization condition (4.62). As a consequence, the renormalized vacuum energy given by eqn (4.57) is unique, i.e. it does not depend either on the regularization chosen or on the parameter μ. This normalization condition was discussed by Bordag (2000). However, as mentioned there, this condition is meaningful for massive fields only. In the case of a massless field, this approach is not applicable, and there is as yet no known way to give a satisfactory renormalization condition independent of the classical model (unless, of course, the corresponding heat kernel coefficients are zero).
Next, we draw special attention to the divergent contribution resulting from the heat kernel coefficient a 0. On the one hand, in this simple model, we do not have a classical counterpart that has the same structure (proportional to the volume V, which is infinite here). On the other hand, this contribution does not depend on the background field. This is clearly the contribution which would be present in empty space, i.e. in the absence of the background field. Because of this, we do not relate it to the vacuum energy resulting from the background, and therefore drop it. This is the same case as when one considers only the response of the vacuum energy to a change in the background field. There is, however, one scenario where this is not possible. Namely, when we consider quantum fluctuations in a gravitational background, we cannot drop this contribution, because it is the source of the gravitational field. But in that case there exists a structure for renormalization, namely the term containing the cosmological constant, which needs to be renormalized in the same way as the gravitational constant.
4.3.2 Singular background fields and boundary conditions
The situation described in the preceding subsection changes when nonsmooth background fields are considered. From a formal point of view, one first observes that the heat kernel coefficients ak /2 become infinite starting from some k. This is because the coefficients contain powers of the background field and its derivatives in increasing order. For instance, when the interaction potential in the model (4.47) becomes proportional to a delta function on a sphere of radius R, i.e.
The situation is different when one starts from an already singular background. Here, the only case which has been investigated so far in any detail is a one-dimensional delta function potential on a spherical surface. In that case the spectral problem for the fluctuations is well defined and all the heat kernel coefficients exist. For example, with eqn (4.63), the heat kernel coefficients are (Bordag et al. 1999a)
In the above, we have written down only those coefficients which are relevant to the renormalization. It is a characteristic of this singular background that the coefficient a 3/2, with a half-integer number, appears [for a more general discussion see Bordag and Vassilevich (1999, 2004)].
Finally, we consider the vacuum energy in the presence of boundary conditions. This case is the most relevant for the Casimir effect. For simplicity, we restrict ourselves to the case of a sphere with Dirichlet or Neumann boundary conditions. The heat kernel coefficients are given by eqn (4.25). Since no background field exists, one needs to introduce other classical parameters in order to accommodate the renormalization. Blau et al. (1988) suggested the geometric structure
(p.71) However, a situation may arise where there is no classical system available for the justification of the renormalization. In that case one could take the normalization condition (4.62), provided the quantum field has a mass. If the field is massless and the corresponding heat kernel coefficients do not vanish, one cannot give the vacuum energy a satisfactory interpretation. The most important examples are that of a conducting sphere of finite thickness and that of a dielectric ball. These will be discussed in Sections 9.3.3 and 9.3.4.
4.3.3 Finiteness of the Casimir force between separate bodies
We have seen in Section 4.2.2 that the divergent part of the vacuum energy follows from the heat kernel coefficients a 0 through a 2. These coefficients are represented by eqn (4.21) as integrals over local quantities: the background potential and the coefficients of the second fundamental form (4.19), including their derivatives and powers. This locality is a fundamental property of the heat kernel coefficients that holds under very general assumptions. It is believed that it is related to the locality of the ultraviolet divergences in quantum field theory.
With respect to the Casimir effect, the local nature of the coefficients determining the divergent part of the vacuum energy has a far-reaching consequence. The definition of the heat kernel coefficients presented in Section 4.2.2 is of a rather general character. It refers both to simply connected manifolds (a compact body with some finite volume restricted by a boundary surface S) and to nonsimply connected manifolds. As an example of the latter, let us consider two separate, i.e. nonintersecting bodies with volumes V 1 and V 2 and surfaces S 1 and S 2 having no common points. We also assume that there is no background field. It follows from the latter that all of the local heat kernel coefficients bk /2 in eqn (4.22), excepting b 0, are equal to zero. Thus, none of the global heat kernel coefficients ak /2 in eqn (4.21) with k ≥ 1 contain a volume contribution. They are given by
Here the local coefficients and are defined by eqn (4.22) for the respective parts S 1 and S 2 of the surface S. These coefficients need not be the same. Even the boundary conditions on S1 and S 2 may be different.
In the case of two separate interacting bodies, it is reasonable to consider the spatial region V − V 1−V 2, where V is the infinite volume of the entire three-dimensional space, restricted by the boundary surface S consisting of S 1 and S 2. In doing so, we change the sign of the direction of the local normal vector n to the surface. This leads to an opposite sign for the coefficients of the second fundamental form given in eqn (4.19). For example, in the case of the Casimir interaction between two spheres with radii R 1 and R 2, and As a result, from eqns (4.21) and (4.22) we obtain
The divergent part of the vacuum energy (4.30) is then determined by the coefficients (4.66) with 1 ≤ k ≤ 4. As is seen from the structure of eqn (4.66), the ak /2 do not contain any information about the relative location of the parts S 1 and S 2 of the boundary surface S. In other words, the heat kernel coefficients do not depend on the distance between the interacting bodies under consideration. If we now insert the ak /2 into the divergent part (4.30) or (4.32) or the pole part (4.28) of the vacuum energy in any regularization, that part is also found to be independent of the distance. The distance dependence is contained only in the finite renormalized vacuum energy defined in eqn (4.57). We emphasize that information about the distance dependence of cannot be obtained from the heat kernel expansion. It is contained in the finite part of the energy remaining after subtraction of the divergent part. As the divergent part is independent of the separation, the force between two separate bodies is always finite.