Local Zeta Regularization and the Scalar Casimir Effect

The following sections are included:

• Preface
• Acknowledgements
• Contents

In the present chapter and in the related Appendix A, we introduce the general framework for a neutral scalar field on a d-dimensional spatial domain Ω, including a brief discussion of the stress-energy tensor with its conformal and non-conformal parts. This is an occasion to fix the attention on the basic elliptic operator A := −Δ+V on Ω, where Δ is the Laplacian and V : Ω → R is an external potential. Next, we proceed to introduce zeta regularization in the formulation first given in [58]: the basic idea is to replace the quantized field $\widehat{\varphi }$ with its regularized version ${\widehat{\varphi }}^u:={\left(A/k^2\right)}^{-u/4}\widehat{\varphi },$ where u ∈ C is the regulating parameter (or regulator) and k > 0 is a mass parameter introduced for dimensional reasons. Formally, ${\widehat{\varphi }}^u$ becomes $\widehat{\varphi }$ for u = 0; the zeta approach implements this idea in terms of analytic continuation. This means that for any local or global observable (say, the VEV of either the stress-energy tensor or of the total energy), after introducing a regularized version of it based on ${\widehat{\varphi }}^u$, the corresponding renormalized value is defined via the analytic continuation at u = 0. We distinguish between two versions of this prescription: the restricted zeta approach, in which the regularized observable is analytic at u = 0, and the extended approach, where a singularity appears at u = 0 and is eliminated by a minimal subtraction prescription, removing the negative powers of u in the corresponding Laurent expansion.

In this chapter and in the related Appendices C and D, we introduce a number of integral kernels associated to the operator A := −Δ+V. The Dirichlet kernel $D_s\left(\text{x,y}\right):=〈{\delta }_{\text{x}}|A^{-s}{\delta }_{\text{y}}〉$ (s in a suitable complex domain, x, y ∈ Ω, δ_x, δ_y the Dirac delta functions at these points) is closely related to the stress-energy VEV. More precisely, the regularized stress-energy VEV built using ${\widehat{\varphi }}^u$ can be expressed in terms of the Dirichlet kernel D_s(x, y) (and of its spatial derivatives) at y = x, s = (u±1)/2; so, the renormalized version of this VEV is determined by the analytic continuation of D_s near s = ±1/2. We show that the continuation of the Dirichlet kernel can be computed algorithmically relating it to the heat kernel $K\left(\text{t;x,}\text{y}\right):=〈{\delta }_{\text{x}}|e^{-\text{t}A}{\delta }_{\text{y}}〉$ or to other kernels (among which is the so-called cylinder kernel $T\left(\text{t;x,}\text{y}\right):=〈{\delta }_{\text{x}}|e^{-\text{t}\sqrt{A}}{\delta }_{\text{y}}〉,$ often considered by Fulling [56, 68]). This procedure ultimately gives a set of mechanical rules for zeta regularization.

Let us refer to the general framework of Chapter 1, where a quantized scalar field on a spatial domain Ω and the associated stress-energy VEV are considered; as usual, A indicates the fundamental operator −Δ + V acting in L²(Ω). In this chapter and in the associated Appendix E we relate to the previous framework the total energy, the pressure and the total forces on the boundary. We also take the occasion to prove (at the regularized level) the equivalence between two alternative definitions of the pressure, often assumed without proof in the literature: pressure as the action of the stress-energy VEV on the normal to the boundary, and pressure as the functional derivative of the bulk energy with respect to deformations of the spatial domain Ω.

In this chapter and in the related Appendix F we consider some variations of the general framework of Chapters 1–3 concerning the following situations: (i) the case where 0 is either an isolated or non-isolated point of the spectrum of the fundamental operator A := −Δ+ V ; (ii) the case where the at spatial domain Ω is described via curvilinear coordinates or, more generally, the case where Ω is a (possibly non-flat) Riemannian manifold equipped with arbitrary coordinates (or an open subset of it, with prescribed boundary conditions). Some of these variations will be used in the applications of Part 2.

Part 2 of this book shows in action the mechanical procedures for analytic continuation presented in the preceding Part 1. In the present Chapter 5 we start from a very simple case, namely, a one-dimensional model with a massless scalar field living on a segment, with no background potential. The formal description of this setting is presented in Section 5.1 (see, in particular, Eq. (5.1)); in the subsequent Sections 5.2–5.5, we describe the general methods that can be used to compute the renormalized VEVs of the stress-energy tensor, of the total energy and of the force on the end-points. Sections 5.6–5.9 take into account several types of boundary conditions, deriving for each one of them explicit expressions for the above mentioned VEVs. The final Section 5.10 gives some details about the comparison with some previous works which are related to the setting under analysis here…

We shall now pass to consider the case of a massless field confined between two parallel (hyper-)planes, for several kinds of boundary conditions; we restrict the attention to the case of odd spatial dimension, for technical reasons to be discussed later on. The formal description of this setting is given in Section 6.1; therein it is pointed out that the analysis of this problem can be reduced to that of a one-dimensional problem, analogous to the one already considered in the preceding Chapter 5. Sections 6.2–6.5 present the general techniques which allow to determine the analytic continuation of the regularized VEVs for several observables. In the conclusive Sections 6.6–6.9 we derive explicit expressions for the renormalized versions of these quantities in the physically interesting case of spatial dimension d = 3…

In this chapter we consider a massive field confined by an arbitrary number of perpendicular (hyper-)planes, fulfilling either Dirichlet or Neumann boundary conditions on each one of them. Also in this case, after giving a formal description of this setting (see Section 7.1), we show how to use the general methods of Part 1 to compute the analytic continuation of the regularized stress-energy VEV and pressure (see Sections 7.2–7.6). In Sections 7.7–7.9 we specialize the previous general results, determining the renormalized observables in two cases: a single plane and two perpendicular planes in spatial dimension d = 3. In the conclusive Section 7.10 we consider the zero mass limit in these two particular settings…

In the present chapter we consider a massless field confined within a wedge of arbitrary width in spatial dimension d = 3, for several types of boundary conditions. We also consider a variation of this configuration, which corresponds essentially to identify the sides of the wedge; this is the so-called case of the “cosmic string” (see Section 8.9). As usual, in the first Section 8.1 we give a formal description of the general configuration we are going to consider. In the subsequent Sections 8.2–8.5 we use the general techniques of Part 1 to determine the analytic continuation of the regularized observables VEVs, for several boundary conditions; the renormalized counterparts of these quantities are computed in Sections 8.6–8.9…

Hereafter we proceed to describe a more engaging application of the general techniques developed in Part 1 of this book, which requires some numerical computations. The results reported here were first derived in a previous work of ours [60]; they concern the case of a massless scalar field living on the whole space R^d, and confined by a background harmonic potential. For simplicity, the latter is assumed to be isotropic, i.e. to be proportional to the squared radius |x|² (x ∈ R^d); nevertheless, the approach described here could be extended with little effort to anisotropic harmonic potentials and to the case of a massive field theory…

In this final chapter we consider the case of a massless field confined within a d-dimensional rectangular spatial domain. We summarize here the results obtained in [61]; following this reference, for the sake of simplicity we restrict attention to the case where the field fulfills Dirichlet conditions on the boundary. Nevertheless, the techniques we employ could be generalized with little effort to treat different settings including, e.g. the case of Neumann or periodic boundary conditions…

Consider the stress-energy tensor operator introduced in Eq. (1.15) of Chapter 1. As pointed out therein, this object is obtained via second-quantization of a classical formula for the stress-energy tensor [18, 30, 33, 115], which we review in the present appendix for completeness…

We refer to an operator A in L²(Ω) possessing the features described by Eqs. (2.7), (2.8) or (2.9) (recall that the situation (2.7) is more general than (2.8), and (2.8) is more general than (2.9)). In this appendix we present a number of results which are useful in relation to several integral kernels associated to A. Proving these results would require a heavy use of functional analysis, which is not among the purposes of the present book; therefore, hereafter we just sketch the basic ideas that are presented with more details (and greater mathematical rigor) in [62]…

In the present appendix we show how to derive the identity in Eq. (2.109) (stated in Subsection 2.8.3 of Chapter 2), allowing to express the Mellin transform of a suitable function in terms of an integral along the Hankel contour in the complex plane…

We refer to the framework of Section 2.10 about the slab configuration where $\text{Ω}={\text{Ω}}_1\times {\text{R}}^{d_2};$ in the present appendix we retain all the assumptions and notations of the cited section and show how to derive Eqs. (2.131)–(2.133)…

As in the final part of Subsection 3.2.1 and in Section 3.3, we work on a domain Ω and assume Dirichlet boundary conditions are prescribed on ∂Ω.

Consider the framework developed in Section 4.2 for a non-negative operator 𝒜 = −Δ+ V, such that σ(𝒜)⊂ [0, + ∞) and 0 is in the continuous spectrum of 𝒜. In the present appendix we use the deformed operator

The following sections are included:

• Bibliography
• Index