Casimir energy is calculated for the 5D electromagnetism and 5D scalar theory in the { \it warped} geometry. It is compared with the flat case(arXiv:0801.3064). A new regularization, called { \it sphere lattice regularization}, is taken. In the integration over the 5D space, we introduce two boundary curves (IR-surface and UV-surface) based on the { \it minimal area principle}. It is a { \it direct} realization of the geometrical approach to the { \it renormalization group }. The regularized configuration is { \it closed-string like}. We do { \it not} take the KK-expansion approach. Instead, the position/momentum propagator is exploited, combined with the { \it heat-kernel method}. All expressions are closed-form (not KK-expanded form). The { \it generalized} P/M propagators are introduced. We numerically evaluate $\La$ (4D UV-cutoff), $\om$ (5D bulk curvature, warp parameter) and $T$ (extra space IR parameter) dependence of the Casimir energy. We present two { \it new ideas} in order to define the 5D QFT: 1) the summation (integral) region over the 5D space is { \it restricted} by two minimal surfaces (IR-surface, UV-surface) ; or 2) we introduce a { \it weight function} and require the dominant contribution is given by the { \it minimal surface}. Based on these, 5D Casimir energy is { \it finitely} obtained after the { \it proper renormalization procedure.} The { \it warp parameter} $\om$ suffers from the { \it renormalization effect}. We examine the meaning of the weight function and reach a { \it new definition} of the Casimir energy where { \it the 4D momenta(or coordinates) are quantized} with the extra coordinate as the Euclidean time (inverse temperature). We comment on the cosmological constant term and present an answer to the problem at the end. Dirac's large number naturally appears.
英文摘要：
Casimir energy is calculated for the 5D electromagnetism and 5D scalar theory in the {\it warped} geometry. It is compared with the flat case(arXiv:0801.3064). A new regularization, called {\it sphere lattice regularization}, is taken. In the integration over the 5D space, we introduce two boundary curves (IR-surface and UV-surface) based on the {\it minimal area principle}. It is a {\it direct} realization of the geometrical approach to the {\it renormalization group}. The regularized configuration is {\it closed-string like}. We do {\it not} take the KK-expansion approach. Instead, the position/momentum propagator is exploited, combined with the {\it heat-kernel method}. All expressions are closed-form (not KK-expanded form). The {\it generalized} P/M propagators are introduced. We numerically evaluate $\La$(4D UV-cutoff), $\om$(5D bulk curvature, warp parameter) and $T$(extra space IR parameter) dependence of the Casimir energy. We present two {\it new ideas} in order to define the 5D QFT: 1) the summation (integral) region over the 5D space is {\it restricted} by two minimal surfaces (IR-surface, UV-surface) ; or 2) we introduce a {\it weight function} and require the dominant contribution is given by the {\it minimal surface}. Based on these, 5D Casimir energy is {\it finitely} obtained after the {\it proper renormalization procedure.} The {\it warp parameter} $\om$ suffers from the {\it renormalization effect}. We examine the meaning of the weight function and reach a {\it new definition} of the Casimir energy where {\it the 4D momenta(or coordinates) are quantized} with the extra coordinate as the Euclidean time (inverse temperature). We comment on the cosmological constant term and present an answer to the problem at the end. Dirac's large number naturally appears.
在膜世界宇宙学中，膨胀的宇宙被认为是一个在扭曲的高维时空中运动的膜。就像一面移动的镜子会从真空涨落中产生光子一样，一个移动的膜会导致引力子的产生。我们表明，非常一般地，卡鲁扎-克莱因（KK）粒子的尺度像刚性物质与宇宙的膨胀，因此不能代表暗物质在一个扭曲的braneworld。本文给出了在五维反德西特空间中弹跳膜产生无质量和KK引力子的结果。我们发现，对于一个现实的反弹，从所产生的引力子的反作用将是最有可能相关。这封信总结了主要的结果和结论，从数值模拟中详细介绍了在一个长的文件[M。Durrer，Phys. Rev. D 76，104014（2007），arXiv：0704.0790]
In the present paper, we investigate regularization of the one-loop quantum corrections with infinite Kaluza-Klein (KK) states and evaluate Casimir vacuum energy from extra dimensions. The extra dimensional models always involve the infinite massless or massive Kaluza-Klein states, and therefore, the regularization of the infinite KK corrections is highly problematic. In order to avoid the ambiguity, we adopt the proper time integrals and the Riemann zeta function regularization in evaluating the summations of infinite KK states. In the calculation, we utilize the KK regularization method with…
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 sufficien…
We propose a procedure for the renormalization of Casimir energy, that is based on the implicit versions of standard steps of renormalization procedure --- regularization, subtraction and removing the regularization. The proposed procedure is based on the calculation of a set of convergent sums, which are related to the original divergent sum for the non-renormalized Casimir energy. Then one constructs a linear equations system, that connects this set of convergent sums with the renormalized Casimir energy, which is a solution to this equations system. This procedure slightly reduces the indet…
Applying the general framework for local zeta regularization proposed in Part I of this series of papers, we renormalize the vacuum expectation value of the stress-energy tensor (and of the total energy) for a scalar field in presence of an external harmonic potential.
Casimir Energy of 5D Electromagnetism and New Regularization Based on Minimal Area Principle
We examine the Casimir energy of 5D electromagnetism in the recent standpoint. The bulk geometry is flat. Z $_2$ symmetry and the periodic property, for the extra coordinate, are taken into account. After confirming the consistency with the past result, we do new things based on a { \it new regularization}. In the treatment of the divergences, we introduce IR and UV cut-offs and { \it restrict} the (4D momentum, extra coordinate)-integral region. The regularized configuration is the { \it sphere lattice}, in the 4D continuum space, which changes along the extra coordinate. The change (renormal…
In this article, we present the zero and first-order radiative correction to the Dirichlet Casimir energy for massive and massless scalar field confined in a rectangle. This calculation procedure was conducted in two spatial dimensions and for the case of the first-order correction term is new. The renormalization program that we have used in this work, allows all influences from the dominant boundary conditions (e.g. the Dirichlet boundary condition) be automatically reflected in the counterterms. This permission usually makes the counterterms position-dependent. Along with the renormalizatio…
Casimir Energy of the Universe and New Regularization of Higher Dimensional Quantum Field Theories
Casimir energy is calculated for the 5D electromagnetism and 5D scalar theory in the { \it warped} geometry. It is compared with the flat case. A new regularization, called { \it sphere lattice regularization}, is taken. In the integration over the 5D space, we introduce two boundary curves (IR-surface and UV-surface) based on the { \it minimal area principle}. It is a { \it direct} realization of the geometrical approach to the { \it renormalization group }. The regularized configuration is { \it closed-string like}. We do { \it not} take the KK-expansion approach. Instead, the position/momen…
We extend previous work on the vacuum energy of a massless scalar field in the presence of singular potentials. We consider a single sphere denoted by the so-called "delta-delta prime" interaction. Contrary to the Dirac delta potential, we find a nontrivial one-parameter family of potentials such that the regularization procedure gives an unambiguous result for the Casimir self-energy. The procedure employed is based on the zeta function regularization and the cancellation of the heat kernel coefficient a_2. The results obtained are in agreement with particular cases, such as the Dirac delta o…
In Part I of this series of papers we have described a general formalism to compute the vacuum effects of a scalar field via local (or global) zeta regularization. In the present Part II we exemplify the general formalism in a number of cases which can be solved explicitly by analytical means. More in detail we deal with configurations involving parallel or perpendicular planes and we also discuss the case of a three-dimensional wedge.
In this paper, we presented the zero- and first-order radiative corrections to the Casimir energy for a massive scalar field confined with Dirichlet boundary condition in an open-ended rectangular waveguide. In the calculation procedure, we applied a systematic renormalization program that allows all influences imposed by dominant boundary conditions in a problem be automatically reflected in the counterterms, leading the counterterms to be obtained in a position-dependent manner. To remove the appeared divergences in the computation task, the Box Subtraction Scheme as a regularization techniq…
Applying the general framework for local zeta regularization proposed in Part I of this series of papers, we compute the renormalized vacuum expectation value of several observables (in particular, of the stress-energy tensor and of the total energy) for a massless scalar field confined within a rectangular box of arbitrary dimension.
We quantitatively estimate the effect of the UV physics on the Casimir energy in a five-dimensional (5D) model on $S^1/Z_2$. If the cutoff scale of the 5D theory is not far from the compactification scale, the UV physics may affect the low energy result. We work in the cutoff regularization scheme by introducing two independent cutoff scales for the spatial momentum in the non-compact space and for the Kaluza-Klein masses. The effects of the UV physics are incorporated as a damping effect of the contributions to the vacuum energy around the cutoff scales. We numerically calculate the Casimir e…
We study $d$-dimensional Conformal Field Theories (CFTs) on the cylinder, $S^{d-1}\times \mathbb{R}$, and its deformations. In $d=2$ the Casimir energy (i.e. the vacuum energy) is universal and is related to the central charge $c$. In $d=4$ the vacuum energy depends on the regularization scheme and has no intrinsic value. We show that this property extends to infinitesimally deformed cylinders and support this conclusion with a holographic check. However, for $\mathcal{N}=1$ supersymmetric CFTs, a natural analog of the Casimir energy turns out to be scheme independent and thus intrinsic. We gi…
The Casimir effect for rectangular boxes has been studied for several decades. But there are still some points unclear. Recently, there are new developments related to this topic, including the demonstration of the equivalence of the regularization methods and the clarification of the ambiguity in the regularization of the temperature-dependent free energy. Also, the interesting quantum spring was raised stemming from the topological Casimir effect of the helix boundary conditions. We review these developments together with the general derivation of the Casimir energy of the $p$-dimensional ca…
In this paper, the first order radiative correction to the Casimir energy for a massive scalar field in the $ϕ^4$ theory on a spherical surface with $S^2$ topology was calculated. In common methods for calculating the radiative correction to the Casimir energy, the counter-terms related to free theory are used. However, in this study, by using a systematic perturbation expansion, the obtained counter-terms in renormalization program were automatically position-dependent. We maintained that this dependency was permitted, reflecting the effects of the boundary conditions imposed or background sp…
The unitary Fermi gas, by virtue of its description as a nonrelativistic conformal field theory, has proven an interesting system by which the quantum properties of CFT can be held to experimental verification. Here, we examine the structure of conformal dimensions of charge-Q operators in nonrelativistic CFT, in the large-Q regime, from the non-linear sigma model perspective. We discuss in detail the renormalization of edge divergences using dimensional regularization, elucidating the presence of $\log(Q)$ terms in the large-charge expansion. Finally we use dimensional regularization to compu…
This is the first one of a series of papers about zeta regularization of the divergences appearing in the vacuum expectation value (VEV) of several local and global observables in quantum field theory. More precisely we consider a quantized, neutral scalar field on a domain in any spatial dimension, with arbitrary boundary conditions and, possibly, in presence of an external classical potential. We analyze, in particular, the VEV of the stress-energy tensor, the corresponding boundary forces and the total energy, thus taking into account both local and global aspects of the Casimir effect. In …
In this paper we discuss the computation of Casimir energy on a quantum computer. The Casimir energy is an ideal quantity to calculate on a quantum computer as near term hybrid classical quantum algorithms exist to calculate the ground state energy and the Casimir energy gives physical implications for this quantity in a variety of settings. Depending on boundary conditions and whether the field is bosonic or fermionic we illustrate how the Casimir energy calculation can be set up on a quantum computer and calculated using the Variational Quantum Eigensolver algorithm with IBM QISKit. We compa…
We study the Casimir energy of four-dimensional supersymmetric gauge theories in the context of the rigid limit of new minimal supergravity. Firstly, revisiting the computation of the localized partition function on $S^1\times S^3$, we recover the supersymmetric Casimir energy. Secondly, we consider the same theories in the Hamiltonian formalism on $\mathbb{R}\times S^3$, focussing on the free limit and including a one-parameter family of background gauge fields along $\mathbb{R}$. We compute the vacuum expectation value of the canonical Hamiltonian using zeta function regularization, and show…
