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 (renormalization flow) is specified by the { \it minimal area principle}, hence this regularization configuration is string-like. We do the analysis not in the Kaluza-Klein expanded form but in a { \it closed} form. We do { \it not} use any perturbation. The formalism is based on the heat-kernel approach using the { \it position/momentum propagator}. Interesting relations between the heat-kernels and the P/M propagators are obtained, where we introduce the { \it generalized} P/M propagators. A useful expression of the Casimir energy, in terms of the P/M propagator, is obtained. The restricted-region approach is replaced by the weight-function approach in the latter-half description. Its meaning, in relation to the { \it space-time quantization}, is argued. { \it Finite} Casimir energy is numerically obtained. The compactification-size parameter (periodicity) suffers from the renormalization effect. Numerical evaluation is exploited. Especially the minimal surface lines in the 5D flat space are obtained both numerically using the Runge-Kutta method and analytically using the general solution.
英文摘要：
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 (renormalization flow) is specified by the {\it minimal area principle}, hence this regularization configuration is string-like. We do the analysis not in the Kaluza-Klein expanded form but in a {\it closed} form. We do {\it not} use any perturbation. The formalism is based on the heat-kernel approach using the {\it position/momentum propagator}. Interesting relations between the heat-kernels and the P/M propagators are obtained, where we introduce the {\it generalized} P/M propagators. A useful expression of the Casimir energy, in terms of the P/M propagator, is obtained. The restricted-region approach is replaced by the weight-function approach in the latter-half description. Its meaning, in relation to the {\it space-time quantization}, is argued. {\it Finite} Casimir energy is numerically obtained. The compactification-size parameter (periodicity) suffers from the renormalization effect. Numerical evaluation is exploited. Especially the minimal surface lines in the 5D flat space are obtained both numerically using the Runge-Kutta method and analytically using the general solution.
By the thermofield dynamics (TFD) formalism we obtain the energy-momentum tensor for the Electromagnetism with Lorentz Breaking Even term of the Standard Model Extended (SME) Sector in a topology $S^{1}\times S^{1}\times R^{2}$. We carry out the compactification by a generalized TFD-Bogoliubov transformation that is used to define a renormalized energy-momentum tensor, and the Casimir energy and pressure at finite temperature are then derived. A comparative analysis with the electromagnetic case is developed, and we remark the influence of the background in the traditional Casimir effect.
We discuss Stueckelberg massive electromagnetism on an arbitrary four-dimensional curved spacetime (gauge invariance of the classical theory and covariant quantization, wave equations for the massive spin-1 field $A_μ$, for the auxiliary Stueckelberg scalar field $Φ$ and for the ghost fields $C$ and $C^\ast$, Ward identities, Hadamard representation of the various Feynman propagators and covariant Taylor series expansions of the corresponding coefficients). This permits us to construct, for a Hadamard quantum state, the expectation value of the renormalized stress-energy tensor associated with…
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…
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 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…
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…
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…
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…
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…
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…
A diffraction problem for a flat Chern-Simons layer at plane boundary of a dielectric half space is solved. The Casimir energy of two dielectric half spaces with Chern-Simons layers at plane-parallel boundaries separated by a vacuum slit is derived. Crossing from the repulsive to the attractive Casimir force is analyzed for two Au and two Si half spaces with boundary Chern-Simons layers. Boundary quantum Hall layers in external magnetic field lead to Casimir repulsion at nanoscales. We discuss features that make systems with boundary quantum Hall layers unique for force measurements and search…
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…
Casimir Energy of 5D Warped System and Sphere Lattice Regularization
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, t…
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.
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.
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.
In this work, we obtain the Casimir energy for the real scalar field and the Elko neutral spinor field in a field theory at a Lifshitz fixed point (LP). We analyze the massless and the massive case for both fields using dimensional regularization. We obtain the Casimir energy in terms of the dimensional parameter and the LP parameter. Particularizing our result, we can recover the usual results without LP parameter in (3+1) dimensions presented in the literature. Moreover, we compute the effects of the LP parameter in the thermal corrections for the massless scalar field.
The zeta function regularization technique is used to study the Casimir effect for a scalar field of mass $m$ satisfying Dirichlet boundary conditions on a spherical surface of radius $a$. In the case of large scalar mass, $ma\gg1$, simple analytic expressions are obtained for the zeta function and Casimir energy of the scalar field when it is confined inside the spherical surface, and when it is confined outside the spherical surface. In both cases the Casimir energy is exact up to order $a^{-2}m^{-1}$ and contains the expected divergencies, which can be eliminated using the well established …
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…
