scholarly journals Gravitons in a Casimir box

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Francesco Alessio ◽  
Glenn Barnich ◽  
Martin Bronte
Keyword(s):  
Partition Function ◽  
Boundary Conditions ◽  
Chemical Potential ◽  
Finite Size ◽  
Neumann Boundary ◽  
Scalar Fields ◽  
Casimir Energy ◽  
Massless Scalar ◽  
Analytical Control ◽  
Slab Geometry

Abstract The partition function of gravitons with Casimir-type boundary conditions is worked out. The simplest box that allows one to achieve full analytical control consists of a slab geometry with two infinite parallel planes separated by a distance d. In this setting, linearized gravity, like electromagnetism, is equivalent to two free massless scalar fields, one with Dirichlet and one with Neumann boundary conditions, which in turn may be combined into a single massless scalar with periodic boundary conditions on an interval of length 2d. When turning on a chemical potential for suitably adapted spin angular momentum, the partition function is modular covariant and expressed in terms of an Eisenstein series. It coincides with that for photons. At high temperature, the result provides in closed form all sub-leading finite-size corrections to the standard (gravitational) black body result. More interesting is the low-temperature/small distance expansion where the leading contribution to the partition function is linear in inverse temperature and given in terms of the Casimir energy of the system, whereas the leading contribution to the entropy is proportional to the area and originates from gravitons propagating parallel to the plates.

On the vacuum stress induced by uniform acceleration or supporting the ether

1977 ◽  
Vol 354 (1676) ◽  
pp. 79-99 ◽  
Keyword(s):  
Scalar Field ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Scalar Fields ◽  
Black Body ◽  
Dirichlet Boundary Conditions ◽  
Massless Scalar ◽  
Perfect Conductor ◽  
Infinite Plane ◽  
Uniform Acceleration

The stresses induced in the vacuum by the uniform acceleration of an infinite plane conductor are computed for the massless scalar and electromagnetic fields. Both Dirichlet and Neumann boundary conditions are considered for the scalar field; far from the conductor it is found, independently of the boundary condition, that the vacuum stress is ‘local’ and corresponds to the absence from the vacuum of black body radiation. Approaching the conductor, the energy density in the Dirichlet case is slightly lower than the ‘local’ term, and in the Neumann case slightly higher. At very small distances it again has the same asymptotic form for both scalar fields. For the electromagnetic field the results are similar to those for the scalar field with Dirichlet boundary conditions. Far from the conductor the spectrum is again black-body, though not Planckian. In all cases the acausal nature of ‘ perfect conductor ’ boundary conditions prevents the stress tensor from being finite on the conductor.

scholarly journals Modular invariance in finite temperature Casimir effect

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Francesco Alessio ◽  
Glenn Barnich
Keyword(s):  
Partition Function ◽  
Chemical Potential ◽  
Casimir Effect ◽  
Temperature Inversion ◽  
Massless Scalar ◽  
Partition Functions ◽  
Parallel Plates ◽  
Modular Transformations ◽  
Real Analytic ◽  
Set Up

Abstract The temperature inversion symmetry of the partition function of the electromagnetic field in the set-up of the Casimir effect is extended to full modular transformations by turning on a purely imaginary chemical potential for adapted spin angular momentum. The extended partition function is expressed in terms of a real analytic Eisenstein series. These results become transparent after explicitly showing equivalence of the partition functions for Maxwell’s theory between perfectly conducting parallel plates and for a massless scalar with periodic boundary conditions.

scholarly journals Notes on massless scalar field partition functions, modular invariance and Eisenstein series

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Francesco Alessio ◽  
Glenn Barnich ◽  
Martin Bonte
Keyword(s):  
Scalar Field ◽  
Partition Function ◽  
Low Temperature ◽  
Eisenstein Series ◽  
Chemical Potential ◽  
Proper Time ◽  
Series Representation ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Spacetime Manifold

Abstract The partition function of a massless scalar field on a Euclidean spacetime manifold ℝd−1 × 𝕋2 and with momentum operator in the compact spatial dimension coupled through a purely imaginary chemical potential is computed. It is modular covariant and admits a simple expression in terms of a real analytic SL(2, ℤ) Eisenstein series with s = (d + 1)/2. Different techniques for computing the partition function illustrate complementary aspects of the Eisenstein series: the functional approach gives its series representation, the operator approach yields its Fourier series, while the proper time/heat kernel/world-line approach shows that it is the Mellin transform of a Riemann theta function. High/low temperature duality is generalized to the case of a non-vanishing chemical potential. By clarifying the dependence of the partition function on the geometry of the torus, we discuss how modular covariance is a consequence of full SL(2, ℤ) invariance. When the spacetime manifold is ℝp × 𝕋q+1, the partition function is given in terms of a SL(q + 1, ℤ) Eisenstein series again with s = (d + 1)/2. In this case, we obtain the high/low temperature duality through a suitably adapted dual parametrization of the lattice defining the torus. On 𝕋d+1, the computation is more subtle. An additional divergence leads to an harmonic anomaly.

Stationary bound states of massless scalar fields around black holes and black hole analogues

2015 ◽  
Vol 24 (09) ◽  
pp. 1542018 ◽  
Author(s):  
Carolina L. Benone ◽  
Luís C. B. Crispino ◽  
Carlos A. R. Herdeiro ◽  
Eugen Radu
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Spatial Distribution ◽  
Scalar Field ◽  
Bound States ◽  
Neumann Boundary ◽  
Mass Term ◽  
Scalar Fields ◽  
Massless Scalar ◽  
Field System

We discuss stationary bound states, a.k.a. clouds, for a massless test scalar field around Kerr black holes (BHs) and spinning acoustic BH analogues. In view of the absence of a mass term, the trapping is achieved via enclosing the BH — scalar field system in a cavity and imposing Dirichlet or Neumann boundary conditions. We discuss the variation of these bounds states with the discrete parameters that label them, as well as their spatial distribution, complementing results in our previous work [C. L. Benone, L. C. B. Crispino, C. Herdeiro and E. Radu, Phys. Rev. D91 (2015) 104038].

scholarly journals The Casimir energy for scalar field with mixed boundary condition

2018 ◽  
Vol 15 (10) ◽  
pp. 1850172 ◽  
Author(s):  
M. A. Valuyan
Keyword(s):  
Scalar Field ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Scalar Fields ◽  
Casimir Energy ◽  
Massless Scalar ◽  
Regularization Technique ◽  
Massive Scalar Field ◽  
Subtraction Scheme ◽  
First Order

In this study, the first-order radiative correction to the Casimir energy for massive and massless scalar fields confined with mixed boundary conditions (BCs) (Dirichlet–Neumann) between two points in [Formula: see text] theory was computed. Two issues in performing the calculations in this work are essential: to renormalize the bare parameters of the problem, a systematic method was employed, allowing all influences from the BCs to be imported in all elements of the renormalization program. This idea yields our counterterms appeared in the renormalization program to be position-dependent. Using the Box Subtraction Scheme (BSS) as a regularization technique is the other noteworthy point in the calculation. In this scheme, by subtracting the vacuum energies of two similar configurations from each other, regularizing divergent expressions and their removal process were significantly facilitated. All the obtained answers for the Casimir energy with the mixed BC were consistent with well-known physical grounds. We also compared the Casimir energy for massive scalar field confined with four types of BCs (Dirichlet, Neumann, mixed of them and Periodic) in [Formula: see text] dimensions with each other, and the sign and magnitude of their values were discussed.

Stress tensor and conformal anomalies for massless fields in a Robertson–Walker universe

1977 ◽  
Vol 356 (1687) ◽  
pp. 569-574 ◽  
Keyword(s):  
Stress Tensor ◽  
Scalar Fields ◽  
Casimir Energy ◽  
Massless Scalar ◽  
Einstein Universe ◽  
Massless Fields

We derive the unique, local vacuum stress tensor for electromagnetic, neutrino and massless scalar fields propagating in a Robertson─Walker background spacetime. The result is used to compute the numerical coefficients of the conformal trace anomalies from the known values of the Casimir energy in the Einstein universe.

scholarly journals Casimir effect of a Lorentz-violating scalar in magnetic field

2020 ◽  
Vol 35 (31) ◽  
pp. 2050209
Author(s):  
Andrea Erdas
Keyword(s):  
Magnetic Field ◽  
Scalar Field ◽  
Boundary Conditions ◽  
Casimir Effect ◽  
Lorentz Invariance ◽  
Neumann Boundary ◽  
Casimir Energy ◽  
Neumann Boundary Conditions ◽  
Parallel Plates ◽  
Massive Scalar Field

In this paper, I study the Casimir effect caused by a charged and massive scalar field that breaks Lorentz invariance in a CPT-even, aether-like manner. I investigate the case of a scalar field that satisfies Dirichlet or mixed (Dirichlet–Neumann) boundary conditions on a pair of very large plane parallel plates. The case of Neumann boundary conditions is straightforward and will not be examined in detail. I use the [Formula: see text]-function regularization technique to study the effect of a constant magnetic field, orthogonal to the plates, on the Casimir energy and pressure. I investigate the cases of a timelike Lorentz asymmetry, a spacelike Lorentz asymmetry in the direction perpendicular to the plates, and a spacelike asymmetry in the plane of the plates and, in all those cases, derive simple analytic expressions for the zeta function, Casimir energy and pressure in the limits of small plate distance, strong magnetic field and large scalar field mass. I discover that the Casimir energy and pressure, and their magnetic corrections, all strongly depend on the direction of the unit vector that implements the breaking of the Lorentz symmetry.

scholarly journals Casimir effect with one large extra dimension

2021 ◽  
Author(s):  
Andrea Erdas
Keyword(s):  
Scalar Field ◽  
Boundary Conditions ◽  
Extra Dimension ◽  
Casimir Effect ◽  
Neumann Boundary ◽  
Casimir Energy ◽  
Neumann Boundary Conditions ◽  
Three Dimensions ◽  
Parallel Plates ◽  
Large Extra Dimension

In this work, I study the Casimir effect of a massive complex scalar field in the presence of one large compactified extra dimension. I investigate the case of a scalar field confined between two parallel plates in the macroscopic three dimensions, and examine the cases of Dirichlet and mixed (Dirichlet–Neumann) boundary conditions on the plates. The case of Neumann boundary conditions is uninteresting, since it yields the same result as the case of Dirichlet boundary conditions. The scalar field also permeates a fourth compactified dimension of a size that could be comparable to the distance between the plates. This investigation is carried out using the [Formula: see text]-function regularization technique that allows me to obtain exact expressions for the Casimir energy and pressure. I discover that when the compactified length of the extra dimension is similar to the plate distance, or slightly larger, the Casimir energy and pressure become significantly different than their standard three-dimensional values, for either Dirichlet or mixed boundary conditions. Therefore, the Casimir effect of a quantum field that permeates a compactified fourth dimension could be used as an effective tool to explore the existence of large compactified extra dimensions.

Distribution of quantum states in enclosures of finite size: II

1991 ◽  
Vol 69 (11) ◽  
pp. 1342-1361
Author(s):  
J. Hugo Souto ◽  
A. N. Chaba
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Finite Size ◽  
Neumann Boundary ◽  
Summation Formula ◽  
Quantum States ◽  
Dirichlet Boundary Conditions ◽  
Actual Number ◽  
Mathematical Functions ◽  
Poisson’S Summation Formula

By making use of the modified form of Poisson's summation formula, we calculate the expression for the number of eigenstates, N(K), with eigenvalues [Formula: see text] of a particle in spherical and cylindrical enclosures of finite size, and with its wave-function subject to Dirichlet boundary conditions and Neumann boundary conditions at the walls of the container. We also obtain the oscillatory terms in addition to the important nonoscillatory terms already known and compare our results with the actual number of such states computed from the tables of the zeros of the relevant special mathematical functions. The inclusion of these oscillatory terms improves the accuracy of the expressions in all cases, especially in the case of the cylinder, where these are quite significant. Some possible applications of the results obtained here are also indicated.

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