Zeta Function Regularization Method Research Articles

Casimir energy for spinor fields with δ-shell potentials

This work analyzes the Casimir energy of a massive spinor field propagating in flat space endowed with a spherically symmetric δ-function potential. By utilizing the spectral zeta function regularization method, the Casimir energy is evaluated after performing a suitable analytic continuation. Explicit numerical results are provided for specific cases in which the Casimir energy is unambiguously defined. The results described in this work represent a generalization of the MIT bag model for spinor fields.

Resolution of challenging problems in quantum cosmology with electromagnetic radiation

We investigate the quantum cosmology of a closed spatially homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker (FLRW) minisuperspace model with electromagnetic radiation as matter content. We solve the corresponding Wheeler–DeWitt equation by utilizing Riemann's zeta function regularization method. We demonstrate that the regularized vacuum energy of the electromagnetic field can overcome factor ordering, boundary conditions, and singularity problems.

Generalized quantum spring

Recently, it was found that after imposing a helix boundary condition on a scalar field, the Casimir force coming from the quantum effect is linearly proportional to r, which is the ratio of the pitch to the circumference of the helix. This linear behavior of the Casimir force is just like that of the force obeying Hooke’s law on a spring. In this paper, inspired by some complex structures that live in the cells of human body like DNA, protein, collagen, etc., we generalize the helix boundary condition to a more general one, in which the helix consists of a tiny helix structure, and makes up a hierarchy of helix. After imposing this kind of boundary condition on a massless and a massive scalar, we calculate the Casimir energy and force by using the so-called zeta function regularization method. We find that Hooke’s law with the generalized helix boundary condition is not exactly the same as the usual one. In this case, the force is proportional to the cube of r instead. So, we regard it as a generalized Hooke’s law, which is complied by a generalized quantum spring.

The Casimir effect for thick pistons

In this work, we analyze the Casimir energy and force for a thick piston configuration. This study is performed by utilizing the spectral zeta function regularization method. The results we obtain for the Casimir energy and force depend explicitly on the parameters that describe the general self-adjoint boundary conditions imposed. Numerical results for the Casimir force are provided for specific types of boundary conditions and are also compared to the corresponding force on an infinitely thin piston.

Casimir effect in the presence of external fields

In this work the Casimir effect is studied for scalar fields in the presence of boundaries and under the influence of arbitrary smooth potentials of compact support. In this setting, piston configurations are analyzed in which the piston is modeled by a potential. For these configurations, analytic results for the Casimir energy and force are obtained by employing the zeta function regularization method. Also, explicit numerical results for the Casimir force are provided for pistons modeled by a class of compactly supported potentials that are realizable as delta-sequences. These results are then generalized to higher dimensional pistons by considering additional Kaluza–Klein dimensions.

Casimir energy for a spherical boundary within the surface impedance approach: Obtaining negative values

We compute the Casimir Energy of a spherical region using a Surface Impedance approach. We characterize the Surface Impedance of the boundary using plasma model. Exact analytical formulae are obtained by means of the zeta function regularization method and all the divergencies are explicitly computed. We find that it is possible to have negative Casimir energy for some range of values of the relevant parameter. Limits of applicability of the model are discussed.

ZETA FUNCTION REGULARIZATION IN CASIMIR EFFECT CALCULATIONS AND J. S. DOWKER's CONTRIBUTION

A summary of relevant contributions, ordered in time, to the subject of operator zeta functions and their application to physical issues is provided. The description ends with the seminal contributions of Stephen Hawking and Stuart Dowker and collaborators, considered by many authors as the actual starting point of the introduction of zeta function regularization methods in theoretical physics, in particular, for quantum vacuum fluctuation and Casimir effect calculations. After recalling a number of the strengths of this powerful and elegant method, some of its limitations are discussed. Finally, recent results of the so-called operator regularization procedure are presented.

ZETA FUNCTION REGULARIZATION IN CASIMIR EFFECT CALCULATIONS AND J. S. DOWKER'S CONTRIBUTION

A summary of relevant contributions, ordered in time, to the subject of operator zeta functions and their application to physical issues is provided. The description ends with the seminal contributions of Stephen Hawking and Stuart Dowker and collaborators, considered by many authors as the actual starting point of the introduction of zeta function regularization methods in theoretical physics, in particular, for quantum vacuum fluctuation and Casimir effect calculations. After recalling a number of the strengths of this powerful and elegant method, some of its limitations are discussed. Finally, recent results of the so called operator regularization procedure are presented.

One-loop kink mass shifts: A computational approach

In this paper we develop a procedure to compute the one-loop quantum correction to the kink masses in generic ( 1 + 1 ) -dimensional one-component scalar field theoretical models. The procedure uses the generalized zeta function regularization method helped by the Gilkey–de Witt asymptotic expansion of the heat function via Mellinʼs transform. We find a formula for the one-loop kink mass shift that depends only on the part of the energy density with no field derivatives, evaluated by means of a symbolic software algorithm that automates the computation. The improved algorithm with respect to earlier work in this subject has been tested in the sine-Gordon and λ ( ϕ ) 2 4 models. The quantum corrections of the sG-soliton and λ ( ϕ 4 ) 2 -kink masses have been estimated with a relative error of 0.00006% and 0.00007% respectively. Thereafter, the algorithm is applied to other models. In particular, an interesting one-parametric family of double sine-Gordon models interpolating between the ordinary sine-Gordon and a re-scaled sine-Gordon model is addressed. Another one-parametric family, in this case of ϕ 6 models, is analyzed. The main virtue of our procedure is its versatility: it can be applied to practically any type of relativistic scalar field models supporting kinks.

Conical Casimir pistons with hybrid boundary conditions

In this paper, we compute the Casimir energy and force for massless scalar fields endowed with hybrid boundary conditions, in the setting of the bounded generalized cone. By using spectral zeta function regularization methods, we obtain explicit expressions for the Casimir energy and force in arbitrary dimensions in terms of the zeta function defined on the piston. Our general formulas are, subsequently, specialized to the case in which the piston is modeled by a d-dimensional sphere. In this particular situation, explicit results are given for d = 2, 3, 4, 5.

ONE-LOOP QUANTUM COSMOLOGICAL CORRECTION TO THE GRAVITATIONAL CONSTANT IN THE CLOSED FRIEDMANN–ROBERTSON–WALKER UNIVERSE

In this paper, we calculate the one-loop quantum cosmological corrections to the kink energy in the closed Friedmann–Robertson–Walker universe in which the fluctuation potential V″ has a shape invariance property. We use the generalized zeta-function regularization method to implement our setup for describing quantum kink-like states. It is conjectured that the corrections lead to the renormalized gravitational constant.

REPORT ON THE DETAILED CALCULATION OF THE EFFECTIVE POTENTIAL IN SPACETIMES WITH S1 × Rd TOPOLOGY AND AT FINITE TEMPERATURE

In this paper, we review the calculations that are needed to obtain the bosonic and fermionic effective potential at finite temperature and volume (at one loop). The calculations at finite volume correspond to S1 × Rd topology. These calculations appear in the calculation of the Casimir energy and of the effective potential of extra dimensional theories. In the case of finite volume corrections, we impose twisted boundary conditions and obtain semi-analytic results. We mainly focus in the details and validity of the results. The zeta function regularization method is used to regularize the infinite summations. Also the dimensional regularization method is used in order to renormalize the UV singularities of the integrations over momentum space. The approximations and expansions are carried out within the perturbative limits. After the end of each section, we briefly present applications associated to the calculations. Particularly the calculation of the effective potential at finite temperature for the standard model fields, the effective potential for warped and large extra dimensions, and the topological mass creation. In the end, we discuss on the convergence and validity of one of the obtained semi-analytic results.

The cosmological constant as an eigenvalue of f(R)-gravity Hamiltonian constraint

In the framework of ADM formalism, it is possible to find out eigenvalues of the WDW equation with the meaning of vacuum states, i.e. cosmological constants, for f(R) theories of gravity, where f(R) is a generic analytic function of the Ricci curvature scalar R. The explicit calculation is performed for a Schwarzschild metric where one-loop energy is derived by the zeta function regularization method and a renormalized running Λ0 constant is obtained.

Riemann–Liouville and Weyl fractional oscillator processes

Two types of oscillator processes can be obtained as solutions to fractional Langevin equation based on Riemann–Liouville and Weyl fractional integro-differential operators. The relation between these fractional oscillator processes and the corresponding fractional Brownian motion is considered. Generalization of the Weyl fractional oscillator process to positive temperature can be carried out and its partition function can be calculated using the zeta function regularization method.

Quantum oscillations of self-dual Abrikosov-Nielsen-Olesen vortices

The mass shift induced by one-loop quantum fluctuations on self-dual ANO vortices is computed using heat kernel/generalized zeta function regularization methods. The quantum masses of super-imposed multi-vortices with vorticity lower than five are given. The case of two separate vortices with a quantum of magnetic flux is also discussed.

Casimir effect for gauge fields in spaces with negative constant curvature

We consider gauge theories based on abelian $p-$forms on real compact hyperbolic spaces. Using the zeta-function regularization method and the trace tensor kernel formula, we determine explicitly an expression for the vacuum energy (Casimir energy) corresponding to skew-symmetric tensor fields. It is shown that the topological component of the Casimir energy for co-exact forms on even-dimensional spaces, associated with the trivial character, is always negative. We infer on the possible cosmological consequences of this result.

Quantum corrections to the mass of self-dual vortices

The mass shift induced by one-loop quantum fluctuations on self-dual ANO vortices is computed using heat kernel/generalized zeta function regularization methods.

Normalization of off-shell boundary state,g-function and zeta function regularization

We consider the model in two dimensions with boundary quadratic deformation (BQD), which has been discussed in tachyon condensation. The partition function of this model (BQD) on a cylinder is determined, using the method of zeta function regularization. We show that, for closed channel partition function, a subtraction procedure must be introduced in order to reproduce the correct results at conformal points. The boundary entropy (g-function) is determined from the partition function and the off-shell boundary state. We propose and consider a supersymmetric generalization of BQD model, which includes a boundary fermion mass term, and check the validity of the subtraction procedure.

Semi-classical mass of quantum k-component topological kinks

We use the generalized zeta function regularization method to compute the one-loop quantum correction to the masses of the TK1 and TK2 kinks in a deformation of the O( N) linear sigma model on the real line.

Generalized zeta functions and one-loop corrections to quantum kink masses

A method for describing the quantum kink states in the semi-classical limit of several (1+1)-dimensional field theoretical models is developed. We use the generalized zeta function regularization method to compute the one-loop quantum correction to the masses of the kink in the sine-Gordon and cubic sinh-Gordon models and another two P( φ) 2 systems with polynomial self-interactions.