Zeta Function Regularization Method Research Articles

Ground state energy of massive scalar field inside a spherical region in the global monopole background

Using the zeta function regularization method we calculate the ground state energy of scalar massive field inside a spherical region in the space–time of a pointlike global monopole. Two cases are investigated: (i) First, we calculate the Casimir energy inside a sphere of radius R and analyze the obtained result. We observe that this energy may be positive or negative depending on metric coefficient α and nonconformal coupling ξ. In the limit R→∞, this energy vanishes: (ii) In the second model, we surround the monopole by an additional sphere of radius r0<R and consider the scalar field confined in the region between these two spheres. In this case, the ground state energy presents an additional contribution due to boundary at r0 which is divergent for small radius. Additional comments about renormalization are considered.

Finite-temperature Casimir effect on the radius stabilization of noncommutative torus

The one-loop correction to the spectrum of Kaluza-Klein system for the $\phi^3$ model on $R^{1,d}\times (T_\theta^2)^L$ is evaluated in the high temperature limit, where the $1+d$ dimensions are the ordinary flat Minkowski spacetimes and the $L$ extra two-dimensional tori are chosen to be the noncommutative torus with noncommutativity $\theta$. The corrections to the Kaluza-Klein mass formula are evaluated and used to compute the Casimir energy with the help of the Schwinger perturbative formula in the zeta-function regularization method. The results show that the one-loop Casimir energy is independent of the radius of torus if L=1. However, when $L>1$ the Casimir energy could give repulsive force to stabilize the extra noncommutative torus if $d-L$ is a non-negative even integral. This therefore suggests a possible stabilization mechanism of extra radius in high temperature, when the extra spaces are noncommutative.

Expansion of effective action in curved spacetime with large anisotropy

A simple perturbative method is used to evaluate the renormalized effective action of a free scalar field in the Bianchi type I spacetime with large anisotropy. With the help of the zeta function regularization method the action is expanded to the second order in the Schwinger perturbative formula. We apply the method of iterative reduction to reduce the generalized Einstein equation to a second-order equation to obtain the self-consistent solution of the semiclassical gravity theory. The reduced equation shows that the space anisotropy, which will be smoothed out during the evolution of Universe, may play an important role in the dynamics of early Universe. We thus quantize the corresponding minisuperspace model to investigate the behavior of space anisotropy in the early Universe. From the wavefunction of the Wheeler–De Witt equation we see that the state probability of the Bianchi type I spacetime with large anisotropy is less than that with a small anisotropy.

GROUND STATE ENERGY OF THE MODIFIED NAMBU–GOTO STRING

We calculate, using zeta function regularization method, semiclassical energy of the Nambu–Goto string supplemented with the boundary, Gauss–Bonnet term in the action and discuss the tachyonic ground state problem.

Relative Zeta Functions, Relative Determinants and Scattering Theory

We use the method of zeta function regularization to regularize the ratio det A det A0$ of the determinants of two elliptic self-adjoint operators A, A0 satisfying certain natural assumptions. This is of interest, especially, if the regularized determinants of the individual operators don't exist as, for example, in the case of elliptic operators on a noncompact manifold.

Partition functions for the rigid string and membrane at any temperature.

Exact expressions for the partition functions of the rigid string and membrane at any temperature are obtained in terms of hypergeometric functions. By using zeta function regularization methods, the results are analytically continued and written as asymptotic sums of Riemann-Hurwitz zeta functions, which provide very good numerical approximations with just a few first terms. This allows to obtain systematic corrections to the results of Polchinski et al., corresponding to the limits $T\rightarrow 0$ and $T\rightarrow \infty$ of the rigid string, and to analyze the intermediate range of temperatures. In particular, a way to obtain the Hagedorn temperature for the rigid membrane is thus found.

On zeta function regularization for operators with continuous spectra

We present the generalization of the method of zeta function regularization for operators with continuous spectra where one cannot interchange the order of integration and limit processes. A sample calculation is presented.

Axial Anomaly for Gravitino and Unconventional FP Ghosts

The axial anomaly for the gravitino in a background gravitational field is obtained by using the new formalism of Hata and Kugo for supergravity with unconventional Faddeev-Popov (FP) ghosts. Despite the new FP ghosts have essentially different features from the ordinary ones it is shown that the anomaly leads to the same value as the ordinary one which is -21 times that of a spin 1/2 Majorana Fermion. This means that in the new formalism the part which contributes to the anomaly is the Lagrangian multiplier for gauge fixing and not FP ghosts in contrast to the case of the ordinary formalism. The relation between the canonical formalism of Hata-Kugo and the functional integral formalism is clarified. It is also shown that the axial anomaly is obtained within the framework of the path integral formalism by adopting the zeta-function regularization method.

Approaches to the gravitational [formula omitted] axial anomaly

The spin- 3 2 axial anomaly is discussed from various points of view. Two consistent theories for a spin- 3 2 field interacting with gravity are considered: supergravity and a real quantized spin- 3 2 field in a classical gravitational background. The Feynman graph method, the zeta function regularization method, the point splitting method and the topological method all yield the same result for the latter theory, in agreement with that first found by Christensen and Duff.