Andrei A. Bytsenko

Quantum fields and extended objects in space-times with constant curvature spatial section

Abstract The heat-kernel expansion and ζ-regularization techniques for quantum field theory and extended objects on curved space-times are reviewed. In particular, ultrastatic space-times with spatial section consisting in manifold with constant curvature are discussed in detail. Several mathematical results, relevant to physical applications are presented, including exact solutions of the heat-kernel equation, a simple exposition of hyperbolic geometry and an elementary derivation of the Selberg trace formula. With regard to the physical applications, the vacuum energy for scalar fields, the one-loop renormalization of a self-interacting scalar field theory on a hyperbolic space-time, with a discussion on the topological symmetry breaking, the finite-temperature effects and the Bose-Einstein condensation, are considered. Some attempts to generalize the results to extended objects are also presented, including some remarks on path-integral quantization, asymptotic properties of extended objects and a novel representation for the one-loop (super) string free energy.

Ray-Singer torsion for a hyperbolic 3-manifold and asymptotics of Chern-Simons-Witten invariant

Abstract The Ray-Singer torsion for a compact smooth hyperbolic 3-dimensional manifold H 3 is expressed in terms of Selberg zeta functions, making use of the associated Selberg trace formulae. An application to the evaluation of the semiclassical asymptotics of Wittens invariant for the Chern-Simons theory with gauge group SU (2) as well as to the sum over topologies in 3-dimensional quantum gravity are presented.

Finite-temperature effects for massive fields in D-dimensional Rindler-like spaces

The first quantum corrections to the free energy for massive fields in D-dimensional space-times of the form R × R+ × MN−1, where D = N + 1 and MN−1 is a constant curvature manifold, is investigated by means of the ξ-function regularization. It is suggested that the nature of the divergences, which are present in the thermodynamical quantities, might be better understood making use of the conformal related optical metric and associated techniques. The general form of the horizon divergences of the free energy is obtained as a function of the free energy densities of fields having negative square masses (absence of the gap in the Laplace operator spectrum) on ultrastatic manifolds with hyperbolic spatial section HN−2n and of the Seeley-DeWitt coefficients of the Laplace operator on the manifold MN−1. Furthermore, recurrence relations are found relating higher and lower dimensions. The cases of Rindler space, where MN−1 = RN-1 and very massive D-dimensional black holes, where MN−1 = SN−1 are treated as examples. The renormalization of the internal energy is also discussed.

Determinant of the Laplacian on a non-compact three-dimensional hyperbolic manifold with finite volume

The functional determinant of Laplace-type operators on a three-dimensional non-compact hyperbolic manifold with invariant fundamental domain of finite volume is expressed via the Selberg zeta function related to the Picard group .

Quantum fields in hyperbolic spacetimes with finite spatial volume

The 1-loop effective action for a massive self-interacting scalar field is investigated in four-dimensional ultrastatic spacetime , where is a non-compact hyperbolic manifold with finite volume. Making use of the Selberg trace formula, the -function related to the small disturbance operator is constructed. For an arbitrary gravitational coupling, it is found that has a simple pole at s = 0. The 1-loop effective action is analysed by means of proper-time regularizations and the 1-loop divergences are found explicitly. It is pointed out that, in this special case, -function regularization also requires a divergent counterterm, which however is not necessary in the free massless conformal invariant coupling case. Finite-temperature effects are studied and a high-temperature expansion is presented. A possible application to the problem of the divergences of the entanglement entropy for a free massless scalar field in a Rindler-like spacetime is briefly discussed.

ARE p-BRANES ASYMPTOTICALLY BLACK HOLES?

An attempt is made to compare the asymptotic state density of twisted p-branes and the related state density of mass level M of a D-dimensional neutral black holes. To this aim, the explicit form of the twisted p-brane total level degeneracy is calculated. The prefactor of the degeneracy, in contrast to the leading behavior, is found to depend on the winding number of the p-brane.

SELF-INTERACTING SCALAR FIELDS ON SPACE-TIME WITH COMPACT HYPERBOLIC SPATIAL PART

We calculate the one-loop effective potential of a self-interacting scalar field on the space-time of the form ℝ2×H2/Γ. The Selberg trace formula associated with a co-compact discrete group Γ in PSL(2, ℝ) (hyperbolic and elliptic elements only) is used. The closed form for the one-loop unrenormalized and renormalized effective potentials is given. The influence of non-trivial topology on curvature induced phase transitions is also discussed.

ZETA-FUNCTION REGULARIZATION APPROACH TO FINITE TEMPERATURE EFFECTS IN KALUZA-KLEIN SPACE-TIMES

In the framework of heat-kernel approach to zeta-function regularization, the one-loop effective potential at finite temperature for scalar and spinor fields on Kaluza-Klein space-time of the form , where MP is p-dimensional Minkowski space-time is evaluated. In particular, when the compact manifold is , the Selberg trace formula associated with discrete torsion-free group Γ of the n-dimensional Lobachevsky space Hn is used. An explicit representation for the thermodynamic potential valid for arbitrary temperature is found. As a result a complete high temperature expansion is presented and the roles of zero modes and topological contributions is discussed.

MASSLESS SCALAR CASIMIR EFFECT IN A CLASS OF HYPERBOLIC KALUZA-KLEIN SPACE-TIMES

In the framework of heat-kernel approach to zeta-function regularization we calculate the one-loop effective potential (Casimir effect) massless scalar field on Kaluza-Klein space-time of the form RD−n×Hn/Γ(2≤n<D). In addition the Selberg trace formula associated with discrete torsion-free group Γ of the n-dimensional Lobachevsky space Hn is used. A negative Casimir effect related to trivial line bundle with character χ=1 is found. A comparison of the results obtained and Casimir effect for massless field on torus backgrounds is also presented.

SEMICLASSICAL APPROXIMATION FOR A CLASS OF QUANTUM p-BRANE MODELS

A class of quantum p-brane models is considered. An action that is linear in the determinant of the world-metric and equivalent to Nambu-Goto-Dirac p-brane action is proposed. A semiclassical approximation to path integral quantization is presented for the bosonic sector when the closed p-brane sweeps out an n=(p+1) dimensional compact hyperbolic manifold Hn/Γ, Γ being a strictly hyperbolic subgroup of isometries of the Lobachevsky space Hn. The computation of the related Laplace operator determinant is presented.