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Quantum Fields and Extended Objects in Space-Times with Constant Curvature Spatial Section

A. A. Bytsenko, G. Cognola, L. Vanzo, S. Zerbini

TL;DR

The work analyzes quantum fields and extended objects in space-times with spatial sections of constant curvature, emphasizing ultrastatic manifolds. It develops and applies ζ-function regularization and heat-kernel techniques to compute one-loop determinants, vacuum energies, and finite-temperature potentials, including exact Selberg-trace-formula-based results on compact hyperbolic spaces. The authors extend these methods to strings and p-branes, deriving density-of-states asymptotics and examining Casimir energies and symmetry-breaking effects induced by topology and curvature. Finite-temperature analyses reveal rich thermodynamic structures, including Bose–Einstein condensation in hyperbolic spaces and curvature/topology-dependent phase behavior. Collectively, the paper provides a comprehensive framework for exact and semiclassical quantum analyses on curved backgrounds, with explicit results for toroidal, spherical, and hyperbolic geometries and for extended objects in such spaces.

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 regards 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.

Quantum Fields and Extended Objects in Space-Times with Constant Curvature Spatial Section

TL;DR

The work analyzes quantum fields and extended objects in space-times with spatial sections of constant curvature, emphasizing ultrastatic manifolds. It develops and applies ζ-function regularization and heat-kernel techniques to compute one-loop determinants, vacuum energies, and finite-temperature potentials, including exact Selberg-trace-formula-based results on compact hyperbolic spaces. The authors extend these methods to strings and p-branes, deriving density-of-states asymptotics and examining Casimir energies and symmetry-breaking effects induced by topology and curvature. Finite-temperature analyses reveal rich thermodynamic structures, including Bose–Einstein condensation in hyperbolic spaces and curvature/topology-dependent phase behavior. Collectively, the paper provides a comprehensive framework for exact and semiclassical quantum analyses on curved backgrounds, with explicit results for toroidal, spherical, and hyperbolic geometries and for extended objects in such spaces.

Abstract

The heat-kernel expansion and -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 regards 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.

Paper Structure

This paper contains 122 sections, 6 theorems, 533 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Notations.
  3. Path integral and regularization techniques in curved space-times
  4. The path integral
  5. The $\zeta$-function regularization
  6. Complex powers and heat kernel of elliptic operators
  7. The $\zeta$-function
  8. Zero-modes.
  9. The factorization property.
  10. Example: $S^1\times\cal M^N$.
  11. Example: $\hbox{$I\!\! R$}^p\times\cal M^N$.
  12. Other regularization techniques
  13. Regularizations in $\hbox{$I\!\! R$}^D$.
  14. The one-loop effective action and the renormalization group equations
  15. Renormalization group equations.
  16. ...and 107 more sections

Key Result

Theorem 1

Let $\kappa$ be non zero and $a>0$ defined by $\kappa=ea^{-2}$, $e=\pm1$ and where $g_{ij}$ has signature $(n,N-n)$ and $x^i=(x^0,...,x^{N-1})$ are rectilinear coordinates on $\hbox{$I\!\! R$}^N$. Then every $\Sigma_n^N$, endowed with the induced metric, is a complete pseudo-Riemannian manifold of constant curvature $\kappa$ and signature $(n,N-n-1)$ if $e=1$ or $(n-1,N-n)$

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3: Thurston
  • Lemma 1
  • Theorem 4: Meinardus
  • Theorem 5: Mc Kean