Multi-loop zeta function regularization and spectral cutoff in curved spacetime

We emphasize the close relationship between zeta function methods and arbitrary spectral cutoff regularizations in curved spacetime. This yields, on the one hand, a physically sound and mathematically rigorous justification of the standard zeta function regularization at one loop and, on the other hand, a natural generalization of this method to higher loops. In particular, to any Feynman diagram is associated a generalized meromorphic zeta function. For the one-loop vacuum diagram, it is directly related to the usual spectral zeta function. To any loop order, the renormalized amplitudes can be read off from the pole structure of the generalized zeta functions. We focus on scalar field theories and illustrate the general formalism by explicit calculations at one-loop and two-loop orders, including a two-loop evaluation of the conformal anomaly.     

Smoothing regularization and particle creation in curved spacetime

A regularization procedure is given for the stress tensor of a quantized field in a background metric. This regularization is shown to be equivalent to a covariant renormalization of constants in the generalized Einstein equations. An example of the massive spinor field in Robertson-Walker universe is considered. Regular values of the stress tensor near the cosmological singularity are found.     

自由粒子配分函数的zeta函数正规化处理

zeta函数正规化技术在现代物理中有着重要应用.本文探讨其在大学物理中的应用,即采用zeta函数正规化处理自由粒子的配分函数,并导出相应的热力学量.考虑粒子被囚禁在体积有限的立方容器中,给出了热力学量在有限体积下的一级修正.     

Zeta function regularization, anomaly and complex mass term

If zeta function regularization is used and a complex mass term considered for fermions, the phase does not appear in the fermion determinant. This is not a drawback of the regularization, which can recognize the phase through source terms, as demonstrated by the anomaly equation which is explicitly derived here for a complex mass term.     

Bosonization in curved spacetime

The bosonization of a pair of two-dimensional Weyl fermions, of opposite chiralities, is considered. When the fermions are in different metrics the bosonization procedure gives a unique expression for the gravitational anomaly.     

Some remarks on zeta function regularization in curved space-times

The question of to what extent zeta function regularization respects the invariances of a quantum field theory in a background gravitational field is investigated. It is shown that zeta function regularization provides a generalization to curved space-time of analytic propagator regularization which is known not to respect gauge invariance. Furthermore, a study of the regularized stress tensor of a conformally invariant scalar field indicates that both conformai and general coordinate invariance are violated.     

Particle production in curved spacetime

Particle production in curved spacetime has been discussed through the method of complex time WKB approximation. We consider Dirac equation in non-flat spacetime to understand particle production as particle-antiparticle rotation. The method is also generalized to understand particle production through parametric resonance. To understand the method of CWKB we consider particle production in Kasner spacetime as an example.     

An accurate power series expansion for path integrals on curved manifolds

A general and simple framework for treating path integrals on curved manifolds is presented. The crucial point is expanding the exponent of the propagator of general diffusion processes in a power series in time. The expansion coefficients are determined by recursive relations and can be analytically evaluated to any desired level of accuracy int. The treatment is both theoretically and numerically advantageous with respect to the other path integral methods known in the literature. Its power is illustrated on two exactly solvable models. The propagator obtained is shown to be much more accurate over a broad range oft than the standard short time approximation. In view of its numerical application this means significant reducing the number of time steps that are required to evaluate a path integral.     

Quantum field theory in curved spacetime

Quantum field theory predicts a number of unusual physical effects in non-Minkowskian manifolds (flat or curved) that have no immediate analogs in Minkowski spacetime. The following examples are reviewed: (1) The Casimir effect; (2) Radiation from accelerating conductors; (3) Particle production in manifolds with horizons, including both stationary black holes and black holes formed by collapse. In the latter examples curvature couples directly to matter through the stress tensor and induces the creation of real particles. However, it also induces serious divergences in the vacuum stress. These divergences are analyzed, and methods for handling them are reviewed.     

On quantum radiation in curved spacetime

In the context of quantum field theories in curved spacetime, we compute the effective action of the transition amplitude from vacuum to vacuum in the presence of an external gravitational field. The imaginary part of the resulted effective action determines the probability of vacuum decay via a quantum tunneling process, giving the rate and spectrum of particle creations. We show that (i) the gravitational field polarizes the vacuum and discretizes its spectrum; (ii) vacuum gains gravitational energy by such a polarization. On the basis of gravitational vacuum polarization, we discuss the quantum origin of vacuum decay in curved spacetime as pair-creations of particles and anti-particles. The thermal spectrum of particle creations is attributed to (i) the CPT invariance of pair-creations (annihilations) from (into) vacuum and (ii) vacuum acts as a reserve with the temperature determined by gravitational energy-gain.     

Dimensional regularization of massless quantum electrodynamics in spherical spacetime

The quantization and renormalization of massless electrodynamics in a spacetime of constant curvature are discused. A formalism is presented which is valid in an arbitrary number of dimensions and therefore allows the use of dimensional regularization. In the discussion of the photon propagator it is found that anomalous mass terms dependent on the curvature arise, although these vanish in four dimensions. Further, the gauge-fixing term in the Lagrangian has the unconventional feature of not being a perfect square. The renormalizability of the theory is then demonstrated to one loop order, and the renormalization constants are shown to retain their flat spacetime values. Finally, expansions for the renormalized electron and photon propagators in terms of appropriate spherical harmonics are derived.