Vacuum polarization on the spinning circle

Abstract

Vacuum polarization of a massive scalar field in the background of a two-dimensional version of a spinning cosmic string is investigated. It is shown that when the "radius of the universe" is such that spacetime is globally hyperbolic the vacuum fluctuations are well behaved, diverging though on the "chronology horizon". Naive use of the formulas when spacetime is nonglobally hyperbolic leads to unphysical results. It is also pointed out that the set of normal modes used previously in the literature to address the problem gives rise to two-point functions which do not have a Hadamard form, and therefore are not physically acceptable. Such normal modes correspond to a locally (but not globally) Minkowski time, which appears to be at first sight a natural choice of time to implement quantization.PACS numbers: 04.70. Dy, 04.62.+v The study of quantum fields around cosmic strings is a pertinent issue since such defects may play a role in the cosmological scenario [1]. Most of the literature concerns spinless cosmic strings (see Ref.[1] and references therein), and only a few works have considered quantum mechanics and quantum field theory around spinning cosmic strings [2,3,4,5,6,7].The locally flat spacetime around an infinitely thin spinning cosmic string [1] is characterized by the line elementand by the identification (τ, ρ, ϕ, z) ∼ (τ, ρ, ϕ + 2π, z), where 0 < α ≤ 1 is the cone parameter and S ≥ 0 is the spin density (clearly the Minkowski spacetime corresponds to S = 0 and α = 1). As the region for which ρ < S/α contains closed timelike contours the spacetime is not globally hyperbolic. In other words a global time is not available, so that it is not clear whether quantum theory makes sense in this background [8].The study of a relativistic quantum scalar particle moving on the spinning cone [the corresponding threedimensional line element is obtained from Eq. (1) by setting dz = 0] has shown that a nonvanishing S spoils unitarity [2]. It has been speculated that this sort of first quantized pathology could be eliminated in the second quantized approach. However, Ref. [7] seems to frustrate this possibility by showing that the vacuum fluctuations of a massless scalar field diverge on concentric cylindrical shells around the spinning cosmic string. These pathological results have been attributed to the nonglobally hyperbolic nature of the background.In order to exhibit clearly aspects of global hyperbolicity (and related issues) in actual calculations, this work will consider a toy model which consists of a quantum scalar field existing in a two dimensional spacetime whose line element is obtained by truncating Eq. (1) as [5](α was dropped since it can be removed by redefining the parameter ρ) and observingIt is clear that S = 0 corresponds to a cylindrical spacetime of periodicity length 2πρ. The main pedagogical feature of this toy model is that, for a given "radius of the universe" ρ > 0, one can tune the spin S such that the background is globally hyperbolic (ρ > S) or otherwise (ρ ≤ S). Assuming ρ > S a...

Renormalized scalar propagator around a dispiration

“…This result should not be extended to all vacuum fluctuations without caution, since some of them are obtained from the renormalized propagator by applying prescriptions which may eliminate the dominant contribution in Eq. (12) [see e.g. Eqs.…”

“…(17) kills off the dominant contribution in Eq. (12), with the result that the subleading contribution yields two nonvanishing off-diagonal components,…”

“…Such a helical time structure poses serious difficulties when implementing quantization, since the corresponding spacetime is not globally hyperbolic [5]. In fact, by insisting on implementing quantization with the usual procedures, one ends up with observables presenting pathological behavior [15,16,29]. [ φ 2 (r) around a spinning cosmic string can be obtained from Eq.…”

“…[12], and investigations on quantum theory in related backgrounds have been carried out in Refs. [13,14,15,16]. In Ref.…”

Vacuum stress around a topological defect

“…We have recently obtained D (α,κ) (x, x ′ ) (classical propagators have been considered in Ref [8]) by using the Schwinger proper time prescription combined with the completeness relation of the eigenfunctions of the d'Alembertian operator [9]. Such eigenfunctions have the form R(r)χ(ϕ) exp{i(νZ − ωt)} which, by observing Eq.…”