Fock quantization of a scalar field with time dependent mass on the three-sphere: Unitarity and uniqueness

Abstract

We study the Fock description of a quantum free field on the three-sphere with a mass that depends explicitly on time, also interpretable as an explicitly time dependent quadratic potential. We show that, under quite mild restrictions on the time dependence of the mass, the specific Fock representation of the canonical commutation relations which is naturally associated with a massless free field provides a unitary dynamics even when the time varying mass is present. Moreover, we demonstrate that this Fock representation is the only acceptable one, up to unitary equivalence, if the vacuum has to be SO(4)-invariant (i.e., invariant under the symmetries of the field equation) and the dynamics is required to be unitary. In particular, the analysis and uniqueness of the quantization can be applied to the treatment of cosmological perturbations around Friedmann-Robertson-Walker spacetimes with the spatial topology of the three-sphere, like e.g. for gravitational waves (tensor perturbations).In addition, we analyze the extension of our results to free fields with a time dependent mass defined on other compact spatial manifolds. We prove the uniqueness of the Fock representation in the case of a two-sphere as well, and discuss the case of a three-torus.

Unique Fock quantization of scalar cosmological perturbations

“…In comparison with the situation discussed in Ref. [20], the asymptotic corrections to the equations of motion provided by the functions p n , q n , r n , or the O(n −2 ) contributions to s n , are subdominant enough so as to preserve the unitary implementability of the dynamics.…”

“…[20], leads one to conclude that its solutions are at most of the order of n −1 . Neglecting explicitly subdominant terms in the right-hand side of Eq.…”

“…transform with the same irreducible representation of SO(4) [20], so creation variables do not mix with annihilation variables under the kind of transformations that we are considering.…”

“…In order to make this manuscript self contained, we now summarize the master lines of the proof discussed in Ref. [20], and explain the modifications needed for our specific model.…”

“…[20] for the Fock quantization of a Klein-Gordon field with a time-dependent mass on S 3 . Despite the differences between the two situations, the analysis can be adapted without major difficulties.…”