Unique Fock quantization of scalar cosmological perturbations

Abstract

We investigate the ambiguities in the Fock quantization of the scalar perturbations of a FriedmannLemaître-Robertson-Walker model with a massive scalar field as matter content. We consider the case of compact spatial sections (thus avoiding infrared divergences), with the topology of a threesphere. After expanding the perturbations in series of eigenfunctions of the Laplace-Beltrami operator, the Hamiltonian of the system is written up to quadratic order in them. We fix the gauge of the local degrees of freedom in two different ways, reaching in both cases the same qualitative results. A canonical transformation, which includes the scaling of the matter field perturbations by the scale factor of the geometry, is performed in order to arrive at a convenient formulation of the system. We then study the quantization of these perturbations in the classical background determined by the homogeneous variables. Based on previous work, we introduce a Fock representation for the perturbations in which: (a) the complex structure is invariant under the isometries of the spatial sections and (b) the field dynamics is implemented as a unitary operator. These two properties select not only a unique unitary equivalence class of representations, but also a preferred field description, picking up a canonical pair of field variables among all those that can be obtained by means of a time-dependent scaling of the matter field (completed into a linear canonical transformation). Finally, we present an equivalent quantization constructed in terms of gauge-invariant quantities. We prove that this quantization can be attained by a mode-by-mode time-dependent linear canonical transformation which admits a unitary implementation, so that it is also uniquely determined.PACS numbers: 98.80. Qc, 04.62.+v, 98.80.Cq

Uniqueness of the Fock quantization of scalar fields and processes with signature change in cosmology

“…Finally, let us emphasize that this particle production is due to the quantum evolution of the vacuum in the Euclidean regime, a phenomenon which is apparently independent of the amplification of modes [40] that has also been confirmed recently in the effective description of inhomogeneous models in hybrid LQC [41], systems in which the cosmological background always remains Lorentzian and a bounce in the expansion occurs.…”

Hybrid quantization of an inflationary universe

“…Fortunately, the recent uniqueness results for fields in non-stationary spacetimes [25] motivate us to pick up the family of Fock representations in which the vacuum state is invariant under the group of spatial isometries -namely, the SO(4) group-and where the classical dynamics of the field (obtained after deparametrization) is implemented by a unitary quantum operator. It has been possible to prove [26] that this family is indeed a unique unitary equivalence class, as in the situations treated in Ref. [25].…”

“…(3.23). In this description, the massless representation permits a unitary quantum implementation of the field dynamics [26]. To perform a standard Fock quantization, it is hence convenient to rewrite these modes in terms of the corresponding annihilation and creation-like variables, namely a fn and their complex conjugate a * fn , with…”

“…[31], as well as to apply the results of Ref. [26], we employ here the variables α, ϕ, and their corresponding momenta π α and π ϕ , to describe the homogeneous sector. In the absence of inhomogeneities (i.e., in the unperturbed system), the relation with the homogeneous variables used in Sec.…”

“…[25,26], regarding the quantization of fields in non-stationary scenarios after deparametrization. Specifically, any SO(4)-invariant Fock representation that implements the dynamics unitarily is only compatible with a particular choice of variables on the phase space of the field, as any genuinely time-dependent linear canonical transformation of those variables prevents the simultaneous fulfillment of both properties in another representation [33].…”