Criteria for the determination of time dependent scalings in the Fock quantization of scalar fields with a time dependent mass in ultrastatic spacetimes

Abstract

We consider the quantization of scalar fields in spacetimes such that, by means of a suitable scaling of the field by a time dependent function, the field equation can be regarded as that of a field with a time dependent mass propagating in an auxiliary ultrastatic static background. For Klein-Gordon fields, it is well known that there exist an infinite number of nonequivalent Fock representations of the canonical commutation relations and, therefore, of inequivalent quantum theories. A context in which this kind of ambiguities arises and prevents the derivation of robust results is, e.g., in the quantum analysis of cosmological perturbations. In these situations, typically, a suitable scaling of the field by a time dependent function leads to a description in an auxiliary static background, though the nonstationarity still shows up in a time dependent mass. For such a field description, and assuming the compactness of the spatial sections, we recently proved in three or less spatial dimensions that the criteria of a natural implementation of the spatial symmetries and of a unitary time evolution are able to select a unique class of unitarily equivalent vacua, and hence of Fock representations. In this work, we succeed to extend our uniqueness result to the consideration of all possible field descriptions that can be reached by a time dependent canonical transformation which, in particular, involves a scaling of the field by a function of time. This kind of canonical transformations modify the dynamics of the system and introduce a further ambiguity in its quantum description, exceeding the choice of a Fock representation. Remarkably, for any compact spatial manifold in less than four dimensions, we show that our criteria eliminate any possible nontrivial scaling of the field other than that leading to the description in an auxiliary static background. Besides, we show that either no time dependent redefinition of the field momentum is allowed or, if this may happen -something which is typically the case only for one-dimensional spatial manifolds-, the redefinition does not introduce any Fock representation that cannot be obtained by a unitary transformation.

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

“…A detailed demonstration of these results can be found in Refs. [5,28]. The particular case of compact flat spatial topology, which deserves a careful study owing to some specific peculiarities and given its importance for physical applications (recall the apparent flatness of the observed universe), has been analyzed in Ref.…”

“…Let us also comment that, with our change in time reparametrization and scaling, there is no guarantee a priori that the transformed squared mass m 2 be positive. However, this poses no obstruction for the application of our uniqueness criterion [5,28]. On other hand, as we mentioned in the previous section, the new mass must have a second derivative that is integrable in every compact subinterval of the time domain.…”

“…Following the line of reasoning of Ref. [28], one can show that the criterion of spatial invariance and unitary time evolution also removes this ambiguity and determines a unique canonical pair for the field among all those related by the mentioned transformations. The main points in the proof of this result are based on the calculation of the Bogoliubov coefficients associated with the dynamics of the new annihilation and creationlike variables, introduced in terms of the original ones, and of the coefficients corresponding to the canonical transformation (K) between two invariant complex structures.…”

“…A general discussion can be found in Refs. [5,28]. It was proven there that the selection of the Fock representation is unique -up to unitary transformations-if one demands invariance of the vacuum under the group of spatial symmetries of the field equations and unitary implementability of the dynamics in the quantum theory.…”

Uniqueness of the Fock quantization of Dirac fields in 2+1 dimensions

“…This question has an affirmative answer for the case of a real Klein-Gordon (KG) field minimally coupled to geometries with homogeneous and isotropic compact spatial sections of dimension smaller than four [7,8], which in particular are relevant in cosmology [9]. In fact, in such scenarios, the absent time-translation invariance is replaced with a demand of unitary implementability of the dynamics in Fock space.…”