Uniqueness of the Fock quantization of a free scalar field onS1with time dependent mass

Abstract

We analyze the quantum description of a free scalar field on the circle in the presence of an explicitly time dependent potential, also interpretable as a time dependent mass. Classically, the field satisfies a linear wave equation of the formξ − ξ ′′ + f (t)ξ = 0. We prove that the representation of the canonical commutation relations corresponding to the particular case of a massless free field (f = 0) provides a unitary implementation of the dynamics for sufficiently general mass terms, f (t). Furthermore, this representation is uniquely specified, among the class of representations determined by S 1 -invariant complex structures, as the only one allowing a unitary dynamics. These conclusions can be extended in fact to fields on the two-sphere possessing axial symmetry. This generalizes a uniqueness result previously obtained in the context of the quantum field description of the Gowdy cosmologies, in the case of linear polarization and for any of the possible topologies of the spatial sections.

Quantum unitary dynamics in cosmological spacetimes

“…Moreover, in the present context of free fields in a (spatially) homogeneous and isotropic cosmological background, that requirement (together with invariance under spatial symmetries) is certainly a valid criterion to select the quantum representation. In this respect, let us recall that it was proven in [1][2][3][4]7,6,8] that: (i) Unitary implementability of the dynamics is impossible to achieve by means of any other rescaling different from φ → χ = aφ, including the formulation in terms of the original unscaled field φ (in this respect, see also [26]); and (ii) Once the formulation in terms of the field χ is chosen, there is a unique equivalence class of Fock representations such that a unitary implementation of the dynamics is possible (notice that, in proving these results, it is always assumed that the CS which defines the Fock quantization is invariant under spatial isometries; in terms of the mode decomposition, this restriction is already encoded in the fact that the creation-annihilation operators do not mix different modes [13]). …”

“…as β k = O(1/k 2(r+1) ) in the ultraviolet regime. 3 Thus, finiteness of the sum  ⃗ k |β k | 2 amounts to the summability of the sequence {g k /k 4(r+1) }, where g k is the degeneracy of each eigenspace of the Laplace operator on the 3-torus. Since, for large k, g k grows at most like k 2 (see [8] for details), it follows that {g k /k 4(r+1) } is indeed a summable sequence, ∀r ≥ 0.…”

Unique Fock quantization of scalar cosmological perturbations

“…Actually, this relation depends on the initial values of the complex function Θ n . One can suitably fix these, without loss of generality, so that Θ n (η 0 ) = 0 andΘ n (η 0 ) = −i [33]. Then, one gets…”

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

“…Precisely this combined criteria of spatial symmetry invariance and unitary dynamics have been used to determine a unique Fock quantization for certain scalar fields describing gravitational waves [16,17,[23][24][25][26], in the context of inhomogeneous cosmologies of the Gowdy type. The criteria have been proven to apply as well to scalar fields with a generic time dependent mass defined on d-spheres, with d = 1, 2, 3 [27,28], including the commented (dimensionally reduced) description of the Gowdy fields as particular cases. More recently, it has been possible to extend the result of the uniqueness of the Fock quantization of scalar fields satisfying a KG equation with time varying mass to fields defined on any compact spatial manifold in three or less dimensions [29].…”