Abstract
In this work, we present an overview of uniqueness results derived in recent years for the quantization of Gowdy cosmological models and for (test) Klein-Gordon fields minimally coupled to Friedmann-Lemaître-Robertson-Walker, de Sitter, and Bianchi I spacetimes. These results are attained by imposing the criteria of symmetry invariance and of unitary implementability of the dynamics. This powerful combination of criteria allows not only to address the ambiguity in the representation of the canonical commutation relations, but also to single out a preferred set of fundamental variables. For the sake of clarity and completeness in the presentation (essentially as a background and complementary material), we first review the classical and quantum theories of a scalar field in globally hyperbolic spacetimes. Special emphasis is made on complex structures and the unitary implementability of symplectic transformations.
Highlights
As it is well known, the quantization of systems with field-like degrees of freedom involves choices that generically lead to inequivalent theories within the standard Hilbert space approach [1].In contrast with the situation found for mechanical systems with a finite number of degrees of freedom, where the Stone-von Neumann theorem guarantees the unitary equivalence between strongly continuous, irreducible, and unitary representations of the Weyl relations [2], in quantum field theory no general uniqueness theorem exists and “physical results” depend on the representation adopted, a fact that brings into question their significance
If the field theory corresponds to a Klein-Gordon (KG) field, Poincaré invariance, adapted to the dynamics of the considered theory, selects a complex structure, which is the mathematical object encoding the ambiguity in the representation of the canonical commutation relations (CCRs), and determines the vacuum state of the Fock representation
In order to achieve a quantum description that satisfies the requirements of invariance under the spatial symmetries and of a unitary dynamics, and inspired by the case of free scalar fields propagating in FLRW spacetimes, we introduce a time dependent transformation of the field canonical pairs, regarded as variables that change with the considered section of constant time t
Summary
As it is well known, the quantization of systems with field-like degrees of freedom involves choices that generically lead to inequivalent theories within the standard Hilbert space approach [1]. The vacuum state be invariant under the isometries of the spatial manifold Σ, the dynamics dictated by the field Equation (1) be unitarily implementable, In order to single out a unique preferred Fock representation for the system These combined criteria of symmetry invariance and of unitary dynamics select a unique preferred field description as well, specifying (in a certain context) a canonical pair of field variables, so that they remove the two kinds of ambiguities present in the quantization of the field system. The second part of the work is entirely dedicated to an overview of the uniqueness results obtained by imposing the criteria of invariance under spatial symmetries and of unitary implementability of the dynamics, applied to the quantization of Gowdy models and (test) KG fields in cosmological spacetimes.
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