Uniqueness of the Fock representation of the Gowdy
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Cortez, Jerónimo; Marugán, Guillermo A. Mena; Velhinho, José M.

doi:10.1088/0264-9381/25/10/105005

Cited by 25 publications

(36 citation statements)

References 16 publications

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“…Such symmetry reductions correspond to the families of Gowdy spacetimes. Thus, our analysis includes the recent treatments of the quantum Gowdy cosmologies: specifically, the linearly polarized T 3 model [2-6] is described by a scalar field ξ on S 1 with f (t) = 1/(2t) 2 , whereas the Gowdy S 1 × S 2 and S 3 models [10][11][12] admit a description in terms of an axisymmetric field ξ on S 2 with f (t) = (1 + csc 2 t)/4. With the present unified treatment, we expect to have contributed both to a better understanding of the previously obtained results, and to an extension of them which can find applications in other symmetry reductions of General Relativity or, more generally, in quantum field theory on curved backgrounds.…”

Section: Discussionmentioning

confidence: 99%

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

et al. 2009

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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.

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Section: Discussionmentioning

confidence: 99%

“…(12). Both Γ and S are symplectic linear spaces, with the respective symplectic structures (independent of the choice of time section)…”

Section: Fock Quantization Of the Modelmentioning

confidence: 99%

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

et al. 2009

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“…In both cases, the complex structure that would be natural if the field were massless allows for a unitary quantum implementation of the field dynamics and, besides, shares the symmetry of the spatial sections of the auxiliary spacetime (which the vacuum state inherits). In fact, these two properties select a unique unitary equivalence class of Fock representations: all the representations with these attributes are unitarily equivalent [16][17][18]. Furthermore, if we change the field description by performing a linear canonical transformation that scales the field by a time-dependent factor, unitarity is lost: there is no Fock representation for the new canonical pair of field variables in which dynamics is represented in terms of a unitary operator [19].…”

Section: Introductionmentioning

confidence: 99%

Unique Fock quantization of scalar cosmological perturbations

et al. 2012

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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

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“…In addition, in the lack of a time symmetry, it sounds reasonable to demand at least that the dynamical evolution be implemented as a family of unitary transformations. 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.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2012

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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.

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Uniqueness of the Fock representation of the Gowdy S ¹ × S ² and S ³ models

Cited by 25 publications

References 16 publications

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

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

Unique Fock quantization of scalar cosmological perturbations

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

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Uniqueness of the Fock representation of the Gowdy S 1 × S 2 and S 3 models

Cited by 25 publications

References 16 publications

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

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

Unique Fock quantization of scalar cosmological perturbations

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

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Uniqueness of the Fock representation of the Gowdy S ¹ × S ² and S ³ models