113 research outputs found
Functorial Aspects of the Space of Generalized Connections
We give a description of the category structure of the space of generalized
connections, an extension of the space of connections that plays a central role
in loop quantum gravity.Comment: 7 pages. To appear in Proceedings of the Lusofona Workshop on Quantum
Gravity and Noncommutative Geometry, Lisbon, July 200
Comments on a Full Quantization of the Torus
Gotay showed that a representation of the whole Poisson algebra of the torus
given by geometric quantization is irreducible with respect to the most natural
overcomplete set of observables. We study this representation and argue that it
cannot be considered as physically acceptable. In particular, classically
bounded observables are quantized by operators with unbounded spectrum.
Effectively, the latter amounts to lifting the constraints that compactify both
directions in the torus.Comment: 10 pages. New "Discussion" section. References added. To appear in
IJMP
Uniqueness of the Fock quantization of the Gowdy model
After its reduction by a gauge-fixing procedure, the family of linearly
polarized Gowdy cosmologies admit a scalar field description whose
evolution is governed by a Klein-Gordon type equation in a flat background in
1+1 dimensions with the spatial topology of , though in the presence of a
time-dependent potential. The model is still subject to a homogeneous
constraint, which generates -translations. Recently, a Fock quantization
of this scalar field was introduced and shown to be unique under the
requirements of unitarity of the dynamics and invariance under the gauge group
of -translations. In this work, we extend and complete this uniqueness
result by considering other possible scalar field descriptions, resulting from
reasonable field reparameterizations of the induced metric of the reduced
model. In the reduced phase space, these alternate descriptions can be obtained
by means of a time-dependent scaling of the field, the inverse scaling of its
canonical momentum, and the possible addition of a time-dependent, linear
contribution of the field to this momentum. Demanding again unitarity of the
field dynamics and invariance under the gauge group, we prove that the
alternate canonical pairs of fieldlike variables admit a Fock representation if
and only if the scaling of the field is constant in time. In this case, there
exists essentially a unique Fock representation, provided by the quantization
constructed by Corichi, Cortez, and Mena Marugan. In particular, our analysis
shows that the scalar field description proposed by Pierri does not admit a
Fock quantization with the above unitarity and invariance properties.Comment: 14 page
Quantum unitary dynamics in cosmological spacetimes
We address the question of unitary implementation of the dynamics for scalar
fields in cosmological scenarios. Together with invariance under spatial
isometries, the requirement of a unitary evolution singles out a rescaling of
the scalar field and a unitary equivalence class of Fock representations for
the associated canonical commutation relations. Moreover, this criterion
provides as well a privileged quantization for the unscaled field, even though
the associated dynamics is not unitarily implementable in that case. We discuss
the relation between the initial data that determine the Fock representations
in the rescaled and unscaled descriptions, and clarify that the S-matrix is
well defined in both cases. In our discussion, we also comment on a recently
proposed generalized notion of unitary implementation of the dynamics, making
clear the difference with the standard unitarity criterion and showing that the
two approaches are not equivalent.Comment: 18 page
A uniqueness criterion for the Fock quantization of scalar fields with time dependent mass
A major problem in the quantization of fields in curved spacetimes is the
ambiguity in the choice of a Fock representation for the canonical commutation
relations. There exists an infinite number of choices leading to different
physical predictions. In stationary scenarios, a common strategy is to select a
vacuum (or a family of unitarily equivalent vacua) by requiring invariance
under the spacetime symmetries. When stationarity is lost, a natural
generalization consists in replacing time invariance by unitarity in the
evolution. We prove that, when the spatial sections are compact, the criterion
of a unitary dynamics, together with the invariance under the spatial
isometries, suffices to select a unique family of Fock quantizations for a
scalar field with time dependent mass.Comment: 11 pages, version accepted for publication in Classical and Quantum
Gravit
Quantum Gowdy model: A uniqueness result
Modulo a homogeneous degree of freedom and a global constraint, the linearly
polarised Gowdy cosmologies are equivalent to a free scalar field
propagating in a fixed nonstationary background. Recently, a new field
parameterisation was proposed for the metric of the Gowdy spacetimes such that
the associated scalar field evolves in a flat background in 1+1 dimensions with
the spatial topology of , although subject to a time dependent potential.
Introducing a suitable Fock quantisation for this scalar field, a quantum
theory was constructed for the Gowdy model in which the dynamics is implemented
as a unitary transformation. A question that was left open is whether one might
adopt a different, nonequivalent Fock representation by selecting a distinct
complex structure. The present work proves that the chosen Fock quantisation is
in fact unique (up to unitary equivalence) if one demands unitary
implementation of the dynamics and invariance under the group of constant
translations. These translations are precisely those generated by the global
constraint that remains on the Gowdy model. It is also shown that the proof of
uniqueness in the choice of complex structure can be applied to more general
field dynamics than that corresponding to the Gowdy cosmologies.Comment: 28 pages, minor changes, version accepted for publication in
Classical and Quantum Gravit
Criteria for the determination of time dependent scalings in the Fock quantization of scalar fields with a time dependent mass in ultrastatic spacetimes
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, the redefinition does not introduce any Fock representation
that cannot be obtained by a unitary transformation.Comment: 37 pages. Modified title. Improved discussion concerning the spatial
symmetry group. New section (section VI
Uniqueness of the Fock quantization of a free scalar field on with time dependent mass
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 form . We prove that the representation of the
canonical commutation relations corresponding to the particular case of a
massless free field () provides a unitary implementation of the dynamics
for sufficiently general mass terms, . Furthermore, this representation
is uniquely specified, among the class of representations determined by
-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.Comment: 13 pages, typos corrected, version accepted for publication in
Physical Review
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