32 research outputs found
Local Properties of Measures in Quantum Field Theory and Cosmology
We show that measure theoretical results concerning the Ashtekar-Lewandowski
measure in the space of generalized connections have direct analogues in the
context of the Bohr compactification of the line and associated Haar measure.
We present also a characterization of the support of the measure associated
with the canonical quantization of the free massive scalar field, following
closely well known analogous results concerning the Euclidean path integral
measure
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
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
Uniqueness of the Fock quantization of Dirac fields in 2+1 dimensions
We study the Fock quantization of a free Dirac field in 2+1-dimensional
backgrounds which are conformally ultrastatic, with a time-dependent conformal
factor. As it is typical for field theories, there is an infinite ambiguity in
the Fock representation of the canonical anticommutation relations. Different
choices may lead to unitarily inequivalent theories that describe different
physics. To remove this ambiguity one usually requires that the vacuum be
invariant under the unitary transformations that implement the symmetries of
the equations of motion. However, in non-stationary backgrounds, where time
translation is not a symmetry transformation, the requirement of vacuum
invariance is in general not enough to fix completely the Fock representation.
We show that this problem is overcome in the considered scenario by demanding,
in addition, a unitarily implementable quantum dynamics. The combined
imposition of these conditions selects a unique family of equivalent Fock
representations. Moreover, one also obtains an essentially unique splitting of
the time variation of the Dirac field into an explicit dependence on the
background scale factor and a quantum evolution of the corresponding creation
and annihilation operators.Comment: 24 pages. Document replaced to match published versio
Unitary evolution and uniqueness of the Fock quantization in flat cosmologies with compact spatial sections
We study the Fock quantization of scalar fields with a time dependent mass in
cosmological scenarios with flat compact spatial sections. This framework
describes physically interesting situations like, e.g., cosmological
perturbations in flat Friedmann-Robertson-Walker spacetimes, generally
including a suitable scaling of them by a background function. We prove that
the requirements of vacuum invariance under the spatial isometries and of a
unitary quantum dynamics select (a) a unique canonical pair of field variables
among all those related by time dependent canonical transformations which scale
the field configurations, and (b) a unique Fock representation for the
canonical commutation relations of this pair of variables. Though the proof is
generalizable to other compact spatial topologies in three or less dimensions,
we focus on the case of the three-torus owing to its relevance in cosmology,
paying a especial attention to the role played by the spatial isometries in the
determination of the representation.Comment: 23 pages. New section 4.2. Added references. Published in EJT
Unitary evolution and uniqueness of the Fock representation of Dirac fields in cosmological spacetimes
We present a privileged Fock quantization of a massive Dirac field in a
closed Friedmann-Robertson-Walker cosmology, partially selected by the criteria
of invariance of the vacuum under the symmetries of the field equations, and
unitary implementation of the dynamics. When quantizing free scalar fields in
homogeneous and isotropic spacetimes with compact spatial sections, these
criteria have been shown to pick out a unique Fock representation (up to
unitary equivalence). Here, we employ the same criteria for fermion fields and
explore whether that uniqueness result can be extended to the case of the Fock
quantization of fermions. For the massive Dirac field, we start by introducing
a specific choice of the complex structure that determines the Fock
representation. Such structure is invariant under the symmetries of the
equations of motion. We then prove that the corresponding representation of the
canonical anticommutation relations admits a unitary implementation of the
dynamics. Moreover, we construct a rather general class of representations that
satisfy the above criteria, and we demonstrate that they are all unitarily
equivalent to our previous choice. The complex structures in this class are
restricted only by certain conditions on their asymptotic behavior for modes in
the ultraviolet sector of the Dirac operator. We finally show that, if one
assumes that these asymptotic conditions are in fact trivial once our criteria
are fulfilled, then the time-dependent scaling in the definition of the
fermionic annihilation and creation-like variables is essentially unique.Comment: 24 page
Uniqueness of the Fock quantization of scalar fields in spatially flat cosmological spacetimes
We study the Fock quantization of scalar fields in (generically) time
dependent scenarios, focusing on the case in which the field propagation occurs
in --either a background or effective-- spacetime with spatial sections of flat
compact topology. The discussion finds important applications in cosmology,
like e.g. in the description of test Klein-Gordon fields and scalar
perturbations in Friedmann-Robertson-Walker spacetime in the observationally
favored flat case. Two types of ambiguities in the quantization are analyzed.
First, the infinite ambiguity existing in the choice of a Fock representation
for the canonical commutation relations, understandable as the freedom in the
choice of inequivalent vacua for a given field. Besides, in cosmological
situations, it is customary to scale the fields by time dependent functions,
which absorb part of the evolution arising from the spacetime, which is treated
classically. This leads to an additional ambiguity, this time in the choice of
a canonical pair of field variables. We show that both types of ambiguities are
removed by the requirements of (a) invariance of the vacuum under the
symmetries of the three-torus, and (b) unitary implementation of the dynamics
in the quantum theory. In this way, one arrives at a unique class of unitarily
equivalent Fock quantizations for the system. This result provides considerable
robustness to the quantum predictions and renders meaningful the confrontation
with observation.Comment: 15 pages, version accepted for publication in JCA
Uniqueness of the Fock quantization of scalar fields in a Bianchi I cosmology with unitary dynamics
The Fock quantization of free scalar fields is subject to an infinite
ambiguity when it comes to choosing a set of annihilation and creation
operators, choice that is equivalent to the determination of a vacuum state. In
highly symmetric situations, this ambiguity can be removed by asking vacuum
invariance under the symmetries of the system. Similarly, in stationary
backgrounds, one can demand time-translation invariance plus positivity of the
energy. However, in more general situations, additional criteria are needed.
For the case of free (test) fields minimally coupled to a homogeneous and
isotropic cosmology, it has been proven that the ambiguity is resolved by
introducing the criterion of unitary implementability of the quantum dynamics,
as an endomorphism in Fock space. This condition determines a specific
separation of the time dependence of the field, so that this splits into a very
precise background dependence and a genuine quantum evolution. Furthermore,
together with the condition of vacuum invariance under the spatial Killing
symmetries, unitarity of the dynamics selects a unique Fock representation for
the canonical commutation relations, up to unitary equivalence. In this work,
we generalize these results to anisotropic spacetimes with shear, which are
therefore not conformally symmetric, by considering the case of a free scalar
field in a Bianchi I cosmology.Comment: 23 pages. Updated to match published versio
Uniqueness of the Fock representation of the Gowdy and models
After a suitable gauge fixing, the local gravitational degrees of freedom of
the Gowdy and cosmologies are encoded in an axisymmetric
field on the sphere . Recently, it has been shown that a standard field
parametrization of these reduced models admits no Fock quantization with a
unitary dynamics. This lack of unitarity is surpassed by a convenient
redefinition of the field and the choice of an adequate complex structure. The
result is a Fock quantization where both the dynamics and the SO(3)-symmetries
of the field equations are unitarily implemented. The present work proves that
this Fock representation is in fact unique inasmuch as, up to equivalence,
there exists no other possible choice of SO(3)-invariant complex structure
leading to a unitary implementation of the time evolution.Comment: 10 pages, minor changes, version accepted for publication in
Classical and Quantum Gravit