arXiv subjects

C. G. Torre

Publications and source records attributed to C. G. Torre.

33 records · Page 2Linked to original sources

arXiv1999

Functional Evolution of Free Quantum Fields

We consider the problem of evolving a quantum field between any two (in general, curved) Cauchy surfaces. Classically, this dynamical evolution is represented by a canonical transformation on the phase space for the field theory. We show that this canonical transformation cannot, in general, be unitarily implemented on the Fock space for free quantum fields on flat spacetimes of dimension greater than 2. We do this by considering time evolution of a free Klein-Gordon field on a flat spacetime (with toroidal Cauchy surfaces) starting from a flat initial surface and ending on a generic final surface. The associated Bogolubov transformation is computed; it does not correspond to a unitary transformation on the Fock space. This means that functional evolution of the quantum state as originally envisioned by Tomonaga, Schwinger, and Dirac is not a viable concept. Nevertheless, we demonstrate that functional evolution of the quantum state can be satisfactorily described using the formalism of algebraic quantum field theory. We discuss possible implications of our results for canonical quantum gravity.

hep-th↗

arXiv1998

Quantum Fields at Any Time

The canonical quantum theory of a free field using arbitrary foliations of a flat two-dimensional spacetime is investigated. It is shown that dynamical evolution along arbitrary spacelike foliations is unitarily implemented on the same Fock space as that associated with inertial foliations. It follows that the Schrodinger picture exists for arbitrary foliations as a unitary image of the Heisenberg picture for the theory. An explicit construction of the Schrodinger picture image of the Heisenberg Fock space states is provided. The results presented here can be interpreted in terms of a Dirac constraint quantization of parametrized field theory. In particular, it is shown that the Schrodinger picture physical states satisfy a functional Schrodinger equation which includes a slice-dependent c-number quantum correction, in accord with a proposal of Kuchar. The spatial diffeomorphism invariance of the Schrodinger picture physical states is established. Fundamental difficulties arise when trying to generalize these results to higher-dimensional spacetimes.

hep-th↗

arXiv1998

Midi-superspace Models of Canonical Quantum Gravity

A midi-superspace model is a field theory obtained by symmetry reduction of a parent gravitational theory. Such models have proven useful for exploring the classical and quantum dynamics of the gravitational field. I present 3 recent classes of results pertinent to canonical quantization of vacuum general relativity in the context of midi-superspace models. (1) I give necessary and sufficient conditions such that a given symmetry reduction can be performed at the level of the Lagrangian or Hamiltonian. (2) I discuss the Hamiltonian formulation of models based upon cylindrical and toroidal symmetry. In particular, I explain how these models can be identified with parametrized field theories of wave maps, thus a natural strategy for canonical quantization is available. (3) The quantization of a parametrized field theory, such as the midi-superspace models considered in (2), requires construction of a quantum field theory on a fixed (flat) spacetime that allows for time evolution along arbitrary foliations of spacetime. I discuss some recent results on the possibility of finding such a quantum field theory.

gr-qc↗

arXiv1997

Local cohomology in field theory (with applications to the Einstein equations)

This is an introductory survey of the theory of $p$-form conservation laws in field theory. It is based upon a series of lectures given at the Second Mexican School on Gravitation and Mathematical Physics held in Tlaxcala, Mexico from December 1--7, 1996. Proceedings available online at http://kaluza.physik.uni-konstanz.de/2MS/ProcMain.html.

hep-th↗

arXiv1996

Asymptotic conservation laws in field theory

A new, general, field theoretic approach to the derivation of asymptotic conservation laws is presented. In this approach asymptotic conservation laws are constructed directly from the field equations according to a universal prescription which does not rely upon the existence of Noether identities or any Lagrangian or Hamiltonian formalisms. The resulting general expressions of the conservation laws enjoy important invariance properties and synthesize all known asymptotic conservation laws, such as the ADM energy in general relativity.

hep-th↗

arXiv1995

Spinors, Jets, and the Einstein Equations

Many important features of a field theory, {\it e.g.}, conserved currents, symplectic structures, energy-momentum tensors, {\it etc.}, arise as tensors locally constructed from the fields and their derivatives. Such tensors are naturally defined as geometric objects on the jet space of solutions to the field equations. Modern results from the calculus on jet bundles can be combined with a powerful spinor parametrization of the jet space of Einstein metrics to unravel basic features of the Einstein equations. These techniques have been applied to computation of generalized symmetries and ``characteristic cohomology'' of the Einstein equations, and lead to results such as a proof of non-existence of ``local observables'' for vacuum spacetimes and a uniqueness theorem for the gravitational symplectic structure.

gr-qc↗

arXiv1994

Some Remarks on Gravitational Analogs of Magnetic Charge

Existing mathematical results are applied to the problem of classifying closed $p$-forms which are locally constructed from Lorentzian metrics on an $n$-dimensional orientable manifold $M$ ($0<p<n$). We show that the only closed, non-exact forms are generated by representatives of cohomology classes of $M$ and $(n-1)$-forms representing $n$-dimensional (with $n$ even) generalizations of the conservation of ``kink number'', which was exhibited by Finkelstein and Misner for $n=4$. The cohomology class that defines the kink number depends only on the diffeomorphism equivalence class of the metric, but a result of Gilkey implies that there is no representative of this cohomology class which is built from the metric, curvature and covariant derivatives of curvature to any finite order.

gr-qc↗

arXiv1994

Natural Symmetries of the Yang-Mills Equations

We define a natural generalized symmetry of the Yang-Mills equations as an infinitesimal transformation of the Yang-Mills field, built in a local, gauge invariant, and Poincaré invariant fashion from the Yang-Mills field strength and its derivatives to any order, which maps solutions of the field equations to other solutions. On the jet bundle of Yang-Mills connections we introduce a spinorial coordinate system that is adapted to the solution subspace defined by the Yang-Mills equations. In terms of this coordinate system the complete classification of natural symmetries is carried out in a straightforward manner. We find that all natural symmetries of the Yang-Mills equations stem from the gauge transformations admitted by the equations.

hep-th↗

arXiv1994

The Problems of Time and Observables: Some Recent Mathematical Results

We present 2 recent results on the problems of time and observables in canonical gravity. (1) We cannot use parametrized field theory to solve the problem of time because, strictly speaking, general relativity is not a parametrized field theory. (2) We show that there are essentially no local observables for vacuum spacetimes.

gr-qc↗

arXiv1994

Classification of Generalized Symmetries for the Vacuum Einstein Equations

A generalized symmetry of a system of differential equations is an infinitesimal transformation depending locally upon the fields and their derivatives which carries solutions to solutions. We classify all generalized symmetries of the vacuum Einstein equations in four spacetime dimensions. To begin, we analyze symmetries that can be built from the metric, curvature, and covariant derivatives of the curvature to any order; these are called natural symmetries and are globally defined on any spacetime manifold. We next classify first-order generalized symmetries, that is, symmetries that depend on the metric and its first derivatives. Finally, using results from the classification of natural symmetries, we reduce the classification of all higher-order generalized symmetries to the first-order case. In each case we find that the generalized symmetries are infinitesimal generalized diffeomorphisms and constant metric scalings. There are no non-trivial conservation laws associated with these symmetries. A novel feature of our analysis is the use of a fundamental set of spinorial coordinates on the infinite jet space of Ricci-flat metrics, which are derived from Penrose's ``exact set of fields'' for the vacuum equations.

gr-qc↗

arXiv1993

Gravitational Observables and Local Symmetries

Using a recent classification of local symmetries of the vacuum Einstein equations, it is shown that there can be no observables for the vacuum gravitational field (in a closed universe) built as spatial integrals of local functions of Cauchy data and their derivatives.

gr-qc↗

arXiv1993

Symmetries of the Einstein Equations

Generalized symmetries of the Einstein equations are infinitesimal transformations of the spacetime metric that formally map solutions of the Einstein equations to other solutions. The infinitesimal generators of these symmetries are assumed to be local, \ie at a given spacetime point they are functions of the metric and an arbitrary but finite number of derivatives of the metric at the point. We classify all generalized symmetries of the vacuum Einstein equations in four spacetime dimensions and find that the only generalized symmetry transformations consist of: (i) constant scalings of the metric (ii) the infinitesimal action of generalized spacetime diffeomorphisms. Our results rule out a large class of possible ``observables'' for the gravitational field, and suggest that the vacuum Einstein equations are not integrable.

gr-qc↗

arXiv1992

Covariant Phase Space Formulation of Parametrized Field Theories

Parametrized field theories, which are generally covariant versions of ordinary field theories, are studied from the point of view of the covariant phase space: the space of solutions of the field equations equipped with a canonical (pre)symplectic structure. Motivated by issues arising in general relativity, we focus on: phase space representations of the spacetime diffeomorphism group, construction of observables, and the relationship between the canonical and covariant phase spaces.

hep-th↗

arXiv1992

Is General Relativity an 'Already Parametrized' Field Theory?

Beginning with the work of Dirac and Arnowitt, Deser, Misner in the late fifties and early sixties, and then after subsequent development by Kucha\v r, the canonical dynamical structure of general relativity has often been viewed as that of a parametrized field theory in which the many-fingered spacetime variables are hidden amongst the geometrodynamical field variables. This paradigm of general relativity as an ``already parametrized theory'' forms the basis for one of the most satisfactory resolutions of the problems of time and observables in classical and quantum gravity. However, despite decades of effort, no identification of many-fingered spacetime variables has ever been satisfactorily obtained for vacuum general relativity. We point out that there is an obstruction to identifying the constraint surface of general relativity (for the case of a closed universe) with that of any parametrized theory. Therefore, strictly speaking, general relativity cannot be viewed as a parametrized field theory. We discuss implications for the canonical quantization program.

hep-th↗

arXiv1991

A Deformation Theory of Self-Dual Einstein Spaces

The self-dual Einstein equations on a compact Riemannian 4-manifold can be expressed as a quadratic condition on the curvature of an $SU(2)$ (spin) connection which is a covariant generalization of the self-dual Yang-Mills equations. Local properties of the moduli space of self-dual Einstein connections are described in the context of an elliptic complex which arises in the linearization of the quadratic equations on the $SU(2)$ curvature. In particular, it is shown that the moduli space is discrete when the cosmological constant is positive; when the cosmological constant is negative the moduli space can be a manifold the dimension of which is controlled by the Atiyah-Singer index theorem.

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