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Full-Text Articles in Physics
Observables For The Polarized Gowdy Model, Charles G. Torre
Observables For The Polarized Gowdy Model, Charles G. Torre
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We give an explicit characterization of all functions on the phase space for the polarized Gowdy 3-torus spacetimes which have weakly vanishing Poisson brackets with the Hamiltonian and momentum constraint functions.
Coherent State Path Integral For Linear Systems, Charles G. Torre
Coherent State Path Integral For Linear Systems, Charles G. Torre
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We present a computation of the coherent state path integral for a generic linear system using "functional methods'' (as opposed to discrete time approaches). The Gaussian phase space path integral is formally given by a determinant built from a first-order differential operator with coherent state boundary conditions. We show how this determinant can be expressed in terms of the symplectic transformation generated by the (in general, time-dependent) quadratic Hamiltonian for the system. We briefly discuss the conditions under which the coherent state path integral for a linear system actually exists. A necessary -- but not sufficient -- condition for existence …
Cosmology, Cohomology, And Compactification, Charles G. Torre
Cosmology, Cohomology, And Compactification, Charles G. Torre
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Ashtekar and Samuel have shown that Bianchi cosmological models with compact spatial sections must be of Bianchi class A. Motivated by general results on the symmetry reduction of variational principles, we show how to extend the Ashtekar-Samuel results to the setting of weakly locally homogeneous spaces as defined, e.g., by Singer and Thurston. In particular, it is shown that any m-dimensional homogeneous space G/K admitting a G-invariant volume form will allow a compact discrete quotient only if the Lie algebra cohomology of G relative to K is non-vanishing at degree m.
Quantum Dynamics Of The Polarized Gowdy T3 Model, Charles G. Torre
Quantum Dynamics Of The Polarized Gowdy T3 Model, Charles G. Torre
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The polarized Gowdy T3 vacuum spacetimes are characterized, modulo gauge, by a “point particle” degree of freedom and a function φ that satisfies a linear field equation and a nonlinear constraint. The quantum Gowdy model has been defined by using a representation for φ on a Fock space F. Using this quantum model, it has recently been shown that the dynamical evolution determined by the linear field equation for φ is not unitarily implemented on F. In this paper, (1) we derive the classical and quantum model using the “covariant phase space” formalism, (2) we show that time evolution is …
The Principle Of Symmetric Criticality In General Relativity, Mark E. Fels, Charles G. Torre
The Principle Of Symmetric Criticality In General Relativity, Mark E. Fels, Charles G. Torre
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We consider a version of Palais' principle of symmetric criticality (PSC) that is applicable to the Lie symmetry reduction of Lagrangian field theories. Given a group action on a space of fields, PSC asserts that for any group-invariant Lagrangian, the equations obtained by restriction of Euler–Lagrange equations to group-invariant fields are equivalent to the Euler–Lagrange equations of a canonically defined, symmetry-reduced Lagrangian. We investigate the validity of PSC for local gravitational theories built from a metric and show that there are two independent conditions which must be satisfied for PSC to be valid. One of these conditions, obtained previously in …
Group Invariant Solutions In Mathematical Physics And Differential Geometry, Ian M. Anderson, Mark E. Fels, Charles G. Torre
Group Invariant Solutions In Mathematical Physics And Differential Geometry, Ian M. Anderson, Mark E. Fels, Charles G. Torre
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This is a brief overview of our work on the theory of group invariant solutions to differential equations. The motivations and applications of this work stem from problems in differential geometry and relativistic field theory. The key feature in our theory is that we allow for non-transverse symmetry group actions, which are very common in applications.
Group Invariant Solutions Without Transversality, Ian M. Anderson, Mark E. Fels, Charles G. Torre
Group Invariant Solutions Without Transversality, Ian M. Anderson, Mark E. Fels, Charles G. Torre
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We present a generalization of Lie's method for finding the group invariant solutions to a system of partial differential equations. Our generalization relaxes the standard transversality assumption and encompasses the common situation where the reduced differential equations for the group invariant solutions involve both fewer dependent and independent variables. The theoretical basis for our method is provided by a general existence theorem for the invariant sections, both local and global, of a bundle on which a finite dimensional Lie group acts. A simple and natural extension of our characterization of invariant sections leads to an intrinsic characterization of the reduced …
Functional Evolution Of Free Quantum Fields, Charles G. Torre, Madhavan Varadarajan
Functional Evolution Of Free Quantum Fields, Charles G. Torre, Madhavan Varadarajan
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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 …
Midisuperspace Models Of Canonical Quantum Gravity, Charles G. Torre
Midisuperspace Models Of Canonical Quantum Gravity, Charles G. Torre
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A midisuperspace 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 three recent classes of results pertinent to canonical quantization of vacuum general relativity in the context of midisuperspace 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 …
Quantum Fields At Any Time, Charles G. Torre, Madhavan Varadarajan
Quantum Fields At Any Time, Charles G. Torre, Madhavan Varadarajan
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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 …
Asymptotic Conservation Laws In Field Theory, Ian M. Anderson, Charles G. Torre
Asymptotic Conservation Laws In Field Theory, Ian M. Anderson, Charles G. Torre
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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 Arnowitt-Deser-Misner energy in general relativity.
Some Remarks On Gravitational Analogs Of Magnetic Charge, Charles G. Torre
Some Remarks On Gravitational Analogs Of Magnetic Charge, Charles G. Torre
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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
Natural Symmetries Of The Yang-Mills Equations, Charles G. Torre
Natural Symmetries Of The Yang-Mills Equations, Charles G. Torre
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A natural generalized symmetry of the Yang–Mills equations is defined 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 a spinorial coordinate system is introduced 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. It is found that all natural symmetries …
Gravitational Observables And Local Symmetries, Charles G. Torre
Gravitational Observables And Local Symmetries, Charles G. Torre
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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.
Covariant Phase Space Formulation Of Parametrized Field Theories, Charles G. Torre
Covariant Phase Space Formulation Of Parametrized Field Theories, Charles G. Torre
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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 space‐time diffeomorphism group, construction of observables, and the relationship between the canonical and covariant phase spaces.
Is General Relativity An "Already Parametrized" Theory?, Charles G. Torre
Is General Relativity An "Already Parametrized" Theory?, Charles G. Torre
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Beginning with the work of Dirac and of Arnowitt, Deser, and Misner in the late 1950s and early 1960s, and then after subsequent development by Kuchař, 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 among 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 …