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Full-Text Articles in Physics
Uniqueness Of Solutions To The Helically Reduced Wave Equation With Sommerfeld Boundary Conditions, Charles G. Torre
Uniqueness Of Solutions To The Helically Reduced Wave Equation With Sommerfeld Boundary Conditions, Charles G. Torre
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We consider the helical reduction of the wave equation with an arbitrary source on (n+1)-dimensional Minkowski space, n ≥ 2. The reduced equation is of mixed elliptic-hyperbolic type on Rn. We obtain a uniqueness theorem for solutions on a domain consisting of an n-dimensional ball B centered on the reduction of the axis of helical symmetry and satisfying ingoing or outgoing Sommerfeld conditions on ∂B ≈ Sn−1. Nonlinear generalizations of such boundary value problems (with n = 3) arise in the intermediate phase of binary inspiral in general relativity.
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.
The Helically Reduced Wave Equation As A Symmetric Positive System, Charles G. Torre
The Helically Reduced Wave Equation As A Symmetric Positive System, Charles G. Torre
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Motivated by the partial differential equations of mixed type that arise in the reduction of the Einstein equations by a helical Killing vector field, we consider a boundary value problem for the helically-reduced wave equation with an arbitrary source in 2+1 dimensional Minkowski spacetime. The reduced equation is a second-order partial differential equation which is elliptic inside a disk and hyperbolic outside the disk. We show that the reduced equation can be cast into symmetric-positive form. Using results from the theory of symmetric-positive differential equations, we show that this form of the helically-reduced wave equation admits unique, strong solutions for …
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 …
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 …