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Classical solutions and the energy-momentum tensor
ANNALS
OF PHYSICS130,249-250
Abstracts
Numerical
Calculations
(1980)
of Papers
in Elementary
to Appear
Quantum
Mechanics
in Future
Using
Feynman
Issues
Path
Integrals.
GARY
SCHER, Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, and Department of Physics, Harvard University, Cambridge, Massachusetts 02138; MALCOLM SMITH, Department of Physics, Northeastern University, Boston, Massachusetts 02115; and MICHEL BARANGER, Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. We show that it is possible to do numerical calculations in elementary quantum mechanics using Feynman path integrals. Our method involves discretizing both time and space, and summing paths through matrix multiplication. We give numerical results for various one-dimensional potentials. The calculations of energy levels and wave functions take approximately 100 times longer than with standard methods, but there are other problems for which such an approach should be more efficient. Equation with Instantaneous Harmonic Oscillator Exchange. LEVERE C. HOSTLER, Department of Physics, Wilkes College, Wilkes-Barre, Pennsylvania 18766; AND WAYNE W. REPKO, Department of Physics, Michigan State University, East Lansing, Michigan 48824.
Bethe-Salpeter
The Bethe-Salpeter equation in the form due to Cung et al. (Ann. Phys. (N. Y.) 98 (1976), 516) is investigated for the special case of instantaneous harmonic oscillator exchange. An exact reduction to a pair of coupled ordinary differential equations for the radial excitations of the 3(J rt l), modes is achieved. The equations in the mass zero case are brought to a form which is quite close to Whittaker’s equation. This similarity to Whittaker’s equation is exploited in a computer study of the level structure as a function of the quark mass. This study covers the region from a highly relativistic spectrum depending only upon J to the nonrelativistic regime where the spectrum depends only upon L. An expression for the leptonic width of a %I state in terms of the Bethe-Salpeter wave function is derived and applied to the #-family. The effect of relativistic corrections is to reduce the predicted value of the leptonic width compared to the value calculated by assuming nonrelativistic kinematics. It is also shown that the relativistic treatment allows a 3D, state to couple directly to a virtual photon. as an Alternative to Goldstone’s Theorem. A. .I. MCKANE AND M. STONE. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, England.
Localization
We study the field theory approach to Anderson localization paying special attention to the way in which localized states avoid Goldstone’s theorem. We provide a formal derivation of Wegner’s non-linear c model for the localization transition and discuss those properties which are unusual, but necessary, if it is to give a correct description of physics near the mobility edge. Classical
Solutions
Dortmund,
and the Energy-Momentum
4600 Dortmund
Tensor.
ALFRED ACTOR. Abteilung Physik, Universitat
50, West Germany.
A hyperelliptic two-meron solution of the massless scalar @v theory in n = 2N/(N - 2) Euclidean dimensions are given. This solution (which interpolates between the two-meron solution and the instanton solution of this theory) is used to illustrate several theory-independent statements which can be made about the energy-momentum tensor for instanton, meron and elliptic meron solutions of all scale invariant classical field theories.
249 Copyright $3 1980 by Academic Press, Inc. All rights of reproduction in any form reserved.
OF PHYSICS130,249-250
Abstracts
Numerical
Calculations
(1980)
of Papers
in Elementary
to Appear
Quantum
Mechanics
in Future
Using
Feynman
Issues
Path
Integrals.
GARY
SCHER, Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, and Department of Physics, Harvard University, Cambridge, Massachusetts 02138; MALCOLM SMITH, Department of Physics, Northeastern University, Boston, Massachusetts 02115; and MICHEL BARANGER, Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. We show that it is possible to do numerical calculations in elementary quantum mechanics using Feynman path integrals. Our method involves discretizing both time and space, and summing paths through matrix multiplication. We give numerical results for various one-dimensional potentials. The calculations of energy levels and wave functions take approximately 100 times longer than with standard methods, but there are other problems for which such an approach should be more efficient. Equation with Instantaneous Harmonic Oscillator Exchange. LEVERE C. HOSTLER, Department of Physics, Wilkes College, Wilkes-Barre, Pennsylvania 18766; AND WAYNE W. REPKO, Department of Physics, Michigan State University, East Lansing, Michigan 48824.
Bethe-Salpeter
The Bethe-Salpeter equation in the form due to Cung et al. (Ann. Phys. (N. Y.) 98 (1976), 516) is investigated for the special case of instantaneous harmonic oscillator exchange. An exact reduction to a pair of coupled ordinary differential equations for the radial excitations of the 3(J rt l), modes is achieved. The equations in the mass zero case are brought to a form which is quite close to Whittaker’s equation. This similarity to Whittaker’s equation is exploited in a computer study of the level structure as a function of the quark mass. This study covers the region from a highly relativistic spectrum depending only upon J to the nonrelativistic regime where the spectrum depends only upon L. An expression for the leptonic width of a %I state in terms of the Bethe-Salpeter wave function is derived and applied to the #-family. The effect of relativistic corrections is to reduce the predicted value of the leptonic width compared to the value calculated by assuming nonrelativistic kinematics. It is also shown that the relativistic treatment allows a 3D, state to couple directly to a virtual photon. as an Alternative to Goldstone’s Theorem. A. .I. MCKANE AND M. STONE. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, England.
Localization
We study the field theory approach to Anderson localization paying special attention to the way in which localized states avoid Goldstone’s theorem. We provide a formal derivation of Wegner’s non-linear c model for the localization transition and discuss those properties which are unusual, but necessary, if it is to give a correct description of physics near the mobility edge. Classical
Solutions
Dortmund,
and the Energy-Momentum
4600 Dortmund
Tensor.
ALFRED ACTOR. Abteilung Physik, Universitat
50, West Germany.
A hyperelliptic two-meron solution of the massless scalar @v theory in n = 2N/(N - 2) Euclidean dimensions are given. This solution (which interpolates between the two-meron solution and the instanton solution of this theory) is used to illustrate several theory-independent statements which can be made about the energy-momentum tensor for instanton, meron and elliptic meron solutions of all scale invariant classical field theories.
249 Copyright $3 1980 by Academic Press, Inc. All rights of reproduction in any form reserved.