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High field magnetoresistance of inhomogeneous semiconductors V: Quantum mechanical theory
ANNALS
OF PHYSICS:
Abstracts
33, 362-366
of
(1965)
Papers
to Appear
in Future
Issues
Peratization of Unrenormalizable Field Theories. W. G~~TTINGER, R.. PENZL, AND E. PFAFFELHUBER, Institute of Theoretical Physics, University of Miinchen, Miinchen, Germany. There exists a cutoff-free, built-in prescription in the formulation of unrenormalizable field theories of weak interactions, making them accessible to diagram and perturbation techniques, and producing finite and physically significant results at each step of approximation, even if the exact solutions are nonanalytic in the coupling constant. Being an extension of renormalization theory, the formalism provides a simple and rigorous basis for the peratization of weak interactions without suffering from the shortcomings of the heuristic method of Feinberg and Pais. The relation between the Bethe-Salpeter scattering equation and singular potentials is discussed. Some theorems concerning the essent,ial singularities on the light cone of the Bethe-Salpeter wave function are proved, and some examples are discussed. Approximation formulas for the scattering amplitude are deduced that, although originating from perturbation techniques, reveal precisely the nonanalytic structure in the coupling constant of the exact amplitude. High Field Magnetoresistance of Inhomogeneous Semiconductors V: Quantum Mechanical Theory. J. MCKENNA AND H. L. FRISCH, Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey. In recent work by Herring and by Frisch and Morrison, a classical model of a high tsmperature, spatially inhomogeneous semiconductor has been used to study the nonsaturat.ion of the magnetoresistance at high magnetic fields. We have studied a quantum mechanical reformulation of this model which is valid in the region where the ratio, a, of the Landau level spacing to the mean electron kinetic energy is less than unity. The electrons, still treated in the effective mass approximation, are statistically described by a quantum mechanical transport equation for the one-electron density matrix. The effective potential arising from a more or less random distribution in space of inhomogeneities, whose scale may now extend to atomic dimensions, is included in the electron Hamiltonian. The electron-phonon interaction is described by an appropriate stochastic, relaxation type collision term first employed by Gross and Lebowitz to study dipole relaxation. To obtain the magnetoresistance we require only the local current and density of electrons. These are obtained from a positive, quasi-classical electron distribution function which arises when we consider the expectation value of the density matrix (Husimi transform) in a state in which both position and velocity of the electron are highly localized. We show explicitly, at least for the case of a stratified medium, that to leading terms in inverse powers of the magnetic field strength the current and thus the mzgnetoresistance, given previously by Frisch and Morrison are correct to order a. We briefly discuss and illustrate by examples certain aspects of the use of Husimi transforms in related problems. Variational Solutions of Perturbation Theory Equations. JEREMY I. MUSHER, The Rockefeller Institute, New York, New York. A procedure is demonstrated to obtain the solution of the partial differential equations of perturbation theory by choosing the functionals to be made stationary directly from the equations themselves. This procedure is shown to be simpler in many-particle problems than 362
OF PHYSICS:
Abstracts
33, 362-366
of
(1965)
Papers
to Appear
in Future
Issues
Peratization of Unrenormalizable Field Theories. W. G~~TTINGER, R.. PENZL, AND E. PFAFFELHUBER, Institute of Theoretical Physics, University of Miinchen, Miinchen, Germany. There exists a cutoff-free, built-in prescription in the formulation of unrenormalizable field theories of weak interactions, making them accessible to diagram and perturbation techniques, and producing finite and physically significant results at each step of approximation, even if the exact solutions are nonanalytic in the coupling constant. Being an extension of renormalization theory, the formalism provides a simple and rigorous basis for the peratization of weak interactions without suffering from the shortcomings of the heuristic method of Feinberg and Pais. The relation between the Bethe-Salpeter scattering equation and singular potentials is discussed. Some theorems concerning the essent,ial singularities on the light cone of the Bethe-Salpeter wave function are proved, and some examples are discussed. Approximation formulas for the scattering amplitude are deduced that, although originating from perturbation techniques, reveal precisely the nonanalytic structure in the coupling constant of the exact amplitude. High Field Magnetoresistance of Inhomogeneous Semiconductors V: Quantum Mechanical Theory. J. MCKENNA AND H. L. FRISCH, Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey. In recent work by Herring and by Frisch and Morrison, a classical model of a high tsmperature, spatially inhomogeneous semiconductor has been used to study the nonsaturat.ion of the magnetoresistance at high magnetic fields. We have studied a quantum mechanical reformulation of this model which is valid in the region where the ratio, a, of the Landau level spacing to the mean electron kinetic energy is less than unity. The electrons, still treated in the effective mass approximation, are statistically described by a quantum mechanical transport equation for the one-electron density matrix. The effective potential arising from a more or less random distribution in space of inhomogeneities, whose scale may now extend to atomic dimensions, is included in the electron Hamiltonian. The electron-phonon interaction is described by an appropriate stochastic, relaxation type collision term first employed by Gross and Lebowitz to study dipole relaxation. To obtain the magnetoresistance we require only the local current and density of electrons. These are obtained from a positive, quasi-classical electron distribution function which arises when we consider the expectation value of the density matrix (Husimi transform) in a state in which both position and velocity of the electron are highly localized. We show explicitly, at least for the case of a stratified medium, that to leading terms in inverse powers of the magnetic field strength the current and thus the mzgnetoresistance, given previously by Frisch and Morrison are correct to order a. We briefly discuss and illustrate by examples certain aspects of the use of Husimi transforms in related problems. Variational Solutions of Perturbation Theory Equations. JEREMY I. MUSHER, The Rockefeller Institute, New York, New York. A procedure is demonstrated to obtain the solution of the partial differential equations of perturbation theory by choosing the functionals to be made stationary directly from the equations themselves. This procedure is shown to be simpler in many-particle problems than 362