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Birkhäuser Boston eBooks

On a New Basis for a Generalized Clifford Algebra and its Application to Quantum Mechanics

Alex Granik, Marvin Franklin Ross

Published Jan 1, 1996• 3 citations
Publisher

Abstract

A generalized Clifford algebra is formulated in terms of a simple Euclidean carrier space. Using this space the eigenvalues, shifting operators, and the Casimir operator are easily derived for a few selected quantum mechanical problems.
DOI
10.1007/978-1-4615-8157-4_6

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Abstract Quantum mechanical problems capable of exact solution are traditionally solved in a few instances only (such as the harmonic oscillator and angular momentum) by operator methods, but mainly by means of Schrodinger's wave mechanics. The present volume shows that a large range of one- and three-dimensional problems, including certain relativistic ones, are solvable by algebraic, representation-independent methods using commutation relations, shift operators, the virial, hypervirial, and Hellman-Feyman theorems. Applications of these operator methods to the calculation of eigenvalues, matrix elements, and wavefunctions are discussed in detail. This book is an introduction to the use of operator methods in quantum mechanics and also a reference work with numerous problmes solved. It is suitable for use by students of intermediate quantum mechanics and also more advanced postgraduate students who wish to study the algebraic method of solving quantum mechanical problems.
We review our approach to quantum mechanics adding also some new interesting results. We start by giving proof of two important theorems on the existence of the and Clifford algebras. This last algebra gives proof of the von Neumann basic postulates on the quantum measurement explaining thus in an algebraic manner the wave function collapse postulated in standard quantum theory. In this manner we reach the objective to expose a self-consistent version of quantum mechanics. We give proof of the quantum like Heisenberg uncertainty relations, the phenomenon of quantum Mach Zender interference as well as quantum collapse in some cases of physical interest We also discuss the problem of time evolution of quantum systems as well as the changes in space location. We also give demonstration of the Kocken-Specher theorem, and also we give an algebraic formulation and explanation of the EPR . By using the same approach we also derive Bell inequalities. Our formulation is strongly based on the use of idempotents that are contained in Clifford algebra. Their counterpart in quantum mechanics is represented by the projection operators that are interpreted as logical statements, following the basic von Neumann results. Using the Clifford algebra we are able to invert such result. According to the results previously obtained by Orlov in 1994, we are able to give proof that quantum mechanics derives from logic. We show that indeterminism and quantum interference have their origin in the logic.
We suggest Clifford algebra as a useful simplifying language for present quantum dynamics. Clifford algebras arise from representations of the permutation groups as they arise from representations of the rotation groups. Aggregates using such representations for their permutations obey Clifford statistics. The vectors supporting the Clifford algebras of permutations and rotations are plexors and spinors, respectively. Physical spinors may actually be plexors describing quantum ensembles, not simple individuals. We use Clifford statistics to define quantum fields on a quantum space–time, and to formulate a quantum dynamics-field-space–time unity that evades the compactification problem. The quantum bits of history regarded as a quantum computation seem to obey a Clifford statistics.
An extension of Fourier analysis for the nnn-torus in the magnetic field and its application to spectral analysis of the magnetic Laplacian
We solved the Schr{\\"o}dinger equation for a particle in a uniform magnetic field in the $ n $-dimensional torus. We obtained a complete set of solutions for a broad class of problems; the torus $ T^n = \\R^n / {\\mit\\Lambda} $ is defined as a quotient of the Euclidean space $ \\R^n $ by an arbitrary $ n $-dimensional lattice $ {\\mit\\Lambda} $. The lattice is not necessary either cubic or rectangular. The magnetic field is also arbitrary. However, we restrict ourselves within potential-free problems; the Schr{\\"o}dinger operator is assumed to be the Laplace operator defined with the covariant derivative. We defined an algebra that characterizes the symmetry of the Laplacian and named it the magnetic algebra. We proved that the space of functions on which the Laplacian acts is an irreducible representation space of the magnetic algebra. In this sense the magnetic algebra completely characterizes the quantum mechanics in the magnetic torus. We developed a new method for Fourier analysis for the magnetic torus and used it to solve the eigenvalue problem of the Laplacian. All the eigenfunctions are given in explicit forms.