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Geometric algebra and the causal approach to multiparticle quantum mechanics

Shyamal Somaroo, Anthony Lasenby, Chris Doran

1999
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Journal of Mathematical Physics
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It is argued that geometric algebra, in the form of the multiparticle spacetime algebra, is well suited to the study of multiparticle quantum theory, with advantages over conventional techniques both in ease of calculation and in providing an intuitive geometric understanding of the results. This is illustrated by comparing the geometric algebra approach for a system of two spin-1/2 particles with the nonrelativistic approach of Holland �Phys. Rep. 169, 294 �1988��. © 1999 American Institute of
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... erican Institute of Physics. �S0022-2488�99�00907-X� I. INTRODUCTION Geometric �Clifford� algebra is a powerful algebraic tool with applications throughout the fields of physics and engineering. The geometric algebra of space-time-the spacetime algebra or STA-is well suited to describing many aspects of classical and quantum relativistic physics 1-4 including gravitation. 5 In Refs. 2 and 6 the multiparticle spacetime algebra �MSTA� was introduced and applied to relativistic multiparticle quantum theory. In the present paper the algebraic advantages of the MSTA approach are demonstrated through a comparison with work on a causal approach to nonrelativistic multiparticle quantum theory based on the Pauli equation. 7, 8 We show that the MSTA elucidates a number of features of the multiparticle causal theory and, in particular, clarifies its geometric content. The causal, or Bohmian, approach to quantum mechanics is an interpretation in which the statistical results of quantum theory are recovered from an ensemble of deterministically evolving systems. The approach is based on establishing a connection between the wave equation and a deterministic model that is supposed to underlie the quantum process. In the case of one spin-1/2 particle, this model consists of a classical spinning rigid body under the additional influence of a quantum potential. 7 In this way physical properties can be associated with the quantum particle, and equations for their evolution obtained from the conventional wave equation. Furthermore, the variables �including spin� on which the wave function depends are consistently interpreted as the spatial position and orientation of the particle through this model. In n-particle nonrelativistic quantum theory the wave function depends on a dynamical configuration space of dimension 3n, as well as on a temporal parameter t. To apply a causal approach to this system one must first associate the wave function in configuration space with a set of physical properties. These are then interpreted as the properties of the individual particles in the ensemble making up the system under consideration. Equations describing the evolution of these properties are then derived from the conventional n-particle Pauli wave equation. Holland 7,8 has addressed the problem of extracting a set of physical properties from the n-particle wave function. His method is to construct a set of tensor variables from quadratic combinations of the spinorial wave function. These tensor variables are more easily associated with a set physical properties than the underlying spinorial degrees of freedom. Here we show that the MSTA formulation of multiparticle quantum theory considerably simplifies the task of extracting these physical variables. Its lack of redundant mathematical a� Electronic mail: C.

doi:10.1063/1.532890
fatcat:u37pkupuvvgh3ihmvzpur3wlly