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Locality, Complex Numbers, and Relativistic Quantum Theory

Published online by Cambridge University Press:  28 February 2022

Simon W. Saunders
Simon W. Saunders*
Affiliation:
Harvard University
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Extract

What is relativistic quantum theory? How does it differ from non-relativistic quantum mechanics? Is there something more involved than the transition from the Galilean group to the Lorentz group?

These questions are evidently of great importance to the understanding of relativistic quantum theory (RQT), in particular to the distinction between the basic principles of quantum theory, and the characterization of a particular class of dynamical evolutions. Within non-relativistic quantum mechanics (NRQM) a reasonably precise response is possible (differences concern the number of degrees of freedom, the existence of internal symmetries, and the form of the Hamiltonian as a function of the canonical variables); what appears characteristic of RQT is the appearance of a dynamics of a qualitatively different character (involving the creation and annihilation of particles), involving new kinds of constraints (renormalization theory: anomalies). Nevertheless, it seems we must know how to pass from the inhomogeneous Galilean group (IHGG) to the Lorentz group (IHLG), if only because we have successfully constructed a RQT, surely, then, we know what it is that we hold invariant in this transition.

Type
Part X. Quantum Theory II
Information
PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association , Volume 1992 , Issue 1: Volume One: Contributed Papers 1992 , 1992 , pp. 365 - 380
Copyright
Copyright © 1992 by the Philosophy of Science Association

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Footnotes

1

Such operators (properly speaking, bilinear forms) have received remarkably little attention in NRQM. What understanding we have is due to Wan and his collaborators (see Wan 1984 and references therein). The identity that holds between local qnumber densities, and (the second quantization of) 1-particle bilinear forms, as also the formal similarity between the former and c-number 1-particle probability densities (the configuration-space expectation-values of the latter), is a consequence of the identity of the classical field solutions with the 1-particle states. To formulate a comparable equivalence in the relativistic case one must interpret the negative-frequency solutions as states (related to this, one must interpret delta-functions in terms of the states). This is essentially what is provided by the Segal approach.

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