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PSA Proceedings of the Biennial Meeting of the Philosophy of Science Association

Locality, Complex Numbers, and Relativistic Quantum Theory

Simon W. Saunders

Published Jan 1, 1992• 19 citations
Publisher

Abstract

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.

DOI
10.1086/psaprocbienmeetp.1992.1.192768

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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.
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Relativistic quantum theories are equipped with a background Minkowski spacetime and non‐relativistic quantum theories with a Galilean space‐time. Traditional investigations have distinguished their distinct space‐time structures and have examined ways in which relativistic theories become sufficiently like Galilean theories in a low velocity approximation or limit. A different way to look at their relationship is to see that both kinds of theories are special cases of a certain five‐dimensional generalization involving no limiting procedures or approximations. When one compares them, striking features emerge that bear on philosophical questions, including the ontological status of the wave function and time reversal invariance.
Non-relativistic quantum mechanics is shown to emerge from classical mechanics through the requirement of a relativity principle based on special transformations acting on position and momentum uncertainties. These transformations keep the Heisenberg inequalities invariant and form a group. They are related to dilatations of space variables provided the quantum potential is added to the classical Hamiltonian functional. The Schrödinger equation appears to have a nonunitary and nonlinear companion acting in another time variable. Evolution in this time seems related to the state vector reduction.
Nonrelativistic Galilean quantum mechanics and the standard transition to relativistic Poincar\'e quantum mechanics is analyzed in terms of group theory. Special emphasis is given to the discussion of the relation between dynamics and geometry. Certain unsatisfactory features are pointed out and a new relativistic group G5{\mathcal{G}}_{5}G5​ is suggested as the symmetry group of dynamics. G5{\mathcal{G}}_{5}G5​ contains both the nonrelativistic Galilei group and the Poincar\'e group as subgroups, and it is a group extension of the restricted Lorentz group. For use in relativistic quantum mechanics, the central extension of G5{\mathcal{G}}_{5}G5​ by a phase group must be employed. The Lie algebra of this relativistic quantum-mechanical Galilei group ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ contains an acceptable covariant space-time position operator and a nontrivial relativistic mass operator. The latter also serves to describe dynamical development. The irreducible unitary projective representations of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ correspond to infinite towers of states with increasing spin.
Let me begin by briefly explaining the concept of ‘interpretation’ relevant to this discussion. Certain physical theories postulate abstract structural constraints which events are held to satisfy. Such theories are termed ‘principle theories’. Interpretations of principle theories aim to explain their relation to the theories they replace. Interpretations are therefore concerned with the nature of the transitions between theories. Theories of space-time structure provide the most accessible illustration of principle theories. For example, Newtonian mechanics in the absence of gravitation represents the 4-dimensional geometry of space-time by the inhomogeneous Galilean group, which acts transitively in the class of free motions, i.e. the inhomogeneous Galilean group is the symmetry group of the free motions: it is a subgroup of the symmetry group of every mechanical system, and the largest such subgroup. Einstein’s special principle of relativity is the hypothesis that the symmetry group of the free motions is the Poincaré group.
Up to this point, our study of quantum mechanics has concerned itself with the behaviour of particles that inhabit a Galilean spacetime. For many purposes, in atomic, molecular and condensed matter physics, this theory is quite adequate. We saw in earlier chapters, however, that our actual spacetime has a structure which is much closer to that of the Minkowski spacetime of special relativity and that more general structures must be considered when gravitational phenomena are significant. From a purely theoretical point of view, it is therefore important to formulate quantum theory in a way which is consistent with these more general spacetimes. The benefits of constructing a relativistic quantum theory actually go far beyond the aesthetic satisfaction of making our geometrical and quantummechanical reasoning compatible. For one thing, we shall discover that the relativistic theory provides a deeper understanding of spin and the distinction between fermions and bosons, which in the non-relativistic theory appear simply as facts of life that we must strive to accommodate. Also, of course, there are many situations in which relativistic effects become observable, for which nonrelativistic theory provides no explanation. The most obvious are high-energy scattering experiments, in which particles acquire kinetic energies comparable with or greater than their rest energies mc2, and the correct 4-momentum (3.34) must be used. There are, however, more subtle effects, such as the spin-orbit coupling that is essential for interpreting atomic spectra, which are also of relativistic origin.