Primary manifestation of symmetry. Origin of quantal indeterminacy

Abstract

Quantal physics is established as a manifestation of symmetry more far-reaching than hitherto appears to have been recognized. In this primary manifestation, the coordinate transformations of spacetime invariance are themselves the elementary variables, which define their own properties without appeal to an assumed quantal formalism. In irreducible representations, the symmetry variables are inherently indeterminate, and the probabilistic laws invoked in the interpretation of traditional quantum physics are found to originate in geometric relations between these variables. Completeness is, therefore, not an issue, and the quantum of action is not part of the theory of symmetry variables. Quantal physics thus emerges as but an implication of relativistic invariance, liberated from a substance to be quantized and a formalism to be interpreted. A symmetry variable appears in a measurement with one of its eigenvalues, but does not have a value (cannot be represented by a number) in an irreducible representation, which combines sets of eigenvalues. It is this generalized significance of a measurement that allows for correlations that cannot arise for classical variables. The observation of symmetry variables is illustrated by an interferometer experiment measuring reflection symmetry and by the equivalent coincidence experiment registering the polarization of two quanta. The measurement process becomes a matter of following the state of affairs of the symmetry variables in their unitary evolution. For the resolution of the dilemmas that quantal phenomena have been felt to pose, it appears crucial to recognize that indeterminacy, as an inherent property of a symmetry variable in a multidimensional representation, is not affected by subsequent observations. A position variable and the canonical commutator with momentum, which are basic elements of nonrelativistic quantum mechanics, emerge from spacetime symmetry, but require the link between space and time of relativistic invariance. The transition to the classical regime is analyzed in terms of a quenching of nonlocality in the state of affairs of the multidimensional symmetry variables. While the elementary variables constitute individual quanta in irreducible representations, product representations of spacetime symmetry describe systems of bosons and fermions, which form local fields with canonical properties. The discussion is focused on spacetime invariance (noninteracting quanta), but gauge invariance is itself a primary manifestation of symmetry and is as such encompassed by the theory of symmetry variables.