Group Representational Clues to a Theory Underlying Quantum Mechanics

TL;DR

This paper suggests that quantum mechanics' structure can be derived from an underlying linear equation invariant under a group of transformations, providing a group-theoretic foundation for the theory.

Contribution

It proposes a new underlying theoretical framework based on a linear differential equation invariant under a combined group, explaining the group structure of quantum mechanics.

Findings

Quantum states can be represented as functions of independent variables with group labels.

The underlying theory is based on a linear, invariant differential operator.

Provides insights into gauge theory and internal symmetries.

Abstract

The current form of quantum mechanics is very successful and is almost certainly correct. It is remarkable, however, that the entire structure-from the mass, spin and charge labels on particlelike states to antisymmetry to broken internal symmetries to gauge transformations to the equations of motion-is built upon concepts from group representation theory. That is, the theory is constructed exactly as if it were a representational form of an underlying theory. Our proposed form for the underlying theory is that it is based on a linear equation, OF(V)=0. F is a function of some set of independent, currently unknown variables V, with O being a linear, partial differential operator in those variables. The operator is assumed to be invariant under a group of transformations of the Vs, homomorphic to the direct product of the inhomogeneous Lorentz group and the internal symmetry group. In such a theory, a state vector, denoted by a ket with group-theoretic labels, would represent a function of the independent variables. In addition to explaining the group representational structure of quantum mechanics, an underlying theory offers insight into gauge theory.