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the nature of quantum theory

Moffat, James

Published Jan 1, 2018• 0 citations
Publisher

Abstract

A discussion of the fundamental nature of non-relativistic and relativistic Quantum Theory, including Dirac spinors and the Dirac Operator.
DOI
10.13140/rg.2.2.11200.17923

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We first give a geometrical description of the action of the parity operator ( ˆ P) on non relativistic spin 1 2 Pauli spinors in terms of bundle theory. The relevant bundle, SU(2) ⊙Z2 → O(3), is a non trivial extension of the universal covering group SU(2) → SO(3). ˆ P is the non relativistic limit of the corresponding Dirac matrix operator P = iγ0 and obeys ˆP 2 = −1. Then, from the direct product of O(3) by Z2, naturally induced by the structure of the galilean group, we identify, in its double cover, the time reversal operator ( ˆ T) acting on spinors, and its product with ˆ P. Both, ˆ P and ˆ T, generate the group Z4 × Z2. As in the case of parity, ˆT is the non relativistic limit of the corresponding Dirac matrix operator T = γ 3 γ 1, and obeys ˆ T 2 = −1. Key words: parity transformation; time reversal; non relativistic spinors; group theory. 1.
The axiomatic foundations of quantum mechanics, while operationally successful, obscure the underlying physical reality. This work introduces a mechanical topological ontology intended to provide a causal basis for the theory. We propose a single postulate: that a fundamental fermion is the 0-dimensional boundary of a 1-dimensional, massive, string-like object topologically anchored to a dynamic vacuum field. From this postulate, we argue that the core structures of single-particle relativistic quantum mechanics can be understood as derivable theorems. First, we construct the object’s state space by identifying two independent, binary properties—a local, vibrational/topological duality (spin) and a global, orientational duality (matter/antimatter)—whose tensor product naturally yields the 4-component Dirac spinor. Second, we show that the requirement of relativistic consistency uniquely determines the algebraic operators (gamma matrices and their Clifford algebra) and the lowest-order dynamical law, leading to the Dirac equation as a theorem of consistency. Third, we demonstrate that the non-relativistic Schrödinger and Pauli equations emerge as its correct low-energy limits, where the quantum potential V (x) is identified with a local variation in the object’s rest mass. Finally, we argue that the foundational tenets of quantum measurement—the spin-1/2 property, the probabilistic Born rule, and the correlations of entanglement—can be reframed as necessary consequences of the object’s intrinsic topology and interaction mechanics. This framework thus offers a path toward demoting the axioms of quantum theory to derivable theorems, aiming to resolve its long-standing paradoxes by providing a clear and coherent physical picture.