Spatial Quantization in Three Dimensions

Gary R. Gruber

Jeff Maki, CARLOS R. ORDÓÑEZ

In this paper we explore the transport properties of three-component Fermi gases confined to one spatial dimension, interacting via a three-body interaction, in the high temperature limit. At the classical level, the three-body interaction is scale invariant in one dimension. However, upon quantization, an anomaly appears which breaks the scale invariance. This is very similar to the physics of two-component fermions in two spatial dimensions, where the two-body interaction is also anomalous. Previous studies have already hinted that the physics of these two systems are intimately related. Here we expand upon those studies by examining the thermodynamic properties of this anomalous one-dimensional system in the high temperature limit. We show there is an exact mapping between the traditional two-body anomalous interaction in two dimensions, to that of three-body interaction in one dimension. This result is valid in the high temperature limit, where the thermodynamics can be understood in terms of few-body correlations.

D. J. R. Lloyd-Evans

Ai Chen, (Anthropic) Claude Sonnet, (Anthropic) Claude Opus

The standard formulation of quantum mechanics relies on the Schrödinger equation and the Born rule but lacks a unified principle explaining why discreteness appears so universally. This paper proposes the Phase Quantization Principle: in any finite system, the total phase accumulated over a closed process must be a rational multiple of π. This is the necessary and sufficient condition for quantization. The special status of π does not arise from convention but from two structural facts: the topology of the three-dimensional rotation group SO(3) — whose fundamental group is Z₂ — and the spin-statistics of fermions, which acquire a phase of π under 2π rotation. π is therefore the minimum non-trivial physical phase in three spatial dimensions. The paper's primary new contribution is a phase-stability account of decoherent state selection: not only why superpositions disappear, but why the surviving definite states are those specific values. Four classes of quantum phenomena — energy quantization, geometric (Berry) phase, particle statistics, and post-decoherence state selection — are shown to be expressions of the same phase-closure condition.

Heimo G. Latal, W. Schweiger

Subir Ghosh, Bibhas Ranjan Majhi

We study complexified harmonic oscillator models in two and three dimensions. Our work is a generalization of the work of Smilga (2007 Preprint 0706.4064 ( J. Phys. A: Math. Theor. at press)) who initiated the study of these Crypto-gauge invariant models that can be related to PT -symmetric models. We show that rotational symmetry in higher spatial dimensions naturally introduces more constraints (in contrast to Smilga (2007 Preprint 0706.4064 ( J. Phys. A: Math. Theor. at press)) where one deals with a single constraint) with a much richer constraint structure. Some common as well as distinct features in the study of the same Crypto-oscillator in different dimensions are revealed. We also quantize the two dimensional Crypto-oscillator.