Scale Invariance Breaking and Discrete Phase Invariance in Few-Body Problems

• URL: http://arxiv.org/abs/2601.09266v1
• Date: Wed, 14 Jan 2026 08:00:00 GMT
• Title: Scale Invariance Breaking and Discrete Phase Invariance in Few-Body Problems
• Authors: Satoshi Ohya,
• Abstract summary: Scale invariance in quantum mechanics can be broken in several ways.<n>We show that continuous scale invariance can be broken to discrete phase invariance in a small window of coupling constant.<n>We present three examples of few-body problems that exhibit discrete phase invariance.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Scale invariance in quantum mechanics can be broken in several ways. A well-known example is the breakdown of continuous scale invariance to discrete scale invariance, whose typical realization is the Efimov effect of three-body problems. Here we discuss yet another discrete symmetry to which continuous scale invariance can be broken: discrete phase invariance. We first revisit the one-body problem on the half line in the presence of an inverse-square potential -- the simplest example of nontrivial scale-invariant quantum mechanics -- and show that continuous scale invariance can be broken to discrete phase invariance in a small window of coupling constant. We also show that discrete phase invariance manifests itself as circularly distributed simple poles on Riemann sheets of the S-matrix. We then present three examples of few-body problems that exhibit discrete phase invariance. These examples are the one-body Aharonov-Bohm problem, a two-body problem of nonidentical particles in two dimensions, and a three-body problem of nonidentical particles in one dimension, all of which contain a codimension-two ``magnetic'' flux in configuration spaces.

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