Fluctuation Operators and Spontaneous Symmetry Breaking

TL;DR

This paper introduces a new approach to analyzing fluctuation operators and spontaneous symmetry breaking, focusing on limits of correlation functions and employing Fourier and spectral methods to clarify different fluctuation regimes.

Contribution

It presents an alternative, simplified method for studying fluctuation operators, complementing existing approaches by directly analyzing correlation functions and using smooth spatial averaging.

Findings

Correlation functions vanish for l≥3 under proper scaling

Normal and poor clustering fluctuation regimes are characterized

Spontaneous symmetry breaking and Goldstone phenomena are analyzed

Abstract

We develop an alternative approach to this field, which was to a large extent developed by Verbeure et al. It is meant to complement their approach, which is largely based on a non-commutative central limit theorem and coordinate space estimates. In contrast to that we deal directly with the limits of $l$-point truncated correlation functions and show that they typically vanish for $l\geq 3$ provided that the respective scaling exponents of the fluctuation observables are appropriately chosen. This direct approach is greatly simplified by the introduction of a smooth version of spatial averaging, which has a much nicer scaling behavior and the systematic developement of Fourier space and energy-momentum spectral methods. We both analyze the regime of normal fluctuations, the various regimes of poor clustering and the case of spontaneous symmetry breaking or Goldstone phenomenon.