The behavior of renormalization and related observables

TL;DR

This paper introduces new reference observables for the renormalization group, analyzes their behavior in the 1D Ising model, and establishes a scaling formula, with implications for both static and dynamic systems.

Contribution

It develops a new set of observables and a corresponding renormalization scaling equation, extending the understanding of correlation decay and system independence.

Findings

Two point observables decay exponentially away from criticality

New observables align with physical renormalization procedures

Results extend to finite point observables and are system-parameter independent

Abstract

In this paper, we introduce new reference observables to establish a scaling formula in the renormalization group equation. Using the transfer matrix method, we calculate the two point observables of the one dimensional Ising model without an external field under general boundary conditions. The results indicate that the two point observables exhibit exponential decay as the distance between these two sites tends to infinity, except at the critical point. Corresponding to the renormalization procedure underlying the correlation function, we establish a similar procedure for new observables, which aligning with findings in physics. Additionally, from the dynamic point of view, we construct a random system using the stochastic quantization method. We calculate the new observables of this random system under the initial distribution that satisfies Dobrushin Lanford Ruelle(DLR) equations. Furthermore, we formulate a new renormalization scaling equation with respect to the two point observables. Finally, these results can be extended to a more general case of finite point observables, and demonstrating independence from the choice of system parameters.