Properties of the Hamiltonian Renormalisation and its application to quantum mechanics on the circle

TL;DR

This paper studies how different embedding choices affect the Hamiltonian renormalisation group flow in discretised quantum theories on a circle, exploring continuum limits, operator algebras, and equivalence notions.

Contribution

It analyzes the impact of embedding maps on RG flow and continuum limits, proposing prescriptions for operator algebras and a weaker form of distributional equivalence.

Findings

Different embedding maps lead to unitarily inequivalent continuum limits.

Preferred renormalisation prescriptions can ensure algebraic relations in the continuum.

Weak distributional equivalence can relate inequivalent continuum limits.

Abstract

We consider the Hamiltonian renormalisation group flow of discretised one-dimensional physical theories. In particular, we investigate the influence the choice of different embedding maps has on the RG flow and the resulting continuum limit, and show in which sense they are, and in which sense they are not equivalent as physical theories. We are furthermore elucidating the interplay of the RG flow and the algebras operators satisfy, both on the discrete and the continuum. Further, we propose preferred renormalisation prescriptions for operator algebras guaranteeing to arrive at preferred algebraic relations in the continuum, if suitable extension properties are assumed. Finally, we introduce a weaker form of distributional equivalence, and show how unitarily inequivalent continuum limits, which arise due to a choice of different embedding maps, can still be weakly equivalent in that sense.