Effective and exact holographies from symmetries and dualities

TL;DR

This paper explores the theoretical foundations of holographic principles in physics, deriving inequalities and dualities that explain effective and exact dimensional reduction across various systems, with implications for quantum information storage.

Contribution

It introduces a unified framework linking symmetries and dualities to holographic dimensional reduction, including new algebraic methods for exact dualities in quantum systems.

Findings

Derived inequalities linking different dimensional quantum systems.

Demonstrated invariance of local density of states under certain transformations.

Applied bond algebra techniques to topologically ordered systems.

Abstract

The theoretical basis of the phenomenon of effective and exact dimensional reduction, or holographic correspondence, is investigated in a wide variety of physical systems. We first derive general inequalities linking quantum systems of different spatial (or spatio-temporal) dimensionality, thus establishing bounds on arbitrary correlation functions. These bounds enforce an {\em effective} dimensional reduction and become most potent in the presence of certain symmetries. {\em Exact} dimensional reduction can stem from a duality that (i) follows from properties of the local density of states, and/or (ii) from properties of Hamiltonian-dependent algebras of interactions. Dualities of the first type (i) are illustrated with large-$n$ vector theories whose local density of states may remain invariant under transformations that change the dimension. We argue that a broad class of examples of dimensional reduction may be understood in terms of the functional dependence of observables on the local density of states. Dualities of the second type (ii) are obtained via {\em bond algebras}, a recently developed algebraic tool. We apply this technique to systems displaying topological quantum order, and also discuss the implications of dimensional reduction for the storage of quantum information.