Unified approach to Quantum and Classical Dualities

TL;DR

It is shown how classical and quantum dualities, as well as duality relations that appear only in a sector of certain theories (emergent dualities), can be unveiled, and systematically established, and new self-dualities for four-dimensional Abelian gauge field theories are obtained.

Abstract

We show how classical and quantum dualities, as well as duality relations that appear only in a sector of certain theories ("emergent dualities"), can be unveiled, and systematically established. Our method relies on the use of morphisms of the "bond algebra" of a quantum Hamiltonian. Dualities are characterized as unitary mappings implementing such morphisms, whose even powers become symmetries of the quantum problem. Dual variables -which were guessed in the past- can be derived in our formalism. We obtain new self-dualities for four-dimensional Abelian gauge field theories.

Figures (3)

• Figure 1: The plaquette variable $\Delta\theta_n^3=A^2_{n+e_1}-A^2_n-A^1_{n+e_2}+A^1_{n}$.
• Figure 2: Schematic of the bond algebra mapping \ref{['dual_mapping']}. Associated with each lattice site $n$, there are three $\Pi$ fields and three $\Delta \theta$ plaquettes. The $\Delta \theta$ plaquettes at site $n$ map by \ref{['dual_mapping']} to displaced $\Pi$s, each colored plaquette mapping to the $-\Pi$ at the correspondingly colored site.
• Figure 3: The effect of the exchange duality of \ref{['dual_mapping']} on the three $\Pi$ fields at site $n$.