Unified Approach to Quantum and Classical Dualities

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 have been guessed in the past, can be derived in our formalism. We obtain new self-dualities for four-dimensional Abelian gauge field theories.