Unified approach to classical and quantum dualities

          @misc{ pirsa_PIRSA:09110001,
            doi = {10.48660/09110001},
            url = {https://pirsa.org/09110001},
            author = {Ortiz, Gerardo},
            keywords = {Quantum Information},
            language = {en},
            title = {Unified approach to classical and quantum dualities},
            publisher = {Perimeter Institute},
            year = {2009},
            month = {nov},
            note = {PIRSA:09110001 see, \url{https://pirsa.org}}
          }
          

Abstract

Dualities appear in nearly all disciplines of physics and play a central role in statistical mechanics and field theory. I will discuss in a pedagogical way our recent findings motivated by a quest for a simple unifying framework for the detection and treatment of dualities. I will explain how classical and quantum dualities, as well as duality relations that appear only in a sector of certain theories (i.e. 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 (non-local mappings between the elementary degrees of freedom of the theory) which were guessed in the past can be derived in our formalism. New self-dualities for four-dimensional Abelian gauge field theories will be discussed.