Spacetime and Physical Equivalence
In this paper I begin to lay out a conceptual scheme for: (i) analysing dualities as cases of theoretical equivalence; (ii) assessing when cases of theoretical equivalence are also cases of physical equivalence. The scheme is applied to gauge/gravity dualities. I expound what I argue to be the contribution of gauge/gravity dualities to questions about: (iii) the nature of spacetime in quantum gravity; (iv) broader philosophical and phyiscal discussions of gauge/gravity dualities. (i) (ii) proceed by analysing duality through four contrasts. A duality will be a suitable isomorphism between models: and the four relevant contrasts are as follows: (a) Bare theory: a triple of states, quantities, and dynamics endowed with appropriate structures and symmetries; vs. interpreted theory: which is endowed with, in addition, a suitable pair of interpretative surjections from the triples to a domain in the world. (b) Extendable vs. unextendable theories: which can, respectively cannot, be extended as regards their domains of applicability. (c) External vs. internal intepretations: which are constructed by coupling the theory to another interpreted theory, respectively from within the theory itself. (d) Theoretical vs. physical equivalence: which distinguish formal equivalence from the equivalence of fully interpreted theories. I also discuss three meshing conditions: between symmetries and duality, between symmetries and interpretation, and between duality and interpretation. The meshing conditions lead to a characterisation of symmetries as redundant or as physical, and to a characterisation of physical equivalence as a commutativity condition between two maps. I will apply the above scheme to answering questions (iii) (iv) for gauge/gravity dualities. I will argue that the things that are physically relevant are those that stand in a bijective correspondence under gauge/gravity duality: the common core of the two models. I therefore conclude that most of the mathematical and physical structures that we are familiar with in these models (the dimension of spacetime, tensor fields, Lie groups, and the classical quantum distinction) are largely, though crucially never entirely, not part of that common core. Thus, the interpretation of dualities for theories of quantum gravity compels us to rethink the roles that spacetime, and many other tools in theoretical physics, play in theories of spacetime.