Implementation of the Quantum Equivalence Principle

The quantum equivalence principle says that, for any given point, it is possible to find a quantum coordinate system with respect to which we have definite causal structure in the vicinity of that point. It is conjectured that this principle will play a similar role in the construction of a theory of Quantum Gravity to the role played by the equivalence principle in the construction of the theory of General Relativity. To actually implement the quantum equivalence principle we need a suitable notion of quantum coordinate systems - setting up a framework for these is the main purpose of the present paper. First we introduce a notion of extended states consisting of a superposition of terms (labeled by $u$) where each term corresponds to a manifold, $\mathcal{M}_u$, with fields defined on it. A quantum coordinate system consists of an identification of points between some subsets, $\mathcal{O}_u\subseteq \mathcal{M}_u$, of these manifolds along with a coordinate, $x$, that takes the same value on those points identified. We also introduce a notion of quantum coordinate transformations (which can break the identification map between the manifolds) and show how these can be used to attain definite causal structure in the vicinity of a point. We discuss in some detail how the quantum equivalence principle might form a starting point for an approach to constructing a theory of Quantum Gravity that is analogous to way the equivalence principle is used to construct General Relativity.