Considering a superposition of classical reference frames

A ubiquitous feature of quantum mechanical theories is the existence of states of superposition. This is expected to be no different for a quantum gravity theory. Guided by this consideration and others we consider a framework in which classical reference frames may be in superposition relative to one another. Mirroring standard quantum mechanics we introduce a complex-valued wavefunctional, which takes as input the transformations between the coordinates, $\Psi[x(x')]$, with the interpretation that an interaction between the reference frames may select a particular transformation with probability distribution given by the Born rule - $P[x(x')] = \text{probability distribution functional} \equiv \vert \Psi[x(x')] \vert^2$. The cases of two and three reference frames in superposition are considered explicitly. It is shown that the set of transformations is closed. A rule for transforming wavefunctions from one system to another system in superposition is proposed and consistency with the Schrodinger equation is demonstrated.