A New Approach to Quantum Gravity from a Model of an Elastic Solid

Abstract

We show that the dynamics of an elastic solid embedded in a Minkowski space consist of a set of coupled equations describing a spin-1/2 field, $\Psi$, obeying Dirac's equation, a vector potential, $A_\mu$, obeying Maxwell's equations and a metric, $g_{\mu\nu}$, which satisfies the Einstein field equations. The combined set of Dirac's, Maxwell's and the Einstein field equations all emerge from a simple elastic model in which the field variables $\Psi$, $A_\mu$ and $g_{\mu\nu}$ are each identified as derived quantities from the field displacements of ordinary elasticity theory. By quantizing the elastic field displacements, a quantization of all of the derived fields are obtained even though they do not explicitly appear in the Lagrangian. We demonstrate the approach in a three dimensional setting where explicit solutions of the Dirac field in terms of fractional derivatives are obtained. A higher dimensional version of the theory would provide an alternate approach to theories of quantum gravity.