Correspondence between 3<i>BF</i> and Einstein–Cartan formulations of quantum gravity

Abstract We construct a correspondence between the quantized constrained 3 BF theory and the quantized Einstein–Cartan theory with contact spin–spin interaction, both of which describe the Standard Model coupled to Einstein–Cartan gravitational field. First we introduce the expectation values of observables using the path integral formalism for both theories, and then by integrating out some configuration space variables in the quantum 3 BF theory we obtain the definition of the corresponding observable in the quantum Einstein–Cartan theory with contact interaction. The correspondence is a rather general result, since it can be established without actually performing the detailed quantization of either theory. Finally, we discuss the differences in the predictions of the two theories on the example of the 4-volume density of spacetime, and on the example of gravitational waves.