Nonuniqueness of gravity induced fermion interaction in the Einstein-Cartan theory

The problem of nonuniqueness of minimal coupling procedure for Einstein-Cartan (EC) gravity with matter is investigated. It is shown that the predictions of the theory of gravity with fermionic matter can radically change if the freedom of the addition of a divergence to the flat space matter Lagrangian density is exploited. The well-known gravity-induced four-fermion interaction is shown to reveal unexpected features. The solution to the problem of nonuniqueness of minimal coupling of EC gravity is argued to be necessary in order for the theory to produce definite predictions. In particular, the EC theory with fermions is shown to be indistinguishable from usual the general relativity on the effective level, if the flat space fermionic Lagrangian is appropriately chosen. Hence, the solution to the problem of nonuniqueness of minimal coupling procedure is argued to be necessary if EC theory is to be experimentally verifiable. It could also enable experimental tests of theories based on EC, such as the loop approach to quantization of gravitational field. Some ideas of how the arbitrariness incorporated in EC theory could be restricted or even eliminated are presented.