The difference between Einstein's general relativity and its Cartan extension is analyzed within the scenario of asymptotic safety of quantum gravity. In particular, we focus on the four-fermion interaction which distinguishes the Einstein-Cartan theory from its Riemannian limit. 1. Introduction In the coupling of gravity to Dirac type spinor fields [1], it is at times surmised that the Einstein-Cartan (EC) theory [2] is superior to standard General Relativity (GR), inasmuch as the involved torsion tensor of Cartan [3, 4] can accommodate the spin of fundamental Fermions of electrons and quarks in gravity. However, classically,the effects of spin and torsion cannot be detected by Lageos or Gravity Probe B [5] and would be significant only at densities of matter that are very high but nevertheless smaller than the Planck density at which quantum gravitational effects are believed to dominate. It was even claimed [6] that EC theory may avert the problem of singularities in cosmology, but for a coupling to Dirac fields, the opposite happens [7–9]. Recently, it has been stressed by Weinberg [10–12] that the Riemann-Cartan (RC) connection , a one-form, is just a deformation of the Christoffel connection by the (con-)tortion tensor-valued one-form , at least from the field theoretical point of view. Although algebraically complying with [13], this argument has been refuted [14] on the basis of the special geometrical interpretation [15, 16] of Cartan’s torsion. It is well-known [17, 18] that EC theory coupled to the Dirac field is effectively GR with an additional four-fermion (FF) interaction. However, such contact interactions are perturbatively nonrenormalizable in without Chern-Simons (CS) terms [19], which was one of the reasons for giving up Fermi’s theory of the beta decay. Since GR with a cosmological constant appears to be asymptotically safe, in the scenario [20] first devised by Weinberg [21], one may ask [22] what the situation in EC theory is, where Cartan’s algebraic equation relates torsion to spin, that is, to the axial current in the case of Dirac fields, on dimensional grounds coupled with gravitational strength. 2. Dirac Fields in Riemann-Cartan Spacetime In our notation [13, 23–25], a Dirac field is a bispinor-valued zero-form for which denotes the Dirac adjoint and is the exterior covariant derivative with respect to the RC connection one-form , providing a minimal gravitational coupling. In the manifestly Hermitian formulation, the Dirac Lagrangian is given by the four-form where is the Clifford algebra-valued coframe, obeying , and is
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