The Poincaré Gauge Theory of Gravity and the Immirzi Parameter

Abstract

The minimal coupling method proved to yield definite and correct physical predictions when applied to fundamental fermions within the framework of Yang-Mills theories of Standard Model. Similarly, the possibility of formulating gravity as the Poincaré gauge theory gives the opportunity to produce definite predictions for fermions in the presence of gravitational field. The minimal coupling procedure, however, cannot be applied naively but rather needs to be modified slightly such that it is unambiguous. Application of the corrected coupling method to fermions, together with the inclusion of the Holst term in the gravitational part of the action, leads to the conclusion that the Immirzi parameter is in principle classically measurable, in agreement with the result of Perez and Rovelli.All the relativistic theories in the absence of gravity are invariant under the (global) action of the Poincaré group. As shown by by Kibble [1], it is possible to obtain a reasonable theory of gravity by simply localizing this global symmetry. This approach necessitates first order formalism for general relativity, with metric (but non-symmetric) connection, as the set of Yang-Mills fields has to consist of ten independent one-forms. Interestingly, this formalism appears to simplify the canonical analysis and quantization of the theory and therefore it is employed in Loop Quantum Gravity (LQG), although in standard LQG the time gauge is imposed at the very beginning that breaks the gauge group effectively to SU (2).The natural way to introduce the interaction within the spirit of Yang-Mills theory is to use the minimal coupling procedure (MCP). Indeed, in the standard model of particle physics this procedure is followed on the fundamental level, leading to predictions that agree with experimental results with great accuracy. Also, in GR the principle of equivalence, which states that the effects of gravitation can be locally 'turned off' by a suitable choice of a reference frame, necessitates minimal coupling [2]. However, in the Poincaré gauge theory of gravity MCP appears to be ambiguous. The ambiguity is of importance for the standard EC theory with fermions [3], as well as the theory modified by the addition of the Holst term [5]. The predictions concerning fermions in the Ashtekar-Barbero-Immirzi formalism obtained in [4] can be radically changed if the freedom of adding divergence of a vector field to the initial fermionic Lagrangian density is exploited [5].Luckily, the corrected unambigoues coupling procedure has been proposed [8] which makes the predictions of the theory unique. They appear to agree with those derived by Perez and Rovelli. II. THE POINCARÉ GAUGE THEORY A. Yang-Mills theoriesThe leading idea of standard Yang-Mills theories is that any interaction can be associated with a symmetry group. LetG be a Lie group and letrepresent the action of a field theory of a matter field φ in Minkowski space M . Here L m is a Lagrangian density and L m a Lagrangian four-form. Assume that V is a (finite dimensional) linear spac...