Comparison of Two Formulas for Metric Connections in the Bundle of Dirac Spinors

Dirac spinors play crucial role in modern particle physics. However, this crucial application of Dirac spinors is based mostly on the special relativity, where the base manifold M is the flat Minkowski space. Passing to the general relativity, we get a little bit more complicated theory of spinors. LetM be a space-time manifold of the general relativity. It is a four-dimensional orientable manifold equipped with a pseudo-Euclidean Minkowski-type metric g and with a polarization. The polarization ofM is responsible for distinguishing the Future light cone from the Past light cone at each point p ∈ M (see [1] for more details). Let’s denote by DM the bundle of Dirac spinors over M (see [2] and [3] for detailed description). In addition to the metric tensor g inherited from M , the Dirac bundle DM is equipped with four other basic spin-tensorial fields: