On the Dirac Equation in a Gravitation Field and the Secondary Quantization

The Dirac equation for massive free electrically neutral spin 1/2 particles in a gravitation field is considered. The secondary quantization procedure is applied to it and the Hilbert space of multiparticle quantum states is constructed. 1. The Dirac equation and its current. Let M be a space-time manifold. It is a four-dimensional orientable manifold equipped with a pseudo-EuclideanMinkowski-type metric g and with a polarization. The polarization of M 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 of this bundle). In addition to the metric tensor g inherited from M , the Dirac bundle DM is equipped with four other basic spin-tensorial fields: Symbol Name Spin-tensorial type g Metric tensor (0, 0|0, 0|0, 2) d Skew-symmetric metric tensor (0, 2|0, 0|0, 0) H Chirality operator (1, 1|0, 0|0, 0) D Dirac form (0, 1|0, 1|0, 0) γ Dirac γ-field (1, 1|0, 0|1, 0) (1.1) As we see in the table (1.1), the metric tensor g is interpreted as a spin-tensorial field of the type (0, 0|0, 0|0, 2). The Dirac bundle is a complex bundle over a real manifold. For this reason spin-tensorial bundles produced from DM are equipped with the involution of complex conjugation τ : τ −−−−→ D β D̄ γT n M D γD̄ βT n M. ←−−−− τ (1.2) Note that two fields g and D in (1.1) are real fields: τ(g) = g, τ(D) = D. 2000 Mathematics Subject Classification. 81T20, 83C47.