Quantization in the neighborhood of a classical solution in the theory of a Fermi field

The quantization of a Fermi–Bose field system in the neighborhood of a classical solution of the equations of motion that contains both bosonic and spinor components is considered. The latter is regarded as an absolutely anticommuting (Grassmann) component of a fermion field. On account of the transport of the fermion number, such an object mixes the fermionic and bosonic and fermionic and antifermionic degrees of freedom already at the level of the single-particle states (in the approximation of quadratic forms). Explicit expressions are obtained for the operator of the $S$ matrix, which describes such transport processes, and the total Hamiltonian and total fermion charge of the system in this approximation.