We developed a quantum field theory of spinors based on the algebra of canonical anticommutation relations (CAR algebra) [1]. The proposed approach combines and expands the approaches of algebraic quantum field theory [2] and theory of algebraic spinors [3]. It is based on the use of Grassmann densities in the momentum space and derivatives with respect to them [4-5] and the construction from these densities of both basic Clifford vectors of spacetime and spinor vacuum [5].

P, C, and T transformations are defined as operators that change basis Clifford vectors, but do not change components of spinors and vectors. We have shown that, with this approach, C and T are Clifford complex conjugation and Clifford transposition operators invariant to the choice of matrix or other linear representation of the Clifford group. And that they can be exact symmetries only in phenomena in which tensor quantities appear or in those where only spinors or only conjugated spinors are involved. A symmetry operator iQ also exists for electrically charged spinors. It is a reflection operator of two basic Clifford vectors corresponding to the internal degrees of freedom of spinors.

We have shown that P, CT, iQ, and CTP can be exact symmetries of spinors, and that CTP is a generalized Dirac conjugation [4] operator.

1. Gårding L., Wightman A. Proc. Nat. Acad. Sci. USA, 1954, v. 40, p. 617.
2. Haag R., Kastler D. Journal of Mathematical Physics, 1964, v. 5, p. 848.
3. Lounesto P. Clifford algebras and spinors. Cambridge University Press, 2001.
4. Monakhov V. V. Theoretical and Mathematical Physics, 2019, v. 200, p. 1026.
5. Monakhov V. Universe, 2019, v. 5, p. 162.