From Grassmann to Clifford

Abstract

The aims of this note are to convince the readers in as elementary as possible way that: 1. Clifford algebra is the particularcase of the Grassmann algebra which is the most fundamental from the point of view of the physics, geometry and analysis. Grassmann algebra with the (pseudo-) Riemannian structure (including degenerate case), or with the (pre-) symplectic structure, or with the Hermitian (Hilbert) structure, contain the corresponding Riemannian or symplectic or Hermitian Clifford algebra. 2. Grassmann algebra is the tautology of the formalism of the (multi-) fermion and antifermion creation and annihilation operators. This relation does not need at all to fix any kind of the Riemannian (Euclidean) or Hilbertian, Hermitian, etc, structures. 3. Clifford algebra is the tautology of the formalism of the quasi-particles in nuclear physics and therefore can be viewed as the Bogoliubov transformation of the Grassmann algebra. 4. The last two statements open wide possibilities of the applications of the Grassmann and Hermitian Clifford algebras particularly to nucleons and quarks in the nuclear shell theory. 5. The Clifford product can always be reexpressed in terms of the fermion creation and annihilation operators. 6. It is interesting that symplectic Clifford algebras has not yet been explored in the classical Lagrangian and Hamiltonian mechanics.