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.