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In this paper, we will use the fact that geometric algebra can be formulated in terms of a Grassmann algebra [9]. We will show that in this context the geometric product can be made explicit as a fermionic star product. It is then straight forward to translate classical mechanics described with geometric algebra into a version where it is described in terms of fermionic deformed super analysis. The fermionic part of the formalism represents hereby the basis vector structure of the space on which the theory is formulated, i.e., the three-dimensional space, the phase space or the spacetime. In all cases we consider only the case of flat spaces. In a second step one can then go over to quantum mechanics, where we use here deformation quantization, while in [9] canonical and path integral quantization was used. Combining in this way geometric algebra formulated with a fermionic star product with the bosonic star product of deformation quantization one arrives at a supersymmetric star product formalism that allows to describe quantum mechanics with spin in a unified manner. Moreover by using star products one can immediately give the classical → 0 limit and see how the spin as a physical observable vanishes. Furthermore, one can see that classical mechanics can be described as a half deformed theory, while quantum mechanics is a totally deformed theory, i.e., in classical mechanics the star product acts only on the fermionic basis vector part of the formalism, while for > 0 there exists also a bosonic star product that acts on the coefficients of the basis vectors.

In Section 2 we will very shortly review the bosonic and fermionic star product formalism and show how quantum mechanics with spin can be described in this context. Then we will show how geometric algebra can be formulated with the fermionic star product. We will therefore formulate well-known results of geometric algebra in the formalism of fermionic deformed superanalysis. Afterwards in Sections 5 and 6 we will extend the formalism to the case of non-relativistic quantum mechanics and Dirac theory by using the bosonic Moyal product.

2. The star product formalism

We first want to introduce the star product formalism in bosonic and fermionic physics with the example of the harmonic oscillator [5]. The bosonic oscillator with the Hamilton function , can be quantized by using the Moyal product

The star product replaces the conventional product between functions on the phase space and it is so constructed that the star anticommutator, i.e., the antisymmetric part of first order, is the Poisson bracket:

This relation is the principle of correspondence. The states of the quantized harmonic oscillator are described by the Wigner functions . The Wigner functions and the energy levels E_n of the harmonic oscillator can be calculated with the help of the star exponential

where Hⁿ _M=H _M… _MH is the n-fold star product of H. The star exponential fulfills the analogue of the time dependent Schrödinger equation

The energy levels and the Wigner functions fulfill the -genvalue equation

and for the harmonic oscillator one obtains and

where the L_n are the Laguerre polynomials. The Wigner functions are normalized according to and the expectation value of a phase space function f can be calculated as

The same procedure can now be used for the grassmannian case [6]. The simplest system in grassmannian mechanics [8] is a two-dimensional system with Lagrange function

With the canonical momentum

the Hamilton function is given by

Together with Eq. (2.9) this Hamiltonian suggests that the fermionic oscillator describes rotation. Indeed, calculating the fermionic angular momentum, which corresponds to the spin, leads to

so that the Hamiltonian in (2.10) can also be written as H = ωS₃. As a vector the angular momentum points out of the θ₁-θ₂-plane. Therefore, we consider the two-dimensional fermionic oscillator as embedded into a three-dimensional fermionic space with coordinates θ₁, θ₂, and θ₃. Note that we choose both for the fermionic space and momentum coordinates the units .

Quantizing the fermionic oscillator [6] involves a star product that is given by

We will call this star product the Clifford star product because it leads to a cliffordization of the Grassmann algebra of the θ_i. This can be seen by considering the star-anticommutator that is given by

Since the Grassmann variables

fulfill the relations

with [σⁱ,σ^j] _C=σⁱ _Cσ^j-σ^j _Cσⁱ, they correspond to the Pauli matrices. From equations Figs. (2.11) and (2.14) it follows that and . Note that {1, σ¹, σ², σ³} is a basis of the even subalgebra of the Grassmann algebra and that this space is also closed under _C multiplication.

In the space of Grassmann variables there exists an analogue of complex conjugation, which is called the involution. As in [8] it can be defined as a mapping , satisfying the conditions

where c is a complex number and its complex conjugate. For the generators θ_i of the Grassmann algebra we assume , so that for σⁱ defined in (2.14) the relation holds true. This corresponds to the fact that the 2 × 2 Pauli matrices are hermitian.

We now define the Hodge dual for Grassmann numbers with respect to the metric δ_ij. The Hodge dual maps a Grassmann monomial of grade r into a monomial of grade d−r, where d is the number of Grassmann basis elements (which is in our case three):

With the help of the Hodge dual one can define a trace as

The integration is given by the Berezin integral for which we have ∫dθ_iθ_j= δ_ij, where the on the right-hand side is due to the fact that the variables θ_i have units of . The only monomial with a non-zero trace is 1, so that by the linearity of the integral we obtain the trace rules

With the fermionic star product (2.12) one can—as in the bosonic case—calculate the energy levels and the -eigenfunctions of the fermionic oscillator. This can be done with the fermionic star exponential

where the Wigner functions are given by

The fulfill the -genvalue equation for the energy levels . The Wigner functions are complete, idempotent and normalized with respect to the trace, i.e., they fulfill the equations

respectively. Furthermore, they correspond to spin up and spin down states since (2.21) corresponds to the spin projectors and the expectation values of the angular momentum are

where the spin was used with components of as defined in (2.14).

In the fermionic θ-space the spin is the generator of rotations, which are described by the star exponential

where we used the definition with rotation angle and a rotation axis given by the unit vector . The vector transforms passively according to

with being the well-known SO (3) rotation matrix. The axial vector transforms in the same way. Note that the passive transformation (2.25) of the θ_i amounts to an active transformation of the components x_i in the vector .

3. Geometric algebra and the Clifford star product

Starting point for geometric algebra [1] and [3] is an n-dimensional vector space over the real numbers with vectors a,b,c, … A multiplication, called geometric product, of vectors can then be denoted by juxtaposition of an indeterminate number of vectors so that one gets monomials A, B, C, … These monomials can be added in a commutative and associative manner: A + B = B + A and (A + B) + C = A + (B + C), so that they form polynomials also denoted by capital letters. The so obtained polynomials can be multiplied associatively, i.e., A (BC) = (AB) C and they fulfill the distributive laws (A + B) C = AC + BC and C (A + B) = CA + CB. Furthermore, there exists a null vector a0 = 0 and the multiplication with a scalar λa = aλ, with . The connection between scalars and vectors can be given if one assumes that the product ab is a scalar iff a and b are collinear, so that is the length of the vector a. These axioms define now the Clifford algebra Cℓ (V) and the elements A, B, C, … of Cℓ (V) are called Clifford or c-numbers.

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