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Next, we examine the integration for f_n (x). Consider first the region

(A.137) can be written as

Introduce

and

When n = 0, we set

From (A.150), or more conveniently by using (x|G|z) given by (A.130), one can readily verify that, for α < x < γ,

etc. These solutions can also be readily derived by directly using the differential equation satisfied by ψ_n (x) = χ (x)f_n (x):

where in accordance with (A.14), . For α < x < γ, we have

and therefore

Introduce S_n (ξ) and C_n (ξ) to be polynomials in ξ, with

From Figs. (A.153), (A.157) and (A.158), we find

where the dot denotes , so that , etc. At x = γ, we have ξ = 0, and therefore

For n = 0, S₀ (ξ) = 1 and C₀ (ξ) = 0. Therefore, for n = 1, (A.159) becomes

Assuming S₁ and C₁ to be both polynomials of ξ, we can readily verify that S₁ is a constant and C₁ is proportional to ξ. Using (A.161) and the boundary condition (A.160), we can establish the first equation in (A.155), and likewise the other equations for n > 1.

To understand the structure of v₁ (ξ), v₂ (ξ), v₃ (ξ), … , we may turn to the exact solution ψ (x) given by (A.123). In analogy to (A.153), we define v (ξ) through

Thus, for α < x < γ,

From Figs. (A.13), (A.14) and (A.118), we have

In terms of

we write

with ξ given by (A.151), as before. It is straightforward to expand v (ξ) as a power series in :

To compare the above series with v_n (ξ) of (A.155), we can neglect O ( ^{n + 1}) in (A.167). The replacements of all linear -terms by _n, ²-terms by _{n − 1 n}, ³-terms by _{n−2 n − 1 n}, etc. lead from (A.167) to v_n (ξ). It is of interest to note that the expansion (A.167) of v (ξ) in power of has a radius of convergence

On the other hand, the iterative sequence {v_n (ξ)} is always convergent, on account of the Hierarchy Theorem. The main difference between Figs. (A.155) and (A.167) is that in (A.155) each iterative _n is determined by the fraction (A.143).

In a similar way, we can derive ψ_n (x) in other regions, −α < x < α and −γ < x < −α. The results for n = 1 are given in Table 2. The functions ₁ (x) and ξ (x) are discontinuous from region to region. The constants κ_II and ρ_II are determined by requiring ψ₁ (x) and and to be continuous at x = α. In region I, when x = α+, we have

and

where the constant

with

In region II, when x = α−

and

where the constant

The constants κ_II and ρ_II are determined by

Likewise, the constants κ_III and ρ_III are given by

and the constants κ_IV and E₁ are determined by

Table 2.

The n = 1 iterative solution ψ₁ (x) in the four regions: I (α < x < γ), II (0 < x < α), III (−α < x < 0), and IV (−γ < x < −α)

The constants , κ_II, κ_III, κ_IV, ρ_II, and ρ_III are given by Figs. (A.176), (A.177) and (A.178).

Star products and geometric algebra

Peter Henselder , Allen C. Hirshfeld ^, and Thomas Spernat

Fachbereich Physik, Universität Dortmund 44221, Dortmund, Germany

Received 10 August 2004; accepted 20 September 2004. Available online 1 February 2005.

Abstract

The formalism of geometric algebra can be described as deformed super analysis. The deformation is done with a fermionic star product, that arises from deformation quantization of pseudoclassical mechanics. If one then extends the deformation to the bosonic coefficients of superanalysis one obtains quantum mechanics for systems with spin. This approach clarifies on the one hand the relation between Grassmann and Clifford structures in geometric algebra and on the other hand the relation between classical mechanics and quantum mechanics. Moreover it gives a formalism that allows to handle classical and quantum mechanics in a consistent manner.

PACS: 03.65; 03.40.D; 03.65.F
Keywords: Star products; Geometric algebra; Deformation quantization

Article Outline

1. Introduction

2. The star product formalism

3. Geometric algebra and the Clifford star product

4. Geometric algebra and classical mechanics

5. Non-relativistic quantum mechanics

6. Spacetime algebra and Dirac theory

7. Conclusions

References

1. Introduction

Geometric algebra goes back to early ideas of Hamilton, Grassmann, and Clifford. But it was first developed into a full formalism by Hestenes in [1] and [2]. The formalism of geometric algebra is based on the definition of the geometric or Clifford product. This product is for vectors defined as the sum of the scalar and the wedge product and equips the vector space with the algebraic structure of a Clifford algebra. This structure then proved to be a very powerful tool, that allows to describe and generalize the structures of vector analysis, of complex analysis and of the theory of spin in a unified and clear formalism. The formalism can then be used to describe classical mechanics in the realm of geometric algebra instead of linear algebra, which is advantageous in many respects [3] and [4]. It is also possible to generalize the formalism from the algebra of space to the algebra of spacetime in order to describe electrodynamics and special relativity [1] and [4].

In quantum mechanics the Clifford structures of the σ- and the γ-matrices correspond to the structures of geometric algebra. So by formulating classical physics and quantum physics with geometric algebra one achieves a formal unification of both areas on a geometric level. Nevertheless this formulation is conceptually not totally unified, because classical mechanics is still formulated on the phase space while quantum mechanics is formulated in Hilbert space. To achieve a totally unified formulation we will here combine geometric algebra with the star product formalism. The star product formalism [5] appears in the context of deformation quantization where one describes the non-commutativity that enters physics in quantum mechanics not by using non-commuting objects like operators, but by introducing a non-commutative product on the phase space that replaces the conventional product of functions. This star product is so constructed that the quantized star product of two phase space functions corresponds to the operator product of the quantized factors, which then allows to do quantum mechanics on phase space. To include spin in this formalism we used in [6] and [7] fermionic star products that result directly from deformation quantization of pseudoclassical mechanics [8]. Fermionic star products were already discussed in [5], where it was noticed that they lead to a cliffordization of the underlying Grassmann algebra. So it is possible to describe a Clifford algebra as a deformed Grassmann algebra, where this deformation is nothing else than Chevalley cliffordization [7].

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