A discrete Dirac-K\"{a}hler equation using a geometric discretisation scheme

Discrete models of the Dirac-K\"{a}hler equation and the Dirac equation in the Hestenes form are discussed. A discrete version of the plane wave solutions to a discrete analogue of the Hestenes equation is established.


Introduction
We study a discrete model of the Dirac-Kähler equation in which key geometric aspects of the continuum counterpart are captured. We pay special attention to the description of some method of construction a discretization scheme based on the use of the differential forms calculus. The aim of this paper is to establish some discrete version of the plane wave solutions to a discrete Dirac equation in the Hestenes form. To construct these discrete solutions we introduce a Clifford product acting on the space of discrete inhomogeneous forms. This work is a direct continuation of that described in my previous papers [12,13,14,15]. In [15], a correspondence between a discrete Dirac-Kähler equation and a discrete analogue of the Hestenes equation was studied. There are several approaches to study of discrete versions of the Dirac-Kähler equation based on the use a discrete Clifford calculus framework on lattices. For a review of discrete Clifford analysis, we refer the reader to [4,5,6,10,16]. It is beyond the scope of this paper to fully discuss differences and intersections between approaches.
We first briefly review some definitions and basic facts on the Dirac-Kähler equation [9,11] and the Dirac equation in the spacetime algebra [7,8]. Let M = R 1,3 be Minkowski space with metric signature (+, −, −, −). Denote by Λ r (M ) the vector space of smooth differential r-forms, r = 0, 1, 2, 3, 4. We consider Λ r (M ) over C. Let ω, ϕ ∈ Λ r (M ). The inner product is defined by where ∧ is the exterior product and * is the Hodge star operator * : Λ r (M ) → where r ω ∈ Λ r (M ). The Dirac-Kähler equation for a free electron is given by where i is the usual complex unit and m is a mass parameter. It is easy to show that Eq. (1.2) is equivalent to the set of equations Denote by Ω ev and by Ω od the even and odd parts of Ω. It is clear that The operator d + δ is the analogue of the gradient operator in Minkowski spacetime where γ µ is the Dirac gamma matrix. Think of {γ 0 , γ 1 , γ 2 , γ 3 } as a vector basis in spacetime. Then the gamma matrices γ µ can be considered as generators of the Clifford algebra of spacetime Cℓ(1, 3) [1,2]. This algebra Hestenes [8] is called the Hestenes form of the Dirac equation [7,8].
It should be noted that the grade algebra Λ(M ) endowed with the Clifford multiplication is an example of the the Clifford algebra. In this case the basis covectors e µ = dx µ of spacetime are considered as generators of the Clifford algebra. Then the Hestenes equation (1.4) can be rewritten in terms of inhomogeneous forms as

Discretization Scheme
The starting point for consideration is a combinatorial model of Euclidean space. The proposed approach was originated by Dezin [3]. For the convenience of the reader we briefly repeat the relevant material from [13] without proofs, thus making our presentation self-contained.
Let the tensor product of a 1-dimensional complex be a combinatorial model of Euclidean space R 4 . The 1-dimensional complex C is defined in the following way. Introduce the sets {x κ } and {e κ }, κ ∈ Z. Let C 0 and C 1 be the free abelian groups of 0-dimensional and 1-dimensional chains generated by {x κ } and {e κ }. The free abelian group is understood as the direct sum of infinity cyclic groups generated by {x κ }, {e κ }. Arbitrary elements (chains) a ∈ C 0 and b ∈ C 1 can be written as the formal sums It is convenient to introduce the shift operator τ in the set of indices by We define the boundary operator ∂ : The definition (2.2) is extended to arbitrary chains by linearity.
The direct sum C = C 0 ⊕ C 1 with the boundary operator ∂ defines the 1-dimensional complex. It is known that a free abelian group is an abelian group with basis. One can regard the sets {x κ }, {e κ } as sets of basis elements of the groups C 0 and C 1 . Geometrically we can interpret the 0-dimensional basis elements x κ as points of the real line and the 1dimensional basis elements e κ as open intervals between points. We call the complex C a combinatorial real line.
Multiplying the basis elements x κ , e κ in various way we obtain basis elements of C(4). Let s (r) k be an arbitrary r-dimensional basis element C(4). Then we have s where s kµ is either x kµ or e kµ and k = (k 0 , k 1 , k 2 , k 3 ), k µ ∈ Z. The dimension r of a basis element s  can be written as and where a ∈ C(p) and b ∈ C(q) and r is the dimension of the chain a. It is easy to check that ∂∂a = 0 for any a ∈ C(4).
Suppose that the combinatorial model of Minkowski space has the same structure as C(4). We will use the index k 0 to denote the basis elements of C which correspond to the time coordinate of M . Hence, the indicated basis elements will be written as x k 0 , e k 0 .
For example, for the 1-and 2-dimensional basis elements we have * c e 0 k =ẽ 123 k , The operation (2.4) is linearly extended to chains.
Proof. The proof consists in applying the operation * c for basis elements.
Let us now consider a dual complex to C(4). We define it as the complex of cochains K(4) with complex coefficients. The complex K(4) has a similar structure, namely K(4) = K ⊗ K ⊗ K ⊗ K, where K is a dual complex to the 1-dimensional complex C. Let x κ and e κ , κ ∈ Z, be the 0-and 1dimensional basis elements of K. Then an arbitrary r-dimensional basis element of K(4) can be written as s k (r) = s k 0 ⊗ s k 1 ⊗ s k 2 ⊗ s k 3 , where s kµ is either x kµ or e kµ and k = (k 0 , k 1 , k 2 , k 3 ). We will call cochains forms, emphasizing their relationship with differential forms. Denote by K r (4) the set of all r-forms. Then K(4) can be expressed by where K ev (4) = K 0 (4) ⊕ K 2 (4) ⊕ K 4 (4) and K od (4) = K 1 (4) ⊕ K 3 (4). The complex K(4) is a discrete analogue of Λ(M ). Let r ω ∈ K r (4), then we have where e k µ , e k µν and e k ιµν are the 1-, 2-and 3-dimensional basis elements of K(4), and We define the pairing (chain-cochain) operation for any basis elements ǫ k ∈ C(4), s k ∈ K(4) by the rule The operation (2.5) is linearly extended to arbitrary chains-cochains. The coboundary operator d c : where a r+1 ∈ C(4) is an r + 1 dimensional chain. The operator d c is an analog of the exterior differential. From the above it follows that d c 4 ω = 0 and d c d c r ω = 0 for any r.
Let the difference operator ∆ µ be defined by where ω (r) k ∈ C is a component of r ω ∈ K r (4) and τ µ is the shift operator which acts as (2.10) LetK(4) be a complex of the cochains over the double complexC(4), with the coboundary operator d c defined in it by (2.6). Hence,K(4) has the same structure as K(4). This means that the corresponding forms ω ∈ K(4) andω ∈K(4) have the same components.
Let us introduce a discrete version of the Hodge star operator by using the double complex construction. Define the operation * : K r (4) →K 4−r (4) for the basis element s k (r) ∈ K r (4) by the rule * s k where This definition makes sense because the formula (2.11) preserves the Lorentz signature of metric on K(4). More explicitly, we have * x k =ẽ k , Proof. The operation * is linear. It is easy to check that by definition, the composition of * with itself gives * * s k for any basis element s k (r) ∈ K r (4).
Let us consider the 4-dimensional finite chain e n ⊂ C(4) of the form: where n µ ∈ N is a fixed number for each µ = 0, 1, 2, 3. This finite sum of 4-dimensional basis elements of C(4) imitates a domain of M . We set k , k µ = 1, 2, ..., n µ .
For example, For any r-forms ϕ, ω ∈ K r (4) the inner product over the set e n (2.13) is defined by the rule (ϕ, ω) en = V, ϕ ⊗ * ω , (2.14) where ω denotes the complex conjugate of the form ω. For forms of different degrees the product (2.14) is set equal to zero. It is clear that It should be noted that the definition of the inner product correctly imitates the continual case (1.1) and the Lorentz metric structure is still captured here. The inner product (2.14) makes it possible to define the adjoint of d c , denoted δ c . and * −1 is the inverse operation of * .
Proof. The proof is a computation. See Proposition 4 in [13].
The relation (2.15) is a discrete analog of the Green formula. From (2.12) we infer Putting this in (2.16) we obtain This makes it clear that the operator δ c : K r+1 (4) → K r (4) is a discrete analog of the codifferential δ. From (2.17) it follows that δ c 0 ω = 0 and δ c δ c r ω = 0 for any r.
Using the definitions of d c and * we can calculate The linear map is called a discrete analogue of the Laplacian. It is clear that

Discrete Dirac-Kähler and Hestenes Equations
Let us introduce a discrete inhomogeneous form as follows where i is the usual complex unit and m is a positive number (mass parameter). We can write this equation more explicitly by separating its homogeneous components as Let us define the Clifford multiplication in K(4) by the following rules: (a) x k x k = x k , x k e k µ = e k µ x k = e k µ , (b) e k µ e k ν + e k ν e k µ = 2g µν x k , g µν = diag(1, −1, −1, −1), (c) e k µ 1 · · · e k µs = e k µ 1 ···µs for 0 ≤ µ 1 < · · · < µ s ≤ 3, supposing the product to be zero in all other cases. The operation is linearly extended to arbitrary discrete forms. For example, for 1 ω, Consider the following unit forms Note that the unit 0-form x plays a role of the unit element in K(4), i.e. for any r-form Proposition 3.1. The following holds: e µ e ν + e ν e µ = 2g µν x, µ, ν = 0, 1, 2, 3.
Proof. By the rule (b) it is obvious.
Proposition 3.2. For any inhomogeneous form Ω ∈ K(4) we have where ∆ µ is the difference operator which acts on each component of Ω by the rule (2.7).
Clearly, the discrete Dirac-Kähler equation can be rewritten in the form Let Ω ev ∈ K ev (4) be a real-valued even inhomogeneous form, i.e. Ω ev = In [15,Proposition 5], it is proven that by a solution of the discrete Dirac-Kähler equation (3.3) four independent real-valued solutions of the discrete Hestenes equation (3.4) are constructed. This is a discrete version of the well-known result for corresponding continuum equations [2].
It should be noted that the components Ψ ± k can be represented as where ψ ± k = ψ ± k (p 0 , p 1 , p 2 , p 3 ) and φ ± k = φ ± k (p 0 , p 1 , p 2 , p 3 ). Hence Ψ ± are inhomogeneous real-valued even forms of the form We wish to find a solution of the discrete Hestenes equation (3.4) of the form Ω ± = AΨ ± , (4.5) where A ∈ K ev (4) is a constant real-valued form. Hence A can be expanded as where α 0 , α µν , α 4 ∈ R and x, e µν , e are the unit forms given by (3.2). We have Substituting this into the discrete Hestenes equation (3.4) we obtain ∓ 3 µ=0 e µ p µ AΨ ± e 12 e 12 = mAΨ ± e 0 .
Since Ψ ± e 0 = e 0 Ψ ± , the equation above reduces to It is clear that the same is true for AΨ + in place AΨ − , i.e. if we take Eq. (4.7) we obtain the condition (4.10) again.
Let p = {p 0 , p 1 , p 2 , p 3 } be the energy-momentum vector of a particle with (proper) mass m. Then the relation (4.10) is the energy-momentum relation. It is known that in the continuum theory the Hestenes equation (1.4) admits the plane wave solutions of the form Thus the forms Ω ± = AΨ ± are discrete versions of the plane-wave solutions Φ ± .
Thus for given p µ , µ = 1, 2, 3, there are four linearly independent solutions of the form (4.5) for each positive and negative p 0 .