Iterants, Majorana Fermions and the Majorana-Dirac Equation

: This paper explains a method of constructing algebras, starting with the properties of discrimination in elementary discrete systems. We show how to use points of view about these systems to construct what we call iterant algebras and how these algebras naturally give rise to the complex numbers, Clifford algebras and matrix algebras. The paper discusses the structure of the Schrödinger equation, the Dirac equation and the Majorana Dirac equations, ﬁnding solutions via the nilpotent method initiated by Peter Rowlands.


Introduction
The square root of minus one can be seen as an oscillation between plus and minus one. With this viewpoint, a simplest discrete system corresponds directly to the imaginary unit. This aspect of the square root of minus one as an iterant is explained below. By starting with a discrete time series, one has non-commutativity of observations and this non-commutativity can be formalized in an iterant algebra as defined in Section 3 of this paper. Iterant algebra generalizes matrix algebra and we shall see that it can be used to formulate the Lie algebra su(3) for the Standard Model for particle physics and the Clifford algebra for Majorana Fermions. The present paper is a sequel to [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and it uses material from these papers. The present paper represents a synthesis of these papers and contains new material about the relationships of these algebras with the Majorana-Dirac equation.
Distinction and processes arising from distinction are at the base of the described world. Distinctions are elemental bits of awareness. The world is composed not of things but processes and observations. It is the purpose of this paper to explore, from this perspective, a source of algebraic structures that have been important for the development of both mathematics and physics. We will discuss how basic Clifford algebra comes from very elementary processes such as an alternation of · · · + − + − + − · · · and the fact that one can think of √ −1 itself as a temporal iterant, a product of an and an η where the is the · · · − + − + − + · · · and the η is a time shift operator. Clifford algebra is at the base of this mathematical world, and the fermions are composed of these things.
View Figure 1. The discrete process · · · − + − + − + · · · can be seen as an iteration of [+1, −1]  Iterants formalize the intuition that i is a ± oscillation that interacts with itself through a delay of one time-step.
[-1,+1] [+1,-1] Section 2 is a discussion of the discrete Schrödinger equation and its relationship with iterants and with the complex numbers. We see how a discrete variant of the diffusion equation gives rise at once to both the complex numbers (as a summary of the iterant behaviour of this discretization of diffusion) and the Schrödinger equation as we know it. The section serves as an introduction to the key ideas in the rest of the paper. Sections 3 and 4 are an introduction to the algebra of iterants and its relation with the square root of minus one. Section 4 shows how iterants produce 2 × 2 matrix algebra and the split quaternions. Section 5 studies iterants of arbitrary period. We generalize the iterant construction to arbitrary finite groups G. We show that by rearranging the multiplication table of the group so that the identity element appears on the diagonal, we get a set of permutation matrices representing the group faithfully as n × n matrices. Section 6 discusses the relationship of iterants with the Artin braid group. Each element of the braid group has an associated permutation. We generalize iterants so that the group acting on the vector part of the iterant is the Artin braid group. This leads to relationships with framed braids and we end the section with a description of a braided iterant formulation of the particle interaction model of Sundance Bilson-Thompson [16]. Section 7 is about Majorana Fermion operators and their associated anyonic braiding. This section is key for this paper in relation to the later sections on the Majorana Dirac equation, as we show in the later sections that the Clifford algebras of Majorana operators are fundamental for the structure of solutions to the Majorana-Dirac equation. It is a main point of this paper to make this connection in the full context of iterants. The connection between solutions of the Majorana-Dirac equation and Majorana operators will be the subject of our further research.
Section 8 gives an iterant interpretation of the su(3) Lie algebra for the Standard Model. Section 9 discusses the Dirac equation and how the nilpotent and the Majorana operators work in this context. This section provides a link between our work, the work of Peter Rowlands [17] and with our joint work [1]. We end this section with an expression in split quaternions for the the Majorana Dirac equation in one dimension of time and three dimensions of space. The Majorana Dirac equation can be written as follows: where η and are the simplest generators of iterant algebra with η 2 = 2 = 1 and η + η = 0, andˆ ,η make a commuting copy of this algebra. Combining the simplest Clifford algebra with itself is the underlying structure of Majorana Fermions, forming indeed the underlying structure of all Fermions. The Majorana-Dirac equation is expressed entirely with real matrices so that it can have solutions over the real numbers. It was Majorana's conjecture that such solutions would correspond to particles that were their own anti-particles. Sections 10 and 11 go on to study the original Majorana-Dirac equation and variations of involving the algebraic approach in this paper. We show how the nilpotent method described in Section 9 gives rise to solutions to this equation, hence to fundamental structures related to Majorana Fermions. This part of the paper reviews our work in [1] and sets the stage for future work.

Iterants and the Schrödinger Equation
We begin with the Diffusion Equation We reformulate this equation as a difference equation in space and time. In writing it as a difference equation, I shall use dt for a finite increment in time and dx for a finite increment in space. ( This is equivalent to where κ = dt (dx) 2 τ, since for the continuum limit to exist we need to assume that dt (dx) 2 is constant as dt and dx go to zero. We shall use dt = 1 for convenience. Then the above equation becomes Consider the possibility of putting a "plus or minus" ambiguity into this equation, like so: The ± coefficient should be lawful not random, for then we can follow an algebraic formulation of the process behind the equation. We shall take ± to mean the alternating sequence ± = · · · + − + − + − + − · · · and time will be discrete. Then the new equation will become a difference equation in space and time We must consider the continuum limit. But in that limit there is no direct meaning for the parity of the number of time steps In the discrete model the wave function ψ divides into parts with even time index ψ e and parts with odd time index ψ o . So we can write (thinking of these as the corresponding discrete equations or as the continuum limits).
x ψ e . We take the limit of ψ e and ψ o separately.
Then one can interpret the {±} as the complex number i. Recall that the complex number i has the property that i 2 = −1 so that when A and B are real numbers, i = −1/i, and so if i = 1 then i = −1, and if i = −1 then i = 1. So i can be interpreted as oscillating between +1 and −1, and so we shall regard i as a definition of ±1.
In fact, when we multiply ii = (±1)(±1), we get −1 because (using this temporal interpretation) i takes a duration to oscillate and when the second term multiplies the first term, they are shifted by one step, and so we get either (+1)(−1) = −1 or (−1)(+1) = −1. We formalize this point of view later in the paper. Now i = ±1 behaves according to these rules, and we can write We have deduced the complex form of the Schrödinger equation as the limit of these discrete systems. In these systems there is a mutual dependency where the temporal variation of ψ e is mediated by the spatial variation of ψ o and the temporal variation of ψ o is mediated by the spatial variation of ψ e . We arrive at the Schrödinger equation in the context of i = ± as an iterant.
Remark 1. The discrete recursion, just discussed, can be implemented to approximate solutions to the Schrödinger equation. A further study of this recursion is intended. This way of thinking about the Schrödinger equation shows that it is intimately connected with a generalization of the discrete diffusion process with a parity oscillation that becomes i in the limit. The temporal interpretation of i indicated here will be given an algebraic context in the body of this paper.

Iterants and Idempotents
In this section we give a general context and formalization for the idea that the square root of negative unity can be regarded as an oscillation between +1 and −1 that is phase shifted with respect to itself via a time-step in the course of interacting with itself. We have used this idea in the previous section to motivate a discrete model of the Schrödinger equation. Here we will take an ordered pair [1, −1] to represent the oscillation and a permutation operator η to represent the time-step. The permutation operator will have order two and the property that The details of this construction are given below. The general form of the construction involves vectors and permutations. We will use Greek letters for permuation operators. They are not to be confused with any Greek letters in the previous section.
An iterant is a sum of elements of the form [a 1 , a 2 , ..., a n ]σ where [a 1 , a 2 , ..., a n ] is a vector of scalars (real or complex numbers in most cases) and σ is an element of the permutation group on n letters. The vectors are sums of elementary vectors of the form [0, 0, ...0, 1, 0, ..., 0]σ = e i σ where the 1 is in the i-th place. The elements e i are the basic idempotents that generate the iterants with the help of the permutations. If a = [a 1 , a 2 , ..., a n ], then we let a σ denote the vector with its elements permuted by the action of σ.
If a and b are vectors then ab denotes the vector where (ab) i = a i b i , and a + b denotes the vector where (a + b) i = a i + b i . Note that we define the usual sum of vectors and also the product of vectors as term-by-term combinations. Thus, for example, Then, with vectors combined as above and the usual product (composition) of the permutations, we define products and sums of iterants as shown below.
Iterant algebra is generated by the elements where e i is a vector with a 1 in the i-th place and zeros elsewhere, and σ is an abritrary element of the symmetric group S n . We have, by definition, that where σ(j) = i. In this way, multiplication of iterants is defined in terms of the action of the symmetric group on the vectors. For example, if σ ∈ S 3 is the cyclic permutation such that for the cyclic permutation σ in this paragraph. By themselves, the elements e i are idempotent (e 2 i = 1 for each i)and we have The iterant algebra is generated by these combinations of idempotents and permutations. For example, if η is the order two permutation of two elements, then [a, b] η = η[a, b]η = [b, a]. Define the "shift" operator η on iterants by the equation with η 2 = 1. Think of η as a delay operator, since it shifts the waveform · · · ababab · · · by one internal time step. We can define i = [−1, 1]η and then Complex numbers emerge from iterants. Interpret [−1, 1] as an oscillation between −1 and +1 and η as a temporal shift operator. Then i = [−1, 1]η is time sensitive and its selfinteraction equals minus one. Iterants are a formalization of elementary discrete processes.
We can write a = [1, 0], b = [0, 1] and A = aη, B = bη where η denotes the transposition so that [x, y]η = η[y, x] and η 2 = 1. Then we have This is the mixed idempotent and permutation algebra for n = 2. Then we have This is the beginning of the relationships between idempotents, iterants and Clifford algebras.
Note also that the non-commuting of and η is directly related to the interaction of the idempotents and the permutations.

Iterants, Discrete Processes and Matrix Algebra
In this section we relate iterants to matrix algebra. An elementary iterant is a periodic time series · · · abababababab · · · .
We now show how the iterant algebra is related to matrix algebra. In order to keep track of this patterning, lets write Matrix multiplication is isomorphic with iterant multiplication.
Ordinary matrix multiplication can be written in a concise form using the following rules: where Q is any two element iterant. Note the correspondence This means that [a, d] corresponds to a diagonal matrix.
and [b, c]η corresponds to the product of a diagonal matrix and the permutation matrix.
Note also that This is the matrix interpretation of the equation A two by two matrix is combinatorially the union of the identity pattern (the diagonal) and the interchange pattern (the antidiagonal). These correspond to the operators 1 and η for iterants.
* @ @ * In the case of complex numbers we represent The square root of minus one takes the form of the matrix If we identify the ordered pair (a, b) with a + ib, then this means taking the identification In iterant terms we have and this corresponds to the matrix equation More generally, we see that writing the 2 × 2 matrix algebra as a system of hypercomplex numbers. Note that The formula on the right equals the determinant of the matrix. Thus we define the conjugate of Z = A + Bη by the formula and we have the formula Note also that we assume that a, b, c, d are in a commutative base ring.
Note also that for Z as above, This is the classical adjoint of the matrix Z.
We leave it to the reader to check that for matrix iterants Z and W, Note also that η = −η, We can prove that That WW is in the commutative base ring allows us to remove it from in between the appearance of Z and Z. Iterants as 2 × 2 matrices form a direct non-commutative generalization of the complex numbers. The split quaternions are the system {±1, ± , ±η, ±i}.
The quaternions arise directly from the split quaternions once we construct an extra square root of minus one that commutes with them. Call this extra root of minus one √ −1. Then the quaternions are generated by In the next section we give a number of other ways to construct the quaternions, and we show how the iterant point of view is related to matrix representations of the quaternions such as matrices over the complex numbers in SU(2).

Iterants of Arbirtarily High Period
As a next example, consider a waveform of period three.

· · · abcabcabcabcabcabc · · ·
Here we see three viewpoints (depending upon whether one starts at a, b or c).
The appropriate shift operator is given by the formula and S 3 = 1. With this we obtain a closed algebra of iterants whose general element is of the in this formalism a, b, c, d, e, f , g, h, k are real or complex numbers. The algebra is denoted Vect 3 (R) with the scalars in a commutative ring with unit F. For matrices, M 3 (F) is the 3 × 3 matrix algebra over F.
The pattern behind the 3 × 3 matrices is held by the symbolic matrix T occupies positions in the matrix corresponding to a permutation matrix. The letter S occupies the positions corresponding to its permutation matrix. The 1's occupy the diagonal for the an identity matrix. In this case the matrices form a permutation representation of the cyclic group of order 3, It should be clear to the reader that this construction generalizes directly for iterants of any period and hence for a set of operators forming a cyclic group of any order. In fact we can generalize further to any finite group G. See [13] for more information about these generalizations.

1.
In this example we consider the group G = C 2 × C 2 , often called the "Klein 4-Group." Thus G has the multiplication table, which is also its G- Table for Vect 4 (G, F).
Thus we have the permutation matrices that I shall call E, A, B, C whose entries are obtained from the matrix above by writing 1 for the places occupied by the corresponding letter and 0 for the other places. For example, The reader will verify that And let I = αA, J = βB, K = γC.
Then we have The quaternions are iterants in relation to the Klein Four Group. Figure 2 illustrates these quaternion generators with string diagrams for the permutations. The reader can check that the permutations correspond to the permutation matrices constructed for the Klein Four Group.

2.
The set of matrices of the form a1 + bI + cJ + dK with a 2 + b 2 + c 2 + d 2 = 1 is isomorphic to the group SU (2). To see this, note that SU (2) is the set of matrices with complex entries z and w with determinant 1 so that zz + ww = 1.
Letting z = a + bi and w = c + di, we have With a commutative i = √ −1 we obtain as described in the previous section. This construction shows how the structure of the quaternions comes directly from the non-commutative structure of period two iterants. In other, words, quaternions can be represented by 2 × 2 matrices. This is the way it has been presented in standard language as in the group SU(2).

3.
Here represents a Hermitian 2 × 2 matrix and hence an observable for quantum processes mediated by SU (2). Hermitian matrices have real eigenvalues.
Take a = T + X, b = T − X, c = Y, d = Z, then we obtain an iterant representation for a point in Minkowski spacetime.
Note that we have the formula The eigenvalues of H are T ± H can observe the time and the invariant spatial distance from the origin of the event (T, X, Y, Z). Here quantum mechanics and special relativity are reconciled.

4.
Iterants generate Hamilton's Quaternions. We express them algebraically as shown below.
The permutations are products of transpositions (ij). One can verify that Then GG = F(GG) and every F in D has a fixed point. This is the method of the lambda calculus of Church and Curry [4]. Constructions for fixed points that extend given domains is a way of thinking about the nature of our constructions in this paper. This theme is the subject of other work of the author [26]. Here i is an eigenform for R(x) = −1/x. Indeed, each generating quaternion is an eigenform for the transformation R(x) = −1/x. The richness of the quaternions comes from the closed algebra that arises with its infinity of eigenforms that satisfy the equation U 2 = −1 : U = aI + bJ + cK where a 2 + b 2 + c 2 = 1.

5.
Clifford algebras C n generated by elements {e 1 , e 2 , · · · , e n } with e 2 i = 1 and e i e j = −e j e i when i = j occur very often in both mathematics and physics. These algebras are often used as part of what is called geometric algebra [27]. It is worth noting that these algebras fit naturally into the iterants framework via their self-action. That is, if we take C n as a vector space over a field K, then it has basis consisting of all the ordered products of the form e i 1 e i 2 · · · e i k for k = 1, 2, · · · , n and i 1 < i 2 < · · · < i k . We can list the basis and obtain signed permutation matrices that represent the left action of the algebra on itself, just as we have done with the group representations in this section. For example, when n = 2 we have the basis list (1, e 1 , e 2 , e 1 e 2 ) and e 1 (1, e 1 , e 2 , e 1 e 2 ) = (e 1 , 1, e 1 e 2 , e 2 ) while e 2 (1, e 1 , e 2 , e 1 e 2 ) = (e 2 , −e 1 e 2 , 1, −e 2 ). Thus if σ = (12)(34) and τ = (13)(24), then we can represent e 1 = [1, 1, 1, 1]σ and e 2 = [1, −1, 1, −1]τ as iterants. In the case n = 2 we have already given a simpler iterant representation of this algebra at the beginning of the paper, using e 1 = [−1, +1] and e 2 = η where η denotes the transposition in S 2 . It is interesting how this representation appears doubled in the one we deduced from the multiplication table of the algebra. It is of interest to carry out corresponding calculations for higher values of n.

Iterants Associated with the Framed Braid Group
The Symmetric Group S n has presentation S n = (T 1 , · · · T n−1 |T 2 The Artin Braid Group B n has presentation Thus there is a natural homomorphism from the Artin Braid Group to the Symmetric Group. In Figure 3 are shown the generators σ 1 , σ 2 , σ 3 of the 4-strand braid group with the topological relation σ 1 σ 2 σ 1 = σ 2 σ 1 σ 2 and the commuting relation σ 1 σ 3 = σ 3 σ 1 . The elementary braid generators σ i correspond to the interchange of the i-th strand iwith the i + 1-th strand. The homomorphism π : B n −→ S n defined on generators by π(σ i ) = T i . It is natural to generalize iterants to braided iterants by first generalizing the braid group to the framed braid group. In this generalization, we associate integers t a to the top of each braid strand. One can replace each braid strand by a ribbon and interpret t a as a 2πa twist in the ribbon. In Figure 4 it is shown how to multiply two framed braids. The braids A and B are given by the formulas The framed braid group on three strands is denoted FB 3 . As the Figure 4 illustrates, there is the formula where v is a vector of the form v = [t a , t b , t c ] (for n = 3) and v π(σ) is the result of permuting the vector by the permutation associated with the braid. We can form an algebra Alg[FB n ] by taking formal sums of framed braids of the form ∑ c k v k g k where c k is a scalar, v k is a framing vector and g k is an element of the Artin Braid group B n . This algebra is a generalization of iterant algebra, based on the action of the Artin Braid Group. The representation π : B n −→ S n induces a map of algebraŝ π : Alg[FB n ] −→ Alg[FS n ] where we recognize Alg[FS n ] as exactly an iterant algebra based in the symmetric group S n .
In [16] Fermions are represented as framed braids. See Figure 5. The positron and the electron are given by the framed braids  Here we use [t a , t b , t c ] for the framing numbers (a, b, c). Products of framed braids catalog particle interactions. The electron and the positron are algebraic inverses. In Figure 6 are bosons, including a photon γ. Figure 7 illustrates the muon decay The muon decay is a multiplicative identity in the braid algebra:

Fermions, Majorana Fermions and Anyonic Braiding
In this section we consider the algebras of Femions and Majorana fermions. The generators of this Clifford algebra represent fermions that are their own anti-particles. For a long time it has been conjectured that neutrinos may be Majorana fermions. More recently, it has been suggested that Majorana fermions may occur in collective electronic phenomena and in subtle correlations in nano-wires and in two dimensional anyonic physics [28][29][30][31].
In order to explain this association, we first give a short exposition of the algebra of fermion operators. In a standard collection of fermion operators m 1 , · · · , m k one has that each m i is a linear operator on a Hilbert space with an adjoint operator m † i (corresponding to the anti-particle for the particle created by m i ) and relations There is another brand of Fermion algebra where we have generators c 1 , · · · c k and c Here is an example that shows how topology is related to the Majorana Fermion operators. Let x, y, z be three Majorana fermions. Let A = yx, B = zy, C = xz. We have already seen that A, B, C represent the quaternions. Now define It is easy to see that σ i and σ j satisfy the braiding relation for any i = j. For example, here is the verification for i = 1, j = 2. Similarly, Thus and so a natural braid group representation arises from the Majorana fermions. This braid group representation is significant for possible applications in topological quantum computing. For the purpose of this discussion, the braid group representation shows that the Clifford algebraic representation for knot sets is related to topology at more than one level. The relation x 2 = 1 for generators makes the individual sets, taken as products of generators, invariant under the Reidemeister moves (up to a global sign). But braiding invariance of certain linear combinations of sets is a relationship with knotting at a second level. This multiple relationship certainly deserves more thought. We will make one more remark here, and reserve further analysis for a subsequent paper. The braiding operators act on the complex vector space spanned by the fermions x, y, z.
It follows that T(x) = y and T(y) = −x. In Figure 8 where we show an interpretation for the braiding of two fermions. In the interpretation the two fermions are joined by a belt. On particle interchange, the belt is twisted by 2π. A twist of 2π corresponds to a phase change of −1. See [32]. It may not be evident which particle should receive the phase change. Topology alone tells only the relative change of phase. The Clifford algebra makes a specific choice and so fixes the representation of the braiding.

Iterants and the Standard Model
Here we give an iterant interpretation for the Lie algebra of the special unitary group SU(3). The Lie algebra su(3) is generated by the following eight Gell Man Matrices [33].
The group SU(3) consists in the matrices U( 1 , · · · , 8 ) = e i ∑ a a λ a where 1 , · · · , 8 are real numbers and a ranges from 1 to 8. The Gell Man matrices satisfy the relations: We sum over repeated indices. tr is matrix trace, [A, B] = AB − BA. δ ab equals 1 when a = b and equals 0 otherwise. Structure coefficients f abc have the non-zero values shown below.
An iterant representation for the Gell Man matrices that is based on the pattern   as we have previously described. We use the cyclic group of order three to represent all 3 × 3 matrices as iterants based on the permutation matrices it is easy to verify the formulas for the Gell Mann Matrices in the iterant format: Letting F a = λ a /2, the Lie algebra rewrites as iterants of the form [a, b, c]G where G is cyclic. Compare with [34]. Let Then we have the specific iterant formulas Thus the basic su(3) Lie algebra reduces to iterants.

The Dirac Equation and Majorana Fermions
We construct the Dirac equation. The speed of light is equal to 1 by convention. Energy E, momentum p and mass m are related by the relativisitic equation We obtain Dirac's operator by first taking the case where p is a scalar (one dimension of space and one dimension of time). Let E = αp + βm where α and β are elements of a a possibly non-commutative, associative algebra. Then Hence we will satisfiy E 2 = p 2 + m 2 if α 2 = β 2 = 1 and αβ + βα = 0. This is our familiar Clifford algebra pattern and we can use the iterant algebra generated by e and η if we wish. Then, because the quantum operator for momentum is −i∂/∂x and the operator for energy is i∂/∂t, we have the Dirac equation i∂ψ/∂t = −iα∂ψ/∂x + βmψ.

Now note that
and let This nilpotent element leads to a (plane wave) solution to the Dirac equation as follows: We have shown that Oψ = ∆ψ for ψ = e i(px−Et) . It then follows that is a (plane wave) solution to the Dirac equation.
In fact, this calculation suggests that we should multiply the operator O by βα on the right, obtaining the operator and the equivalent Dirac equation Dψ = 0.
In fact for the specific ψ above we will now have D(Ue i(px−Et) ) = U 2 e i(px−Et) = 0. This idea of reconfiguring the Dirac equation in relation to nilpotent algebra elements U is due to Peter Rowlands [17]. We see that U = βαE + βp − αm with U 2 = 0. .
If we letψ = e i(px+Et) (reversing time), then we have giving a definition of U † corresponding to the anti-particle for Uψ.
We have Note that here we have We have that The decomposition of U and U † into the corresponding Majorana Fermion operators corresponds to E 2 = p 2 + m 2 . Dividing by 2E we have Fermion creation and annihilation algebra arises naturally in the nilpotent formulation.
We then have the following form of the Dirac equation.

that the Dirac equation takes the form
Oψ(x, t) = 0. Let Apply the Dirac operator to this ψ. For nilpotency, the modified Dirac operator is So U 2 = 0 and Uψ is a solution to the modified Dirac Equation.
We have the structure of the Fermion operators and Majorana Fermion operators.

Majorana Fermions
We now make a Dirac algebra distinct from the one generated by α, β, σ 1 , σ 2 , σ 3 and obtain an equation that can have real solutions. Majorana [30] followed this strategy to construct his new equation. A real equation may have solutions invariant under complex conjugation. Such solutions correspond to particles that are their own anti-particles. We construct the Majorana algebra in terms of the split quaternions and η. We will use the matrix representation given below. It can be formulated in iterants as we have discussed.
To see that this is a correct Dirac equation, note that E = α xpx + α ypy + α zpz + βm (Here the "hats" denote the quantum differential operators corresponding to the energy and momentum.) will satisfyÊ 2 =p x 2 +p y 2 +p z 2 + m 2 if the algebra generated by α x , α y , α z , β satisfies the conditions: Each generator has square one. Each distinct pair of generators anti-commute. The general Dirac equation occurs by replacingÊ by i∂/∂t, andp x with −i∂/∂x (and same for y, z).

Nilpotents and the Majorana-Dirac Equation
Let D = (∂/∂t +ηη∂/∂x + ∂/∂y +ˆ η∂/∂z −ˆ ηηm). In the last section we have shown how D can be taken as the Majorana operator for which we can look for real solutions to the Dirac equation. Letting ψ(x, t) = e i(p•r−Et) , we have and The element U is nilpotent, U 2 = 0, and we have that and A and B the Majorana operators described above. Note how the Majorana Fermion algebra generated by A and B comes into play in the construction of these solutions. We take it as quite significant that the Majorana algebra is directly involved in these solutions. In other work [2,5,13,32] we review the main features of recent applications of the Majorana algebra and its relationships with representations of the braid group and with topological quantum computing. We are now in a position to assess the relationship of the Majorana algebra with actual solutions to the Majorana-Dirac equation.

Spacetime in 1 + 1 Dimensions
Using the method of this section and spacetime with one dimension of space (x), we can write a real Majorana Dirac operator in the form where, the matrix representation is now two dimensional with We obtain a nilpotent operator, D by multiplying by iη : Note that A 2 = E 2 − p 2 = m 2 and B 2 = m 2 , from which it is easy to see that A + iB is nilpotent. A and B are the Majorana operators for this decomposition. Multiplying out, we find (A + iB)ψ = (A + iB)(cos(θ) + isin(θ)) = (Acos(θ) − Bsin(θ)) + i(Bcos(θ) + Asin(θ)) where θ = px − Et. We now examine the real part of this expression, as it will be a real solution to the Dirac equation. The real part is Each column vector is a solution to the original Dirac equation corresponding to the operator ∇ = ∂/∂t + ∂/∂x + ηm written as a 2 × 2 matrix differential operator. We can see this in an elegant way by changing to light-cone coordinates: (Recall that we take the speed of light to be equal to 1 in this discussion.) Then and the Dirac equation becomes the pair of equations ∂ψ 1 /∂l = mψ 2 , ∂ψ 2 /∂r = −mψ 1 .
Note that these equations are satisfied by exactly when E 2 = p 2 + m 2 as we have assumed. It is quite interesting to see these direct solutions to the Dirac equation emerge in this 1 + 1 case. The solutions are fundamental and they are distinct from the usual solutions that emerge from the Feynman Checkerbooad Model [7,35]. It is the above equations that form the basis for the Feynman Checkerboard model that is obtained by examining paths in a discrete Minkowski plane generating a path integral for the Dirac equation. We will investigate the relationship of this approach with the Checkerboard model in a subsequent paper.

Spacetime Algebra
Another way to put the Dirac equation is to formulate it in terms of a spacetime algebra. By a spacetime algebra we mean a Clifford algebra with generators {e 1 , e 2 , e 3 , e 4 } such that e 2 1 = e 2 2 = e 2 3 = 1, e 2 4 = −1 and e i e j + e j e i = 0 for i = j. Thus the generators of the algebra fit the Minkowski metric and we can represent a point in space time by p = xe 1 + ye 2 + ze 3 + te 4 so that p 2 = x 2 + y 2 + z 2 − t 2 corresponds to the spacetime metric with the speed of light c = 1. (The reader may wish to compare this approach with Hestenes [27].) Since the Dirac algebra demands {α 1 , α 2 , α 3 , β} with all elements squaring to 1 and anti-commuting, we see that spacetime algebra is interchangeable with Dirac algebra via the translation: Furthermore, we can see that the following lemma gives us a guide to constructing nilpotent formulations of the Dirac equation.
It follows that This completes the proof.

Example 1.
Before proceeding to the Majorana structure, consider the standard Dirac algebra. Here we have σ 1 , σ 2 , σ 3 with σ 2 i = 1 for each i = 1, 2, 3 and each pair of distinct operators anticommutes. This can be taken to be the Pauli algebra and is represented by matrices over the complex numbers. We take α and β as before to generate a Clifford algebra that commutes with the Pauli algebra and is independent of it. Then the associated spacetime algebra has generators e 1 = ασ 1 , e 2 = ασ 2 , e 3 = ασ 3 , e 4 = √ −1β and the nilpotency corresponds to the fact that these generators, multiplied by βα, yield another spacetime algebra. This is given by The corresponding nilpotent Dirac operator is D = µ∂/∂t + e 1 ∂/∂x + e 2 ∂/∂y + e 3 ∂/∂z + e 4 m.
Applying this operator to ψ = e √ −1(p•r−Et) we obtain the nilpotent This can be replaced by the nilpotent U = −βαE + βσ 1 p x + βσ 2 p y + βσ 3 p z − αm by factoring out the common square root of minus one. This is the same nipotent that we have previously derived. Note that in relation to this standard Dirac algebra we have the conjugate nilpotent U † = −βαE + βσ 1 p x + βσ 2 p y + βσ 3 p z − αm, and that U + U † = 2(βσ 1 p x + βσ 2 p y + βσ 3 p z − αm) This is as we have derived earlier in the paper. The decomposition into Clifford operators follows these lines, giving Clifford elements that square to E. When we work with the real spacetime algebras (below) that correspond to the Majorana Dirac equation, the decomposition into Clifford algebras takes a different pattern, centering on the mass m rather than the energy E.

Example 4.
We now give a number of examples of spacetime algebras. For this purpose it is useful to change notation. We will use I = , J = η, i =ˆ , j =η.
Thus I 2 = J 2 = i 2 = j 2 = 1 and I J + J I = 0 and ij + ji = 0. We will indicate a spacetime algebra as a 4-tuple (e 1 , e 2 , e 3 , e 4 ) where we require that the e i anti-commute and that the squares of the first three e i are 1 while e 2 4 = −1. The following are spacetime algebras. It is easy to see that A, B, C and D are nilpotent. Note that (up to signs) B is obtained from A by interchanging i, j with I, J and then interchanging i and j. C is obtained from A by interchanging i and j directly. To see that A is nilpotent, multiply by I J. The algebra D is also nilpotent, via multiplying by ijJ.
The General Case. We are now in a position to prove the following Theorem. {ijI J, I, J, I Ji}.
Here the notation of types of algebras is as we have explained in the previous examples. The proof will proceed in the form of the discussion below. In subsequent work we shall return to this result and its possible physical consequences, since each spacetime algebra gives a Dirac equation that can be studied both for its physics and for its mathematics.
Let ι = √ −1, a square root of negative unity that commutes with all algebra elements. Applying D to ψ = e ι(p•r−Et) we obtain the nilpotent A = ι(−µE + e 1 p x + e 2 p y + e 3 p z ) + e 4 m.
The nilpotent A is directly decomposed into its two (Majorana) Clifford parts as the real and imaginary parts of A, just as in our previous discussion of a special case. Other examples lead to real solutions to the Majorana Dirac equation just as we have done above. Note that the Clifford parts are ρ = −µE + e 1 p x + e 2 p y + e 3 p z and τ = e 4 m with ρ 2 = τ 2 = −m 2 and ρ and τ anticommute. It is of interest to note that the Clifford algebra is collapsed when the mass is equal to zero. Consider that the fourth elememt of a spacetime algebra has square −1. Up to symmetries the possibilities are ij and I Ji. Take each of these cases in turn. First suppose that e 4 = ij. Then consider first all square one elements. These are S = {i, j, I, J, ijI J, iI, iJ, jI, jJ}.
The subset of elements of S that anti-commute with ij is The first is nilpotent via the multiplier ij and the second is nilpotent via the multiplier I Jj.
Up to symmetries these are all the cases and so we have proved the result In a subsequent paper we shall follow up the consequences of this result.
Funding: Kauffman's work was supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (contract no. 14.Y26.31.0025 with the Ministry of Education and Science of the Russian Federation).