The Dirac Equation and the Majorana Dirac Equation
aa r X i v : . [ phy s i c s . g e n - ph ] S e p THE DIRAC EQUATION ANDTHE MAJORANA DIRAC EQUATION
Louis H Kauffman
Department of Mathematics, Statistics and Computer Science851 South Morgan StreetUniversity of Illinois at ChicagoChicago, Illinois 60607-7045andDepartment of Mechanics and MathematicsNovosibirsk State UniversityNovosibirsk, Russia < kauff[email protected] > Peter Rowlands
Physics DepartmentUniversity of LiverpoolOliver Lodge LaboratoryOxford StreetLiverpool L69 7ZE < [email protected] > We discuss the structure of the Dirac equation and how the nilpo-tent and the Majorana operators arise naturally in this context.This provides a link between Kauffman’s work on discrete physics,iterants and Majorana Fermions [4, 5, 6, 7, 11, 8, 10, 9] and the workon nilpotent structures and the Dirac equation of Peter Rowlands[17, 18, 19, 20, 21, 22]. We give an expression in split quaternionsfor the Majorana Dirac equation in one dimension of time and threedimensions of space.
Louis H Kauffman , Peter Rowlands
In [13] Majorana discovered a version of the Dirac equationthat can be expressed entirely over the real numbers. This ledhim to speculate that the solutions to his version of the Diracequation would correspond to particles that are their own anti-particles. It is the purpose of this paper to examine the structureof this Majorana-Dirac Equation, and to find basic solutions toit by using the nilpotent technique. We succeed in this aim anddescribe our results. The Majorana-Dirac equation can be writtenas follows:( ∂/∂t + ˆ ηη∂/∂x + ǫ∂/∂y + ˆ ǫη∂/∂z − ˆ ǫ ˆ ηηm ) ψ = 0where η and ǫ are the simplest generators of iterant algebra with η = ǫ = 1 and ηǫ + ǫη = 0 , and ˆ ǫ, ˆ η form a copy of this al-gebra that commutes with it. This combination of the simplestClifford algebra with itself is the underlying structure of MajoranaFermions, forming indeed the underlying structure of all Fermions.We show how to make nilpotent formulations for Majorana Diracequations and consequently how to solve these equations via Ma-jorana operators.Here is a concise background about Fermions that will be of usefor the rest of this paper. The operator algebra for a Fermion isgiven by creation and annihilation operators U † and U satisfyingthe equations U = ( U † ) = 0 and U U † + U † U = 1 [23]. Call an al-gebra generated by U and U † a Fermion algebra if it satisfies theseequations. For this introduction make the following well-known re-mark: Suppose that we are given a Clifford algebra with generators a and b so that s = b = 1 and ab + ba = 0 . It is assumed that a † = a and that b † = b. Then we obtain a Fermion algebra fromthis Clifford algebra by defining U = ( a + ib ) / U † = ( a − ib ) / i = √− . The reader will have no difficulty verifying thisassertion.It has been suggested [12, 3] that electrons or other Fermionsmight behave, under certain circumstances, as if the electron wascomposed of two particles corresponding to this decomposition into he Dirac Equation and the Majorana Dirac Equation a and b. Furthermore, since a and b are invariant un-der conjugation ( † ), it has been suggested that the particles cor-responding to a and b are Majorana Fermions , particles that aretheir own anti-particles. The reason for this nomenclature goesback to the paper of Majorana [13] where he constructed a ver-sion of the Dirac equation based on real Clifford algebra so thesolutions could model particles that were their own anti-particles.It has been a subject of speculation whether such particles exist.The recent suggestion that electrons themselves are composed ofMajorana particles is startling to say the least. Some experimentalevidence is availiable for this hypothesis in terms of the behaviourof electrons in nano-wires [1, 14]. Thus we call the operators a and b the Majorana operators related to the Fermion algebra. Note that a = ( U + U † ) and b = ( U − U † ) /i. While it has been natural to say that the operators a and b are Majorana operators, their relationship to the Majorana Diracequation has hitherto been obscure. One purpose of this paper isto show how indeed there are real solutions to the Majorana Diracequation that are built in terms of the Majorana operators.This paper is organized as follows: Section 2 introduces theDirac equation, its nilpotent reformulation and the appearance ofalgebraic Fermion operators as nilpotent algebra elements support-ing solutions to the Dirac equation. We explain the formulationof the Majorana-Dirac operator as described above. In Section3 we use a nilpotent reformulation of the Majorana-Dirac opera-tor to find real solutions to the Majorana-Dirac equation and weshow how the Clifford algebra of Majorana operators is relatedto these solutions. In a separate section we give real solutions tothe Majorana-Dirac equation specialized to one dimension of spaceand one dimension of time. We rewrite this specialization in termsof light-cone coordinates and compare our results with the Feyn-man Checkerboard model [2, 6]. In Section 4 we reformulate theDirac equation in terms of spacetime algebra by which we mean aClifford algebra generated by elements e , e , e , e where all pairs Louis H Kauffman , Peter Rowlands of distinct generators anti-commute and the first three generatorssquare to 1 while the last ( e ) squares to − . We prove that theDirac operator can be written in the form ∂/∂t + e ∂/∂x + e ∂/∂y + e ∂/∂z + e m and that it can be converted to the nilpotent form if and only ifthere is an element µ such that µ = − µe , µe , µe , µe aregenerators for a new spacetime algebra. We then use this result toclassify all possible spacetime algebras that can be used to makeMajorana-Dirac operators. In Section 5 we discuss the physics ofour results from the point of view of Peter Rowlands. Section 6 isa concluding summary. We construct the Dirac equation. If the speed of light is equal to1 (by convention), then energy E , momentum p and mass m arerelated by the (Einstein) equation E = p + m . Dirac constructed his equation by looking for an algebraic squareroot of p + m so that he could have a linear operator for E thatwould take the same role as the Hamiltonian in the Schroedingerequation. We will get to this operator by first taking the case where p is a scalar (we use one dimension of space and one dimension oftime.). Let E = αp + βm where α and β are elements of a possiblynon-commutative, associative algebra. Then E = α p + β m + pm ( αβ + βα ) . Hence we will satisfiy E = p + m if α = β = 1 and αβ + βα = 0 . This is a familiar Clifford algebra pattern. Note that this algebracan be represented by 2 × α = (cid:18) − (cid:19) he Dirac Equation and the Majorana Dirac Equation β = (cid:18) (cid:19) . Then, because the quantum operator for momentum is − i∂/∂x andthe operator for energy is i∂/∂t, we have the Dirac equation i∂ψ/∂t = − iα∂ψ/∂x + βmψ. Let O = i∂/∂t + iα∂/∂x − βm so that the Dirac equation takes the form O ψ ( x, t ) = 0 . Now note that O e i ( px − Et ) = ( E − αp − βm ) e i ( px − Et ) . We let ∆ = ( E − αp − βm )and let U = ∆ βα = ( E − αp − βm ) βα = βαE + βp − αm, then U = − E + p + m = 0 . This nilpotent element leads to a (plane wave) solution to the Diracequation as follows: We have shown that O ψ = ∆ ψ for ψ = e i ( px − Et ) . It then follows that O ( βα ∆ βαψ ) = ∆ βα ∆ βαψ = U ψ = 0 , from which it follows that ψ = βαU e i ( px − Et ) is a (plane wave) solution to the Dirac equation. Louis H Kauffman , Peter Rowlands
In fact, this calculation suggests that we should multiply theoperator O by βα on the right, obtaining the operator D = O βα = iβα∂/∂t + iβ∂/∂x − αm, and the equivalent Dirac equation D ψ = 0 . In fact for the specific ψ above we will now have D ( U e i ( px − Et ) ) = U e i ( px − Et ) = 0 . This idea of reconfiguring the Dirac equation inrelation to nilpotent algebra elements U is due to Peter Rowlands[17]. Rowlands does this in the context of vector (Clifford) andquaternion algebra. Note that the solution to the Dirac equationthat we have found is expressed in Clifford algebra. It can be articu-lated into specific vector solutions by using a matrix representationof the algebra. We see that U = βαE + βp − αm with U = 0 isthe essence of this plane wave solution to the Dirac equation. Thismeans that a natural non-commutative algebra arises directly andcan be regarded as the essential information in a Fermion. It isnatural to compare this algebra structure with algebra of creationand annihilation operators that occur in quantum field theory. Tothis end we recapitulate and start again in the next subsection. U and U † We start with ψ = e i ( px − Et ) and the operators ˆ E = i∂/∂t andˆ p = − i∂/∂x so that ˆ Eψ = Eψ and ˆ pψ = pψ. The Dirac operatoris O = ˆ E − α ˆ p − βm and the modified Dirac operator is D = O βα = βα ˆ E + β ˆ p − αm, so that D ψ = ( βαE + βp − αm ) ψ = U ψ.
If we let ˜ ψ = e i ( px + Et ) (reversing time), then we have D ˜ ψ = ( − βαE + βp − αm ) ψ = U † ˜ ψ, he Dirac Equation and the Majorana Dirac Equation U † corresponding to the anti-particle for U ψ.
We have U = βαE + βp − αm and U † = − βαE + βp − αm Note that here we have( U + U † ) = (2 βp + αm ) = 4( p + m ) = 4 E , and ( U − U † ) = − (2 βαE ) = − E . We have that U = ( U † ) = 0and U U † + U † U = 4 E . Thus we have a direct appearance of the Fermion algebra corre-sponding to the Fermion plane wave solutions to the Dirac equa-tion. Furthermore, as shall see below, the decomposition of U and U † into the corresponding Majorana Fermion operators correspondsto E = p + m . To see this, normalize by dividing by 2 E we have U = ( A + Bi ) E and U † = ( A − Bi ) E, with A = ( βp + αm ) /E and B = iβα. so that A = B = 1 Louis H Kauffman , Peter Rowlands and AB + BA = 0 . This shows how the Fermion operators are expressed in terms ofthe simpler Clifford algebra of Majorana operators. (See the in-troduction to this paper for a discussion of the role of Majoranaoperators.)
So far, we have written the Dirac equation in one dimension ofspace and one dimension of time. We give here a way to boostthe formalism directly to three dimensions of space. We take anindependent Clifford algebra generated by σ , σ , σ with σ i = 1for i = 1 , , σ i σ j = − σ j σ i for i = j. Now assume that α and β as we have used them above generate an independentClifford algebra that commutes with the algebra of the σ i . Replacethe scalar momentum p by a 3-vector momentum p = ( p , p , p )and let p • σ = p σ + p σ + p σ . We replace ∂/∂x with ∇ =( ∂/∂x , ∂/∂x , ∂/∂x ) and ∂p/∂x with ∇ • p. We then have the following form of the Dirac equation. i∂ψ/∂t = − iα ∇ • σψ + βmψ. Let O = i∂/∂t + iα ∇ • σ − βm so that the Dirac equation takes the form O ψ ( x, t ) = 0 . In analogy to our previous discussion we let ψ ( x, t ) = e i ( p • r − Et ) where p = ( p x , p y , p z ) and r = ( x, y, z ) and • denotes the dot prod-uct. We construct solutions by first applying the Dirac operatorto this ψ. The two Clifford algebras interact to generalize directly he Dirac Equation and the Majorana Dirac Equation D = iβα∂/∂t + β ∇ • σ − αm. And we have that D ψ = U ψ where U = βαE + βp • σ − αm. We have that U = 0 and U ψ is a solution to the modified DiracEquation, just as before. And just as before, we can articulate thestructure of the Fermion operators and locate the correspondingMajorana Fermion operators. We leave these details to the reader.
There is more to do. We will now make a Dirac algebra distinctfrom the one generated by α, β, σ , σ , σ to obtain an equation thatcan have real solutions. This was the strategy that Majorana [13]followed to construct his Majorana Fermions. A real equation canhave solutions that are invariant under complex conjugation andso can correspond to particles that are their own anti-particles. Wewill describe this Majorana algebra in terms of the split quaternions ǫ and η. For convenience we use the matrix representation givenbelow. ǫ = (cid:18) − (cid:19) , η = (cid:18) (cid:19) . Let ˆ ǫ and ˆ η generate another, independent algebra of split quater-nions, commuting with the first algebra generated by ǫ and η. Thena totally real Majorana Dirac equation can be written as follows:( ∂/∂t + ˆ ηη∂/∂x + ǫ∂/∂y + ˆ ǫη∂/∂z − ˆ ǫ ˆ ηηm ) ψ = 0 . Louis H Kauffman , Peter Rowlands
To see that this is a correct Dirac equation, note thatˆ E = α x ˆ p x + α y ˆ p y + α z ˆ p z + βm (Here the “hats” denote the quantum differential operators corre-sponding to the energy and momentum.) will satisfyˆ E = ˆ p x + ˆ p y + ˆ p z + m if the algebra generated by α x , α y , α z , β has each generator ofsquare one and each distinct pair of generators anti-commuting.From there we obtain the general Dirac equation by replacing ˆ E by i∂/∂t , and ˆ p x with − i∂/∂x (and same for y, z ).( i∂/∂t + iα x ∂/∂x + iα y ∂/∂y + iα z ∂/∂y − βm ) ψ = 0 . This is equivalent to( ∂/∂t + α x ∂/∂x + α y ∂/∂y + α z ∂/∂y + iβm ) ψ = 0 . Thus, here we take α x = ˆ ηη, α y = ǫ, α z = ˆ ǫη, β = i ˆ ǫ ˆ ηη, and observe that these elements satisfy the requirements for theDirac algebra. Note how we have a significant interaction betweenthe commuting square root of minus one ( i ) and the element ˆ ǫ ˆ η ofsquare minus one in the split quaternions. This brings us back toour original considerations about the source of the square root ofminus one. Both viewpoints combine in the element β = i ˆ ǫ ˆ ηη thatmakes this Majorana algebra work. Since the algebra appearingin the Majorana Dirac operator is constructed entirely from twocommuting copies of the split quaternions, there is no appearanceof the complex numbers, and when written out in tensor productsof 2 × he Dirac Equation and the Majorana Dirac Equation Let D = ( ∂/∂t + ˆ ηη∂/∂x + ǫ∂/∂y + ˆ ǫη∂/∂z − ˆ ǫ ˆ ηηm ) . In the lastsection we have shown how D can be taken as the Majorana oper-ator for which we can look for real solutions to the Dirac equation.Letting ψ ( x, t ) = e i ( p • r − Et ) , we have D ψ = ( − iE + i (ˆ ηηp x + ǫp y + ˆ ǫηp z ) − ˆ ǫ ˆ ηηm ) ψ. Let Γ = ( − iE + i (ˆ ηηp x + ǫp y + ˆ ǫηp z ) − ˆ ǫ ˆ ηηm )and U = ǫη Γ = ( i ( − ηǫE − ˆ ηǫp x + ηp y − ǫ ˆ ǫp z ) + ǫ ˆ ǫ ˆ ηm ) . The element U is nilpotent, U = 0 , and we have that U = A + iB,AB + BA = 0 ,A = ǫ ˆ ǫ ˆ ηm,B = − ηǫE − ˆ ηǫp x + ηp y − ǫ ˆ ǫp z ,A = − m , and B = − E + p x + p y + p z = − m . Letting ∇ = ǫη D , we have a new Majorana Dirac operator with ∇ ψ = U ψ so that ∇ ( U ψ ) = U ψ = 0 . Letting θ = ( p • r − Et ) , wehave U ψ = ( A + Bi ) e iθ = ( A + Bi )( Cos ( θ ) + iSin ( θ )) =( ACos ( γ ) − BSin ( θ )) + i ( BCos ( θ ) + ASin ( θ )) . Thus we have found two real solutions to the Majorana Dirac Equa-tion: Φ =
ACos ( θ ) − BSin ( θ )2 Louis H Kauffman , Peter Rowlands and Ψ =
BCos ( θ ) + ASin ( θ )with θ = ( p • r − Et )and A and B the Majorana operators described above. Note howthe Majorana Fermion algebra generated by A and B comes intoplay in the construction of these solutions.We take it as quite significant that the Majorana algebra is di-rectly involved in these solutions. In other work [9, 8, 10, 4] wereview the main features of recent applications of the Majorana al-gebra and its relationships with representations of the braid groupand with topological quantum computing. We are now in a posi-tion to assess the relationship of the Majorana algebra with actualsolutions to the Majorana-Dirac equation, and this will be the sub-ject of subsequent work. dimensions. Using the method of this section and spacetime with one dimensionof space ( x ), we can write a real Majorana Dirac operator in theform ∂/∂t + ǫ∂/∂x + ǫηm where, the matrix representation is now two dimensional with ǫ = (cid:18) − (cid:19) , η = (cid:18) (cid:19) , ǫη = (cid:18) −
11 0 (cid:19) . We obtain a nilpotent operator, D by multiplying by iη : D = iη∂/∂t + iηǫ∂/∂x − iǫm. Letting ψ = e i ( px − Et ) , we have D ψ = ( A + iB ) ψ where A = ηE + ǫηp he Dirac Equation and the Majorana Dirac Equation B = − ǫm. Note that A = E − p = m and B = m , from which it iseasy to see that A + iB is nilpotent. A and B are the Majoranaoperators 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
Acos ( θ ) − Bsin ( θ ) = ( ηE + ǫηp ) cos ( θ ) + emsin ( θ )= (cid:18) − msin ( θ ) ( E − p ) cos ( θ )( E + p ) cos ( θ ) msin ( θ ) (cid:19) . Each column vector is a solution to the original Dirac equationcorresponding to the operator ∇ = ∂/∂t + ǫ∂/∂x + ǫηm written as a 2 × r = 12 ( t + x ) , l = 12 ( t − x ) . (Recall that we take the speed of light to be equal to 1 in thisdiscussion.) Then θ = px − Et = − ( E − p ) r − ( E + p ) l. and the Dirac equation( ∂/∂t + ǫ∂/∂x + ǫηm ) (cid:18) ψ ψ (cid:19) = 0becomes the pair of equations ∂ψ /∂l = mψ ,∂ψ /∂r = − mψ . Louis H Kauffman , Peter Rowlands
Note that these equations are satisfied by ψ = − msin ( − ( E − p ) r − ( E + p ) l ) ,ψ = ( E + p ) cos ( − ( E − p ) r − ( E + p ) l )exactly when E = p + m as we have assumed. It is quite inter-esting to see these direct solutions to the Dirac equation emerge inthis 1+1 case. The solutions are fundamental and they are distinctfrom the usual solutions that emerge from the Feynman Checker-booad Model [2, 6]. It is the above equations that form the basisfor the Feynman Checkerboard model that is obtained by examin-ing paths in a discrete Minkowski plane generating a path integralfor the Dirac equation. We will investigate the relationship of thisapproach with the Checkerboard model in a subsequent paper. Another way to put the Dirac equation is to formulate it in termsof a spacetime algebra.
By a spacetime algebra we mean a Cliffordalgebra with generators { e , e , e , e } such that e = e = e = 1, e = − e i e j + e j e i = 0 for i = j. Thus the generators of thealgebra fit the Minkowski metric and we can represent a point inspace time by p = xe + ye + ze + te so that p = x + y + z − t corresponds to the spacetime metric with the speed of light c = 1 . (The reader may wish to compare this approach with Hestenes[15].)Since the Dirac algebra demands { α , α , α , β } with all ele-ments squaring to 1 and anti-commuting, we see that spacetimealgebra is interchangeable with Dirac algebra via the translation: α = e , α = e , α = e , β = − ie where i = √− he Dirac Equation and the Majorana Dirac Equation O ψ = 0where O = i∂/∂t + iα ∂/∂x + iα ∂/∂y + iα ∂/∂z − βm. Thus we can rewrite O as O = i∂/∂t + ie ∂/∂x + ie ∂/∂y + ie ∂/∂z + ie m. Then, multiply the whole Dirac equation by − i and we find theequivalent operator O ′ = ∂/∂t + e ∂/∂x + e ∂/∂y + e ∂/∂z + e m. This point of view makes it clear how to search for Majorana al-gebra since we can search for a spacetime algebra of real matrices.Then the Dirac equation in the form O ′ ψ = 0will be an equation over the real numbers. In fact the algebra thatwe have already written for Majorana is a spacetime algebra: e = ˆ ηη, e = ǫ, e = ˆ ǫη, e = ˆ ǫ ˆ ηη. Furthermore, we can see that the following lemma gives us a guideto constructing nilpotent formulations of the Dirac equation.
Definition 1.
Suppose that { e ′ , e ′ , e ′ , e ′ } generates a spacetimealgebra A and that µ is an element of A with µ = − { e = µe ′ , e = µe ′ , e = µe ′ , e = µe ′ } is also a spacetimealgebra with e = e = e = 1, e = − e i e j + e j e i = 0 for i = j. Under these circumstances, we call the spacetime algebra A nilpotent . Lemma.
Let A be a nilpotent spacetime algebra, with notationas in Definition1 above. Then the operator D = µ∂/∂t + e ∂/∂x + e ∂/∂y + e ∂/∂z + e m Louis H Kauffman , Peter Rowlands generates a nilpotent Dirac equation.
Proof.
We wish to show that if ψ = e i ( p • ( x,y,z ) − Et ) and D ψ = U ψ then U = 0 . Calculating, we find that U = i ( − µE + p • ( e , e , e )) + e m. It ifollows that U = − ( − E + p x + p x + p x ) − m = E − p x − p y − p z − m = 0 . This completes the proof. (cid:3)
Example 1.
Before proceeding to the Majorana structure, consider the stan-dard Dirac algebra. Here we have σ , σ , σ with σ i = 1 for each i = 1 , , α and β as before to gener-ate a Clifford algebra that commutes with the Pauli algebra andis independent of it. Then the associated spacetime algebra hasgenerators e ′ = ασ , e ′ = ασ , e ′ = ασ , e ′ = √− β and the nilpotency corresponds to the fact that these generators,multiplied by βα, yield another spacetime algebra. This is givenby e = µe ′ = βαασ = βσ e = µe ′ = βαασ = βσ e = µe ′ = βαασ = βσ e = µe ′ = βα √− β = −√− α The corresponding nilpotent Dirac operator is D = µ∂/∂t + e ∂/∂x + e ∂/∂y + e ∂/∂z + e m. Hence D = βα∂/∂t + βσ ∂/∂x + βσ ∂/∂y + βσ ∂/∂z − √− αm. he Dirac Equation and the Majorana Dirac Equation ψ = e √− p • r − Et ) we obtain the nilpotent A = − βα √− E + βσ √− p x + βσ √− p y + βσ √− p z −√− αm. This can be replaced by the nilpotent U = − βαE + βσ p x + βσ p y + βσ p z − αm by factoring out the common square root of minus one. This isthe same nipotent that we have previously derived. Note thatin relation to this standard Dirac algebra we have the conjugatenilpotent U † = − βαE + βσ p x + βσ p y + βσ p z − αm, and that U + U † = 2( βσ p x + βσ p y + βσ p z − αm )so that U U † + U † U = ( U + U † ) = 4( p + m ) = 4 E . This is as we have derived earlier in the paper. The decompositioninto Clifford operators follows these lines, giving Clifford elementsthat square to E . When we work with the real spacetime alge-bras (below) that correspond to the Majorana Dirac equation, thedecomposition into Clifford algebras takes a different pattern, cen-tering on the mass m rather than the energy E. Example 2.
In the case we have considered with e ′ = ˆ ηη, e ′ = ǫ, e ′ = ˆ ǫη, e ′ = ˆ ǫ ˆ ηη. We take µ = ǫη and we find e = ǫη ˆ ηη = ǫ ˆ η,e = ǫηǫ = − η,e = ǫη ˆ ǫη = ǫ ˆ ǫ,e = ǫη ˆ ǫ ˆ ηη = ǫ ˆ ǫ ˆ η. Louis H Kauffman , Peter Rowlands
Indeed this gives a spacetime algebra and hence a nilpotentMajorana Dirac operator D = ǫη∂/∂t + ǫ ˆ η∂/∂x − η∂/∂y + ǫ ˆ ǫ∂/∂z + ǫ ˆ ǫ ˆ ηm. Example 3.
Here is another example. We take e ′ = ˆ ǫ, e ′ = ˆ η, e ′ = ǫη ˆ ǫ ˆ η, e ′ = ǫ ˆ ǫ ˆ η and µ = η ˆ ǫ ˆ η and find e = η ˆ ǫ ˆ η ˆ ǫ = − η ˆ η,e = η ˆ ǫ ˆ η ˆ η = η ˆ ǫ,e = η ˆ ǫ ˆ ηǫη ˆ ǫ ˆ η = ǫ,e = η ˆ ǫ ˆ ηǫ ˆ ǫ ˆ η = − ηǫ. This gives a spacetime algebra and hence a nilpotent Dirac operator D = η ˆ ǫ ˆ η∂/∂t − η ˆ η∂/∂x + η ˆ ǫ∂/∂y + ǫ∂/∂z − ηǫm. Example 4.
We now give a number of examples of spacetimealgebras. For this purpose it is useful to change notation. We willuse I = ǫ, J = η, i = ˆ ǫ, j = ˆ η. Thus I = J = i = j = 1 and IJ + JI = 0 and ij + ji = 0 . Wewill indicate a spacetime algebra as a 4-tuple ( e , e , e , e ) wherewe require that the e i anti-commute and that the squares of thefirst three e i are 1 while e = − . The following are spacetimealgebras. A = ( Jj, I, Ji, Jij ) B = ( Ii, j, Ji, IJi ) he Dirac Equation and the Majorana Dirac Equation C = ( iJ, I, jJ, ijJ ) D = ( iJ, I, jJ, IJ )It is easy to see that A , B , C and D are nilpotent. Note that (upto signs) B is obtained from A by interchanging i, j with I, J andthen interchanging i and j. C is obtained from A by interchanging i and j directly. To see that A is nilpotent, multiply by IJ.
Thealgebra D is also nilpotent, via multiplying by ijJ. The General Case.
Now suppose that we are given a nilpotentspacetime algebra specified by { e ′ , e ′ , e ′ , e ′ } and µ with µ = − { e , e , e , e } is also a spacetime algebra with e i = µe ′ i for i = 1 , , , . Then we have the nilpotent Dirac operator associatedwith this algebra: D = µ∂/∂t + e ∂/∂x + e ∂/∂y + e ∂/∂z + e m. Let ι = √−
1, a square root of negative unity that commutes withall algebra elements. Applying D to ψ = e ι ( p • r − Et ) we obtain thenilpotent A = ι ( − µE + e p x + e p x + e p x ) + e m. The nilpotent A is directly decomposed into its two (Majorana)Clifford parts as the real and imaginary parts of A , just as in ourprevious discussion of a special case. Other examples lead to realsolutions to the Majorana Dirac equation just as we have doneabove. Note that the Clifford parts are ρ = − µE + e p x + e p x + e p x and τ = e m with ρ = τ = − m and ρ and τ anticommute. It is of interest tonote that the Clifford algebra is collapsed when the mass is equalto zero.But we need to be systematic here. Consider that the fourth el-ememt of a spacetime algebra has square − . Up to symmetries the0
Louis H Kauffman , Peter Rowlands possibilities are ij and IJi.
Take each of these cases in turn. Firstsuppose that e = ij. Then consider first all square one elements.These are S = { i, j, I, J, ijIJ, iI, iJ, jI, jJ } . The subset of elements of S that anti-commute with ij is S [ ij ] = { i, j, iI, iJ, jI, jJ } , and the (up to order and symmetry) the only triplet in S [ ij ] thatmutually anti-commutes is { i, jI, jJ } . This gives the spacetime algebra { i, jI, jJ, ij } . This algebra is nipotent via multiplication by
IJi.
Now consider the subset of elements of S that anti-commutewith IJi.
This subset is S [ IJi ] = { j, I, J, ijIJ, iI, iJ } . The triplets that anti-commute are { j, iI, iJ } and { ijIJ, I, J } . These give rise to spacetime algebras { j, iI, iJ, IJi } and { ijIJ, I, J, IJi } . The first is nilpotent via the multiplier ij and the second is nilpo-tent via the multiplier IJj.
Up to symmetries these are all the casesand so we have proved the result he Dirac Equation and the Majorana Dirac Equation Theorem.
All real Majorana spacetime algebras are nilpotent and,up to permutations and substitutions, they are of the followingtypes: { i, jI, jJ, ij } , { j, iI, iJ, IJi } , { ijIJ, I, J, IJi } . In a subsequent paper we shall follow up the consequences of thisresult.
To compare the Majorana equation with the Rowlands nilpotentformulation, we take the four algebraic operators ǫ , η and ˆ ǫ , ˆ η ascomponents of a double vector set I , J and i, j . That is, we aregiven that I = J = 1 and that IJ + JI = 0 and we are giventhat i = j = 1 and that ij + ji = 0 . Furthermore, the elements i and j commute with the elements I and J. For translation, we canset ǫ = I, η = J, ˆ ǫ = i, ˆ η = j. We use ι for a commuting square root of negative unity. Note thatwhile we use i, j, I, J for these algebras, all these elements squareto one. We can then write the Majorana equation in the form.( ∂/∂t + Ii∂/∂x + j∂/∂y + Ji∂/∂z − JIim ) ψ = 0 . Note that { e = Ii, e = j, e = Ji, e = JIi } forms a spacetimealgebra, and that if we multiply each element by ij we get a newspacetime algebra: { ije = − jI, ije = i, ije = − jJ, ije = − jJI } = { e ′ = − jI, e ′ = i, e ′ = − jJ, e ′ = − jJI } . Louis H Kauffman , Peter Rowlands
Thus by our previous discussion this will give a nilpotent formu-lation of a version of the Majorana Dirac equation. Multiplyingfrom the left by ιij gives( ιij∂/∂t + ιIj∂/∂x − ιi∂/∂y + ιJj∂/∂z + ιIJjm ) ψ = 0 . Rearranging the symbols leads to( ιij∂/∂t + ιjI∂/∂x − ιi∂/∂y + ιjJ∂/∂z + ιjIJm ) ψ = 0 . To compare with the Rowlands Dirac nilpotent formalism, we applythe operator in the bracket to a free particle wavefunction ψ = e ι ( p • r − Et ) to find D ψ = Aψ with A = ijE − jIp x + ip y − jJp z + ιjIJm. This squares to − E + p + m = 0 . and so is nilpotent. Generally, in the discussion below we will des-ignate such nilpotent factors by the letter A. If we compare this to the Rowlands formalism, where (again rec-ognizing the arbitrary nature of the signs, and also the choice ofsymbols between i , j and I , J ) A can be written in the form A = ijE + jIp x + ιjIJp y + jJp z + im. we see that the effect of changing from Dirac to Majorana is toswitch the status of the terms p y and m. In effect, the space-momentum operator no longer has perfect rotation symmetry be-tween its components. This could be related to the fact that ex-periments so far which have claimed to have detected Majoranabehaviour seem to involve only 1 − or 2 − dimensional systems. Atthe same time the Majorana formalism seems to be suggesting a“mingling” of momentum or angular momentum with mass. If theneutrino is a Majorana particle, then its mass could be generated he Dirac Equation and the Majorana Dirac Equation ∂/∂t + I∂/∂x + J∂/∂y − ijIJ∂/∂z − iIJm ) ψ = 0 . Multiply from the left by jIJ :( jIJ∂/∂t − jJ∂/∂x + jI∂/∂y + i∂/∂z − jim ) ψ = 0 . Here the third component of momentum has exchanged with theenergy term, up to complex factor (rather than the second withmass). Now apply a free particle wavefunction ψ = e ι ( p • r − Et ) tofind A = ι ( − jIJE − jJp x + jIp y + ip z ) + ijm This is still nilpotent. We may note that the same double spacestructure of 5 + 3 occurs as in the standard nilpotents. We willmake a more detailed discussion in a subsequent paper.
The key text for Rowland’s discrete version of nilpotent Dirac is“Zero to Infinity” [17], pages 182-184. “The Foundations of Physi-cal Law” [18] has a more abbreviated version. For a formal creationoperator that doesn’t distinguish between particle and antiparticlemathematically, and can be split into two parts, we could look atthe discrete Dirac equation as Rowlands has been writing it, usingKauffman’s non-commutative discrete derivatives [5, 7, 11]. Row-lands’ discrete version of the nilpotent Dirac equation is of the formbelow with the option of premultiplying by ι (a commuting squareroot of negative unity).( ± ij∂/∂t ± i ∇ )( ± ijE ± iIP ± iIP ± iIP + jm ) . This means that the creation operator, which automatically gener-ates the nilpotent amplitude is the first bracket with no m term.This can be split into two parts which are negatives of each other,4 Louis H Kauffman , Peter Rowlands and so could represent particle and antiparticle. But, with no massterm in the operator, the signs could be reversed arbitrarily bypremultiplying from the left by − . So this operator in this formdoesn’t distinguish particle and antiparticle.Rowlands sees Fermions as the only particles that have a totalnonzero weak charge. The weak charge is the only one associ-ated with chirality. He thinks of weak charges as being associatedwith the pseudoscalar ι and having an ambiguity with regard tosign, (this may be related to the discrete time in the models ofKauffman/Noyes [6] and Garnet Ord [16]) and so being ultimatelyresponsible for zitterbewegung, with a weak charge acting like adipole with its vacuum reflection. The most interesting fermion inthis regard is the neutrino, which only has a weak charge. Ambi-guity over the sign of the weak charge (0 or 2 w or 0 or − w ratherthan just 0) appears in those mesons whose decay involves CP vi-olation. In some sense, the zitterbewegung is mixing particle andantiparticle and we are interested in how the Majorana representa-tion can be used here. We are also interested in how the (1 − γ ) / This is indirectly related to Majorana. The question is: why doesthe neutrino have a mass if it remains chiral as to the weak interac-tion, the only one of the gauge interactions to which it is subject?The nilpotent has the following terms:( ijE + iIp x + iJp y + ιiIJp z + jm )The charges which relate to these terms are: weak ( ijE ), strong( iP ), electric ( jm ). Hypothetically massless Fermions are knownto have two sharply defined helicity states. If the nilpotent( ijE + iIp x + iJp y + ιiIJp z + jm ) he Dirac Equation and the Majorana Dirac Equation ijE + iIp x + iJp y + ιiIJp z )which we can write as ( ijE + iP )then chirality is determined by Pauli exclusion because( − ijE + iP )can exist, but not ( − ijE − iP )or ( ijE − iP ) . Helicity is taken as the relative sign of ijE/iP.
Positive is left-handed, negative is right-handed. So we have LH Fermions and RH anti-Fermions.How do real Fermions get their masses? We take the threegauge interactions separately. Let’s take( ijE + iP + jm )and consider the strong interaction. In Rowlands’ representationof baryons, three have + P and three have − P because each direc-tion of P, (the directions are x, y, z ) introduces a separate nilpotentbracket and the active component switches between the three di-rections. The mass comes from this switching, equivalent to theparity operator ( P ), with maximum achirality. The P term is inthe same position as the strong charge. The two signs of P ensurea large mass for any baryon.How does ( ijE + iP + jm ) become ( ijE − iP + jm ) in such a case?We annihilate ( ijE + iP + jm ) by creating ( − ijE − iP + jm )( LH )and at the same time create the new state ( ijE − iP + jm )( RH ) . Thus, to switch between the two spin states requires the exchangeof a spin 1 boson:( − ijE − iP + im )( ijE − iP + jm )6 Louis H Kauffman , Peter Rowlands
The whole process then becomes( ijE + iP + jm )( − ijE − iP + jm )( ijE − iP + jm ) −→ ( ijE − iP + jm )The first bracket is the Fermion state to be changed; the next twobrackets form the spin 1 boson absorbed, and the bracket on theRHS becomes the final Fermion state. The reverse process wouldbe( ijE − iP + jm )( − ijE + iP + jm )( ijE − iP + jm ) −→ ( ijE + iP + jm )with the spin 1 boson becoming( ijE + iP + jm )( − ijE + iP + jm ) . Of course, the bosons in this case (gluons) are massless but we canleave out the mass if we use the discrete representation of the op-erator. (A spin 0 boson, notably, unlike spin 1, does not change aFermionic state.)Electrons are Fermions with electric charges, and this interac-tion is both LH and RH , unlike the weak, so electrons have RH states which experience electric but not weak interactions. In Row-lands’ view, the electric charge is in the same position as the jm term. We can’t switch jm but we can switch ijE and iP simulta-neously via charge conjugation ( C ). This doesn’t happen directlybecause the boson required would be( − ijE − iP + jm )( − ijE − iP + jm ), which zeros immediately. So, it would have to be a combina-tion of separate E and P switches. Does mass happen in some waythrough this process? The two P states give mass immediately.Certainly this is one of the ways mass will be generated, probablythe main way. If there is any switching between + E and E , it willbe at the level of zitterbewegung (see below).The weak charge is in the same position as the ijE term. Toswitch this alone is the same operation as time reversal ( T ). If theweak charge is a dipole with its vacuum reflection, then we canconsider a zitterbewegung taking place, possibly generating a weak he Dirac Equation and the Majorana Dirac Equation iP. The Fermion state is( ijE + iP + jm )and the vacuum state is( − ijE + iP + jm ) . The zitterbewegung disappears on fixing a real state of the neu-trino by observation. If the neutrino mass is due to zitterbewegungit is possible that the mass is observed without two states of spin.The process of changing the ijE term only requires interactionwith a paired Fermion or paired anti-Fermion spin 0 state, suchas ( − ijE − iP + jm )( − ijE + iP + jm ) . Applying this will change( ijE + iP + jm ) to ( − ijE + iP + jm ) via( ijE + iP + jm )( − ijE − iP + jm )( − ijE + iP + jm ) −→ ( − ijE + iP + jm ) . The reverse procedure will be( − ikE + ip + jm )( ikE − ip + jm )( ikE + ip + jm ) −→ ( − ikE + ip + jm ) . If a small weak mass is a product of zitterbewegung which is notdirectly observed in the neutrino state, then maybe the chiralitycan exist at the same time as mass. Does this mean that the weakmass is not generated in the same way as the others, and so doesn’tgive us a problem with chirality?The heuristic way of saying a massive particle can’t be one-handed is to say that, if a particle has a mass and so can’t travelat the speed of light, we could theoretically overtake it and lookback and see that it had flipped to the opposite handedness. Ifthe zitterbewegung mass is smaller than the mass from switchingparity, then we would not be able to do this since we couldn’t finda particle to travel faster than the neutrino. Maybe overtakingthe neutrino would be the same as making it its antiparticle (cfMajorana). It would be interesting to see this in relation to neu-trino mixing where there are three neutrinos with slightly differentmasses flipping into each other.8
Louis H Kauffman , Peter Rowlands
We would like to relate neutrino oscillation and zitterbewegungin further work. It is notable that there are three possible switches(see the Theorem at the end of Section 4) between the Majoranaand standard Dirac nilpotent algebras. One of these switches thealgebra term of component of momentum with the mass term. An-other one switches the algebra component with the energy term.The third has the appearance of switching (again after multiplica-tion by ι ) the energy term and one of the momentum terms fromthe classical Dirac operator. Perhaps these three modes of switch-ing are analogous to CPT, and responsible for the three neutrinogenerations. We have seen how the nilpotent approach to the Dirac equationsheds new light on the Majorana-Dirac equation and on the struc-ture of Majorana Fermions. This paper marks the beginning of newwork on this subject. There is much that remains to be done andwe will consider the key questions in subsequent papers. The directappearance of Majorana operators (often identified with MajoranaFermions in recent literature) in our real solutions to the Majo-rana Dirac equation suggests a deeper examination of the natureof Majorana Fermions in condensed matter physics and in relationto quantum computing. Our solutions to the Dirac equation inone dimension of space and one dimension of time suggests thatit will be useful to re-examine the Feynman Checkerboard modelfor the Dirac propagator. This should lead to new insight intopath integral formulations for solutions to the Dirac equation inthree dimensions of space and one dimension of time. The mostsignificant possibility for Majorana Fermions outside of condensedmatter is in the neutrino sector, where there is a major problem inreconciling chirality with nonzero neutrino masses. A tentative pro-posal is made here, in the last section, toward a possible resolution. he Dirac Equation and the Majorana Dirac Equation References [1] Beenakker C.W. J. (2012), Search for Majorana Fermions insuperconductors, arXiv: 1112.1950.[2] Feynman R.P and Hibbs A.R. (1965) , “Quantum Mechanicsand Path Integrals” McGraw Hill Companies, Inc, New York.[3] Ivanov D. A. (2001) Non-abelian statistics of half-quantumvortices in p -wave superconductors, Phys. Rev. Lett. 86, 268(2001).[4] Kauffman L.H. (1991,1994,2001,2012), Knots and Physics,
World Scientific Pub.[5] Kauffman L.H. and Noyes H. P. (1996), Discrete Physics andthe Derivation of Electromagnetism from the formalism ofQuantum Mechanics,
Proc. of the Royal Soc. Lond. A , ,pp. 81-95.[6] Kauffman L.H. and Noyes H. P. (1996), Discrete Physics andthe Dirac Equation, Physics Letters A , 218 ,pp. 139-146.[7] Kauffman L.H. (2004), Non-commutative worlds,
New Journalof Physics 6 , 2-46.[8] Kauffman L.H. (2016), Knot logic and topological quantumcomputing with Majorana Fermions. In “Logic and algebraicstructures in quantum computing and information”, LectureNotes in Logic, J. Chubb, J. Chubb, Ali Eskandarian, andV. Harizanov, editors, 124 pages Cambridge University Press(2016).[9] Kauffman L.H. . (2018), Majorana Fermions and represen-tations of the braid group. Internat. J. Modern Phys. A 33(2018), no. 23, 1830023, 28 pp.[10] Kauffman L.H. , Iterants, Entropy (2017), 19, 347;doi:10.3390/e19070347.[11] Kauffman L.H. (2018), Non-Commutative Worldsand Classical Constraints, Entropy 2018, 20, 483;doi:10.3390/e20070483.[12] Kitaev A., Anyons in an exactly solved model and be-yond (2005), Ann. Physics 321 (2006), no. 1, 2111. arXiv.cond-mat/0506438 v1 17 June 2005 .0 Louis H Kauffman , Peter Rowlands [13] Majorana E. (1937), A symmetric theory of electrons andpositrons, I Nuovo Cimento, (1937), pp. 171-184.[14] Mourik V. ,Zuo K., Frolov S. M. ,Plissard S. R. , BakkersE.P.A.M. , Kouwenhuven L.P. (2012), Signatures of Majo-rana Fermions in hybred superconductor-semiconductor de-vices, arXiv: 1204.2792.[15] Hestenes D. (1966), Space-Time Algebras (Gordon and Breach,New York)[16] Ord G. N. (2010). Feynmans corner rule; quantum propaga-tion from special relativity. International Journal of Theoret-ical Physics, 49:25282539, 2010. 10.1007/s10773-010- 0445-8.[17] Rowlands P. (2007), “Zero to Infinity - The Foundations ofPhysics”, Series on Knots and Everything - Volume 41, WorldScientific Publishing Co., 2007.[18] Rowlands P. (2014),
The Foundations of Physical Law (WorldScientific, Singapore, London and Hackensack, NJ)[19] Rowlands P. (2015),
How Schr ¨ o dinger’s Cat Escaped the Boxdinger’s Cat Escaped the Box