Revisiting Special Relativity: A Natural Algebraic Alternative to Minkowski Spacetime

Minkowski famously introduced the concept of a space-time continuum in 1908, merging the three dimensions of space with an imaginary time dimension , with the unit imaginary producing the correct spacetime distance , and the results of Einstein’s then recently developed theory of special relativity, thus providing an explanation for Einstein’s theory in terms of the structure of space and time. As an alternative to a planar Minkowski space-time of two space dimensions and one time dimension, we replace the unit imaginary , with the Clifford bivector for the plane that also squares to minus one, but which can be included without the addition of an extra dimension, as it is an integral part of the real Cartesian plane with the orthonormal basis and . We find that with this model of planar spacetime, using a two-dimensional Clifford multivector, the spacetime metric and the Lorentz transformations follow immediately as properties of the algebra. This also leads to momentum and energy being represented as components of a multivector and we give a new efficient derivation of Compton’s scattering formula, and a simple formulation of Dirac’s and Maxwell’s equations. Based on the mathematical structure of the multivector, we produce a semi-classical model of massive particles, which can then be viewed as the origin of the Minkowski spacetime structure and thus a deeper explanation for relativistic effects. We also find a new perspective on the nature of time, which is now given a precise mathematical definition as the bivector of the plane.

We now aim to describe a form of the Dirac equation in Cliord algebra isomorphic to the conventional Dirac equation @ j i + iq A j i = imj i; (1) where i = p 1 and are the Dirac matrices, using natural units in which c =~= 1. If we reduce the number of spatial dimensions to two, then the Dirac algebra can t within the Pauli algebra, and we can write the Dirac equation as @ t j i + ( 1 @ x + 2 @ y ) j i = i 3 mj i; (2) where 1 ; 2 ; 3 are the Pauli matrices [1]. Naturally the one-dimensional Dirac equation can be found by ignoring the y direction as @ t j i + 1 @ x j i = i 3 mj i.
If we select a spinor mapping to the two dimensional multivector as j i = a 0 + ja 3 a 2 + ja 1 ! 6 = a 0 + a 2 e 1 + a 1 e 2 + a 3 e 1 e 2 ; (3) then we nd the following mapping for the Pauli matrices k j i 6 e k (4) for k = 1; 2 and i 3 j i = 1 2 j i 6 e 1 e 2 (5) using iI = 1 2 3 . Expanding Eq. (2) we nd @ t = @ x 1 + @ y 2 + m 1 2 (6) using the relation i 3 = 1 2 . Mapping this to the multivector dened in Eq. (3) we nd @ t = e 1 @ x + e 2 @ y + me 1 e 2 : Multiplying from the left by e 1 e 2 we nd e 1 e 2 (@ t + r) = m : Hence dening @ = (@ t + r), where = e 1 e 2 , we nd @ = m (9) where the Dirac wavefunction is described by the multivector in Eq. (3). We then nd @ 2 = @ 2 t r 2 the d'Alembertian, thus allowing us to recover the Klein-Gordon equation in two dimensions from Eq. (9). This equation implies we have a Hamiltonian H = r+m = p+m. The one dimensional Dirac equation is also given by Eq. (9) but with the spatial gradient operator r = e 1 @ x . Selecting a dierent permutation of Pauli matrices we can nd an alternate equation (@ t r) = m . The two versions of the Dirac equation corresponds to two possible square roots of the Klein-Gordon equation in two dimensions.
Bilinear observables Given = + E + B, then where~ = + E B is the reversion operation. We therefore dene the probability current as This is in the form of the four-velocity detailed in the main paper. The term = 2 +E 2 +B 2 is a positive denite scalar equivalent to = j j 2 = h j i conventionally calculated for the probability density. Thus this equation relates the Dirac current J to the wavefunction . For 3 S, we nd J = SS y y (12) so that provided SS y = , S represents a gauge transformation. Multiplying from the right by S and remembering that S y S is a scalar, then we nd S = S which implies that S commutes with and hence S = e = cos + sin describing a rotation in the spin plane = e 1 e 2 .
To conrm our denition of probability current, we rstly write the Dirac equation and its reverse @ t = r m (15) @ t~ = ~ r + m~ : Multiplying the rst equation on the left with~ and the second equation on the right with we obtaiñ which conrms our denition of probability current in Eq. (11) and Eq. (13), through comparison with Eq. (14).
We have the vector terms @ t v + 2r + 2E (r ¡ E) 2BrB 2rB 2Br (21) Alternatively, using the Dirac equation and its reverse and multiplying the rst equation on the right with~ and the second equation on the left with we obtain = @ t v + r + 2 (E r) E + 2rB 2Br + 2m (2(E + BE)) using the result that 1 2 rE 2 = (r ¡ E) E (E r) E. We have the Poynting vector s = BE which is equivalent so S = E ¢ B in three dimensions. We have the energy density u = 1 We take a trial solution (X) = Ce K¡X = Ce (k¡x wt) ; (29) where C is some constant multivector, then on substitution into Eq. (61), we nd (@ t + cr) Ce (k¡x wt) = C( w)e K¡X + ce 1  Multiplying from the right by~e K¡X , we nd (~w c~k) C = mc 2 C. We thus need to satisfy (~w c~k) C + mc 2 C = 0: Given the multivector C = a + u + b, we nd C = a + u b. Hence we have the equation For a particle at rest we have E =~w = mc 2 , giving Hence for positive energy solutions we require C = a+b, and for negative energy solutions with E = ~w we require C = u. Hence we have the positive and negative wavefunctions + = (a + b)e wt and = ue wt . The positive wavefunction acting on vector will rotate it clockwise and the negative wave function will rotate a vector in the negative direction in agreement with the de Broglie formula E =~w. Hence this model gives immediately the result that a negative energy, is a negative angular frequency which implies a negative time direction, with an inverted spin, in agreement with Feynman's interpretation of negative energies as particles traveling back in time with an inverted spin.
For the general case we need to equate scalar, vector and bivector components of Eq. (32) to nd cu ¡ p + (~w mc 2 )a = 0 (34) acp + (~w + mc 2 )u + bcp = 0 b(~w mc 2 ) + cu p = 0: Interpreted as an operator we can identify a rotation, reection and a boost. If we assume minimal coupling of the form p = ~r qA and E = ~@ t qV then we nd where A = (V + cA) is the potential multivector corresponding to a four-potential. This denition for the potential is compatible with Maxwell's equations, using where E = rV @A @t and cB = cr A and @V c@t + cr ¡ A = 0 is the Lorenz gauge, which produces Maxwell's equations in terms of electromagnetic potentials as @ 2 A = J, which are a set of three uncoupled Poisson equations. We nd F 2 = E 2 B 2 and J ¡ A = qV qv ¡ A, and so we have the eld Lagrange's equations @ @v @(@A) @v @A = 0. We nd T = F F = E 2 + B 2 + 2BE ¡ = (U + S) , where U = E 2 +B 2 is the total eld energy and S = 2BE is the Poynting vector. We can then write Poynting's energy conservation theorem as @ ¡ T = J ¡ F .

Stationary state solutions
We assume A and V are time-independent and we look for stationary states solutions of the form (X) = (r)e wt = (r)e Et=~; and substituting into the Dirac equation in Eq. (41) we nd c@ = ~(@ t + cr) (r)e wt (44) = ~ (r)( w)e wt + ~cr (r)e wt = mc 2 (r)e wt q(V + cA) (r)e wt : Multiplying from the right with e wt , we nd E +~cr = mc 2 + q(V cA) : Writing = a+v +b = A +e 1 2 = A + B which splits into the even and odd parts of the multivector, where A = a + b and B = e 1 (c + d), then we nd two coupled equations E A +~cr B = (qV + mc 2 ) A qcA B (46) E B +~cr A = (qV mc 2 ) B qcA A ; which correspond to conventional solutions [2]. From the second equation we nd B = qcA A +~cr A E qV + mc 2 = ( qcA +~cr) A E qV + mc 2 ; remembering that commutes with A as it is the even subalgebra. The term in brackets in three dimensions factorizes to (i~r + qA) 2 = ~2r 2 + qA 2 + q~ir ¡A+q~iA¡r, however in two dimensions the pseudoscalar is non-commuting and so this factorization is not possible.
We can select the Coulomb gauge, r ¡ A = 0 and for an electron q = e to give E H A = except for the extra factor of two on the A ¡ r term, where A is the conventional Pauli spinor and is the three-vector of Pauli matrices. This could be an artifact of a two-dimensional form. As is well known the coecient e2 m in front of B z , gives a spin gyromagnetic ratio g s = 2 in close agreement with experiment.
If we select A = 0 then Eq. (53) reduces to ~2 2m r 2 eV the Schr odinger equation in two dimensions. The representation of basic physical equations in a two-dimensional Cliord algebra Cl 2;0 , representing a uniform positive signature, gains signicance from the isomorphism Cl (n+2;0) % Cl (0;n) Cl (2;0) (56) so that higher dimensional Cliord algebra can be constructed from the two-dimensional case. This result also leads to the well known Bott periodicity of period eight for real Cliord algebras [3].
The multivector model for the electron In the main paper we represented a particle as P =~k + ~!0 2c , so that under a boost, the frequency ! 0 will increase to ! = ! 0 , with the radius required to shrink to r = r 0 =, so that the tangential velocity v = rw = r0 w 0 = r 0 w 0 = c, remains at the speed of light. Hence this simplied two-dimensional model in Fig. 1, indicates that under a boost, the de Broglie frequency will increase to ! 0 implying an energy and hence a mass increase m 0 , the frequency increase also implies time dilation, and the shrinking radius producing length contraction, thus producing known relativistic eects. Now because the wave vector term k represents a momentum perpendicular to the direction of motion, we can identify it with the angular momentum of the lightlike particle with p = E=c = ~w 0 =(2c) = mc, remembering r ¡ p = 0, we have the spin angular momentum L = rp = r p = [r; p] = 2 mc (mc) = 2 which is invariant, as expected for a spin- 1 2 particle. Integrating the momentum multivector with respect to the proper time , remembering that dt = d , and dividing by the rest mass m we nd where ! 0 = d0 dt . Inspecting the bivector component, we nd r 0 d 0 = r 0 w 0 dt = 2 mc 2mc 2 dt = cdt; and hence the time can be identied as the circumferential distance r 0 d 0 at half the Compton radius 2mc .

Wave mechanics
We now work within the two-dimensional Cliord multivector to intuitively produce two-dimensional versions of Dirac's and Maxwell's equations. From the de Broglie hypothesis that all matter has an associated wave [4], given by the relations p =~k and E =~w, we nding the wave multivector We now nd the dot product of the wave and spacetime multivectors K ¡ X = k ¡ x wt, giving the phase of a traveling wave. Hence for a plane monochromatic wave we can write = e K¡X = e (k¡x wt) , which leads to the standard substitutions, p = ~r and E = ~@ t and so we dene from the momentum multivector where r = e 1 @ x + e 2 @ y . We therefore nd @ 2 = 1 c 2 @ 2 t r 2 the d'Alembertian in two dimensions, so that @ is the square root of the d'Alembertian. Following Dirac, we therefore write @ = mc ; (61) which is isomorphic to the conventional Dirac equation, and comparable to the Dirac equation previously developed in three dimensional Cliord algebra [5, 6]. Acting from the left a second time with the dierential operator @ we produce the Klein-Gordon equation, 1 c 2 @ 2 t r 2 ¡ = m 2 c 2 2 . For the non-relativistic case, summing the kinetic and potential energy, we nd the total energy E = T + V = p 2 2m + V , and substituting the standard operators for p and E we nd where J is a general multivector. Acting a second time with the spacetime gradient produces and provided @J = 0, we satisfy the massless Klein-Gordon equation. Now @J = @ ¡ J + @ J, and for J = ( + J) representing source currents, where we now switch to natural units with c =~= 1, we nd rstly @ ¡ J = @ t + r ¡ J = 0 which is the requirement of charge conservation. Also @ J = @ t J + r J + r that is also zero for steady currents and curl free sources, and hence for this restricted case Eq. (63) is also a solution to the Klein-Gordon equation. Then writing the electromagnetic eld as the multivector = E + B, we have produced Maxwell's equations, that is, from Eq. (63) we nd that when expanded into scalar, vector and bivector components, gives r ¡ E = , rB @E @t = J, and r E + @B @t = 0 respectively, and noting that in three dimensions r E = r ¢ E and rB = r ¢ B, we see that we have produced Maxwell's equations for the plane. This equation also very naturally expands to produce Maxwell's equations in three dimensions as (@ t + r) (E + iB) = J, where the magnetic eld now becomes a three-vector, with i = e 1 e 2 e 3 the trivector [6].
Hence using a general multivector = + E + B we have in natural units That is, with the assumption of the form of the spacetime gradient @ given in Eq. (60) and the spacetime multivectors, the two simplest rst order dierential equations we can write are Maxwell's equations and the Dirac equation. Also, setting m = 0 in the Dirac equation we nd @ = 0, the Weyl equation for the plane. Hence the two dimensional Cliord multivector provides a natural`sandbox', or simplied setting, within which to explore the laws of physics. As a generalization, Maxwell's source term can be expanded to a full multivector to give J = ( + J + s), and s describes magnetic monopole sources.

Solutions
An elegant solution path is found for the Maxwell and Dirac equation in Eq. (67) through dening the eld in terms of a multivector potential A = ( V + cA + M ), with M describing a possible monopole potential, given by = @A: We then nd Maxwell's equations dened in Eq. (67) in terms of a potential becomes Now, as @ 2 = r 2 1 c 2 @ 2 t is a scalar dierential operator we have succeeded in separating Maxwell's equations into four independent inhomogeneous wave equations, given by the scalar, vector and bivector components of the multivectors, each of which have the well known solution [7] given by where r = jr sj, the distance from the eld point r to the charge at s, and where we calculate values at the retarded time. The eld can then be found from Eq. (68) by dierentiation.
We nd (@ t +r) (u + S) = @ t u r¡S+@ t S +ru rS, therefore we can express the conservation of momentum and energy as @T = J + (E ¡ r + r ¡ E)E + 2rB ¡ E; (71) where the scalar components express the conservation of energy and the vector components the conservation of momentum. The cumbersome (E ¡ r + r ¡ E)E term is typically absorbed into a stress energy tensor [7]. The conservation of charge @ t +r¡J = 0 also follows from Maxwell's equation through taking the divergence of Maxwell's equation @ = J.
We can also dene a Lagrangian L = 1  Multivector model for the electron, consisting of a light-like particle orbiting at the de Broglie angular frequency ! 0 at a radius of r 0 = c =2 in the rest frame, and when in motion described generally by the multivector P e =~k + ~!0 2c . Under a boost, the de Broglie angular frequency will increase to ! 0 , giving an apparent mass increase and time dilation, the electron radius will also shrink by , implying length contraction, thus naturally producing the key results of special relativity.