Sierpinski (<1K) The Dirac Particle

    "This is the kind of idea that can ruin a young scientists career. It appears to be too importanrt to keep quiet about. But if you try and explain it to most phsysicists, they are likely to dismiss you as some kind of crankpot."     --     David Hestenes, 1986

    In this chapter we will establish the 5D Dirac-Hestenes equation Ñpyp   =   (m - qap*)yp     with 5D pseudovector potential ap* º ape12345-1 and yp a <0;3;4>-grade scaled idempotent Â4,1 multivector field for a particle of mass m and charge q.
    There are ten 3-blades in any extended basis for Â4,1 , five plussquare ones involving e4 and five minsusquare "spacial" ones. There are five 4-blades four of which are minussquare. Thus <0;3;4> pensities are formed from six plussquare blades (including the unit scalar) and nine minussquare blades.
    However, by replacing any 3-blade by i=i times its dual we can consider <0;3;4>-grade Â4,1 multivectors as residing in even Clifford algera C4,1 +.

     Â4,1 + contains ten plussquare even blades , or eleven if you include the unit scalar ; and ten minusquare even blades. This is suggestive of the the ten or eleven "dimensions" required by "String Theory". [  Symbol Check: ½ denotes the value half. denotes exponential conjugation. y denotes greek psi. ]

The 4D Dirac Equation
    To encompass a four-velocity in a particle's pensity we must move to a <0;3;4>-pensity, but we will approach this indirectly in a traditional QM manner via the Dirac equation. The traditional ket representor y for a ½-spin particle or Dirac spinor is a 4×4 complex matrix having zeroes everywhere but in the first column, effectively a 4-D complex 1-vector.

The Free Dirac Equation
    Dirac derives the Free Dirac equation applicable in the absence of an electrostatic field via a somewhat obsfucated factorisation of the Â1,3 Free Klein Gordan Equation . The Free Klein Gordan equation in a Â1,3 timespace is ( (/t)2 - Ñ2 + a2)y = 0 where Ñ2 denotes the usual Euclidean 3D Laplacian . We must regard this as Ñ2yp = -a2yp in Â1,3 and as Ñ2yp = a2yp in Â3,1.

    Ñp2 yp = -a2 yp factorises as (iÑp+a)(iÑp-a)yp  =  0       where real scalar a = 2pmch-1 = mch-1 = m in natural units .
    Subequation (iÑp - a)yp = 0 Û  (2p)-1he4(iÑp - a)yp = 0 Û  ( me4 - e14 me1 - e24 me2 + e34 me3 - e4mc)yp = 0 , where mei º hÐei = (2p)-1hi Ðei is the momentum operator traditionally denoted by pi. Dirac wrirtes this as ( me4 -a1 me1 -a2 me2 -a3 me3 -b)y= 0     [ PQM 67.7 ]     , seeking 4x4 complex matrices a1 ,a2 ,a3 ,b which anticommute and have ai2=1 , b2 = mc = m . Matrix representors of e01, e02, e03, and e0 (mc)½ clearly fulfill these requirements given an appropriate geometric i and h º h(2p)-1i = hi .

The Dirac Equation
    Dirac incorporates a 1-potential field ap by following the "classical rule" of replacing me0 with me0 - qc-1 V and mej by mej - qc-1 Aj for jÎ{1,2,3} where V and A are the scalar and 3D 1-vector electrodynamic potentials at p.  The relativistic analogue of this requires replacing md by md - qc-1 d-1¿ap for 1-vector "four-potential" ap.
    The Dirac equation for a particle of mass m and charge q in a electrostatic field is thus, in Dirac's original terminology (but retaining m for momentum)
    ( me0-qV -a1( me1-qA1) -a2( me2-qA2) -a3( me3-qA3) -b )yp = 0     [ PQM 67.11 ] ,

    A more common form of the C4×4 Dirac equation is obtained by setting g0 = (mc)-1b ; gj = -(mc)-1baj   for j=1,2,3 (which also anticommute); and exploiting mek º ih Ðxk = ih (/xk) where h=h(2p)-1 . It is
    åk=03 ( ih gk(/xk) - gkqc-1ak ) y = mcy
where the gk are four p-independant anticommuting 4x4 complex Dirac matrices satisfying g02=1 ; gj2=-1 for j=1,2,3 ; a0=V, aj=-Aj are the real scalar coordinates of a Â1,3 1-vector "four-potential"   ; c is the speed of light ; h is the Planck constant; and i=Ö-1 is an "imaginary" scalar that commutes with everything including the gj.

    Any solution to Ñpy = ±ay is a solution to Ñp2y = a2y and an e4-static origin-centred solution is given by yp = |^e4(p)|-1 b(± a|^e4(p)|) [  Where a º ea denotes exponentiation ] . The + solution is unbounded and so rejected, leaving Yukawa potential
    yp = |P|-1 b (-a|P|)   =   |P|-1 b (-a|P|) , where p = P + p4e4. A rapidly damped function of e4-spacial distance from 0.

Dirac Algebra

    There are various equivalent "suites" of Dirac matrices (aka. gamma matrices) in the QM literature. We adopt a coventional set here (see eg. Hannabuss) and construct them from the 2×2 Pauli matrices as
g0= æ 10 ö ; gj= æ 0 sj ö
è 0-1 ø è - sj0 ø
though RgjR-1 provides another set for any invertible 4×4 matrix R.     We list our gj explicitly for reference.
g0= æ 1000 ö    ; g1= æ 0001 ö    ; g2= æ 000-i ö ; g3= æ 0010 ö    
ç 0100 ÷ ç 0010 ÷ ç 00i0 ÷ ç 000-1 ÷
ç 00-10 ÷ ç 0-100 ÷ ç 0i00 ÷ ç -1000 ÷
è 000-1 ø è -1000 ø è -i000 ø è 0100 ø
We observe that they all have zero trace corresponding geometrically to having zero scalar part. Because we are here interested in Â3,1 spacetime rather than Â1,3 timespace we define matrices f j º igj j=0,1,2,3 . We will also change the "timelike" suffix from 0 to 4 via f4ºf0 , x4=x0.
    For distinct i,j,k Î {1,2,3} the Pauli construction gives
g_jk º gjgk= æ - sj sk0 ö =-eijki æ si0 ö     ;     gijk =eijki æ 0-1 ö     ;     g0123 =i æ 0-1 ö     ;     g5º ig0123 = æ 01 ö
è 0- sj sk ø è 0 si ø è 10 ø è -10 ø è 10 ø
    Note that the primary impotent (aka. primitive idempotent) 4×4 unit corner matrix having 1 as the top left element and zeroes elsewhere is given by
    u  º  ½(1+ig12)½(1+g0)   =   ½(1+g0)½(1+ig12)   =   ¼(1+g0 +ig12 +ig012)
    We can construct matrices having 1 in the second, third, and fourth elements of the first column (and zeroes elsewhere) as ig23u = -g23g12u = g13u  =  -e13u , g3u  =  e1245u, and -g1u  =  -e2345u respectively. Ones in only the second, third, and fourth entries of the top row are given by the matrix-transpose (Hermitian multivector conjugation) of these, specifically iug23 = ug13, ug3, and ug1 respectively.

Dirac Conjugation as Clifford Conjugation

    Dirac conjugation (aka. Dirac adjoint) of a 4×4 matrix y is traditionally defined by y»  º  g0 yT^ g0-1   =   g0 yT^ g0 corresponding to transpose, conjugation, and negation of the off-lead-diagonal blocks [ Lounesto 10.3 ] . For a ket left-column matrix Dirac spinor y, the leftmost g0 is superflous.
    Dirac conjugation preserves 1 and the gj, negates double and triple gj products, and preserves g0123 and so corresponds to Â1,3 reversion §.
    With regard to the f k, it preserves 1, negates the f k and double products, preseverving triple and quadruple products and so corresponds to Clifford conjugation §#. Of course, §#=§ when restricted to even multivectors.

    We can thus, albeit somewhat informally, regard a Dirac spinor ket matrix y as a "product" ypu where yp is an even Â3,1 multivector which we must "recast" into a 4×4 complex matrix via ej ® f j prior to matrix multiplication by 4×4 primitive idempotent u . The corresponding Dirac bra matrix y» is given by uyp§# . We cannot instead cast u into a Â3,1 multivector because we do not have a Â3,1 equivalent for i1.

The Dirac-Hestenes Equation
    We will adopt the natural Â3,1 analog of an Â1,3 approach due to Hestenes. First note that we can express any 4×4 complex matrix as a complex-weighted sum of matrix products of the gj .
[ Proof : Left multiplication by g0 negates the bottom two rows (ie. the bottom two 2×2 "subblocks") while right multiplication by g0 negates the left two columns (ie. the left two subblocks). This enables us to "zero" all but a given 2×2 "corner" of any 4×4 matrix via appropriate left and right multiplications by ½(1±g0) . We can thus construct a matrix having any sj or 1 in either off-lead-diagnonal block, or any i sj or i1 in either lead-diagnonal block, and zeroes in all other blocks. Since complex-weighted summations of the Pauli matrices can generate any 2×2 complex matrix, we can generate y as a complex-weighted sum of the gj  .]
    Because y has nonzeroes only in the leftmost column we can safely right-multiply by the unit corner matrix to obtain
    y =yu =y½(1+ig12)½(1+g0)
so it is logical to factorise y as y'u where 4×4 complex matrix y' may contain nonzero elements outside the leftmost column but is nonetheless expressible as a complex-weighted sum of gj products.

    The ½(1+g0) acts as a source or sink of g0's which commute across the ½(1+ig12) and so we can ensure that y' is a complex-weighted sum of only even gj products.
    Define f j º igj
    Since i(1+ig12) = -g12(1+ig12) , we can replace i weightings in y' by rightmultiplications by -g12=f12 ; thus y' can safely be considered a real-weighted sum of even gj products and hence also a real-weighted sum of even f j products. The Dirac equation thus becomes
     åk=03 [ih(2p)-1 gkÐxk(y'u) - gkqc-1aky'u] = mcy'u
    Adding a unit -i2 factor to the ak term and then converting just one of the i into the right multiplication of y' by -g12=f12 we obtain
     åk=03 [ih(2p)-1 gkÐxk(y')u + igkqc-1aky'f12u] = mcy'u   .     Replacing igk with f k and changing timelike suffix from 0 to 4 we obtain
     åk=14 [h(2p)-1 f kÐxk(y') + f kqc-1aky'f12]u = mcy'u .     Exploiting u=g0u=-if4u=-f4f12u  we have
     åk=14 [h(2p)-1 f kÐxk(y') + f kqc-1aky'f12]u = -mcy'f412u .
    Thus a sufficient (though perhaps not necessary, since u is noninvertible) condition for y to solve the Dirac equation is for real-weighted combination of even f j matrix products y' to solve åk=14 [h(2p)-1 f kÐxk(y') + f kqc-1aky'f12] = -mcy'f412    since the u will absorb the f412

    We can interpret 4×4 complex Hermitian matrices f j as an orthonormal frame of Â3,1 1-vectors. We further observe that as any frame f j will provide a set of Dirac matrices, we are free to interpret f j as ej where ek are the Â3,1 frame with regard to which our xk coordinates are constructed (p=x=åk=14 xkek) without regard to signatures.
    Taking c=1 in accordance with natural units (but retaining h) we can thus write the Dirac equation using the Â3,1 1-gradient operator Ñp as
    h(2p)-1 Ñpyp + qapype12 = -mype412     where yp is y' considered as a Â3,1 + multivector. Since e12=e12; and e4=-e4) we can rearrange as
    Ñpyp = 2ph-1 (-qapyp + mype4)e12
    [ Had we favoured Â1,3 we would have obtained  åk=03 [-h(2p)-1 gkÐxk(y')g12 - gkqc-1aky']u = mcy'g0u
    Þ Ñpyp = (2p)h-1 (qapyp + mypg0)g12 which is known as the Â1,3 Dirac-Hestenes equation. ]

    This result differs in sign of ap from Lounesto [13.4] who asserts Ñpyp = 2ph-1 (qapyp + mype4)e12 for the Â3,1 + Dirac equation.

    Assuming natural units, our final Â3,1+ idempotent-stripped Hestenes-Dirac equation is thus
    Ñpyp = (mype4 - qapyp)e12     with yp Î Â3,1 + . Particles having no mass or charge (such as, perhaps, neutrinos and photons)   thus solve Ñpyp = 0 even in the presence of an electromagnetic field.

    We will assume that our u factor contains "all the idempotency" of y so that Â3,1+ multivector yp is invertible and accordingly expressible canonically as yp = (rpebpe1234)½Rp where rp is a positive scalar , bp is a scalar , unit rotor Rp = (½l(ae1234)c) (ie. an even multivector with RpRp§=1) for unit plussquare bivector c, and e1234 is the Â3,1 unit pseudoscalar (preserved by both » and § and so not taking the QM i role) . There is also the particular case Rp=±(1+a) for null bivector a.
    Scalar bp is known as the Yvon-Takabayasi angle bp with bp=0 giving the Dirac particle having magnitide rp , and bp=p giving the Dirac antiparticle with yp»yp = uR§p§rpe1234»e1234Rpu =-rpu.

4D Dirac-Hestenes Solutions

Particle as spinning frame field
    We examine some geometric kinematic consequences of Ñpyp   =   mype124 - qapype12 when yp§ is grade preserving (bp=0 or p) and preserves orthogonality (a conseqeunce of Rp§Rp=1) . We then have
    Ñp yp§(e1)   =   2(myp§(e24) + qapyp§(e2))    ;     Ñp yp§(e2)   =   -2(myp§(e14) + qapyp§(e1))
    Ñp yp§(e3)   =   -2qap¿yp§(e123)   =   2q(apÙyp§(e4))*     provided i=e1234 commutes with yp§.
    Ñp yp§(e4)   =   -2(myp§(e12) + qap¿yp§(e124))   =   -2myp§(e12) + 2q(apÙyp§(e3))* .
[ Proof : Because we are in Â3,1 our kinemetic rule relates only to scalar parts yielding
    Ñp*ypbyp§   =   2((Ñpyp)byp§)   =   2((mype124-qapype12)byp§)<0>   =   -2q(apype12byp§)<0>   =   2qap¿yp§(b¿e12) .
    Thus Ñp ¿ yp§(e3) = Ñp ¿ yp§(e4) = 0 while Ñp ¿ yp§(e1) = 2qap¿yp(e2) and Ñp ¿ yp§(e2) = -2qap¿yp(e1)
       ÑpÙ(ypbyp§)   =   ((Ñpyp)byp§  - ypb(Ñpyp)§ )<2>
      =   m(ype124byp§  - ypbe124§yp§)<2> - q(apype12byp§  - ypbe12§yp§ap§ )<2>
      =   myp§(e124b+be124)<2> - q(apype12byp§  + ypbe12yp§ap )<2>   =   2myp§(b¿e124) - q(apype12byp§  + ypbe12yp§ap )<2>
    Ñp Ùyp§(e1) = 2(myp§(e24) + qapÙyp§(e2))    ;     Ñp Ùyp§(e2) = 2(-myp§(e14) - qapÙyp§(e1))
    Ñp Ùyp§(e3) = -2q(ap~yp§(e123))<2> = -2qap¿yp§(e123)    ;     Ñp Ùyp§(e4) = -2myp§(e12) -2qap¿yp§(e124)  .]

    Thus when ap=0, spacelike spin current wp=yp§(e3) satisfies Ñpwp = 0 suggesting that it is is more fundamental than timelike probability current jp º yp§(e4) which has Ñp jp = -2mSp where 2-blade Sp º yp§(e12) = -2mrp(bpe1234)Sp~ with unit 2-blade Sp~=R§p(e12) [  Ñpjp = -4ph-1mSp in unnatural units ]
    Ñp Sp   =   -(Ñp mrp)(mrp)-1 - 4mjp + 4qaprp so when ap=0 and rp=r we have Ñp2 jp = 8m2jp (ie. (25/2ph-1m)2 jp ) The "Laplacian eigenvalue" 25/2ph-1m is collosal, and we must consider jp as oscillating extremely rapidly but with a small amplitude about a "time-averaged" timelike four-momentum.
[ Proof : Ñp yp§(e12)   =   Ñp rp-1 yp§(e1) yp§(e2)   =   (Ñp rp-1)rp yp§(e12) + rp-1(Ñpyp§(e1))yp§(e2) - rp-1(Ñpyp§(e2))yp§(e1)
      =   -(Ñp rp) rp-1 yp§(e12) + 2rp-1( (myp§(e24) + qapyp§(e2))yp§(e2) - (-myp§(e14) - qapyp§(e1))yp§(e1))
      =   -(Ñp rp)rp-1 yp§(e12) + 2( -myp§(e4) + qapyp§(1) -(myp§(e4) - qapyp§(1)) )
      =   -(Ñp mrp)(mrp)-1 - 4myp§(e4) + 4qaprp  .]

    jp=yp§(e4) naturally decomposes as ½yp§(e4+e3) + ½yp§(e4-e3) as the sum of two independantly conserved opposite spin forward null currents.
    wp=yp§(e3) naturally decomposes as ½yp§(e4+e3) + ½yp§(-e4+e3) as the sum of two independantly conserved same spin oppositely charged (ie. forward and backward) null currents.

    Suppose ap=0 and we are given only a null field sp º yp§(e3+e4). We can recover mrp via Ñpsp = -2mrpSp~ for unit minussquare 2-blade Sp~.
    Ñp2sp = -(Ñp mrp)(mrp)-1 + 4myp§(e4) then provides mjp = mrpvp and hence mrpwp = mjp¿Sp~* where * denotes dual in e1234.
    But sp = rp(vp+wp) so we can isolate rp from m as rp = -vp¿sp = wp¿sp. It is then natural to regard probability rp as the "scaling" or "strength" of a nullvector and mass m as a constant frame-independant scalar measure of the curvature of the null trajectory.
    Similarly, if we are given jp we can reconstruct Sp and mrp and hence mrpwp from Ñpjp and derive rp as (-jp2)½ .
    If we are given wp = yp§(e3) however, Ñpwp=0 and we must resort to wp2 = rp2 and Ðwp~wp .

    If we are given nullvector field spº yp§(e3+e4) then
    Ñpsp   =   -2myp§(e14+e12)   =   2myp§((e3+e4)e1)   =   2mrp-1spyp§(e1)
    Ñp2sp     =   2m(Ñprp-1)spyp§(e1) + 4m2rp-2spyp§(e1)2 - 4m2rp-1yp§(e34)sp
      =   -2m(Ñprp)rp-2spyp§(e1) + 4m2sp - 4m2yp§(e34(e3+e4))   =   2m(Ñprp)rp-2spyp§(e1) + 8m2sp
      =   2mrp-2(Ñprp)¿(spÙyp§(e1)) + 8m2sp     since sp is orthogonal to yp§(e1) and Ñp2sp must be a 1-vector.

    Thus if rp=r , null 1-vector field sp=yp§(e3+e4) satisfies a Klein Gordan equation Ñp2 sp = (23/2m)2 sp .

Simple solutions
    Solutions to the Dirac equation tend to be oscillatory and multidimensional oscillations can be hard to undertand and secribe. We desrcibe a few here for completeness but the essential point is that they model the classical charged particle when "averaged" over nontiny time durations. The picture is one of wildly accelerating motions constrained into apparant smoothness. What appears to be a continuos timelike trajectory is actually the time-averaged value of and extremely tightly wound lightspeed trajectory, for example.

    We first observe that if yp solves Ñpyp = imyp for a fixed blade i commuting with yp, then so to does rpyp provided either  Ñprp = imrp or Ñprp = 0. The latter can always be atained by means of an e4-invarient spherical harmonic solution rp = YlmP)

    The simplest solution to Ñpyp = mype124 is the planewave yp=(m(p¿e4)e12) y0'     for some arbitary fixed y0' that commutes with e124. This solves Ñp2yp = -m2yp since e122 = -1 [  From Ñx2e(x*a)b = -a<1>2 e(p+x*a)b . ] and has constant magnitude over cotemporal plane p¿e4 = t. It can be viewed as representing a "particle" of exact "momentum" me4 and maximally arbitary (effectively nonexistant) "position" . This is more physically acceptable than solutions such as ((p¿e1)e24)y0' because e242=1 makes such such solutions hyperbolic unbounded with distance whereas ((p¿e4)e12) is trigononometric and bounded with time.

    For a more general solution consider first one factorising as yp=bpqpW2)   =   bpWqpe12)W§ for scalar field qp, even multivector field bp= (rp(bpe1234))½Rp with RpRp§ = 1 , and p-independant unit minussquare bivector W2 = W§(e12) for arbitary p-independant unit rotor W. Assume rp¹0 and define
    vp º R§p(e4)     ;     Sp º R§p(e12)    ;     Wp º R§p(W2) .
    We have  Ñpyp   =   Ñp(bp)(½qpW2) + ½Ñp(qp)ypW2 .
[ Proof : Decompose bp=bp ++bp - where bp + commutes with W2 while bp - anticommutes with W2 . Then
    Ñp(bpe½qpW2)   =   Ñp(bp)e½qpW2 + Ñp(e½qpW2)bp + + Ñp(eqpW2)bp -   =   Ñp(bp)e½qpW2 + Ñpqp)epqp)W2bp + - Ñpqp)epqp)W2bp -
    = Ñp(bp)e½qpW2 + Ñpqp)bp +epqp)W2 - Ñpqp)bp -eqpp)W2
      =   Ñp(bp)e½qpW2 + Ñpqp)bp +e½pqpW2bp + + Ñpqp)bp -qpW2)W2   =   Ñp(bp)(½qpW2) + Ñpqp)bpqpW2)W2  .]

    When bp=b is constant , or more generally when Ñpbp = 0,  we have Ñpyp   =   ½Ñp(qp)ypW2 and so right-multiplying the Dirac-Hestenes equation Ñpyp e21 = 2ph-1 (mype4 - qapyp) by yp§ gives Ñp(qp)ypW2e21yp§   =   4ph-1 (mrpvp - qaprp(bpe1234))     Þ     Ñp(qp) WpSp   =   4ph-1 (qap - mvp(-bpe1234)) .

    When W2=e12 this reduces to   (Ñpqp)   =   -4ph-1 (qap - mvp(-bpe1234)) which implies bp=0 or p and we have
½(Ñpqp) + 2ph-1 qap   =   ±  2ph-1mvp In natural units with 2ph-1=1 we have ½(Ñpqp) + qap   =   ± mvp . Squaring gives Hamilton-Jacobi equationÑpqp +qap)2 = m2 , while applying ÑpÙ gives   qÑpÙap = ± mÑpÙvp . Since vp¿(Ñpvp)= vp¿(ÑpÙvp) = v/t  where t is the natural parameterisation of the vp streamline we have m v/t = ± q vp¿(ÑpÙap) = -/+ q vp¿fp     for pure bivector field fp = -ÑpÙap ,  the standard trajectory equation for a spinless charged particle.

    Thus jp=rpvp looks like a conserved current of spinless charges following Maxwell's equations , but jp alone does not fully describe the flow

    If bp=0 and ap=0 and qp = wm~¿p so that Ñpqp = wm~ for p-independant nonzero w and unit timelike m~ we have   ½wm~   =   mvp Þ vp=m~, mw for a DeBroglie plane wave solution yp = rm~(m~+e4)~ ((m¿p)e12) for arbitary constant real rp=r .

    Now consider solutions factorising as yp = (½qpW1)bp . We can decompose bp=bp ++bp - via bp ± º ½(bp ± We12§(W§bp)) such that W§bp + commutes with e12 while W§bp - anticommutes with e12. If Ñpbp ± = 0 (a stronger condition than Ñpbp=0) then Ñpyp = ½(Ñpqp)W1yp and right-multiplying the Dirac-Hestenes equation by yp§ this time gives (Ñpqp)W1Sp   =   4ph-1 (qap - mvp(-bpe1234)) .
[ Proof : yp ± = Wqpe12)W§bp ± = bp ±(±½qpe12) Þ Ñpyp ±   =   (Ñpbp ±)(±½qpW1) + (Ñp½qp)W1yp ±
Þ Ñpyp = (Ñpbp +)(½qpW1) + (Ñpbp -)(-½qpW1) + (Ñp½qp)W1yp  .]

    Combining these results we have for the more general "singly phased" solution yp = (½qpW1)bpqpW2) that, provided
Ñpbp = Ñp We12§(W§bp)) = 0  where W1 = W§(e12) (certainly the case for constant bp=b) , we have
    Ñpyp = ½(Ñpqp)ypWp     and     (Ñpqp)WpSp   =   4ph-1 (qap - mvp(-bpe1234)) where Wp º W2 + R§p(W1) .

    If qp = wm~¿p so that Ñpqp = wm~ for p-independant nonzero w and unit timelike m~ we have
    wm~WpSp   =   4ph-1   (qap - mvp(-bpe1234) ) Þ wWpSp   =   4ph-1   (mvp(-bpe1234) - qm~ap)     where p-independant four-momentum m = mm~ is in general nonparallel with four-velocity vp .
    This grade decomposes into
wWp¿Sp = 4ph-1 (m¿vp cos(bp) + qm~¿ap) = ± 4ph-1 m¿vp for the free Dirac (anti)particle
wWp×Sp = 4ph-1 (mÙvp(-bpe1234) + qm~Ùap) =± 4ph-1 mÙvp for the free Dirac (anti)particle
wWpÙSp = -4ph-1 (m¿vp) sin(bp)e1234 =0 for the free Dirac (anti)particle

    Taking bp=0 so bp=rp½Rp with Rp<0>=1 , e12*Rp=0 , and Rp§Rp=1 and requiring jp=rpvp to solve conservation equation Ñp¿jp=0 provides
    ÐvpRp º (vp¿Ñp)RpÑ   =   -½(ÑpÙvpÑ)Rp .
[ Proof :  The condition Ñpbp=0 becomes (Ñprp½)Rp + rp½(ÑpRp) = 0 and since vp = Rpe4Rp§ Þ vprp½Rp = rp½Rpe4 we have
Ñp(vprp½Rp) = Ñp(rp½Rpe4)  = (Ñpbp)e4 = 0 . But Ñp(vprp½Rp)   =   (ÑpvpÑ)rp½Rp + 2(vp¿Ñp)(rp½Rp)Ñ - vpÑp(rp½Rp)Ñ and so  
    -½(ÑpvpÑ)rp½Rp   =   (vp¿Ñp)(rp½Rp)Ñ   =   ((vp¿Ñp)rp½Ñ)Rp + rp½(vp¿Ñp)RpÑ   =   ½rp((vp¿Ñp)rpÑ)Rp + rp½(vp¿Ñp)RpÑ .
    Ñp¿jp=0 provides (vp¿Ñp)rpÑ = -rp(Ñp¿vpÑ) hence -½(ÑpvpÑ)rp½Rp   =   -½rp(Ñp¿vpÑ)rpRp + rp½(vp¿Ñp)RpÑ
    Result follows when rp¹0.  .]

    We typically consuider the phase ½qp to vary with p much faster than rp½Rp and consider an orientated particle having timelike four-velocity vp = (Rpqpe12))§(e4) = Rp§(e4) and spacelike unit spin axis wp~ =  (Rpqpe12))§(e3) = Rp§(e3) which rotate in a comparatively leisurely manner in accordance with (vp¿Ñp)RpÑ   =   ± ½qm-1 fpRp while spacelike   (Rpqpe12))§(e1) = Rp§((qpe12)e1) and (Rpqpe12))§(e2) = Rp§((qpe12)e2) are rapidly rotating in the Rp§(e12) 2-plane.
    Spacelike wp = rpwp~ = yp§(e3) and timelike current 1-vector jp = rpvp = yp§(e4) are othogonal and have identical magnitude so it is natural to form the phase-independant null 1-vector sp º jp+wp = rp 2 R§p(e3+e4) . This null 1-vector lies always perpendicular to Sp.

    There is another more kinematically suggestive approach in which we redefine particle velocity and spin fields by
    sp º R§p(e4+e2)     and     Sp º R§p((e4+e2)e1)) = spR§p(e1) = spÙR§p(e1) as null 1 and 2-blades respectively.
    We consider rp to be the (real) probality density of "the particle" being "at" p and, if it is there, directions sp and +Sp provide the instanatanous kinematics.

    Right-multiplying the free Dirac-Hestenes equation Ñpyp e21 = 2ph-1 (mype4 - qapyp) by (1+e24)yp§ gives for our Ñpyp = ½(Ñpqp)Wpyp solutions (Ñpqp)Wpyp(e2+e4)Rp§Rpe1yp§   =   4ph-1 (myp(e2+e4)yp§ - qapyp(1+e24)yp§)
    Þ (Ñpqp)WpSp   =   4ph-1 ((-bpe1234)msp - qap(1+R§p(e24)).
    When qp = wm~¿p for timelike unit m~ we have wWpSp   =   4ph-1 (qm~ap(1+R§p(e24)) - (-bpe1234)msp) .
    For the free Dirac particle we have ap=0 and bp=0 or p giving
    wWpSp   =   -/+ 4ph-1 msp  which we can write as wWp mspÙR§p((¼hp-1 m-1e1))   =   msp . One can interpret mspÙR§p((¼hp-1 m-1e1)) as the null angular momentum due to a null linear momentum msp at half Compton wavelength distance ¼hp-1 m-1c-1 (ie. ½m-1 in natural units) from the vp streamline spinning with angular rate w = 4pmc2h-1 = 2m.

    In this interpretation  we can think of the particle "at p" as being an unorientated point actually at pm-1R§p(e1) traversing an exteremely tightly wound helix at light (null) speed, the axis of the helix being given by the comparatively slowly varying current velocity vp passing through p [  "Current" here being used in the sense of electrical current rather than "contemporeneous" ] The period of this zitterbewebung is (4pmc2)-1h = (2mc2)-1h » 2-70 seconds and the averaged value of null sp over a cycle is timelike unit vp=R§p(e4) while that of Sp is R§p(e12) . Thus the principle "effect" of the zitterbewebung is the magnetic moment due to the acceleration   -w2¼hp-1 m-1 R§p(e1) .
    When Ñpqp = wm~ we have ±m~we12Sp   =   2ph-1 mvp Þ we12Sp   =   -/+ 2ph-1 m~mvp .   =   -/+ 2ph-1 mvp     for constant nonunit timelike "momentum" 1-vector m = mm~ not necessarily parallel to the "current" 1-vector vp . Indeed, the bivector part of we12Sp   =   -/+ 2ph-1 mvp provides mÙvp   =   ±h(2p)-1 we12×Sp .

    Hestenes' Zitterbewegung is a classical helical motion perpendicular to the timelike flow. It is a classical interpretation in that the paticle is merely accelerating, but an (e4+Ve1)~ observer will percieve an apparent charge fluctation about average charge Qe-.

Hydrogen Atom
    Static central potential ap = Vpe4 = V(r)e4 where r=|p-p0| gives Dirac Equation h(2p)-1 Ñpyp = (mype4 - qV(r)e4yp)e12 . Left multiplication by e4 gives e4Ñp[e123]yp + Ðe4yp = (myp + qV(r)e4yp)e12 which we can write in Hamiltonian form
    He4(yp) º h(2p)-1 Ðe4ype21   =   -h(2p)-1 e4Ñp[e123]ype21 + myp + qV(r)e4yp .
    Hestenes associates right multiplication by e12 with QM operator i(a) º ae12 .

    For the traditional model of the hydrogen atom we set ap = Zae- r-1 Qe--1 e4 where Qe- is the (negative) electron charge ; integer Z is the atomic charge; and ae- is the fine structure constant.
    An invarient e4-energy solution is given by He4 = E , ie. by Eyp   =   -h(2p)-1 e4Ñp[e123]ype21 + myp + qV(r)e4yp .

    The hydrogen atom fine structure energy levels are traditionally given by En,l = (8h2e02)-1Me-Qe-4 (1+ae-2n-2((j+½)-1n - 3/4) for principle quantum number n and j=l±½ is total angular momentum for orbit angular momentum number l. This holds even for l=0.

The 5D Particle

    "There are strong reasons for suspecting that the modification of quantum theory that will be needed if some form of [the collapse to eigenstate] is to be made into a real physical process, must involve the effects of gravity in a serious way"     Roger Penrose 1995[6.10]

Geometric Interpretation of the Primary Idempotent
    We can regard u as a C3,1 multivector but not a Â3,1 one, since we must allow "scalar" i as a blade coefficient. We have "geometrised" only the "idempotent-stripped" matrix y' . We can follow Hestenes in associating the i of the Dirac equation with right multiplication by -g12 or e12=e12 with regard to its action on yp,  but a "full bloodied" i still lurks within our primitive idempotent u.
    To fully geometrise the entire Dirac ket y we move into Â4,1 and associate   basis 1-vector e5=e5 with the Hermitian matrix
f5= æ 01 ö = ig0123 .     This anticommutes with the other f k ; squares to 1 ; and has f5»   =    g0f5T^g0   =   g0f5g0   =   -f5 as required.
  è 10 ø
  We then have f12345 = g123g0f5 = -g0123f5 = -i-1f5f5= i1 so we can associate i with 5-blade pseudoscalar e12345=-e12345=e12354 (negative signtaure last) which has negative square, flips sign under », and commutes with everything, as required of an i. We can thus regard ia where a is a Â3,1 multivector as the negative dual of a when regarded as a Â4,1 multivector.

    Here at last is an advantage of Â3,1 spacetime over Â1,3 timespace. f5 is also the appropriate "fifth gamma vector" in Â1,3 yet fails to transform like a gamma vector under » there.
    When representing Â4,1 in C4×4 in this manner, complex matrix conjugation ^ corresponds to a automorphic geometric conjusation that preserves 1 and e2 but negates e1,e3,e4 and e5 and so is equivalent to negated reflection in e2 , a^ = e2ae2 . Physicists refer to g2y^ = ie2yp^ = iype2 as the charge conjugation of yp.

    We can now associate u = ½(1+g0)½(1+ig12) , the 4×4 _complex matrix matrix with 1 in the top left corner and zeroes elsewhere, with
    u º ½(1-ie4)½(1-ie12)   =   ½(1-e1235)½(1-e345)   =   ½(1+e124)½(1-e345)   =   ¼(1 - e345 + e124 - e1235)     where unit dirac real plusquare blades e124, e345, and e1235 all commute and are absorbed by u (with  sign change for e34 and e12345) . Note that only e345 commutes with e1234.  
    u= annihilates all basis 1-vectors but e4; all 2-blades but e12 and e35;    all 3-blades but e345 and e124; all 4-blades but e1235;  but preserves e12345. u= can be regarded as projection into the commutative centralizing algebra _Cent[u] = Algebra[e12,e35,e12345} .

    Any ket au decomposes as au = a'u + au = a'u + ua where a = a + a124e124 + a345e345  a1235e1235)u lies in Central(u) = Algebra[e12,e35,e12345} and a' has ua'u = 0 .
    Note that a»a = |a|+2 + |a124|+2 + | + |+ 2(a»a123)0e124 + 2(a»a345)0e345 + 2(a»a1235)0e1235 +?+ 2(a124»a345)0e1235 +?+ 2(a124»a1235)0e345 +?+ 2(a345»a1235)0e124 has as its <0;N>-component the strictly positive scalar sum of four squared complex amplitudes while a2 has as <0;N>-compoent complex sum a2 + a1242 + a3452 + a12352 .

    A general ket product simplifies as = aubcdg while (au)»bu = ua»bu = u¯u(a»b) = u¯u(a»b + a'»b') = u(a»b + ¯u(a'»b')

    au»bu = ¯u(a»b)u = ¯u(a»b + a'»b + a»b' + a'»b') u = (a»b +  ¯u(a'»b') )u while au(bu)» = aub» = aub» + c
    Thus uau=a'u where a' Î Central(u) = Algebra{e12,e35,e12345} = Algebra{e345,e124,e12345} commutes with u so uau has the form (a4e4 + _au12e12 + _au35e35 + _au345e345 + _au124e124 + _a1235e1235)u = (a + be12 + ge35)e4u where a,b, and g are central "complex" values with i=e12345.
    Letting a' º ¯u(a denote the projection of a into the commuting algebra Algebra{e12,e35,e12345} we have aubucud..uguh = aub'c'd'..g'uh where the b',c',..,g' all commute with eachother and u. Letting A denote ket au we have A»A = ua»au = ¯u(a»a)u where Dirac real ¯u(a»a) Î Algebra{e345,e124} and AA» = aua» In particular, any sequence of densities acting on ket G=gu simplifies as for example (AB»)(CD»)(EF»)G = aub»cud»euf»gu = a(¯u(b»c) ¯u(d»e) ¯u(f»g)u so we can arbitarily reorder the densities providing only that we retain AB» as the leftmost.
    And any sequence of densities acting on density GH»=guh» simplifies as for example (AB»)(CD»)(EF»)GH» = aub»cud»euf»guh» = a(¯u(b»c) ¯u(d»e) ¯u(f»g)uh»

    Now (AB»)2 = aub»aub» = ¯u(b»c)

    Further (au)»bu)cu = u¯u(a»b)c' u .
    Further uau»uau = u(a'»a')u = u(a + be12 + ge35)(a» - b»e12 - g»e35) u = (|a|2 + |b|2 + |g|2 +  2(ab^)<0>e12 + 2(ag^)<0>e35 + 2(bg^)<0>e1235 )u has the form a"u where a" Î Algebra{e12,e35} = Algebra{e12,e1235} has postive scalar part and commutes with u.

    Note that uu§ = u§u = u#u = uu# = 0 due to the (1-e345) factor, and u2 = u§#u = u = u§#. Also note that e12u = ue12 = ue12345 = -u* = -e4u.
    u converts e12 ,  e35, and e4 into i.

    u' º e1234ue1234-1   =   ½(1-e124)½(1-e345)   =   ½(1+e1235)½(1-e345)   =   ¼(1-e345-e124+e1235) represents a particle with the same spin e12 but opposite ("backward") e4, corresponding to the oppositely charged particle (bp=p). It corresponds to the C4×4 matrix having 1 in the third position of the lead diagonal and zero everywhere else. and observe that uu' = 0 and u(ae1234) = cos(a)u + sin(a)e1234u' and hence so u(ae1234)u =   cos(a)u .

    Idempotent u# = u§   =   ½(1+e124)½(1+e345)   =   ¼(1 + e345 - e124 - e1235) represents a particle of opposite charge and opposite spin to u.
    The four Dirac real idempotents ½(1±e124)½(1±e345) are "orthogonal" in that uu'=0 for any distinct pair u and u'. They provide an annihilating idempotents basis for the Abelian (commuting) algebra generated by {e124,e345} , facilitating the computation of logatithms. Note that cp=cp'u' has vanishing ketbra transition density yp(cp») = ypuu'cp'» = 0 when uu'=0.
    u = u for e4-dependent Hermitian conjugation . We can regard the four idempotents as "singling out" a the unique timeaxis e4 for which u = u.

    u acts as a source of e124 , -e345, and -e1235 enabling us to keep the u factored component of yp even within e1234 or mixed within e123 according to taste.

    Operator u» = u= corresponds to the matrix operator u=(A)º uAu = A11u which zeroes all but the top left corner of A. Geometrically, u= zeroes any blade which anticommutes with either e1235 or e345 (or both) but does not preserve grade. The eight basis blades which commute with both are 1, e4, e12,e35, e124,e345,  e1235, and e12345 . Of these, only 1,e124,e345 and e1235 are real (b» = b) and so corespond to urbservables; and only 1 and e12 will be found in an even multivector within e1234.

    Thus when acting on an even multivector a in e1234, u=(a)   =   ¯e12(a)u   =   (a<0> + (a*e21)e12)u   =   (a<0> + (a*e21)e12345)u ; and for 1-vector a we have u=(a)   =   ¯e4(a)u .

    Moving into Â4,1 in this manner admits of another planewave type soultion yp=(m(p¿e4)e12) y0'

The 5D Particle Ket

    Having established u, our full "unstripped" particle ket is
    yp   =   ypu   =   rp½bpe1234)Rp½(1-e1235)½(1-e345)     (for positive scalar amplitude rp, scalar Yvon-Takabayasi angle bp, and even unit rotor Rp in e1234 )
    bp=0 is known as the Dirac particle ket having magnitide rp , while bp=p gives the Dirac particle antiket having negative Dirac magnitude -rp .
    The "phase" factor (½qpe12) introduced on the right of the stripped b=0 or p ket by assuming Rp=Rpqpe12) for even Rp with unit scalar part and zero e12 component, can, by exploiting u as a source of -e345, be converted to a central (all-commuting)  factor (-½qpe12345) giving a final ket of the form
    yp   =   rp½(-½qpe12345)bpe1234)Rp½(1-e1235)½(1-e345) .
    The phase factor can be replaced by a (qpe12) factor between Rp and u if desired.

    Setting scalar sp º ln(rp) for rp>0 we have a partially exponentiated form
    yp   =   (½sp + ½bpe1234 - ½qpe12345) Rp½(1-e1235)½(1-e345)     and if Rp = rp for some even rp in e1234  we have
    yp   =   (½sp + ½bpe1234 - ½qpe12345 + rp) ½(1-e1235)½(1-e345) ,
rp commuting with e1234 and e12345.

    Recalling that e1234§#=e1234  we have y»   =   y§#   =   rp½ubpe1234) Rp§ so that
    y»y   =   rpu(bpe1234) u   =   rp cos(bp)u
    More generally,  y»ay   =   rpu=( (bpe1234)R§p(a<+>) + R§p(a<->)) and, in particular, y»eiy   =   rpu=(R§p(ei))   =   -(ei¿(Rp§e4Rp))e4 .

    y1»y2   =   (r1r2)½uR1§R2u   =   (r1r2)½(R1§*R2 + (e21*(R1§R2))e12345)u so the "complex" valued Dirac inner product of two Dirac particle kets is y1»y2   =   (r1r2)½(R2*R1§ + (R2e21R1§)<0>e12345) . When Rp=Rp(qp) for even Rp in e1234 having unit scalar and zero e12 component we have y1»y2   =    (r1r2)½(q2-q1)

5D Dirac Particle Pensity

    y!   =   yuy»   =   rp((½bpe1234)Rp)§(u)   =   ¼rp(ebpe1234(1-R§p(e345)) + R§p(e124 - e1235) )
    Note that (½bpe1234) commutes with u only for bp=0.

    Rp Î Â3,1 + has a basis comprising six 2-vectors, 1 and e1234 .
    y1!y2!   =   r1r2b1e1234)R1uR1§b1e1234)b2e1234)R2uR2§b2e1234)
      =   r1r2b1e1234)R1 u=( ((½b1b2)e1234)R1§ R2) R2§b2e1234) .
    When ½b1b2 = kp for integer k this reduces to y1!y2!     =   ± r1r2 u=(R1§R2) (½b1e1234)  R1uR2§b2e1234) .
    When R1=R2 we have y1!y2!   =    r1r2   cos(½(b1+b2))(½b1e1234)R1uR1§b2e1234) .#
    Hence y!p2   =   rp cosbp y!p and y1! is orthogonal to y2! if R1=R2 and b1+b2=kp for odd integer k.
     For bp=0 we have Dirac particle pensity rpRp§(u) with y!p2 = rpy!p while for bp=p (and rp signed for positive scakar part) we have the Dirac antiparticle pensity y!p = rpRp§(u') with with y!p2 = -rpy!p .

    We can regard a Dirac particle pensity as a commuting idempotent product u = ½(1±e1235)½(1±e345) being rotated through spacetime by even rotor field Rp in e1234. Plussquare unit 4-blade e1235 dual to minussquare e4 is interpreted as velocity, while plussquare 3-blade e345 dual to spacial 2-blade e12 interpreted as spin.
     ½(1-e1235)½(1±e345) gives electrons of opposite spin, and the corresponding antiparticle (positron) is the equivalently rotated u' = ½(1+e1235)½(1±e345) = ½(1-e124)½(1±e345) = ¼(1-e124±e345-/+e1235)   corresponding to the C4×4  matrix having 1 in the third (or second) positions of the lead diagonal and 0 elsewhere. This "selects" -e4 rather than e4 and corresponds to the particle travelling backwards in time, interpreted as having opposite electrical charge.
      We can similarly construct a pensity based on idempotent ½(1+e124)½(1+e345) = ¼(1+e124+e345+e1235) corresponding to the C4×4 matrix having 1 in the fourth position of the lead diagonal and zeros elsewhere. This is a Dirac particle spinning in the opposite sense about R§p(e3) . Similarly, idempotent ½(1-e124)½(1+e345) = ¼(1-e124+e345-e1235) corresponding to 1 in the second position of the lead diagonal and 0 elsewhere gives a Dirac antiparticle spinning in the opposite sense.

    For the bp=0 kets, we can form the general density of two Dirac particle states y1y2»   =   (r1r2)½R1uR2§ .
    More generally ypayp»   =   rp(bpe1234)R§p(u=(a))(bpe1234)   =   rp ((bpe1234)Rp)§ u= (a) .

Dirac Observables
    The Hestenes formulation allows a kinematic interpretation of the sixteen Dirac Observables aka. bilinear covariants which carries over directly into our Â4,1 model.
    Observables are associated with real (a»=a) multivectors which we can construct from the sixteen Dirac-real blades. These are: the unit scalar ; ten 3-blades (six involving e5) ; and five 4-blades (four involving e5) .  Of these, eight (the four 3-blades not involving e5 and the four 4-blades involving e5) anticommute with e1234 while the remainder commute with e1234. Eight ( the six 3-blades involving e4 , the 4-blade e1235, and 1 ) are plussquare while the remaining eight are minussquare.  

    The expected value of the 1-observable asscoiated with real blade b is
    Ey(b) º y»(by)   =   |y»by|s   =   (y»by)<0,N> u0-1   =   (y»by)<0> u0-1   =   b*y!p u0-1   =   b*(y'uy'§) u0-1  
  =   b* ¼rpRp§((bpe1234)(1+e345) + e124+e1235 ) u0-1  .
     For b commuting with e1234 we have (from the cyclic scalar rule and commutativity of Rp and e1234)
    Ey(b)   =    rpu*((bpe1234) Rp§bRp )   =    b* (rpRpuRp§(bpe1234))
    For b anticommuting with e1234 we remove the (bpe1234) term .

DeBroglie Observable
    The DeBroglie 2-urbservable is associated with observing blades 1 and e1234 and traditionally denoted y»y - y»g0123gkyg0123 corresponding to the <0;4> multivector yp(1+e1234) . If the DeBroglie Observable vanishes (yp(1+e1234)=0) then y is called a singular spinor.

Probability Current
    The probability current is the 1-vector dual of the 4-vector 4-urbservable associated with the e5-involving 4-blades e1*, e2*,e3* and e4*. These anticommute with e1234 and eachother and the expected value of observable ei* is   accordingly
     rp ei* ¿ R§p(u)   =   rp ei* ¿ R§pe1235)   =   ¼rp ei ¿ R§p(e1235*)   =   -¼rp ei ¿ R§p(e4) with the - sign ensuring the expected value |ye4*y|s associated with e4* is posive for rp > 0 .
    Thus it is natural to define the probability current to be the Â3,1 1-vector 4-observable
    jp º ype4yp§ = rp R§p(e4) where yp is the u-stripped ket, although in our Â4,1 model, jp is more properly regarded as the dual of the psuedovector hyperblade observable associated with hyperblade e1235
    [ Probability current is traditionally regarded as a point-dependant four-vector (4-D 1-vector) having coordinates (dropping p for brevity)
     Jk º y»gky  = ug0y'T^g0gky'u = u1(y'T^g0gky')     kÎ{0,1,2,3} . A physicist might write Jk=J . gk and think of J as a "vector of gammas". Jk corresponds to the matrix obtained by zeroing all but the top left element of the "inner matrix" g0y'T^g0gky'   =   -iy'»f ky' which in turn corresponds to -i times the Â3,1 1-vector  yp§ekyp . Because only f4 has nonzero topleft element (i) , Jk corresponds to the f4 coordinate of the "inner" 1-vector. ]

    The kinematic rule gives Ñp¿jp = Ñp¿(ype4yp§) = 2((Ñpyp)e4yp§)<0> which vanishes if yp solves the Dirac equation.
[ Proof :   4ph-1 ((qapyp + mype4)e12e4yp§)<0> = 4ph-1 (-(qyp§apype12)<0> - m(ype12yp§)<0>) = 4ph-1(0-0) = 0 .
    Similarly  ((qap*-m)yp"e4yp"§)<0>   = 0  .]

Quantum Spin Vector
    The four 3-blades not involving e5 are the e1234 duals of e1,e2,e3 and e4. They too anticommute with e1234 and with eachother. The expected value of observable eie1234-1 is thus rp (eie1234-1) ¿(R§p(u)   =    ei¿ rp R§p(e1234-1u)   =    ei¿ ¼rpR§p(e3) . wp º ¼rpR§p(e3) is known as the 4D spin 1-vector . One can also think of spin vector Rp§(e3) as being the Â3,1 dual of trivector Rp§(e124) .
    [   Traditionally, wk º y»ig0123gky is the top left element of the matrix y'»ig0123gky' = -y'»f5gky' = -f5y'»gky' = if5y'»f ky' which is i times the first element of the third row of the matrix y'»f ky' correponding to Â3,1 1-vector yp§ekyp. The only f k matrix having non zero (1,3)rd element  is f3 with f31,3 = (-i) . Thus wk is the coefficient of f3 in yp§ekyp . The cyclic scalar provides wk = ek¿yp(e3) = ek¿yp(e3). ]

    For a given e4, the boolean 3-urbservable associated with geometric operators e145,e245 and e345 provides the spin.
    We associate ïñ with ½(1+e345) and ïñ with ½e145(1+e345) = ½e13(1+e345) having associated pensities ½(1_plusminuse345). .

    Ñp¿wp=0 follows similarly to the jp argument.


    It is natural to form nullcurrent vector sp   =   jp+wp   =   rpR§p(e3+e4) with Ñp¿sp=0 and sp2=0 . "Quantum Current" and "spin" may then be considered as the spacelike and timelike components (perceieved by a particular observer) of a more funadamental nonzero null velocity

Moment Density
       We here define the moment density to be the Â3,1 bivector 6-urbservable associated with 3-blades of the form eij5 for eij Î e1234 .
    The expected value of such a 3-blade eij5 is rp eij5¿(RpuRp§(bpe1234))   =   ¼ eij5¿ R§p( cos(bp)e345 + sin(bp)e125) .   =   ¼ eij¿ R§p( cos(bp)e34 + sin(bp)e12) .
    We associate e345 with the ïñ measurement, e145 with ïñ, and e245 with ï´ñ . For bp=0 or p, observables e125, e235, and e135 have expected value 0 .

    Sp º  ype12yp§ = rpebpe1234 Rp§(e12)     and think of Sjk as the coefficiebnt of ejk in Sp.
    [  The spin bivector is traditionally   expressed (dropping p for brevity) as S=åj<k Sjkg_jk where Sjk º y»ig_jky  = ug0y'T^g0ig_jky'u
    We are thus picking the top left element of the matrix y'»ig_jky' = -iy'»f _jky' corresponding to i times Â3,1 bivector yp§ejkyp . Since the only Â3,1 bivector having a non-zero top left element (-i) in its matrix representation is f12=-g12 , real scalar -iy'»f _jky' is simply the coefficient of e12 in the multivector yp§ejkyp .
    Sjk = e21¿(yp§ejkyp) = -(ype12yp§)¿ejk = -rp(ebpe1234R§p(e12))¿ejk ]

    Lounesto classifies spinors according to Dirac urbservables thus:

5D Idealised Dirac-Hestenes Equation
    Since u commutes with both e12 and e124 we can multiply the idempotent-stripped even Â3,1 + Dirac equation by u to obtain "unstripped" mixed Â4,1 ket solving Dirac equation Ñp[e1234]yp = (-qapyp + mype4)e12     [  Dirac's original equation has no Ðe5 component so we must restrict Ñp to within e1234 ] But Â4,1 ket yp solves yp = ype345 = ype1235 = -ype124 due to the  u factor , so we can reformulate the unstripped 5D Dirac equation as both
    Ñp[e1234]yp   =   myp + qapype12345   =   (m - qap*)yp ; and as
    Ñp[e1234]yp   =   mye1235 + qapype12345
    where * denotes duality in the all-commuting i=e12345=e51234 and we require e5¿ap=0 .
    Note that e4u = ue4 = -ue12345 = u* so that uau commutes with e4 for any multivector a.
    This Dirac equation distinguihes e5 only. The "primacy" of e4 and e12 arise when we fix u. For unnatural units, we multiply the lefthandside by h(2p)-1.

    The Dirac operator Ñp[e1234] + qap* anticommutes with e1234=g5 which physicists refer to as indicating chiral symmetry.

    If yp solves the Dirac Hestenes equation then, for any real scalar field lp , cp = (qilp) yp satisfies Ñp[e1234]cp   =   mcpe1235 + q(ap+Ñp[e1234]lp)i)cp . with the Dirac-real <0;4>-vector m - qap* is acting rather like a "pensity potential" (m - qap*) .
[ Proof :  mcpe1235 + qapcpi + (Ñp[e1234](qilp)yp   =   mcpe1235 + qapcpi + qi(qlp)(Ñp[e1234]lp)yp   =   mcpe1235 + q(ap+Ñp[e1234]lp)i)cp  .]

    By insisting on an ideal solution yp=ypu we have removed the e124 elemenats from the equation but this insistance is unconvincing.

    Having introduced a fourth spacial dimension, it is natural to postulate that yp is defined over a 5D eventspace Â4,1 and extend Ñp accordingly. We also relax e5¿ap=0 to obtain our final Â4,1 Dirac Hestenes equation
    Ñpyp   =   (m - qap*)yp
    with 5D pseudovector potential ap* º ape12345-1 .
    The 4D solution yp = ypu for even yp   =   rp(bpe1234)Rp  in e1234 requires a5=0.
    [  We can confirm this alternate geometric form of the Dirac equation by returning to the traditional gamma matrix Dirac equation
    åk=03 [ih(2p)-1 gkÐxk(y'u) - gkqc-1aky'u] = mcy'u     reexpressing it using f k=igk as  
    åk=03 [h(2p)-1 f kÐxk(y')u - f kqc-1aki-1y'u] = mcy'u     Û (åk=03 f kÐxk)(y')u   =   (2p)h-1 (qc-1api-1 + mc)y'u     with associated geometric form Ñp[e1234]ypu   =   (2p)h-1 (-qc-1ap* + mc)yp'u     assuming a5=0 and associating i with -e12345. ]

    In adding a fifth dimension we aquire a newcomponent ¯e1234(e5Ðpe5yp) in ¯e1234(Ñpyp) and an "antiparticulate" term qa5e1234yp in (m-qap*)yp effecting bp.

    If yp solves the 5D Dirac equation then Ñp2yp   =   ((m-q*ap)2 + q*fp)yp + 2q*(ap¿Ñp)ypÑ     where Ñpap = -fp
[ Proof :  Ñp2yp   =   Ñp(m-qap*)yp   =   mÑpyp - q(Ñpap*Ñ)yp - q(Ñpap*)ypÑ   =   mÑpyp - q*(ÑpapÑ)yp - q*(Ñpap)ypÑ
      =   mÑpyp - q*(ÑpapÑ)yp - q*(apÑp-2(ap¿Ñp))ypÑ   =   (m-qap*)Ñpyp + q*fpyp + 2q*(ap¿Ñp)ypÑ  .]

    One might ask why nature apparently favours <0;4> multivectors for a in Ñpyp = apyp . <0;4;8;12;...> multivectors are specified by a#=a§=a§#=a. We saw in Algebraic Equivalences how either #§ or § is often a geometrically invariant conjugation according to N. <0;4;8;12;...> multivectors are thus invariant under the geometrically invariant conjugation , whatever N , which may be a relevant criteria.

Hydrogen Atom
    Imposing a radial e4-static V(|X-X0|)e4 potential and left multiplying the _Reqals4,1 Direc-Hestenes equation y e4 e4 as in Â3,1 gives
    e4Ñp[e1235]yp + Ðe4yp = me4yp + qV(r)iyp which we can write in unnatural units Hamiltonian form
    h Ðe4yp = -h e4Ñp[e1235]yp + myp + qV(r)yp* .

Multiple Dirac Particles

    Physicists traditionally construct multiparticle alegbraes as multiple copies of the single particle algebra. This author remains unconvinced by the utility of this approach, but the following is included for completeness.

Independant Particles
    Incorporating velocity into our kets with y[i] = (rp½w(1-e1235))[i] = (rp½Rp(1±e345)(1-e1235))[i] gives pensities y![i] = (rpw!(1-Rp§(e1235)))[i] .
    The p subscript must now be regarded as denoting dependance on a NK dimensional 1-vector position configuration (p1,_p2,...,pK)
    For multiple qubits, equating the e5[i] ensured the commutation of   e345[i] and e345[j] but if we do this now, e1235[i] and e1235[j] no longer commute and , regardless, e1235[i] need not commute with Rp[j].
    u[i]×u[j] = e345[i]×e345[j] + e1235[i]×e1245[j] = -2e123[i]Ùe123[j] if e5[i]=e5[j] or 2e345[i]Ùe345[j] otherwise.
    Either way, the × product is a 6-blade so if we are considering the scalar part of the product of an observation blade of grade below 6 with a product involving yp[i] and yp[j] we can safely commute them inside the contraction.
    If we unify e4[i]=e4[1]=e4 then yp[i] and yp[j] commute provided we keep Rp even in e1235.
    We can refrain from unifying the e4[i] or e5[i] and allow each particle ket to exist in a unique 5 dimensional space, exploiting the idempotent u[i] to ensure Rp[i] is even. If it is even within e1234[i] then all factors in P(.) y[i] commute save the (1+e345)[i]=(1+e345[i]) .   We can then "move" all the idempotent u[i] to the right and obtain a product ket whose "order" is determined by the order of the (1+e345)[i] factors, which we can "move" to the extreme right of the product.
    We can assume Rp[i] to be an even rotor in Euclidean space e1235[i] and given this and (rp½)[i] > 0  we can deduce the order of the (1+e345[i]) factors in a given ket product. yp[i] occurs before (to the left of) yp[j] iff the coefficient of 6-blade e345[i]Ùe345[j] is positive.
    We adopt the notation P(.) ab..c º (P(.) ab..c ) º (ab..c)(i1) (ab..c)(i2)... (ab..c)(iK)     where i1,i2,..,iK is a permutation of 1,2,..K .
    We can isolate yp[i]~ from the product P(.) yp as ¯e12345[i](P(.) yp)~ but the amplitude (rp½)[i] cannot be individually recovered from P(.) yp alone since P(.) y = (P(.) rp) (P(.) Rp§(1+e1235)) (P(.) w)

    We can write the pensity product of two normalised Dirac particles as y![1]y![2] = (Rp§(1+e1235))[1] (Rp§(1+e1235))[2]w![1]w![2] .
      We have Rp[1]§(1+e1235[1]) × Rp[2]§(1+e1235[2])   =   Rp[1]§(e123[1])Rp[2]§(e123[2])
[ Proof : Rp[1]§Rp[2]§((1+e1235[1]) × (1+e1235[2]))   =   Rp[1]§Rp[2]§(e1235[1]e1235[2])   =   -Rp[1]§Rp[2]§(e123[1]e123[2])  .]
    Also y[1]×y[2] = w[1]w[2] ((1+e1235[1])×(1+e1235[2]))   =   w[1]w[2] e1235[1]e1235[2]   =   -w[1]w[2] e123[1]e123[2]

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Copyright (c) Ian C G Bell 2003
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