Â_{4,1 +} 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 Â_1,3 Free Klein Gordan Equation . The Free Klein Gordan equation in a Â_1,3 timespace is ( (¶/¶t)² - Ѳ + a²)y = 0 where Ѳ denotes the usual Euclidean 3D Laplacian . We must regard this as Ѳy_p = -a²y_p in Â_1,3 and as Ѳy_p = a²y_p in Â_3,1.

    Ñ_p² y_p = -a² y_p factorises as (iÑ_p+a)(iÑ_p-a)y_p  =  0       where real scalar a = 2pmch^-1 = mch^-1 = m in natural units .
    Subequation (iÑ_p - a)y_p = 0 Û  (2p)^-1he⁴(iÑ_p - a)y_p = 0 Û  ( m_e4 - e₁₄ m_e1 - e₂₄ m_e2 + e₃₄ m_e3 - e₄mc)y_p = 0 , where m_ei º hÐ_ei = (2p)^-1hi Ð_ei is the momentum operator traditionally denoted by p_i. Dirac wrirtes this as ( m_e4 -a₁ m_e1 -a₂ m_e2 -a₃ m_e3 -b)y= 0     _{[ PQM 67.7 ]}     , seeking 4x4 complex matrices a₁ ,a₂ ,a₃ ,b which anticommute and have a_i²=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 a_p by following the "classical rule" of replacing m_e0 with m_e0 - qc^-1 V and m_ej by m_ej - qc^-1 A_j 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 m_d by m_d - qc^-1 d^-1¿a_p for 1-vector "four-potential" a_p.
    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)
    ( m_e0-qV -a₁( m_e1-qA₁) -a₂( m_e2-qA₂) -a₃( m_e3-qA₃) -b )y_p = 0     _{[ PQM 67.11 ]} ,

    A more common form of the C_4×4 Dirac equation is obtained by setting g⁰ = (mc)^-1b ; g^j = -(mc)^-1ba_j   for j=1,2,3 (which also anticommute); and exploiting m_ek º ih Ð_x^k = ih (¶/¶x^k) where h=h(2p)^-1 . It is
    å_k=0³ ( ih g^k(¶/¶x^k) - g^kqc^-1a_k ) y = mcy
where the g^k are four p-independant anticommuting 4x4 complex Dirac matrices satisfying g⁰²=1 ; g^j2=-1 for j=1,2,3 ; a₀=V, a_j=-A_j are the real scalar coordinates of a Â_1,3 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 g^j.

    Any solution to Ñ_py = ±ay is a solution to Ñ_p²y = a²y and an e₄-static origin-centred solution is given by y_p = |^_e4(p)|^-1 b(± a|^_e4(p)|)^↑ [  Where a^↑ º e^a denotes exponentiation ] . The + solution is unbounded and so rejected, leaving Yukawa potential
    y_p = |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	1	0	ö ;	gj=	æ	0	sj	ö
	è	0	-1	ø		è	- sj	0	ø

though Rg^jR^-1 provides another set for any invertible 4×4 matrix R.     We list our g^j explicitly for reference.

g0=	æ	1	0	0	0	ö    ;	g1=	æ	0	0	0	1	ö    ;	g2=	æ	0	0	0	-i	ö ;	g3=	æ	0	0	1	0	ö
	ç	0	1	0	0	÷		ç	0	0	1	0	÷		ç	0	0	i	0	÷		ç	0	0	0	-1	÷
	ç	0	0	-1	0	÷		ç	0	-1	0	0	÷		ç	0	i	0	0	÷		ç	-1	0	0	0	÷
	è	0	0	0	-1	ø		è	-1	0	0	0	ø		è	-i	0	0	0	ø		è	0	1	0	0	ø

We observe that they all have zero trace corresponding geometrically to having zero scalar part. Because we are here interested in Â_3,1 spacetime rather than Â_1,3 timespace we define matrices f ^j º ig^j 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 sk

0

ö

=

-eijki

æ

si

0

ö     ;

gijk =

eijki

æ

0

-1

ö     ;

g0123 =

i

æ

0

-1

ö     ;

g5º ig0123 =

æ

0

1

ö

è

0

- sj sk

ø

è

0

si

ø

è

1

0

ø

è

-1

0

ø

è

1

0

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 y^{T^} g^0-1   =   g⁰ y^{T^} 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 g^j, negates double and triple g^j products, and preserves g⁰¹²³ and so corresponds to Â_1,3 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" y_pu where y_p is an even Â_3,1 multivector which we must "recast" into a 4×4 complex matrix via e^j ® f ^j prior to matrix multiplication by 4×4 primitive idempotent u . The corresponding Dirac bra matrix y^» is given by uy_p^§# . We cannot instead cast u into a Â_3,1 multivector because we do not have a Â_3,1 equivalent for i1.

The Dirac-Hestenes Equation
    We will adopt the natural Â_3,1 analog of an Â_1,3 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 g^j .
[ 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 s_j or 1 in either off-lead-diagnonal block, or any i s_j 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 g^j  .]
    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 g^j 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 g^j products.
    Define f ^j º ig^j
    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 g^j products and hence also a real-weighted sum of even f ^j products. The Dirac equation thus becomes
     å_k=0³ [ih(2p)^-1 g^kÐ_x^k(y'u) - g^kqc^-1a_ky'u] = mcy'u
    Adding a unit -i² factor to the a_k term and then converting just one of the i into the right multiplication of y' by -g¹²=f¹² we obtain
     å_k=0³ [ih(2p)^-1 g^kÐ_x^k(y')u + ig^kqc^-1a_ky'f¹²u] = mcy'u   .     Replacing ig^k with f ^k and changing timelike suffix from 0 to 4 we obtain
     å_k=1⁴ [h(2p)^-1 f ^kÐ_x^k(y') + f ^kqc^-1a_ky'f¹²]u = mcy'u .     Exploiting u=g⁰u=-if⁴u=-f⁴f¹²u  we have
     å_k=1⁴ [h(2p)^-1 f ^kÐ_x^k(y') + f ^kqc^-1a_ky'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 å_k=1⁴ [h(2p)^-1 f ^kÐ_x^k(y') + f ^kqc^-1a_ky'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 Â_3,1 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 e^j where e_k are the Â^3,1 frame with regard to which our x^k coordinates are constructed (p=x=å_k=1⁴ x^ke_k) without regard to signatures.
    Taking c=1 in accordance with natural units (but retaining h) we can thus write the Dirac equation using the Â_3,1 1-gradient operator Ñ_p as
    h(2p)^-1 Ñ_py_p + qa_py_pe¹² = -my_pe⁴¹²     where y_p is y' considered as a Â_{3,1 +} multivector. Since e¹²=e₁₂; and e⁴=-e₄) we can rearrange as
    Ñ_py_p = 2ph^-1 (-qa_py_p + my_pe₄)e₁₂
    [ Had we favoured Â_1,3 we would have obtained  å_k=0³ [-h(2p)^-1 g^kÐ_x^k(y')g¹² - g^kqc^-1a_ky']u = mcy'g⁰u
    Þ Ñ_py_p = (2p)h^-1 (qa_py_p + my_pg₀)g₁₂ which is known as the Â_1,3 Dirac-Hestenes equation. ]
    This result differs in sign of a_p from Lounesto [13.4] who asserts Ñ_py_p = 2ph^-1 (qa_py_p + my_pe₄)e₁₂ for the Â_{3,1 +} Dirac equation.

    Assuming natural units, our final Â_3,1+ idempotent-stripped Hestenes-Dirac equation is thus
    Ñ_py_p = (my_pe₄ - qa_py_p)e₁₂     with y_p Î Â_{3,1 +} . Particles having no mass or charge (such as, perhaps, neutrinos and photons)   thus solve Ñ_py_p = 0 even in the presence of an electromagnetic field.

    We will assume that our u factor contains "all the idempotency" of y so that Â_3,1+ multivector y_p is invertible and accordingly expressible canonically as y_p = (r_pe^{b_pe₁₂₃₄})^½R_p where r_p is a positive scalar , b_p is a scalar , unit rotor R_p = (½l(ae₁₂₃₄)^↑c)^↑ (ie. an even multivector with R_pR_p^§=1) for unit plussquare bivector c, and e₁₂₃₄ is the Â_3,1 unit pseudoscalar (preserved by both ^» and ^§ and so not taking the QM i role) . There is also the particular case R_p=±(1+a) for null bivector a.
    Scalar b_p is known as the Yvon-Takabayasi angle b_p with b_p=0 giving the Dirac particle having magnitide r_p , and b_p=p giving the Dirac antiparticle with y_p^»y_p = uR_^§p^§r_pe₁₂₃₄^»e₁₂₃₄R_pu =-r_pu.

4D Dirac-Hestenes Solutions


Particle as spinning frame field
    We examine some geometric kinematic consequences of Ñ_py_p   =   my_pe₁₂₄ - qa_py_pe₁₂ when y_p§ is grade preserving (b_p=0 or p) and preserves orthogonality (a conseqeunce of R_p^§R_p=1) . We then have
    Ñ_p y_p§(e₁)   =   2(my_p§(e₂₄) + qa_py_p§(e₂))    ;     Ñ_p y_p§(e₂)   =   -2(my_p§(e₁₄) + qa_py_p§(e₁))
    Ñ_p y_p§(e₃)   =   -2qa_p¿y_p§(e₁₂₃)   =   2q(a_pÙy_p§(e₄))^*     provided i=e₁₂₃₄ commutes with y_p^§.
    Ñ_p y_p§(e₄)   =   -2(my_p§(e₁₂) + qa_p¿y_p§(e₁₂₄))   =   -2my_p§(e₁₂) + 2q(a_pÙy_p§(e₃))^* .
[ Proof : Because we are in Â_3,1 our kinemetic rule relates only to scalar parts yielding
    Ñ_p*y_pby_p^§   =   2((Ñ_py_p)by_p^§)   =   2((my_pe₁₂₄-qa_py_pe₁₂)by_p^§)_<0>   =   -2q(a_py_pe₁₂by_p^§)_<0>   =   2qa_p¿y_p§(b¿e₁₂) .
    Thus Ñ_p ¿ y_p§(e₃) = Ñ_p ¿ y_p§(e₄) = 0 while Ñ_p ¿ y_p§(e₁) = 2qa_p¿y_p(e₂) and Ñ_p ¿ y_p§(e₂) = -2qa_p¿y_p(e₁)
       Ñ_pÙ(y_pby_p^§)   =   ((Ñ_py_p)by_p^§  - y_pb(Ñ_py_p)^§ )_<2>
      =   m(y_pe₁₂₄by_p^§  - y_pbe₁₂₄^§y_p^§)_<2> - q(a_py_pe₁₂by_p^§  - y_pbe₁₂^§y_p^§a_p^§ )_<2>
      =   my_p§(e₁₂₄b+be₁₂₄)_<2> - q(a_py_pe₁₂by_p^§  + y_pbe₁₂y_p^§a_p )_<2>   =   2my_p§(b¿e₁₂₄) - q(a_py_pe₁₂by_p^§  + y_pbe₁₂y_p^§a_p )_<2>
    Ñ_p Ùy_p§(e₁) = 2(my_p§(e₂₄) + qa_pÙy_p§(e₂))    ;     Ñ_p Ùy_p§(e₂) = 2(-my_p§(e₁₄) - qa_pÙy_p§(e₁))
    Ñ_p Ùy_p§(e₃) = -2q(a_p~y_p§(e₁₂₃))_<2> = -2qa_p¿y_p§(e₁₂₃)    ;     Ñ_p Ùy_p§(e₄) = -2my_p§(e₁₂) -2qa_p¿y_p§(e₁₂₄)  .]
    Thus when a_p=0, spacelike spin current w_p=y_p§(e₃) satisfies Ñ_pw_p = 0 suggesting that it is is more fundamental than timelike probability current j_p º y_p§(e₄) which has Ñ_p j_p = -2mS_p where 2-blade S_p º y_p§(e₁₂) = -2mr_p(b_pe₁₂₃₄)^↑S_p^~ with unit 2-blade S_p^~=R_^§p(e₁₂) [  Ñ_pj_p = -4ph^-1mS_p in unnatural units ]
    Ñ_p S_p   =   -(Ñ_p mr_p)(mr_p)^-1 - 4mj_p + 4qa_pr_p so when a_p=0 and r_p=r we have Ñ_p² j_p = 8m²j_p (ie. (2^5/2ph^-1m)² j_p ) The "Laplacian eigenvalue" 2^5/2ph^-1m is collosal, and we must consider j_p as oscillating extremely rapidly but with a small amplitude about a "time-averaged" timelike four-momentum.
[ Proof : Ñ_p y_p§(e₁₂)   =   Ñ_p r_p^-1 y_p§(e₁) y_p§(e₂)   =   (Ñ_p r_p^-1)r_p y_p§(e₁₂) + r_p^-1(Ñ_py_p§(e₁))y_p§(e₂) - r_p^-1(Ñ_py_p§(e₂))y_p§(e₁)
      =   -(Ñ_p r_p) r_p^-1 y_p§(e₁₂) + 2r_p^-1( (my_p§(e₂₄) + qa_py_p§(e₂))y_p§(e₂) - (-my_p§(e₁₄) - qa_py_p§(e₁))y_p§(e₁))
      =   -(Ñ_p r_p)r_p^-1 y_p§(e₁₂) + 2( -my_p§(e₄) + qa_py_p§(1) -(my_p§(e₄) - qa_py_p§(1)) )
      =   -(Ñ_p mr_p)(mr_p)^-1 - 4my_p§(e₄) + 4qa_pr_p  .]

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

    Suppose a_p=0 and we are given only a null field s_p º y_p§(e₃+e₄). We can recover mr_p via Ñ_ps_p = -2mr_pS_p^~ for unit minussquare 2-blade S_p^~.
    Ñ_p²s_p = -(Ñ_p mr_p)(mr_p)^-1 + 4my_p§(e₄) then provides mj_p = mr_pv_p and hence mr_pw_p = mj_p¿S_p^~* where ^* denotes dual in e₁₂₃₄.
    But s_p = r_p(v_p+w_p) so we can isolate r_p from m as r_p = -v_p¿s_p = w_p¿s_p. It is then natural to regard probability r_p 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 j_p we can reconstruct S_p and mr_p and hence mr_pw_p from Ñ_pj_p and derive r_p as (-j_p²)^½ .
    If we are given w_p = y_p^§(e₃) however, Ñ_pw_p=0 and we must resort to w_p² = r_p² and Ð_wp^~w_p .

    If we are given nullvector field s_pº y_p§(e₃+e₄) then
    Ñ_ps_p   =   -2my_p§(e₁₄+e₁₂)   =   2my_p§((e₃+e₄)e₁)   =   2mr_p^-1s_py_p§(e₁)
    Ñ_p²s_p     =   2m(Ñ_pr_p^-1)s_py_p§(e₁) + 4m²r_p^-2s_py_p§(e₁)² - 4m²r_p^-1y_p§(e₃₄)s_p
      =   -2m(Ñ_pr_p)r_p^-2s_py_p§(e₁) + 4m²s_p - 4m²y_p§(e₃₄(e₃+e₄))   =   2m(Ñ_pr_p)r_p^-2s_py_p§(e₁) + 8m²s_p
      =   2mr_p^-2(Ñ_pr_p)¿(s_pÙy_p§(e₁)) + 8m²s_p     since s_p is orthogonal to y_p§(e₁) and Ñ_p²s_p must be a 1-vector.

    Thus if r_p=r , null 1-vector field s_p=y_p§(e₃+e₄) satisfies a Klein Gordan equation Ñ_p² s_p = (2^3/2m)² s_p .

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 y_p solves Ñ_py_p = imy_p for a fixed blade i commuting with y_p, then so to does r_py_p provided either  Ñ_pr_p = imr_p or Ñ_pr_p = 0. The latter can always be atained by means of an e₄-invarient spherical harmonic solution r_p = Y_lmP)

    The simplest solution to Ñ_py_p = my_pe₁₂₄ is the planewave y_p=(m(p¿e₄)e₁₂)^↑ y₀'     for some arbitary fixed y₀' that commutes with e₁₂₄. This solves Ñ_p²y_p = -m²y_p since e₁₂² = -1 [  From Ñ_x²e^(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 y_p=b_p(½q_pW₂)^↑   =   b_pW(½q_pe₁₂)^↑W^§ for scalar field q_p, even multivector field b_p= (r_p(b_pe₁₂₃₄)^↑)^½R_p with R_pR_p^§ = 1 , and p-independant unit minussquare bivector W₂ = W_§(e₁₂) for arbitary p-independant unit rotor W. Assume r_p¹0 and define
    v_p º R_^§p(e₄)     ;     S_p º R_^§p(e₁₂)    ;     W_p º R_^§p(W₂) .
    We have  Ñ_py_p   =   Ñ_p(b_p)(½q_pW₂)^↑ + ½Ñ_p(q_p)y_pW₂ .
[ Proof : Decompose b_p=b_{p +}+b_{p -} where b_{p +} commutes with W₂ while b_{p -} anticommutes with W₂ . Then
    Ñ_p(b_pe^½q_pW₂)   =   Ñ_p(b_p)e^½q_pW₂ + Ñ_p(e^½q_pW₂)b_{p +} + Ñ_p(e^-½q_pW₂)b_{p -}   =   Ñ_p(b_p)e^½q_pW₂ + Ñ_p(½q_p)e^{(½p+½q_p)W₂}b_{p +} - Ñ_p(½q_p)e^{(½p-½q_p)W₂}b_{p -}
    = Ñ_p(b_p)e^½q_pW₂ + Ñ_p(½q_p)b_{p +}e^{(½p+½q_p)W₂} - Ñ_p(½q_p)b_{p -}e^{(½q_p-½p)W₂}
      =   Ñ_p(b_p)e^½q_pW₂ + Ñ_p(½q_p)b_{p +}e^{½p+½q_pW₂}b_{p +} + Ñ_p(½q_p)b_{p -}(½q_pW₂)^↑W₂   =   Ñ_p(b_p)(½q_pW₂)^↑ + Ñ_p(½q_p)b_p(½q_pW₂)^↑W₂  .]

    When b_p=b is constant , or more generally when Ñ_pb_p = 0,  we have Ñ_py_p   =   ½Ñ_p(q_p)y_pW₂ and so right-multiplying the Dirac-Hestenes equation Ñ_py_p e₂₁ = 2ph^-1 (my_pe₄ - qa_py_p) by y_p^§ gives Ñ_p(q_p)y_pW₂e₂₁y_p^§   =   4ph^-1 (mr_pv_p - qa_pr_p(b_pe₁₂₃₄)^↑)     Þ     Ñ_p(q_p) W_pS_p   =   4ph^-1 (qa_p - mv_p(-b_pe₁₂₃₄)^↑) .

    When W₂=e₁₂ this reduces to   (Ñ_pq_p)   =   -4ph^-1 (qa_p - mv_p(-b_pe₁₂₃₄)^↑) which implies b_p=0 or p and we have
½(Ñ_pq_p) + 2ph^-1 qa_p   =   ±  2ph^-1mv_p In natural units with 2ph^-1=1 we have ½(Ñ_pq_p) + qa_p   =   ± mv_p . Squaring gives Hamilton-Jacobi equation (½Ñ_pq_p +qa_p)² = m² , while applying Ñ_pÙ gives   qÑ_pÙa_p = ± mÑ_pÙv_p . Since v_p¿(Ñ_pv_p)= v_p¿(Ñ_pÙv_p) = ¶v/¶t  where t is the natural parameterisation of the v_p streamline we have m ¶v/¶t = ± q v_p¿(Ñ_pÙa_p) = -/+ q v_p¿f_p     for pure bivector field f_p = -Ñ_pÙa_p ,  the standard trajectory equation for a spinless charged particle.

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

    If b_p=0 and a_p=0 and q_p = wm^~¿p so that Ñ_pq_p = wm^~ for p-independant nonzero w and unit timelike m^~ we have   ½wm^~   =   mv_p Þ v_p=m^~, m=½w for a DeBroglie plane wave solution y_p = rm^~(m^~+e₄)^~ ((m¿p)e₁₂)^↑ for arbitary constant real r_p=r .

    Now consider solutions factorising as y_p = (½q_pW₁)^↑b_p . We can decompose b_p=b_{p +}+b_{p -} via b_{p ±} º ½(b_p ± We_12§(W^§b_p)) such that W^§b_{p +} commutes with e₁₂ while W^§b_{p -} anticommutes with e₁₂. If Ñ_pb_{p ±} = 0 (a stronger condition than Ñ_pb_p=0) then Ñ_py_p = ½(Ñ_pq_p)W₁y_p and right-multiplying the Dirac-Hestenes equation by y_p^§ this time gives (Ñ_pq_p)W₁S_p   =   4ph^-1 (qa_p - mv_p(-b_pe₁₂₃₄)^↑) .
[ Proof : y_{p ±} = W(½q_pe₁₂)^↑W^§b_{p ±} = b_{p ±}(±½q_pe₁₂)^↑ Þ Ñ_py_{p ±}   =   (Ñ_pb_{p ±})(±½q_pW₁)^↑ + (Ñ_p½q_p)W₁y_{p ±}
Þ Ñ_py_p = (Ñ_pb_{p +})(½q_pW₁)^↑ + (Ñ_pb_{p -})(-½q_pW₁)^↑ + (Ñ_p½q_p)W₁y_p  .]

    Combining these results we have for the more general "singly phased" solution y_p = (½q_pW₁)^↑b_p(½q_pW₂)^↑ that, provided
Ñ_pb_p = Ñ_p We_12§(W^§b_p)) = 0  where W₁ = W_§(e₁₂) (certainly the case for constant b_p=b) , we have
    Ñ_py_p = ½(Ñ_pq_p)y_pW_p     and     (Ñ_pq_p)W_pS_p   =   4ph^-1 (qa_p - mv_p(-b_pe₁₂₃₄)^↑) where W_p º W₂ + R_^§p(W₁) .

    If q_p = wm^~¿p so that Ñ_pq_p = wm^~ for p-independant nonzero w and unit timelike m^~ we have
    wm^~W_pS_p   =   4ph^-1   (qa_p - mv_p(-b_pe₁₂₃₄)^↑ ) Þ wW_pS_p   =   4ph^-1   (mv_p(-b_pe₁₂₃₄)^↑ - qm^~a_p)     where p-independant four-momentum m = mm^~ is in general nonparallel with four-velocity v_p .
    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 b_p=0 so b_p=r_p^½R_p with R_p<0>=1 , e_12*R_p=0 , and R_p^§R_p=1 and requiring j_p=r_pv_p to solve conservation equation Ñ_p¿j_p=0 provides
    Ð_vpR_p º (v_p¿Ñ_p)R_pÑ   =   -½(Ñ_pÙv_pÑ)R_p .
[ Proof :  The condition Ñ_pb_p=0 becomes (Ñ_pr_p^½)R_p + r_p^½(Ñ_pR_p) = 0 and since v_p = R_pe₄R_p^§ Þ v_pr_p^½R_p = r_p^½R_pe₄ we have
Ñ_p(v_pr_p^½R_p) = Ñ_p(r_p^½R_pe₄)  = (Ñ_pb_p)e₄ = 0 . But Ñ_p(v_pr_p^½R_p)   =   (Ñ_pv_pÑ)r_p^½R_p + 2(v_p¿Ñ_p)(r_p^½R_p)_Ñ - v_pÑ_p(r_p^½R_p)_Ñ and so  
    -½(Ñ_pv_pÑ)r_p^½R_p   =   (v_p¿Ñ_p)(r_p^½R_p)_Ñ   =   ((v_p¿Ñ_p)r_p^½_Ñ)R_p + r_p^½(v_p¿Ñ_p)R_pÑ   =   ½r_p^-½((v_p¿Ñ_p)r_pÑ)R_p + r_p^½(v_p¿Ñ_p)R_pÑ .
    Ñ_p¿j_p=0 provides (v_p¿Ñ_p)r_pÑ = -r_p(Ñ_p¿v_pÑ) hence -½(Ñ_pv_pÑ)r_p^½R_p   =   -½r_p^-½(Ñ_p¿v_pÑ)r_pR_p + r_p^½(v_p¿Ñ_p)R_pÑ
    Result follows when r_p¹0.  .]

    We typically consuider the phase ½q_p to vary with p much faster than r_p^½R_p and consider an orientated particle having timelike four-velocity v_p = (R_p(½q_pe₁₂)^↑)_§(e₄) = R_p§(e₄) and spacelike unit spin axis w_p^~ =  (R_p(½q_pe₁₂)^↑)_§(e₃) = R_p§(e₃) which rotate in a comparatively leisurely manner in accordance with (v_p¿Ñ_p)R_pÑ   =   ± ½qm^-1 f_pR_p while spacelike   (R_p(½q_pe₁₂)^↑)_§(e₁) = R_p§((q_pe₁₂)^↑e₁) and (R_p(½q_pe₁₂)^↑)_§(e₂) = R_p§((q_pe₁₂)^↑e₂) are rapidly rotating in the R_p§(e₁₂) 2-plane.
    Spacelike w_p = r_pw