The essence of the Â_{1,3} Free Dirac-Hestenes equation represents
ensuring Ñ_{p}^{2}y_{p} = -`m`^{2}y_{p}
by insisting insisting that Ñ_{p}__y___{p} = i`m`__y___{p}**g ^{0}** and associating i with right-multiplication by e

We end up with Ñ

The direct Â_{3,1} equivalent is
Ñ_{p}_{[e1234]}__y___{p} = `m`__y___{p}ie_{124} for some i commuting with e_{124}
and exploiting e_{124}^{2}=1 to solve Ñ_{p}^{2}y_{p}=-`m`^{2}y_{p}.
We might for example take i=**e _{4}** and consider a planewave
solution
y

The obvious generalisation of this idealised approach is y_{p} = __y___{p}a with
Ñ_{p}__y___{p} = __y___{p}b where b^{2}a=-`m`^{2}a
so that, provided a and b are **p**-independant, we have
Ñ_{p}^{2}y_{p} = __y___{p}b^{2}a = -m^{2}y_{p} .

Taking a=u=½(1±e_{124})½(1±e_{345}) means
that b^{2} can lie within the commuting algebra *A _{lgebra}*[e

** Further Idempotents **

We have associated
idempotents ¼(1-e_{124})(1±e_{345}) corresponding to matrices with 1 in the second and third lead diagnonal
positions with the electrons of spin ±½e_{12}
and ¼(1+e_{124})(1±e_{345}) corresponding to 1 in the first and fourth with the positron.

Because these four idempotents annihilate eachother, we can construct a further six idempotents
with scalar part ½ by adding any two of them, and a further four with scalar part 3/4 by adding any three. Adding all four gives 1 so, excluding unity,
we have fifteen commuting (but not annihilating) Clifford-real
(a^{§}^{#}=a) and **e _{4}**-Hermitian-real (a

These further idempotents do not all satisy Â

Of course, any idempotemt u provides unit plussquare b=1-2u with ub = ub = -u and we have

y

If r

[ Proof : Ñ

Our 5D Dirac-Hestsnes equation is "grade skewed" in that Ñ

Our 5D Dirac Equation formulation is cleaner than the pugugly matrix form, but the **e _{4}** remains irritatingly on the "wrong side" of the y

Almeida restricts attention to the 5D nullsphere

Ñ

Almeida incorporates the 4D 1-potential via a (

This too feels unrelativistic because the Ñ

In the Â_{4,1}^{%} embedding, the tangent to *e*_{0}^{*} at null **p** is
*e*_{¥}Ù(*e*_{0}+**p**)Ù(**p***i*)
= **p**^{*} + *e*_{¥}*i* where ^{*} denotes Â_{5,2} dual .

However, if we are going to restrict differentiations to within the null sphere in Â_{4,1}
it might be more natural to consider the tne 2-blade directed "rotational derivatives"
Ð_{eij}y_{p} = *Lim*_{e ® 0} e^{-1}
(y_{(½eeij)↑§(p)}-y_{p})
which we might seek to relate to the ten dimensions of string theory. We could write
Ñ_{<2>} y_{p} = å_{1£i<j£5}0 e^{ij}Ð_{eij}y_{p}
and consider (Ñ_{<2>} - `q`**a**_{p})y_{p} = 0 for 5D 2-potential as
a form of "bivector Dirac equation".

Subtracting y_{p} at distinct null **p** feels ungeometric, however, and we might instead consider
the "splayed density"
(y_{p}__»__y_{p})^{-1}
å_{1£i<j£5} e^{ij}
*Lim*_{e ® 0} e^{-1} y_{(½eeij)↑§(p)}y_{p}^{»} .
** Beyond the Dirac Equation**

While monogenic spherical harmonics provide the observed hydrogen energy eigenvalue spectrae, Ñ_{p}y_{p} = 0
and indeed **m**^{2}=0 for "massless particles" violate the "No Zeroes" spirit of quantum mechanics.
Accordingly let us posit **m**^{2}=±h^{2} where h is small compared to the coordinates of
momentum (or current) **m**.
We thus think of 5D **m** as being very "nearly null" (with **e _{5}** coordinate

The crucial advantage in nonzero

[ Proof : y

Allowing r_{p} to vary with **p** gives us
Ñ_{p} y_{p} = -(q_{p}i + **m**i)y_{p} where q_{p} = q_{p}(y) = r_{p}^{-1} (Ñ_{p}r_{p})
[ Or, more formally, q_{p} = -r_{p}^{-1} (Ñ_{x} r_{x})ï_{x=p} ]
is a <1;4>-vector Bohm quantum potential
with Ñ_{p} q_{p} = -r_{p}^{-1}(Ñ_{p}^{2}r_{p}) + r_{p}^{-2}(Ñ_{p}r_{p})^{2} central for analytic r_{p}.
The <1;4>-field q_{p} is influenced by y_{p} but only via the "shape" of r_{p} independant of its magnitude,
in that q_{p}(ly) = q_{p}(y) for any nonzero central l, so we expect q_{p} to be highly non-localised with large coordinates
at **p** where y_{p} coordinates and magnitude are small.

Thus we have a second role for r_{p} which we can think of as being a complex probaility amplitude field with additonal
slight perhaps high frequency oscillations that while having negligible effect on the effective probabilites
|r_{p}|_{+}^{2} (and so all but impossible to observe directly) provide a potent "pilot wave"
complex 1-potential q_{p} whose effectateousness is independant of the
relative probabilites, and whose driving effect is not limited by or proportional to the magnitude or "energy"
of the r_{p} field, and so may provide a significant (perhaps dominant) contribution to the "pilot wave" q_{p} = r_{p}^{-1} (Ñ_{p}r_{p})
even at places where the particle is overwhelmingly unlikely to be.

We can thus regard
signed real probable charge r_{p}^{2}
as a "carrier" probability signal with intricate non small oscillations.
Real positive probable squared charge (*ie.* probable mass) |r_{p}|^{2}
has second order discontinuities only at impossible positions where r_{p}^{2}=0 ,
and r_{p} may be very large when |r_{p}|_{+} is small.

Allowing **p**-dependant r_{p} implies positional information so we must also replace **m** with a **p**-dependant momentum **m _{p}**.

y

The (Ñ

The electrodynamic four-potential can now be introduced speculatively in the equation

Ñ_{p} y_{p} = (q_{p}+**a _{p}**+

We might seek to introduce a more conventional electrodynamic potential

A typical ideal spinor solution of interest would then be r_{p}**m _{p}**((

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Copyright (c) Ian C G Bell 2003

Web Source: www.iancgbell.clara.net/maths or www.bigfoot.com/~iancgbell/maths

Latest Edit: 01 Oct 2007.