In this chapter we will establish the 5D Dirac-Hestenes equation Ñ

There are ten 3-blades in any extended basis for Â

However, by replacing any 3-blade by i=

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} - Ñ^{2} + a^{2})y = 0
where Ñ^{2} denotes the usual Euclidean 3D Laplacian . We must regard this as
Ñ^{2}y_{p} = -a^{2}y_{p} in Â_{1,3} and as
Ñ^{2}y_{p} = a^{2}y_{p} in Â_{3,1}.

Ñ_{p}^{2} y_{p} = -a^{2} y_{p} factorises
as
(iÑ_{p}+a)(iÑ_{p}-a)y_{p} = 0
where real scalar a
= 2p`mc`*h*^{-1}
= `mc`__h__^{-1}
= `m` in natural units .

Subequation (iÑ_{p} - a)y_{p} = 0
Û (2p)^{-1}*h***e ^{4}**(iÑ

Dirac incorporates a 1-potential field

The

(

A more common form of the *C*_{4×4} Dirac equation is obtained
by setting **g ^{0}** = (

å

where the

Any solution to Ñ_{p}y = ±ay is a solution to
Ñ_{p}^{2}y = a^{2}y and an **e _{4}**-static origin-centred solution is given by
y

y

There are various equivalent "suites" of Dirac matrices (

g=^{0} | æ | 1 | 0 | ö ; | g=^{j} | æ | 0 | s_{j} | ö |

è | 0 | -1 | ø | è | - s_{j} | 0 | ø |

g=^{0} | æ | 1 | 0 | 0 | 0 | ö ; | g=^{1} | æ | 0 | 0 | 0 | 1 | ö ; | g=^{2} | æ | 0 | 0 | 0 | -i | ö ; | g=^{3} | 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 |

For distinct

g
º ^{_jk}g^{j}g=^{k} | æ | - s _{j}s_{k} | 0 | ö | = | -e_{ijk}i | æ | s_{i} | 0 | ö ; | g =^{ijk} | e_{ijk}i | æ | 0 | -1 | ö ; | g =^{0123} | i | æ | 0 | -1 | ö ; | gº i^{5}g
=^{0123} | æ | 0 | 1 | ö | |

è | 0 | - s _{j}s_{k} | ø | è | 0 | s_{i} | ø | è | 1 | 0 | ø | è | -1 | 0 | ø | è | 1 | 0 | ø |

We can construct matrices having 1 in the second, third, and fourth elements of the first column (and zeroes elsewhere) as i

Dirac conjugation (*aka.* *Dirac adjoint*) of a 4×4 matrix
**y** is traditionally defined by
**y**^{»} º **g ^{0}**

Dirac conjugation preserves

With regard to the

We can thus, albeit somewhat informally, regard a Dirac spinor ket matrix **y** as a "product" y_{p}**u**
where y_{p} is an even Â_{3,1} multivector which we must "recast" into a 4×4 complex matrix
via **e ^{j}** ®

We will adopt the natural Â

[ Proof : Left multiplication by

Because

so it is logical to factorise

The ½(**1**+**g ^{0}**) acts as a source or sink of

Define

Since i(

å

Adding a unit -i

å

å

å

Thus a sufficient (though perhaps not necessary, since

We can interpret 4×4 complex Hermitian matrices **f ^{ j}** as an orthonormal frame of Â

Taking

Ñ

[ Had we favoured Â

Þ Ñ

This result differs in sign of

Assuming natural units, our final Â_{3,1+} idempotent-stripped Hestenes-Dirac equation is thus

Ñ_{p}__y___{p} = (`m`__y___{p}**e _{4}** -

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_{p}*e*^{bpe1234})^{½}R_{p}
where r_{p} is a positive scalar , b_{p} is a scalar , unit rotor R_{p} = (½l(ae_{1234})^{↑}**c**)^{↑}
(*ie.* an even multivector with
R_{p}R_{p}^{§}=1) for unit plussquare bivector **c**, and e_{1234} 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_{p}e_{1234}^{»}e_{1234}R_{p}u
=-r_{p}u.
** 4D Dirac-Hestenes Solutions**

** Particle as spinning frame field **

We examine some geometric kinematic consequences of Ñ_{p}__y___{p} = `m`__y___{p}e_{124} - `q`**a _{p}**

Ñ

Ñ

Ñ

[ Proof : Because we are in Â

Ñ

Thus Ñ

Ñ

=

=

Ñ

Ñ

Thus when

Ñ

[ Proof : Ñ

= -(Ñ

= -(Ñ

= -(Ñ

**j _{p}**=

Suppose **a _{p}**=

Ñ

But

Similarly, if we are given

If we are given

If we are given nullvector field **s _{p}**º

Ñ

Ñ

= -2

= 2

Thus if r_{p}=r , null 1-vector field **s _{p}**=

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_{p}y_{p} = i`m`y_{p} for
a fixed blade i commuting with y_{p}, then so to does r_{p}y_{p} provided
either Ñ_{p}r_{p} = i`m`r_{p} or Ñ_{p}r_{p} = 0.
The latter can always be atained by means of an **e _{4}**-invarient spherical harmonic solution
r

The simplest solution to Ñ_{p}__y___{p} = `m`__y___{p}e_{124} is the *planewave*
__y___{p}=(`m`(**p**¿**e _{4}**)e

For a more general solution consider first one factorising as __y___{p}=b_{p}(½q_{p}**W _{2}**)

We have Ñ

[ Proof : Decompose b

Ñ

= Ñ

= Ñ

When b_{p}=b is constant , or more generally when Ñ_{p}b_{p} = 0, we have
Ñ_{p}__y___{p} = ½Ñ_{p}(q_{p})__y___{p}**W _{2}**
and so right-multiplying the Dirac-Hestenes equation
Ñ

When **W _{2}**=e

½(Ñ

Thus **j _{p}**=r

If b_{p}=0 and **a _{p}**=0 and q

Now consider solutions factorising as __y___{p} = (½q_{p}**W _{1}**)

[ Proof : y

Þ Ñ

Combining these results we have for the more general "singly phased" solution
__y___{p} = (½q_{p}**W _{1}**)

Ñ

Ñ

If q_{p} = w**m**^{~}¿**p** so that Ñ_{p}q_{p} = w**m**^{~} for **p**-independant nonzero w and
unit timelike **m**^{~} we have

w**m**^{~}**W**_{p}**S**_{p} = 4p*h*^{-1} (`q`**a _{p}** -

This grade decomposes into

wW_{p}¿S_{p} | = | 4ph^{-1} (m¿v _{p}cos(b_{p}) + qm^{~}¿a)
_{p} | = | ± 4ph^{-1} m¿v for the free Dirac (anti)particle
_{p} |

wW_{p}×S_{p} | = | 4ph^{-1} (mÙv(-b_{p}_{p}e_{1234})^{↑} + qm^{~}Ùa)
_{p} | = | ± 4ph^{-1} mÙv for the free Dirac (anti)particle
_{p} |

wW_{p}ÙS_{p} | = | -4ph^{-1} (m¿v) _{p}sin(b_{p})e_{1234}
| = | 0 for the free Dirac (anti)particle |

Taking b_{p}=0 so b_{p}=r_{p}^{½}R_{p}
with R_{p}_{<0>}=1 , e_{12}_{*}R_{p}=0 , and R_{p}^{§}R_{p}=1
and requiring **j _{p}**=r

Ð

[ Proof : The condition Ñ

Ñ

-½(Ñ

Ñ

Result follows when r

We typically consuider the phase ½q_{p} to vary with **p** much faster than r_{p}^{½}R_{p}
and consider an orientated particle having timelike four-velocity
**v _{p}** = (R

Spacelike

There is another more kinematically suggestive approach in which we redefine particle

We consider r

Right-multiplying the free Dirac-Hestenes equation
Ñ_{p}__y___{p} e_{21} = 2p*h*^{-1} (`m`__y___{p}**e _{4}** -

Þ (Ñ

When q

For the free Dirac particle we have

w

In this interpretation we can think of the particle "at **p**" as being an unorientated point
actually at **p**+½`m`^{-1}R_{§p}(**e _{1}**)
traversing an exteremely tightly wound helix at light (null) speed, the axis of the helix being
given by the comparatively slowly varying

When Ñ

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 (**e _{4}**+

Static central potential

H

Hestenes associates right multiplication by e

For the traditional model of the hydrogen atom we set **a _{p}** =

An invarient

The hydrogen atom fine structure energy levels are traditionally given by
`E`_{n,l} =
(8*h*^{2}e_{0}^{2})^{-1}`M`_{e-}`Q`_{e-}^{4}
(1+a_{e-}^{2}*n*^{-2}((*j*+½)^{-1}*n* - 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]

We can regard

To fully geometrise the entire Dirac ket

f^{5} | = | æ | 0 | 1 | ö | = ig .
This anticommutes with the other ^{0123}f ; squares to ^{ k}1 ; and has
f^{5}^{»} = g^{0}f^{5}^{T}^{^}g
= ^{0}g^{0}f^{5}g = -^{0}f as required.
^{5} |

è | 1 | 0 | ø |

Here at last is an advantage of Â_{3,1} spacetime over Â_{1,3} timespace.
**f ^{5}** is also the appropriate "fifth gamma vector" in Â

When representing Â

We can now associate
**u** =
½(**1**+**g ^{0}**)½(

u º ½(1-i

u

Any ket au decomposes as
au = a'u + *a*u = a'u + u*a* where
*a* = a + a^{124}e_{124} + a^{345}e_{345} a^{1235}e_{1235})u
lies in ** C_{entral}**(u)
=

Note that

A general ket product simplifies as aubucu...gu = au*bcdg*
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
** C_{entral}**(u) =

Letting a'

And any sequence of densities acting on density GH

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 + be_{12} + ge_{35})(a^{»} - b^{»}e_{12} - g^{»}e35)
u
= (|a|^{2} + |b|^{2} + |g|^{2} +
2(ab^{^})_{<0>}e_{12}
+ 2(ag^{^})_{<0>}e_{35}
+ 2(bg^{^})_{<0>}e_{1235} )u
has the form a"u where a" Î *A _{lgebra}*{e

Note that
uu^{§} =
u^{§}u =
u^{#}u =
uu^{#} = 0
due to the (1-e_{345}) factor, and
u^{2} = u^{§}^{#}u = u = u^{§}^{#}. Also note that
e_{12}u = ue_{12} = ue_{12345} = -u^{*} = -**e _{4}**u.

u converts e

u' º e_{1234}ue_{1234}^{-1} =
½(1-e_{124})½(1-e_{345}) =
½(1+e_{1235})½(1-e_{345}) =
¼(1-e_{345}-e_{124}+e_{1235})
represents a particle with the same spin e_{12} but opposite ("backward") **e _{4}**, corresponding to the oppositely charged
particle (b

Idempotent u^{#} = u^{§}
= ½(1+e_{124})½(1+e_{345})
= ¼(1 + e_{345} - e_{124} - e_{1235})
represents a particle of opposite charge and opposite spin to u.

The four Dirac real idempotents ½(1±e_{124})½(1±e_{345})
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 {e_{124},e_{345}} ,
facilitating the computation of logatithms.
Note that c_{p}=c_{p}'u' has vanishing
ketbra transition density y_{p}(c_{p}^{»}) = __y___{p}uu'c_{p}'^{»} = 0
when uu'=0.

u^{†} = u for **e _{4}**-dependent Hermitian conjugation

u acts as a source of e_{124} , -e_{345}, and -e_{1235} enabling us to
keep the u factored component of y_{p} even within e_{1234} or mixed within e_{123}
according to taste.

Operator u_{»} = u_{=}
corresponds to the matrix operator **u**_{=}(**A**)º **uAu**
= *A _{11}*

Thus when acting on an even multivector a in e_{1234},
u_{=}(a) = ¯_{e12}(a)u = (a_{<0>} + (a_{*}e_{21})e_{12})u
= (a_{<0>} + (a_{*}e_{21})e_{12345})u ; and for 1-vector **a** we have
u_{=}(**a**) = ¯_{e4}(**a**)u .

Moving into Â_{4,1} in this manner admits of another planewave type soultion
__y___{p}=(`m`(**p**¿**e _{4}**)e

Having established u, our full "unstripped" particle ket is

y

b

The "phase" factor (½q

y

The phase factor can be replaced by a (q

Setting scalar s_{p} º *ln*(r_{p}) for r_{p}>0 we have a partially exponentiated form

y_{p} = (½s_{p} + ½b_{p}e_{1234} - ½q_{p}e_{12345})^{↑} R_{p}½(1-e_{1235})½(1-e_{345})
and if R_{p} = r_{p}^{↑} for some even r_{p} in e_{1234} we have

y_{p} = (½s_{p} + ½b_{p}e_{1234} - ½q_{p}e_{12345} + r_{p})^{↑} ½(1-e_{1235})½(1-e_{345})
,

r_{p} commuting with e_{1234} and e_{12345}.

Recalling that e_{1234}^{§}^{#}=e_{1234} we have
y^{»} = y^{§}^{#} =
r_{p}^{½}u
(½b_{p}e_{1234})^{↑}
R_{p}^{§}
so that

y^{»}y
= r_{p}u(b_{p}e_{1234})^{↑} u
= r_{p} *cos*(b_{p})u

More generally, y^{»}ay =
r_{p}u_{=}(
(b_{p}e_{1234})^{↑}R_{§p}(a_{<+>}) + R_{§p}(a_{<->}))
and, in particular, y^{»}**e _{i}**y = r

y_{1}^{»}y_{2}
= (r_{1}r_{2})^{½}uR_{1}^{§}R_{2}u
= (r_{1}r_{2})^{½}(R_{1}^{§}_{*}R_{2} + (e_{21}_{*}(R_{1}^{§}R_{2}))e_{12345})u
so the "complex" valued Dirac inner product of two Dirac particle kets is
y_{1}__»__y_{2}
= (r_{1}r_{2})^{½}(R_{2}_{*}R_{1}^{§} + (R_{2}e_{21}R_{1}^{§})_{<0>}e_{12345})
.
When R_{p}=R_{p}(q_{p})^{↑} for even R_{p} in e_{1234} having unit scalar and zero e_{12} component we have
y_{1}^{»}y_{2} = (r_{1}r_{2})^{½}(q_{2}-q_{1})^{↑}
** 5D Dirac Particle Pensity**

y^{!} = yuy^{»}
= r_{p}((½b_{p}e_{1234})^{↑}R_{p})_{§}(u)
= ¼r_{p}(*e*^{bpe1234}(1-R_{§p}(e_{345})) + R_{§p}(e_{124} - e_{1235}) )

Note that (½b_{p}e_{1234})^{↑} commutes with u only for b_{p}=0.

R_{p} Î Â_{3,1 +} has a basis comprising six 2-vectors, 1 and e_{1234} .

y_{1}^{!}y_{2}^{!} =
r_{1}r_{2}
(½b_{1}e_{1234})^{↑}R_{1}uR_{1}^{§}(½b_{1}e_{1234})^{↑}
(½b_{2}e_{1234})^{↑}R_{2}uR_{2}^{§}(½b_{2}e_{1234})^{↑}

= r_{1}r_{2}
(½b_{1}e_{1234})^{↑}R_{1}
u_{=}( ((½b_{1}+½b_{2})e_{1234})^{↑}R_{1}^{§} R_{2})
R_{2}^{§}(½b_{2}e_{1234})^{↑} .

When ½b_{1}+½b_{2} = *k*p for integer *k* this reduces to
y_{1}^{!}y_{2}^{!}
= ± r_{1}r_{2} u_{=}(R_{1}^{§}R_{2})
(½b_{1}e_{1234})^{↑} R_{1}uR_{2}^{§}(½b_{2}e_{1234})^{↑} .

When R_{1}=R_{2} we have
y_{1}^{!}y_{2}^{!}
= r_{1}r_{2} *cos*(½(b_{1}+b_{2}))(½b_{1}e_{1234})^{↑}R_{1}uR_{1}^{§}(½b_{2}e_{1234})^{↑} .#

Hence y^{!}_{p}^{2} = r_{p} *cos*b_{p} y^{!}_{p} and y_{1}^{!} is orthogonal to y_{2}^{!} if R_{1}=R_{2} and
b_{1}+b_{2}=*k*p for odd integer *k*.

For b_{p}=0
we have Dirac particle pensity r_{p}R_{p}_{§}(u) with
y^{!}_{p}^{2} = r_{p}y^{!}_{p} while for b_{p}=p (and r_{p} signed for positive scakar part)
we have the Dirac antiparticle pensity y^{!}_{p} = r_{p}R_{p}_{§}(u') with
with y^{!}_{p}^{2} = -r_{p}y^{!}_{p} .

We can regard a Dirac particle pensity as a commuting idempotent product
u = ½(1±e_{1235})½(1±e_{345})
being rotated through spacetime by even rotor field R_{p} in e_{1234}.
Plussquare unit 4-blade e_{1235} dual to minussquare **e _{4}** is interpreted as velocity, while
plussquare 3-blade e

½(1-e

We can similarly construct a pensity based on idempotent ½(1+e

For the b_{p}=0 kets, we can form the general density of two Dirac particle states
y_{1}y_{2}^{»}
= (r_{1}r_{2})^{½}R_{1}uR_{2}^{§}
.

More generally y_{p}ay_{p}^{»} =
r_{p}(b_{p}e_{1234})^{↑}R_{§p}(u_{=}(a))(b_{p}e_{1234})^{↑}
= r_{p} ((b_{p}e_{1234})^{↑}R_{p})_{§} 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 Â_{4,1} 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 **e _{5}**) ; and
five 4-blades (four involving

The expected value of the 1-observable asscoiated with real blade b is

*E*_{y}(b) º y__»__(by)
= |y^{»}by|_{s}
= (y^{»}by)_{<0,N>} *u _{0}^{-1}*
= (y

= b

For b commuting with e

For b anticommuting with e

The

The probability current is the 1-vector dual of the 4-vector 4-urbservable associated with the

r

Thus it is natural to define the

[ Probability current is traditionally regarded as a point-dependant four-vector (4-D 1-vector) having coordinates (dropping

The kinematic rule gives Ñ_{p}¿**j _{p}** = Ñ

[ Proof : 4p

Similarly ((

The four 3-blades not involving

[ Traditionally,

For a given **e _{4}**, the boolean 3-urbservable associated with geometric operators e

We associate ï↑ñ with ½(1+e

Ñ_{p}¿**w _{p}**=0 follows similarly to the

It is natural to form *nullcurrent vector* **s _{p}** =

We here define the

The expected value of such a 3-blade e

We associate e

S_{p} º __y___{p}e_{12}__y___{p}^{§} = r_{p}*e*^{bpe1234} R_{p}_{§}(e_{12})
and think of *S ^{jk}* as the coefficiebnt of e

[ The spin bivector is traditionally expressed (dropping

We are thus picking the top left element of the matrix

Lounesto classifies spinors according to Dirac urbservables thus:

- y
_{§p}(1+e_{1234}) ¹ 0 - Dirac spinor -
y
_{§p}(1+e_{1234})=0 ;**w**¹_{p}**0**, S_{p}¹0 -*Flag-dipole spinor*. -
y
_{§p}(1+e_{1234})=0 ;**w**=_{p}**0**, S_{p}¹0 -*Weyl spinor*. -
y
_{§p}(1+e_{1234})=0 ,**w**¹_{p}**0**, S_{p}=0 -*Majorana spinor*.

Since u commutes with both e

Ñ

Ñ

where

Note that

This Dirac equation distinguihes

The *Dirac operator* Ñ_{p}_{[e1234]} + `q`**a _{p}**

If y_{p} solves the Dirac Hestenes equation then,
for any real scalar field l_{p} ,
c_{p} = (`q`*i*l_{p})^{↑} y_{p}
satisfies
Ñ_{p}_{[e1234]}c_{p}
= `m`c_{p}e_{1235} + `q`(**a _{p}**+Ñ

[ Proof :

By insisting on an ideal solution y_{p}=__y___{p}u we have removed the e_{124} elemenats
from the equation but this insistance is unconvincing.

Having introduced a fourth spacial dimension, it is natural to postulate that y_{p} is defined
over a 5D eventspace Â^{4,1} and extend Ñ_{p} accordingly. We also relax **e ^{5}**¿

Ñ

The 4D solution y

[ We can confirm this alternate geometric form of the Dirac equation by returning to the traditional gamma matrix Dirac equation

å

å

In adding a fifth dimension we aquire a newcomponent
¯_{e1234}(**e ^{5}**Ð

If y_{p} solves the 5D Dirac equation then
Ñ_{p}^{2}y_{p}
= ((`m`-`q`^{*}**a _{p}**)

[ Proof : Ñ

=

One might ask why nature apparently favours <0;4> multivectors for a in
Ñ_{p}y_{p} = a_{p}y_{p} . <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 **e _{4}**-static

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.

Incorporating velocity into our kets with y

The

For multiple qubits, equating the

u

Either way, the

If we unify

We can refrain from unifying the

We can assume R

We adopt the notation P

We can isolate y

We can write the pensity product of two normalised Dirac particles as
y^{!}^{[1]}y^{!}^{[2]} =
(R_{p}_{§}(1+e_{1235}))^{[1]}
(R_{p}_{§}(1+e_{1235}))^{[2]}w^{!}^{[1]}w^{!}^{[2]} .

We have
R_{p}^{[1]}_{§}(1+e_{1235}^{[1]}) __×__ R_{p}^{[2]}_{§}(1+e_{1235}^{[2]})
= R_{p}^{[1]}_{§}(e_{123}^{[1]})R_{p}^{[2]}_{§}(e_{123}^{[2]})

[ Proof :
R_{p}^{[1]}_{§}R_{p}^{[2]}_{§}((1+e_{1235}^{[1]}) __×__ (1+e_{1235}^{[2]}))
= R_{p}^{[1]}_{§}R_{p}^{[2]}_{§}(e_{1235}^{[1]}e_{1235}^{[2]})
= -R_{p}^{[1]}_{§}R_{p}^{[2]}_{§}(e_{123}^{[1]}e_{123}^{[2]})
.]

Also y^{[1]}__×__y^{[2]} = w^{[1]}w^{[2]}
((1+e_{1235}^{[1]})__×__(1+e_{1235}^{[2]}))
= w^{[1]}w^{[2]} e_{1235}^{[1]}e_{1235}^{[2]}
= -w^{[1]}w^{[2]} e_{123}^{[1]}e_{123}^{[2]}

Next : Beyond The Dirac Equation

Glossary Contents Author

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.