Dirac-Lanczos equation and math-ph/0201049
Jonathan Scott
Dirac's equation could be formulated in complex four
vector algebra (otherwise known as Pauli Algebra or
biquaternions). The late Pertti Lounesto thought my
formulation was original when he first saw it in
around 1993-4, but I was unable to take up his offer
to publish it. Later, I found that W Baylis had
published a paper in 1995 including this result under
the title "Eigenspinors and Electron Spin". However,
it seems that all of us were mistaken, in that the
following paper says that the first to come up with
this was Cornelius Lanczos in 1929 (the year after
Dirac published his relativistic wave equation)!
arXiv:math-ph/0201049 v1 22 Jan 2002
Comment on Formulating and Generalizing
Dirac’s, Proca’s, and Maxwell’s Equations
with Biquaternions or Clifford Numbers1
Jonathan Scott
Danny Ross Lunsford
> In several threads over the years, I've pointed out that
> Dirac's equation could be formulated in complex four
> vector algebra (otherwise known as Pauli Algebra or
> biquaternions)...
>
> following paper says that the first to come up with
> this was Cornelius Lanczos in 1929 (the year after
> Dirac published his relativistic wave equation)!
> arXiv:math-ph/0201049 v1 22 Jan 2002
> Comment on Formulating and Generalizing
> Dirac', Proca', and Maxwell' Equations
> with Biquaternions or Clifford Numbers1
The problem here is the final form of the equation:
D S = m S* i K
where i is the Gaussian "i", and K is the third quaternion unit.
0) As pointed out before, the explicit appearence of this K shows that
the spacetime inversion properties of the invariants involved have been
implicity violated.
1) A biquaternion equation is certainly possible - but it is not Dirac,
rather Proca with magnetic sources, and without further assumptions
(local duality gauge), the magnetic sources and complex potential may be
duality-rotated away.
2) Finkelstein and Jauch showed that "quaternion quantum mechanics"
could provide a starting point for seeing electromagnetism and weak
interactions from a unified perspective - and they actually had a
dynamics, rather than the algebraic speculations of Gursey, and did not
revert to complexified quaternions.
3) The source of 0) is a misidentification of "reversion" as an
invariant operation. In fact all inversions must be referred to the unit
pseudoscalar in the Dirac algebra, -iy5, which does not appear in
Lanczos' equation. Hestenes made the same error 35 years later. The
"correct" form is
(ym dm) W.(E + iy5 B) = W.(J + iy5 K)
where Wi is the Dirac alpha and is y4 yi. J is the electric current, K
the magnetic. Everything in this equation is a proper spacetime
covariant. The "E+iB" one often sees is in fact the above expression in
terms of the unit spacetime pseudoscalar. The latter provides a natural
"I" in a Clifford algebra.
The lesson here is - the complexified Pauli algebra is nothing but the
natural complex structure on a Clifford algebra combined with the *even*
subalgebra of the Dirac algebra. Any spacetime statements that do not
utilize the full Dirac algebra cannot possibly represent Fermion fields.
To emphasize - the underpinning of spacetime is *not* the quaternions,
but the Dirac algebra.
-drl
Danny Ross Lunsford
>
> (ym dm) W.(E + iy5 B) = W.(J + iy5 K)
>
> where Wi is the Dirac alpha and is y4 yi.
This is a type-fast-o, should be
(ym dm) W.(E + iy5 B) = ym(Jm + iy5 Km)
Notice that this is
vector * bivector = vector + pseudovector
--
-drl
Jonathan Scott
"Danny Ross Lunsford" <antima...@yahoo.com>
wrote in message news:ufAub.1402$kO5...@newssvr22.news.prodigy.com...
> [snip]
> The problem here is the final form of the equation:
>
> D S = m S* i K
>
> where i is the Gaussian "i", and K is the third quaternion unit.
>
> 0) As pointed out before, the explicit appearence of this K shows that
> the spacetime inversion properties of the invariants involved have been
> implicity violated.
As pointed out before, the choice of K here is arbitrary and invariant,
as it effectively represents a choice of "up" in the rest frame of the
particle itself. The overall equation transforms consistently under
Lorentz transformations, leaving the K term unchanged.
> 1) A biquaternion equation is certainly possible - but it is not Dirac,
> rather Proca with magnetic sources, and without further assumptions
> (local duality gauge), the magnetic sources and complex potential may be
> duality-rotated away.
This formulation can be expanded into explicit separate terms to show
that it is exactly equivalent to the Dirac equation in the absence
of potential, and there is similar form which includes the potential
and transforms correctly under gauge transformations (but which
doesn't look exactly like the conventional notation for the Dirac
equation as it includes an explicit post-multiplicative factor of K),
like the RHS above.
> The lesson here is - the complexified Pauli algebra is nothing but the
> natural complex structure on a Clifford algebra combined with the *even*
> subalgebra of the Dirac algebra. Any spacetime statements that do not
> utilize the full Dirac algebra cannot possibly represent Fermion fields.
> To emphasize - the underpinning of spacetime is *not* the quaternions,
> but the Dirac algebra.
It's certainly not the quaternions, but the above equation (plus
potential where required) is formally equivalent to the Dirac equation
in all details so the full Dirac algebra is not required.
For my paper on this subject see:
http://pws.prserv.net/jonathan_scott/physics/diraceqn.pdf
See also a paper "Eigenspinors and Electron Spin" by Prof W Baylis,
University of Windsor, Ontario, Canada which used to be available
via his web page at that University but the link I currently have
for it is broken.
Jonathan Scott - jonatha...@attglobal.net
Danny Ross Lunsford
(moderator - supersedes previous message on this topic)
Jonathan Scott wrote:
> In several threads over the years, I've pointed out that
> Dirac's equation could be formulated in complex four
> vector algebra (otherwise known as Pauli Algebra or
> biquaternions)...
>
> following paper says that the first to come up with
> this was Cornelius Lanczos in 1929 (the year after
> Dirac published his relativistic wave equation)!
>
> arXiv:math-ph/0201049 v1 22 Jan 2002
> Comment on Formulating and Generalizing
> Dirac’s, Proca’s, and Maxwell’s Equations
> with Biquaternions or Clifford Numbers1
The problem here is the final form of the equation:
D S = m S* i K
where i is the Gaussian "i", and K is the third quaternion unit.
0) As pointed out before, the explicit appearence of this K shows that the
spacetime inversion properties of the invariants involved have been
implicity violated.
1) A biquaternion equation is certainly possible - but it is not Dirac,
rather Proca with magnetic sources, and without further assumptions (local
duality gauge), the magnetic sources and complex potential may be
duality-rotated away.
2) Finkelstein and Jauch showed that "quaternion quantum mechanics" could
provide a starting point for seeing electromagnetism and weak interactions
from a unified perspective - and they actually had a dynamics, rather than
the algebraic speculations of Gursey, and did not revert to complexified
quaternions.
3) The source of 0) is a misidentification of "reversion" as an invariant
operation. In fact all inversions must be referred to the unit pseudoscalar
in the Dirac algebra, -iy5, which does not appear in Lanczos' equation.
Hestenes made the same error 35 years later. The "correct" form is
(ym dm) W.(E + iy5 B) = ym(Jm + iy5 Km)
where Wi is the Dirac alpha and is y4 yi. J is the electric current, K the
magnetic. Everything in this equation is a proper spacetime covariant. The
"E+iB" one often sees is in fact the above expression in terms of the unit
spacetime pseudoscalar. The latter provides a natural "I" in a Clifford
algebra.
The lesson here is - the complexified Pauli algebra is nothing but the
natural complex structure on a Clifford algebra combined with the even
subalgebra of the Dirac algebra. Any spacetime statements that do not
utilize the full Dirac algebra cannot possibly represent Fermion fields. To
emphasize - the underpinning of spacetime is not the quaternions, but the
Dirac algebra.
-drl
--
-drl