Re: Question about the electromagnetic representation of the Dirac [Archive]

Danny Ross Lunsford

Apr17-04, 05:08 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>A. Kyriakos wrote:\n\n> In the e-print-archive the results of my investigation of the\n> non-linear electromagnetic field theory are presented\n> (http://arXiv.org/abs/quant-ph/0404088 ). It is shown that the linear\n> approximation of this theory is the quantum electrodynamics (QED).\n\nNo, sorry. Yes, EM can be represented with the Dirac algebra. No, the\nspinor field is not the bivector of EM. Moreover your alpha_1..4 do not\nform a basis of the Dirac algebra (they all anticommute but they don\'t\ngive the right metric), nor can they be regarded as the "alpha"\nrepresentation of Dirac\'s work, because alpha_4 is really gamma_0 or\ngamma_5 depending on the choice of gammas (Dirac or Weyl\nrepresentation). Even if you wanted to use the bivectors and vectors in\nthe same equation, you would need something like\n\nalpha_k (Ek - i gamma_5 Bk) + gamma_mu A_mu\n\nin order to have a Lorentz invariant theory. So your equation is not\nLorentz invariant under change of basis.\n\nFurthermore, the Lagrangian is not parity-invariant (combines FmnFmn and\nF*mnFmn). Since QED is certainly parity-invariant, this settles the\nissue at once.\n\n-drl\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>A. Kyriakos wrote:

> In the e-print-archive the results of my investigation of the
> non-linear electromagnetic field theory are presented
> (http://arXiv.org/abs/http://www.arxiv.org/abs/quant-ph/0404088 ). It is shown that the linear
> approximation of this theory is the quantum electrodynamics (QED).

No, sorry. Yes, EM can be represented with the Dirac algebra. No, the
spinor field is not the bivector of EM. Moreover your \alpha_1..4 do not
form a basis of the Dirac algebra (they all anticommute but they don't
give the right metric), nor can they be regarded as the "\alpha"
representation of Dirac's work, because \alpha_4 is really \gamma_0 or
\gamma_5 depending on the choice of gammas (Dirac or Weyl
representation). Even if you wanted to use the bivectors and vectors in
the same equation, you would need something like

\alpha_k (Ek - i \gamma_5 Bk) + \gamma_mu A_{mu}

in order to have a Lorentz invariant theory. So your equation is not
Lorentz invariant under change of basis.

Furthermore, the Lagrangian is not parity-invariant (combines FmnFmn and
F*mnFmn). Since QED is certainly parity-invariant, this settles the
issue at once.

-drl

A. Kyriakos

Apr22-04, 04:21 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Danny Ross Lunsford <antimatter33@yahoo.nose-pam.com> wrote in message news:<9LRfc.17575\\$S%2.12078@newssvr22.news.prodi gy.com>...\n> A. Kyriakos wrote:\n>\n> > In the e-print-archive the results of my investigation of the\n> > non-linear electromagnetic field theory are presented\n> > (http://arXiv.org/abs/quant-ph/0404088 ). It is shown that the linear\n> > approximation of this theory is the quantum electrodynamics (QED).\n>\n> No, sorry. Yes, EM can be represented with the Dirac algebra. No, the\n> spinor field is not the bivector of EM. Moreover your alpha_1..4 do not\n> form a basis of the Dirac algebra (they all anticommute but they don\'t\n> give the right metric), nor can they be regarded as the "alpha"\n> representation of Dirac\'s work, because alpha_4 is really gamma_0 or\n> gamma_5 depending on the choice of gammas (Dirac or Weyl\n> representation). Even if you wanted to use the bivectors and vectors in\n> the same equation, you would need something like\n>\n> alpha_k (Ek - i gamma_5 Bk) + gamma_mu A_mu\n>\n> in order to have a Lorentz invariant theory. So your equation is not\n> Lorentz invariant under change of basis.\n>\n> Furthermore, the Lagrangian is not parity-invariant (combines FmnFmn and\n> F*mnFmn). Since QED is certainly parity-invariant, this settles the\n> issue at once.\n>\n> -drl\n\n\nI know very well, that the Dirac theory cannot be reduced to the usual\nlinear Maxwell theory (look the paper A. Gsponer from references to my\npaper). Therefore I would not offer a theory, which contradicts to\nthis statement.\n\nMy paper is not about the usual linear Maxwell theory. In the paper\nthe nonlinear electromagnetic theory is submitted. It is substantially\nthe quantum field theory in another description, and the Dirac theory\nis its approximation. This theory is similar to the Maxwell theory\nonly by its form, and consequently it is difficult to get rid of the\nopinion, that we don\'t deal with Maxwell theory. In the nonlinear\ntheory there are no vectors of the linear field. Its functions are\ncurvilinear electromagnetic waves, which in the quantum representation\nare spinors.\n\nThe set of alphas-matrixes and the set of gammas-matrixes work\nidentically, since it is easy to proceed from the one set to the\nother by means of a canonical transformation. Since the Dirac theory\nfollows from the nonlinear theory, all the invariant properties of the\nnonlinear theory are the same as in Dirac theory. I use the alpha-set\nof matrixes because it gives the more simple electromagnetic\nrepresentation of Dirac spinors.\n\nOnce again I highlight, that the electromagnetic representation of\nQED, stated in my paper, is not the Maxwell representation of QED. It\nis the nonlinear electromagnetic field theory, which is concurent to\nthe quantum field theory. The purpose of this paper is to prove, that\nQED is the consequence of the nonlinear electromagnetic field theory.\nIn this theory all mathematical features of the Dirac theory have a\nfull explanation in a nonlinear electromagnetic representation.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Danny Ross Lunsford <antimatter33@yahoo.nose-pam.com> wrote in message news:<9LRfc.17575$S%2.12078@newssvr22.news.prodigy.com>...
> A. Kyriakos wrote:
>
> > In the e-print-archive the results of my investigation of the
> > non-linear electromagnetic field theory are presented
> > (http://arXiv.org/abs/http://www.arxiv.org/abs/quant-ph/0404088 ). It is shown that the linear
> > approximation of this theory is the quantum electrodynamics (QED).
>
> No, sorry. Yes, EM can be represented with the Dirac algebra. No, the
> spinor field is not the bivector of EM. Moreover your \alpha_1..4 do not
> form a basis of the Dirac algebra (they all anticommute but they don't
> give the right metric), nor can they be regarded as the "\alpha"
> representation of Dirac's work, because \alpha_4 is really \gamma_0 or
> \gamma_5 depending on the choice of gammas (Dirac or Weyl
> representation). Even if you wanted to use the bivectors and vectors in
> the same equation, you would need something like
>
> \alpha_k (Ek - i \gamma_5 Bk) + \gamma_mu A_{mu}
>
> in order to have a Lorentz invariant theory. So your equation is not
> Lorentz invariant under change of basis.
>
> Furthermore, the Lagrangian is not parity-invariant (combines FmnFmn and
> F*mnFmn). Since QED is certainly parity-invariant, this settles the
> issue at once.
>
> -drl

I know very well, that the Dirac theory cannot be reduced to the usual
linear Maxwell theory (look the paper A. Gsponer from references to my
paper). Therefore I would not offer a theory, which contradicts to
this statement.

My paper is not about the usual linear Maxwell theory. In the paper
the nonlinear electromagnetic theory is submitted. It is substantially
the quantum field theory in another description, and the Dirac theory
is its approximation. This theory is similar to the Maxwell theory
only by its form, and consequently it is difficult to get rid of the
opinion, that we don't deal with Maxwell theory. In the nonlinear
theory there are no vectors of the linear field. Its functions are
curvilinear electromagnetic waves, which in the quantum representation
are spinors.

The set of alphas-matrixes and the set of gammas-matrixes work
identically, since it is easy to proceed from the one set to the
other by means of a canonical transformation. Since the Dirac theory
follows from the nonlinear theory, all the invariant properties of the
nonlinear theory are the same as in Dirac theory. I use the \alpha-set
of matrixes because it gives the more simple electromagnetic
representation of Dirac spinors.

Once again I highlight, that the electromagnetic representation of
QED, stated in my paper, is not the Maxwell representation of QED. It
is the nonlinear electromagnetic field theory, which is concurent to
the quantum field theory. The purpose of this paper is to prove, that
QED is the consequence of the nonlinear electromagnetic field theory.
In this theory all mathematical features of the Dirac theory have a
full explanation in a nonlinear electromagnetic representation.

Danny Ross Lunsford

Apr24-04, 09:16 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>A. Kyriakos wrote:\n\n> I know very well, that the Dirac theory cannot be reduced to the usual\n> linear Maxwell theory (look the paper A. Gsponer from references to my\n> paper). Therefore I would not offer a theory, which contradicts to\n> this statement.\n\nThis misses the point - the spacetime algebra is equally useful for\nrepresenting either, and in fact any theory with covariance will be\nexpressible in terms of the spacetime algebra.\n\n> My paper is not about the usual linear Maxwell theory. In the paper\n> the nonlinear electromagnetic theory is submitted. It is substantially\n> the quantum field theory in another description, and the Dirac theory\n> is its approximation. This theory is similar to the Maxwell theory\n> only by its form, and consequently it is difficult to get rid of the\n> opinion, that we don\'t deal with Maxwell theory. In the nonlinear\n> theory there are no vectors of the linear field. Its functions are\n> curvilinear electromagnetic waves, which in the quantum representation\n> are spinors.\n\nYou\'ll have to explain this better. If you mean you did the Born-Infeld\ntheory in a representation in terms of spacetime algebra, then what\nadditional physical content has been added to remove the arbitrariness\nof it?\n\n> The set of alphas-matrixes and the set of gammas-matrixes work\n> identically, since it is easy to proceed from the one set to the\n> other by means of a canonical transformation.\n\nThis however is simply incorrect. The alpha matrices are fundamentally\ndifferent than the gammas and occupy 1/2 the bivector sector of the\nspacetime algebra (the other half being the alphas x gamma_5, the unit\npseudoscalar). Dirac\'s original equation was implicitly multiplied\nthrough by gamma_0 because this makes it possible to write the current\nvector simply as\n\npsi* alpha psi\n\n(*=Hermitian conjugate) and the 4th component as\n\npsi*psi\n\nwhich is positive definite. Of couse the actual current is\n\npsibar gamma_mu psi\n\nthat is, invariant objects must be formed with the adjoint spinor, not\nthe direct Hermitian conjugate. That is the price of living under an\nindefinite metric. There cannot be any Lorentz invariant formulation\nthat turns gammas into alphas because they each are independent,\nirreducible geometric objects. Note that the simple fact that the alphas\ntake the same form in representation X as do the spatial gammas in\nrepresentation Y is not an invariant statement - the algebra as a whole\nmust be considered.\n\n> Since the Dirac theory follows from the nonlinear theory...\n\nThis cannot be true. The Dirac theory is a "primitive" one and cannot be\nfurther reduced. It is the simplest possible invariant statement about a\nrelativistic particle in terms of an eigenvalue problem,\n\nPop psi = m psi\n\nwhere m is the rest mass. If you believe any of the underpinning of\nphysics in terms of Lorentz invariance, relativistic mechanics, and\nquantum theory, you have to accept this.\n\n-drl\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>A. Kyriakos wrote:

> I know very well, that the Dirac theory cannot be reduced to the usual
> linear Maxwell theory (look the paper A. Gsponer from references to my
> paper). Therefore I would not offer a theory, which contradicts to
> this statement.

This misses the point - the spacetime algebra is equally useful for
representing either, and in fact any theory with covariance will be
expressible in terms of the spacetime algebra.

> My paper is not about the usual linear Maxwell theory. In the paper
> the nonlinear electromagnetic theory is submitted. It is substantially
> the quantum field theory in another description, and the Dirac theory
> is its approximation. This theory is similar to the Maxwell theory
> only by its form, and consequently it is difficult to get rid of the
> opinion, that we don't deal with Maxwell theory. In the nonlinear
> theory there are no vectors of the linear field. Its functions are
> curvilinear electromagnetic waves, which in the quantum representation
> are spinors.

You'll have to explain this better. If you mean you did the Born-Infeld
theory in a representation in terms of spacetime algebra, then what
additional physical content has been added to remove the arbitrariness
of it?

> The set of alphas-matrixes and the set of gammas-matrixes work
> identically, since it is easy to proceed from the one set to the
> other by means of a canonical transformation.

This however is simply incorrect. The \alpha matrices are fundamentally
different than the gammas and occupy 1/2 the bivector sector of the
spacetime algebra (the other half being the alphas x \gamma_5, the unit
pseudoscalar). Dirac's original equation was implicitly multiplied
through by \gamma_0 because this makes it possible to write the current
vector simply as

\psi* \alpha \psi

(*=Hermitian conjugate) and the 4th component as

\psi*\psi

which is positive definite. Of couse the actual current is

psibar \gamma_mu \psi

that is, invariant objects must be formed with the adjoint spinor, not
the direct Hermitian conjugate. That is the price of living under an
indefinite metric. There cannot be any Lorentz invariant formulation
that turns gammas into alphas because they each are independent,
irreducible geometric objects. Note that the simple fact that the alphas
take the same form in representation X as do the spatial gammas in
representation Y is not an invariant statement - the algebra as a whole
must be considered.

> Since the Dirac theory follows from the nonlinear theory...

This cannot be true. The Dirac theory is a "primitive" one and cannot be
further reduced. It is the simplest possible invariant statement about a
relativistic particle in terms of an eigenvalue problem,

Pop \psi = m \psi

where m is the rest mass. If you believe any of the underpinning of
physics in terms of Lorentz invariance, relativistic mechanics, and
quantum theory, you have to accept this.

-drl

A. Kyriakos

Apr28-04, 02:47 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Danny Ross Lunsford <antimatter33@yahoo.nose-pam.com> wrote in message news:<G7wic.1248\\$Uz2.630@newssvr24.news.prodigy. com>...\n\n\n>This misses the point - the spacetime algebra is equally useful for\n>representing either, and in fact any theory with covariance will be\n>expressible in terms of the spacetime algebra.\n\n\nMy theory, as well as any physical theory, can be written down in a\nset of various mathematical ways, such as space-time algebra,\naccording to the differential geometry, Clifford algebra, quaternion\nalgebra etc., which are the different dialects of one mathematical\nlanguage. But the mathematical description form doesn\'t define\nphysics.\n\n\n\n>You\'ll have to explain this better. If you mean you did the\nBorn-Infeld\n>theory in a representation in terms of space-time algebra, then what\n>additional physical content has been added to remove the\narbitrariness\n>of it?\n\nMy theory is not the Born - Infeld theory in representation of the\nspace-time algebra. But since the Born - Infeld theory has formally\nsolved many problems connected with the electron, the reference to it\nenables the estimation of the opportunities of my theory.\n\nNow I will try to explain the logic of my theory (see paper\nhttp://arXiv.org/abs/quant-ph/0404088 ). I appologise if the\nexplanation will take up a lot of space.\n\nThe first goal of the paper is to show that the Dirac equations of the\nelectron-positron follow from the following reasons:\n1) Consider a pair production of the electron-positron from\ngamma-quantum at the presence of an electromagnetic field of a\nnucleus:\ngamma-quantum => electron + positron.\n\n2) In terms of the mathematical description of the field equations\nthis process can be written in the form:\n\nEM wave equation => Dirac electron equation + Dirac\npositron equation.\n\n3) let\'s prove now the following theorem: the electron and the\npositron appear during the disintegration and twirling of the EM\nphoton wave in a field of a nucleus.\n\nProof:\na) we take the equation of an EM wave (in the quantum form of the\nrecord this equation is the Klein – Gordon equation without\nmass);\nb) we disintegrate it on two linear Maxwell equations, which are the\nequation of the advanced and regarding waves (they have the form of\nthe Dirac equations for electron and positron without mass);\nc) further we “twist" both waves in rings: it can be made, at\nleast, in two ways: either by 1. using the curvilinear metrics, or by\n2. using the differential geometry. As consequence of the twisting\n(twirling) in the Dirac equations the mass-current terms appear and\nthe Dirac equation becomes simultaneously complex.\n\nThus, in the quantum record we receive exactly the Dirac equations,\nand in the electromagnetic record they are the equations of the\ncurvilinear (i.e. nonlinear) waves. The proof is finished.\n\nFurther we check the theorem consequences: we consider the solutions\nof the Dirac equation in the quantum representation and in the\nelectromagnetic representation, and we show, that they really\nrepresent the stationary circular electromagnetic waves.\n\n\nThe second goal of the paper is to find the non-linear equation of the\nelectron and positron, and also the Lagrangian of this equation.\n\nWe automatically find the nonlinear electron (positron) equations and\nits Lagrangian, using the electromagnetic representation of the fields\nof the electron (positron)\n\nThe check of this results is the accordance of the first\napproximations of the nonlinear equation and Lagrangian with known\nresults of Heisenberg, Born - Infeld, Nambu and Jona - Lasinio, etc.\n\nThe consequences of the theory of the twirled (nonlinear) waves are\nmany. We consider some of them in the following paper\n( http://arXiv.org/abs/quant-ph/0404116 ).\n\n\n>This however is simply incorrect. The alpha matrices are\nfundamentally\n>different than the gammas and occupy 1/2 the bivector sector of the\n>.spacetime algebra (the other half being the alphas x gamma_5, the\nunit\n>pseudoscalar). Dirac\'s original equation was implicitly multiplied\n>through by gamma_0 because this makes it possible to write the\ncurrent\n>vector simply as…………………\n>that is, invariant objects must be formed with the adjoint spinor,\nnot\n>the direct Hermitian conjugate. That is the price of living under an\n..indefinite metric. There cannot be any Lorentz invariant formulation\n>that turns gammas into alphas because they each are independent,\n>irreducible geometric objects. Note that the simple fact that the\nalphas take the same form in >representation X as do the spatial\ngammas in representation Y is not an invariant statement - >the\nalgebra as a whole must be considered.\n\n\nThe Dirac matrixes are the elements of the Clifford algebra. The\nelements of the Clifford algebra are defined first of all by their\ncommutative properties. The alpha-set and a gamma-set of matrixes of\nthe Dirac equation have identical commutative properties. You can find\nan example of the canonical transformation of the alpha-matrixes and\nthe physical sense of it in: http://arXiv.org/abs/quant-ph/0404116 ).\n\n\n> This cannot be true. The Dirac theory is a "primitive" one and cannot be further reduced. It is >the simplest possible invariant statement about a relativistic particle in terms of an eigenvalue >problem,\n>Pop psi = m psi\n>where m is the rest mass. If you believe any of the underpinning of\n>physics in terms of Lorentz invariance, relativistic mechanics, and\n>quantum theory, you have to accept this.\n\nI don\'t reduce the Dirac theory. I deduce it from more general\npropositions. My theory does not contradict to the modern\nrepresentations. You have an opportunity to check everything. For this\npurpose everything is written in detail.\n\nA.G.K.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Danny Ross Lunsford <antimatter33@yahoo.nose-pam.com> wrote in message news:<G7wic.1248$Uz2.630@newssvr24.news.prodigy.com>...

>This misses the point - the spacetime algebra is equally useful for
>representing either, and in fact any theory with covariance will be
>expressible in terms of the spacetime algebra.

My theory, as well as any physical theory, can be written down in a
set of various mathematical ways, such as space-time algebra,
according to the differential geometry, Clifford algebra, quaternion
algebra etc., which are the different dialects of one mathematical
language. But the mathematical description form doesn't define
physics.

>You'll have to explain this better. If you mean you did the
Born-Infeld
>theory in a representation in terms of space-time algebra, then what
>additional physical content has been added to remove the
arbitrariness
>of it?

My theory is not the Born - Infeld theory in representation of the
space-time algebra. But since the Born - Infeld theory has formally
solved many problems connected with the electron, the reference to it
enables the estimation of the opportunities of my theory.

Now I will try to explain the logic of my theory (see paper
http://arXiv.org/abs/http://www.arxiv.org/abs/quant-ph/0404088 ). I appologise if the
explanation will take up a lot of space.

The first goal of the paper is to show that the Dirac equations of the
electron-positron follow from the following reasons:
1) Consider a pair production of the electron-positron from
\gamma-quantum at the presence of an electromagnetic field of a
nucleus:
\gamma-quantum => electron + positron.

2) In terms of the mathematical description of the field equations
this process can be written in the form:

EM wave equation => Dirac electron equation + Dirac
positron equation.

3) let's prove now the following theorem: the electron and the
positron appear during the disintegration and twirling of the EM
photon wave in a field of a nucleus.

Proof:
a) we take the equation of an EM wave (in the quantum form of the
record this equation is the Klein – Gordon equation without
mass);
b) we disintegrate it on two linear Maxwell equations, which are the
equation of the advanced and regarding waves (they have the form of
the Dirac equations for electron and positron without mass);
c) further we “twist" both waves in rings: it can be made, at
least, in two ways: either by 1. using the curvilinear metrics, or by
2. using the differential geometry. As consequence of the twisting
(twirling) in the Dirac equations the mass-current terms appear and
the Dirac equation becomes simultaneously complex.

Thus, in the quantum record we receive exactly the Dirac equations,
and in the electromagnetic record they are the equations of the
curvilinear (i.e. nonlinear) waves. The proof is finished.

Further we check the theorem consequences: we consider the solutions
of the Dirac equation in the quantum representation and in the
electromagnetic representation, and we show, that they really
represent the stationary circular electromagnetic waves.

The second goal of the paper is to find the non-linear equation of the
electron and positron, and also the Lagrangian of this equation.

We automatically find the nonlinear electron (positron) equations and
its Lagrangian, using the electromagnetic representation of the fields
of the electron (positron)

The check of this results is the accordance of the first
approximations of the nonlinear equation and Lagrangian with known
results of Heisenberg, Born - Infeld, Nambu and Jona - Lasinio, etc.

The consequences of the theory of the twirled (nonlinear) waves are
many. We consider some of them in the following paper
( http://arXiv.org/abs/http://www.arxiv.org/abs/quant-ph/0404116 ).

>This however is simply incorrect. The \alpha matrices are
fundamentally
>different than the gammas and occupy 1/2 the bivector sector of the
>.spacetime algebra (the other half being the alphas x \gamma_5, the
unit
>pseudoscalar). Dirac's original equation was implicitly multiplied
>through by \gamma_0 because this makes it possible to write the
current
>vector simply as…………………
>that is, invariant objects must be formed with the adjoint spinor,
not
>the direct Hermitian conjugate. That is the price of living under an
..indefinite metric. There cannot be any Lorentz invariant formulation
>that turns gammas into alphas because they each are independent,
>irreducible geometric objects. Note that the simple fact that the
alphas take the same form in >representation X as do the spatial
gammas in representation Y is not an invariant statement - >the
algebra as a whole must be considered.

The Dirac matrixes are the elements of the Clifford algebra. The
elements of the Clifford algebra are defined first of all by their
commutative properties. The \alpha-set and a \gamma-set of matrixes of
the Dirac equation have identical commutative properties. You can find
an example of the canonical transformation of the \alpha-matrixes and
the physical sense of it in: http://arXiv.org/abs/http://www.arxiv.org/abs/quant-ph/0404116 ).

> This cannot be true. The Dirac theory is a "primitive" one and cannot be further reduced. It is >the simplest possible invariant statement about a relativistic particle in terms of an eigenvalue >problem,
>Pop \psi = m \psi
>where m is the rest mass. If you believe any of the underpinning of
>physics in terms of Lorentz invariance, relativistic mechanics, and
>quantum theory, you have to accept this.

I don't reduce the Dirac theory. I deduce it from more general
propositions. My theory does not contradict to the modern
representations. You have an opportunity to check everything. For this
purpose everything is written in detail.

A.G.K.