## Weyl-Dirac equations, explicit violation of parity.

Suppose that in an arbitrary way we dilate the Weyl components
of the coupled dirac equation

Take the Dirac equation in the Weyl representation:

D |L> = m |R>
D |R> = m |L>

and suppose we dilate, in an arbitrary way, the two chiral
components |L> --> kl |L'> |R> -- kr |R'>

The new pair is still Lorentz Invariant, but it is
not parity invariant anymore. Is this transformation
of some use?

R' and L' are still solutions of the Klein Gordon equation with mass
m, but now they are coupled with different mass terms

D |L'> = (m kr/kl) |R'>
D |R'> = (m kl/kr) |L'>

Moreover, if 1/kl+1/kr=2, the sum |L'>+|R'> has the same properties that
the original, mass symmetric, Dirac.

Does anybody remember if this mass-asymmetric formulation is used elsewhere?

Alejandro Rivero

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 "alejandro.rivero" wrote: > > Suppose that in an arbitrary way we dilate the Weyl components > of the coupled dirac equation > > Take the Dirac equation in the Weyl representation: > > D |L> = m |R> > D |R> = m |L> > > and suppose we dilate, in an arbitrary way, the two chiral > components |L> --> kl |L'> |R> -- kr |R'> > > The new pair is still Lorentz Invariant, but it is > not parity invariant anymore. Is this transformation > of some use? > > R' and L' are still solutions of the Klein Gordon equation with mass > m, but now they are coupled with different mass terms > > D |L'> = (m kr/kl) |R'> > D |R'> = (m kl/kr) |L'> > > Moreover, if 1/kl+1/kr=2, the sum |L'>+|R'> has the same properties that > the original, mass symmetric, Dirac. > > Does anybody remember if this mass-asymmetric formulation is used elsewhere? > > Alejandro Rivero You are obviously familiar with Connes' non-commutative geometry approach to mass evolution given your publications. Ashetkar resolved General Relativity into complimentary chiral components, Google "Abhay Ashtekar" chiral 212 hits Whether opposite parity test masses have identical inertial and gravitational masses (within themselves and in comparison) in the real world is a most interesting question. The parity Eotvos experiment experiment, qz.pdf below, is being executed by a respected academic group as you read this. Results are due out 01 August 2005. Theory is nice. A reproducible experimental result falsifying theory is nicer - it leaves less theory to worry about. -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf
 Uncle Al wrote: > "alejandro.rivero" wrote: > > > > and suppose we dilate, in an arbitrary way, the two chiral > > components |L> --> kl |L'> |R> -- kr |R'> > > > > The new pair is still Lorentz Invariant, but it is > > not parity invariant anymore. Is this transformation > > of some use? > > > > R' and L' are still solutions of the Klein Gordon equation with mass > > m, but now they are coupled with different mass terms > > > > D |L'> = (m kr/kl) |R'> > > D |R'> = (m kl/kr) |L'> > > > > (...) > You are obviously familiar with Connes' non-commutative geometry > approach to mass evolution given your publications. Ashetkar resolved > General Relativity into complimentary chiral components, I will check it again; I have always seen Ashtekar as a hard-to-read author, even compared to Connes. > Whether opposite parity test masses have identical inertial and > gravitational masses (within themselves and in comparison) in the real > world is a most interesting question. The parity Eotvos experiment > experiment, qz.pdf below, is being executed by a respected academic > group as you read this. Results are due out 01 August 2005. In Spain we have a phrase, "llevar el agua a mi molino", "to carry the water to my own watermill" :-) But yes, parity and masses are relevant issues in my question! Chirality is not parity; L and R spinors are eigenvectors of the chiral operator. Dirac equation uses two equally connected L and R spinors in order to have a parity invariant equation, so both concepts are connected. How can you give different masses to the L and R spinors? Both are solution of the same Klein-Gordon equation, one with mass sqrt(MM'), being M and M' the mass terms crossing L to R and R to L. It order to have different masses, we would need a pair of mass matrix products MM' and M'M having different eigenvalues, which I believe is not possible if both M and M' are non singular (and even I am unsure about singular matrices). Yours, Alejandro

## Weyl-Dirac equations, explicit violation of parity.

>
> (snip)
>
> I am having a problem here. If you scale |L> --> kl |L'>
> and you want to keep D |L> = m |R> then you need to scale
> R with *the same* scaling factor - unless m=0.

Of course I do not want to keep keep D |L> = m |R>. You can see
it in the (snip)ed part of the message!

I am happy enough with keeping relativistic invariance.

 alejandro.rivero wrote: > > D |L'> = (m kr/kl) |R'> > D |R'> = (m kl/kr) |L'> > > Moreover, if 1/kl+1/kr=2, the sum |L'>+|R'> has the same properties that > the original, mass symmetric, Dirac. > > Does anybody remember if this mass-asymmetric formulation is used elsewhere? Yuri Danoyan points me an amusing relationship between mesons and barions, namely that the mesons seem to be symmetrical respect the proton mass. From this hint, it could be possible to find a use to the relationship above: find a "isochiral" decomposition of the neutron or the proton such that for instance R is connected to L with a mass equal to K0, while L is connected to R with a mass equal to D0. Then the "isochiral" fermion has a mass sqrt(K0*D0)= 962 MeV The process, although, seems to be a bit more complicated that a Higgs, because it needs of intermediate electroweak conversions. On the other hand, the need of electroweak forces could be welcome because it justifies chirality. Alejandro