Weyl-Dirac equations, explicit violation of parity.

alejandro.rivero

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

Uncle Al

"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

arivero@unizar.es

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

arivero@unizar.es

Arkadiusz Jadczyk wrote:
>
> (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.

Al.Rivero@gmail.com

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