Separate normalisation of chiral spinors?

In summary, the question posed is whether it is possible and useful to impose separate normalisation conditions on left and right spinors when solving the Dirac equation for free massive particles. This has been explored in books on weak interactions, but the speaker is interested in studying non-unitary chiral symmetry and the resulting conserved charge. The possibility of dilating spinors and the effects on the Dirac equation are also discussed. It is noted that different values of k can map the Dirac equation to the same bispinorial equation, and the new bispinor still fulfills normalisation-related properties depending on the type of transformations used.
  • #1
arivero
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When solving Dirac equation (for free massive particles) we usually impose normalisation conditions upon the eigenfunction in a single stroke.

I am wondering, Is it possible/useful to impose separate normalisation conditions upon the left and right spinors? Should we still have a resolution of the identity, etc?
 
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  • #2
Yes,of course.See any book on weak interactions (the ones which used to treat neutrinos as being massless),as it deals with normalized solutions of the Weyl equations.I remember seing it in D.Bailin's book on weak interactions.

Daniel.
 
  • #3
Hmm I was thinking in massive particles, not neutrinos.

The think I am pursuing is some kind of study of non unitary chiral symmetry, this is, the one generated with [tex]i\gamma^5[/tex] instead of [tex]\gamma^5[/tex]. Do we get a conserved charge?
 
  • #4
Let me to think aloud.

If we dilate two spinors [tex]|R>,|L>[/tex] so that [tex]|R> \rightarrow k_R |R'>[/tex], [tex]|L> \rightarrow k_L |L'>[/tex]
the new pair [tex]|R'>,|L'>[/tex] does not fulfil Dirac equation but instead
[tex]D |L'> = {k_R \over k _L} m |R'>[/tex]
[tex]D |R'> = {k_L \over k _R} m |L'>[/tex]

Does this new pair has still some physical sense?

Of course it is still a pair of relativistic wavefuntions of mass m. Intriguingly we have that different values of k map Dirac eq to the same bispinorial equation, because the new equation only depends on the quotient.

The new bispinor [tex]\Psi' \equiv |L'> \oplus |R'>[/tex] still fulfils some normalisation-related properties, depending of the kind of transformations.

Particularly if [tex]k_R, k _L[/tex] are real numbers (or diagonal self adjoint matrices) with [tex]k_R k _L=1[/tex] we have
[tex]<\bar \Psi | \Psi> = <\bar \Psi' | \Psi'> [/tex]
 
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Related to Separate normalisation of chiral spinors?

What are chiral spinors?

Chiral spinors are mathematical objects used in physics to describe the properties of particles that have a specific type of spin. They are represented by mathematical equations and are used to study the behavior of these particles in different physical systems.

What is the purpose of separate normalization of chiral spinors?

The main purpose of separate normalization of chiral spinors is to ensure that the mathematical equations used to represent these particles accurately describe their properties. It also helps to simplify calculations and make them more manageable.

What is the difference between separate normalization and standard normalization?

In standard normalization, all spinors are normalized together as a single group. In separate normalization, chiral spinors are normalized separately according to their unique properties. This allows for a more detailed analysis of the particles and their behavior.

Why is it important to use separate normalization for chiral spinors?

Separate normalization is important because it allows for a more accurate and detailed understanding of the properties of chiral spinors. It also helps to avoid potential errors or inconsistencies that may arise from using standard normalization for these particles.

What are some applications of separate normalization of chiral spinors?

Separate normalization of chiral spinors is used in many different areas of physics, including quantum mechanics, particle physics, and cosmology. It is also used in theoretical studies to help understand the fundamental laws of the universe and their implications for the behavior of particles.

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