# Bilinear Expressions

## Main Question or Discussion Point

Hi all,

We've been looking at the bilinear expressions involving the adjoint spinor and the gamma matrices and how the various combinations transform as scalars, vectors etc.

My question is this:

Other than just appreciating the fact that various combinations of these bilinears transform analogously to scalars, pseudoscalars etc - what is the point of them. i.e what is the justification of their existence, why are the useful, when would one use them etc.

Rich

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dextercioby
Homework Helper
Bilinear covariants involving gamma matrices are very useful once u get into discussing weak interactions,which,as i think you already found out,do not conserve parity...This parity violation is described by means of the famous $\gamma_{5}$ matrix...It's definition is no unique,there are 2 possible ways,but's actually irrelevant.
It matters that everytime you wish to approach weak interaction phenomena (such as various decays),you need to take into account this $\gamma_{5}$ matrix and the various other covariants which include bilinear combinations of gamma matrices.
Of course,they appear in the EW interaction,as well.

So there's a point in discussing them...

Daniel.

Thanks!

That was very useful.

I would like to know their physical meaning though.

I mean <psi|psi> is the vacuum expectation value in non-rel QM.

So would we say that the first bilinear <adjoint psi | psi > is some kind of vev.

Then <adjoint psi| gamma5 psi > is some other kind of vev?

I mean, are the covariants taken separately as some kind of VEV or do they come into play somewhere in the dirac eqtn for example?

Thanks again!

Rich

dextercioby
Homework Helper
robousy said:
I mean <psi|psi> is the vacuum expectation value in non-rel QM.
Hold on,you mean
$$\langle \psi|\psi\rangle$$

is a VEV in nonrelativistic QM??There are two objections that i have
1.Who is psi??Is it a vector from the Hilbert space of states??
2.There is no vacuum in QM.There's the fundamental state,that's it.Fock space is not a notion of QM...QM includes only Hilbert spaces...

robousy said:
So would we say that the first bilinear <adjoint psi | psi > is some kind of vev.
You mean the VEV is:
$$\langle 0|\hat{\bar{\Psi}}_{\alpha}\hat{\Psi}^{\alpha}|0\rangle$$
,then i agree.It's the VEV of the product of Dirac spinors...

robousy said:
Then <adjoint psi| gamma5 psi > is some other kind of vev?
In QFT,these spinors are operators,not vectors... :grumpy: Don't make that confusion...The VEV should be:
$$\langle 0|\hat{\bar{\Psi}}_{\alpha}(\gamma_{5})^{\alpha} \ _{\beta}\hat{\Psi}^{\beta}|0\rangle$$

robousy said:
I mean, are the covariants taken separately as some kind of VEV or do they come into play somewhere in the dirac eqtn for example?
Depends on the situation...Since they involve gamma matrices,they could not considered in the absence of Dirac spinors,whether psi,psi bar,or the spinor amplitudes...

Daniel.

Thanks for pointing me straight - I was wrong there.
I need to think about that some more.

:)

Rich

Holy ****e.

You are only 22 dude??!

You some kind of prodigy?
You must have had a good education.

dextercioby
Homework Helper
Nope,i'm rude most of the time... :tongue2: Yeah,i'm only 22. And i used to listen to "Prodigy". :tongue2:

Anyway,i'm glad u got the picture.If u have anymore problems with QFT,feel free to post them here...We've got some people on this forum who know their business...

Daniel.

Sure, will do.

:)
What are you specialising in?