Why does ψ have a ghost number of -1 in BRST symmetry?

[ndung200790]

In BRST symmetry the action is

Inew=I0[$\phi$]+s$\psi$[$\phi$,$\omega$,$\varpi$,h].
Where ω and $\varpi$ is ghost and anti ghost.
If we expand I in series of terms In that satisfy sIn=0(s is BRST operator(Slavnov operator)).
We introduce Hodge operator t=$\varpi$$\delta$/$\delta$h.Then
ψ=-∑tIn/n.
They conclude that ψ has ghost number is -1,but I do not understand why?

 

[Avodyne]

It follows from your first line. Ghost number is conserved (by construction), the action has ghost number zero, the BRST operator s has ghost number +1, and so ψ must have ghost number -1.

 

[ndung200790]

I do not understand why s operator has ghost number 1?Is it(s operator) being the ''charge'' of BRST symmetry?

 

[ndung200790]

Or is s operator being a ''ladder'' operator of ghost number?

 

[Avodyne]

s represents (anti)commutation with the BRST charge, which carries ghost number +1. So acting with s raises ghost number by +1.

Conservation of ghost number is not a fundamental principle, and you could image using a ψ that includes terms of different ghost number (thus breaking ghost number conservation). It's just that it's more convenient to have ghost number conservation as an additional tool to restrict possible counterterms.

 

[ndung200790]

Is it the relation:

[Q,$\Phi$]$_{\pm}$=is$\Phi$?Then how can we see s being ''ladder'' operator of ghost number?

 

[Avodyne]

Let N = ghost number operator.

[N,Q]=+Q
[N,ψ]=-ψ

Compare the harmonic oscillator, N=a⁺a:

[N,a⁺]=+a⁺
[N,a]=-a

 

[ndung200790]

Why we know [N,Q]=+Q?And why by this we know ghost number of s =+1?

 

[Avodyne]

Sorry, I can't help any more. I suggest taking a look at Srednicki's text.

 

[ndung200790]

Thank very much for your kind helping.
At the moment,I have a new look: the full BRST transformation is θs,where θ is fermionic variable,I think the ghost number of θ is -1(?).Because of BRST symmetry θs has ghost number 0(?) then s has ghost number +1.
Please consider my question again.