Constraints for a quantum field theory

[Linker]

Hello,

 

when I have a quantum field theory with action S(psi, a_mu, a*_nu) dependent on fermion field psi and dependent on the two boson fields a_mu and a*_nu. All boson fields are complex numbers; no noncommutative objects like matrices. These boson fields are connected with two conditions. In the whole theory there should be satisfied the following conditions:

(i) a_{mu}a*^{mu}=1

(ii) a^{mu} a*^{nu} d_{mu} a_{nu} = 0

Here, d_{mu} is the ordinary partial derivative in spacetime direction mu. The Feynman propagator reads:

f = e^(iS) delta(a_{mu}a*^{mu}-1) delta(a^{mu} a*^{nu} d_{mu} a_{nu}) (Integration over all psi, a_mu and a*_nu)

The first delta distribution (condition (i)) can be written as an integral over e^(i integral d^4x K(x) (a_{mu}a*^{mu}-1))/(2 pi) by the variable K(x). But how I can express condition (ii) in the propagator? Condition (ii) contains a spacetime derivative. How I can transform delta(a^{mu} a*^{nu} d_{mu} a_{nu}) in the Feynman propagator?

Shall I use faddeev-popov ghosts (despite it is a commutative theory)?

 

 

[ajb]

You have asked a technical question and as such I am not sure how much help people can really be. I don't know how to tackle this from the top of my head. Where does the QFT come from?

[Mordred]

Hello,

 

when I have a quantum field theory with action [latex]S(psi, a_{\mu}, a*_{\nu})[/latex] dependent on fermion field psi and dependent on the two boson fields a_mu and a*_nu. All boson fields are complex numbers; no noncommutative objects like matrices. These boson fields are connected with two conditions. In the whole theory there should be satisfied the following conditions:

(i)[latex] a_{\mu}a*^{\mu}=1[[/latex]

(ii)[latex] a^{\mu} a*^{\nu} d_{\mu} a_{\nu} = 0[/latex]

Here, d_\mu is the ordinary partial derivative in spacetime direction mu. The Feynman propagator reads:

[latex]f = e^(iS) delta(a_{\mu}a*^{\mu}-1) delta(a^{\mu} a*^{\nu} d_{\mu} a_{\nu})[/latex] (Integration over all [latex]psi, a_{\mu} and a*_{\nu})[/latex]

The first delta distribution (condition (i)) can be written as an integral over [latex]e^(i integral d^4x K(x) (a_{\mu}a*^{\mu}-1))/(2 \pi)[/latex] by the variable K(x). But how I can express condition (ii) in the propagator? Condition (ii) contains a spacetime derivative. How I can transform[latex] delta(a^{mu} a*^{nu} d_{mu} a_{nu})[/latex] in the Feynman propagator?

Shall I use faddeev-popov ghosts (despite it is a commutative theory)?

 

 

seeing if I can complete the latex to make it easier to read

 

edit still didn't help me, recognize the forms well enough to see what your doing perhaps you might get better help if they are all properly converted.

Edited May 19, 2014 by Mordred

[`hýsøŕ]

sorry i cannot come close to answering your question, can i ask you how many years it took to get to the level where any of that makes sense? :S

[Linker]

Assuming that [latex]a_{\mu}[/latex] is a gauge function and [latex]a^*_{\mu}[/latex] is the gauge field, then the generalized rule for delta distribution is (applied for condition (ii)):

 

[latex]\delta(a^{* \mu}a^{\nu} \partial_{\nu} a_{\mu}) = \int da'^{\nu} \frac{ \delta(a^{\nu}-a'^{\nu})}{det(Da^{* \mu}a^{\nu} \partial_{\nu} a_{\mu})}[/latex]

 

Here, [latex]a'^{\mu}[/latex] are all the [latex]a^{\mu}[/latex] that obeye condition (ii). The Jacobian for condition (ii) is formed by the operator D that means D varies condition (ii) functionally by [latex]a^{\mu}[/latex]. Am I right from mathematics?

When condition (ii) is varied, one obtains

 

[latex]det(Da^{* \mu}a^{\nu} \partial_{\nu} a_{\mu}) = det((a^{* \beta} \partial^{\nu} a_{\beta} - a^{* \mu}a^{\nu} \partial_{\nu})Da_{\mu})[/latex].

 

How I should proceed?

Edited May 19, 2014 by Linker

[Mordred]

well that definitely helps me realize what your doing is higher than my current QFT, I'm still in studies myself on QFT.

[Linker]

well that definitely helps me realize what your doing is higher than my current QFT, I'm still in studies myself on QFT.

This topic is not a question about a quantum field theory; it is about how to express

[latex]J:= \delta(a^{* \mu}a^{\nu} \partial_{\nu} a_{\mu})[/latex]

in terms of arbritary functions on the spacetime [latex]a_{\mu},a^*_{\mu}[/latex]. There is no need to know in what sense these vector fields are used.

What I should do when I express [latex]J[/latex] in terms of Fadeev-Popov ghost fields?

Edited May 19, 2014 by Linker

[ajb]

This topic is not a question about a quantum field theory; it is about how to express

[latex]J:= \delta(a^{* \mu}a^{\nu} \partial_{\nu} a_{\mu})[/latex]

in terms of arbritary functions on the spacetime [latex]a_{\mu},a^*_{\mu}[/latex]. There is no need to know in what sense these vector fields are used.

What I should do when I express [latex]J[/latex] in terms of Fadeev-Popov ghost fields?

 

 

If your a's are gauge fields then the FP trick could help you. Are you thinking about some less common choice of gauge fixing? I am not sure exactly what you are trying to do and I confess it has been a while since I looked at calculations in any detail.

Edited May 19, 2014 by ajb

[Linker]

The condition (i) reads:

[latex]a^{* \mu}a_{\mu}=1[/latex].

Now, [latex]a^{* \mu}[/latex] is a gauge field (like the photon field). Its gauge transformation is given by:

[latex]a^{* \mu} \rightarrow a^{* \mu}+ a^{* \nu} \partial^{\mu} a_{\nu}[/latex].

Inserting this gauge transformation into condition (i) and requiring gauge fixed one obtains condition (ii)

The field [latex]a_{\nu}[/latex] is the generating field for the gauge. What I'm meaning with generating field: For a photon it is the field [latex]\phi[/latex] with gauge transformation [latex]A_\mu = A_\mu + \partial_\mu \phi[/latex].

Now I have to vary with respect to [latex]a_{\nu}[/latex] (because this is the generating field) when computing [latex]J[/latex]. But how I gain the Faddev-Popov determinant in this case?

Edited May 19, 2014 by Linker

[ajb]

But how I gain the Faddev-Popov determinant in this case?

You try to follow the most general construction that you can find. One problem is that textbooks and reviews tend to only consider covariant gauges, remembering that some non-covariant ones are ghost free. Have you found some good reviews of BRST, BV and the FP ghosts? The right one will help you, I hope.