Quantum Field Theory Purly in Momentum Space?

  • Thread starter AJS2011
  • Start date
  • #1
AJS2011
11
0
Quantum Field Theory Purly in Momentum Space???

Hello,

I have a complicated nonlinear-nonlocal-nonrelativistic-effective action in momentum space and would like to do perturbation theory with that. I need to find propagator and Feynman rules. I can not go to x-space and follow the standard procedure of finding the equation of motion and inverting it to get the propagator.

Has anyone seen any quantum field theory that was purely done in momentum space from the action to the perturbation?

I would appreciate it if you could show me a reference or an idea?

Thanks
 

Answers and Replies

  • #2
Avodyne
Science Advisor
1,396
90


Is there a term quadratic in the fields? If yes, does it conserve momentum? That is, is it of the form

[tex]\int d^3k\,F(k)\phi(k)\phi(-k)[/tex]

where [itex]\phi(k)[/itex] is the field and [itex]F(k)[/itex] is some function?

If yes, then it's easy. The propagator is 1/F(k). And a term in the lagrangian of the form

[tex]{1\over n!}\int d^3k_1\ldots d^3k_n \,G(k_1,\ldots,k_n)\phi(k_1)\ldots\phi(k_n)[/tex]

leads to a vertex that connects [itex]n[/itex] lines with a vertex factor of [itex]G(k_1,\ldots,k_n)[/itex].
 
  • #3
tom.stoer
Science Advisor
5,778
170


Formally you can derive equations of motion in momentum space as well; but of course this could fail due to non-locality. Is there a way to introduce auxiliary fields to get rid of the non-locality?
 
  • #4
AJS2011
11
0


Thanks, Avodyne! My action looks similar to what you suggested. The action I have in (1+1)dimension is of the form:

[tex]
\begin{equation}
\begin{split}
S_E = \int d\omega dq & \Bigl\{ f(q,\omega)(q^2 VV^*
+q\omega VA^*+q\omega V^*A+\omega^2AA^*)\\&+g(q,\omega)(VV^*-i\omega V\varphi^*+i\omega V^*\varphi+\omega^2 \varphi\varphi^*)\\& +
h(q,\omega)(q^2\varphi\varphi^*-q\varphi A^*+q\varphi^*A+ AA^*)
\Bigr\}
\end{split}
\end{equation}
[/tex]

Here \phi scalar field, V electric potential and A magnetic potential.

Now according to what you said the propagator for \phi should read as

[tex]
\frac{1}{\omega^2 g(q,\omega)+q^2 h(q,\omega)}
[/tex]
Is it correct?

2-One more question:
How can I deal with other terms containing one potential and one scalar field? Should I consider them as interactions?

Thanks!
 
  • #5
RedX
970
3


Just to add to what Avodyne said, 1/F(k) is the propagator for a scalar field, but for something like:


[tex]
\int d^3k\,F^{\mu \nu}(k)\phi_\mu(k)\phi_\nu(-k)
[/tex]

then the propagator would be [tex][F^{\mu \nu}(k)]^{-1} [/tex], which will most likely not be invertible, and usually you handle that with the FP determinant/ghosts/gauge-fixing.

Also I think the vertex should also conserve momentum, so there should be a delta function there too. But I guess it doesn't have too, but then you are working with something like the interaction with a semiclassical field.
 
  • #6
A. Neumaier
Science Advisor
Insights Author
8,090
4,009


How can I deal with other terms containing one potential and one scalar field? Should I consider them as interactions?

No. You should perform a Bogoliubov (also spelled Bogolubov) transformation -- such as done in superconductivity, if you haven't seen it before -- that brings the quadratic part into the standard form without mixed terms, and then solve in terms of the resulting quasi-particle fields.
 
  • #7
A. Neumaier
Science Advisor
Insights Author
8,090
4,009


the propagator would be [tex][F^{\mu \nu}(k)]^{-1} [/tex], which will most likely not be invertible, and usually you handle that with the FP determinant/ghosts/gauge-fixing.

It will typically be invertible, except when gauge fields are involved.
 
  • #8
AJS2011
11
0


Thanks to all of you Avodyne, tom.stoer, RedX, A. Neumaier!
 

Suggested for: Quantum Field Theory Purly in Momentum Space?

Replies
17
Views
6K
Replies
7
Views
2K
Replies
12
Views
13K
Replies
4
Views
1K
Replies
4
Views
313
  • Last Post
Replies
4
Views
1K
Replies
152
Views
7K
Replies
13
Views
4K
  • Last Post
Replies
7
Views
3K
Replies
6
Views
2K
Top