# Quantum Field Theory Purly in Momentum Space?

• AJS2011
I've found a reference that does purely in momentum space. It's called "Perturbation theory in momentum space" by M. A. Lax.

#### AJS2011

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

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

$$\int d^3k\,F(k)\phi(k)\phi(-k)$$

where $\phi(k)$ is the field and $F(k)$ is some function?

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

$${1\over n!}\int d^3k_1\ldots d^3k_n \,G(k_1,\ldots,k_n)\phi(k_1)\ldots\phi(k_n)$$

leads to a vertex that connects $n$ lines with a vertex factor of $G(k_1,\ldots,k_n)$.

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?

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

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

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

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

$$\frac{1}{\omega^2 g(q,\omega)+q^2 h(q,\omega)}$$
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!

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

$$\int d^3k\,F^{\mu \nu}(k)\phi_\mu(k)\phi_\nu(-k)$$

then the propagator would be $$[F^{\mu \nu}(k)]^{-1}$$, 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.

AJS2011 said:
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.

RedX said:
the propagator would be $$[F^{\mu \nu}(k)]^{-1}$$, 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.

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