I need to derive the euler-lagrange equations for the following non-local lagrangian density for a complex scalar field ψ(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\mathcal{L} = \partial_{\mu}\psi^* \partial_{\mu}\psi - \lambda \int dy\, f(x,y) \psi^*(y) \psi(y)[/itex]

where λ is the coupling constant, f is a certain real-positive valued function linear in the first argument that satisfies f(x,y)=1/f(y,x) (which also implies f(x,x)=1). The integral is over all spacetime.

Applying the usual euler-lagrange equations shouldn't be correct here. I tried taking the functional derivative of the action S=∫dx L with respect to ψ*and set it equal to zero, and I get

[itex]\partial_{\mu}\partial^{\mu}\psi(x) = -\lambda \int dy\, f(y,x) \psi(x)[/itex]

where indeed we have a non-constant mass term. On the other hand, I used the methods in this paper

http://www.astro.columbia.edu/~lhui/G6047_2012/HowTo.pdf

to derive the feynman rules for the only possible vertex in the theory (this already made me think about a correction to the propagator); I get -iλ∫dxdy f(x,y) which purely depends on f. This result can also be quickly derived with eq. (136) here

http://www2.ph.ed.ac.uk/~egardi/MQFT/mqft_lecture_9_10.pdf [Broken]

The full propagator is therefore one of a free complex scalar field with mass m²= λ∫dxdy f(x,y). At least this is the result I got, and I'd like to confirm it deriving this mass term in the equations of motion.

I also calculated the leading order correction to the transition amplitude between single-particle states in the canonical formalism, and the result agrees with the above procedure.

The final doubt that arises is this: even if the equations of motions lead to the same result, why would the non-locality in the lagrangian be completely gone, turning into a mass term?

I hope at least part of my post makes sense. Thanks in advance for helping. :)

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A (apparently?) non-local quantum field theory

**Physics Forums | Science Articles, Homework Help, Discussion**