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A (apparently?) nonlocal quantum field theory 
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#1
Dec912, 05:05 PM

P: 12

I need to derive the eulerlagrange equations for the following nonlocal lagrangian density for a complex scalar field ψ
[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 realpositive 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 eulerlagrange 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 nonconstant mass term. On the other hand, I used the methods in this paper http://www.astro.columbia.edu/~lhui/...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...cture_9_10.pdf The full propagator is therefore one of a free complex scalar field with mass mē= 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 singleparticle 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 nonlocality 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. :) 


#2
Dec1012, 08:43 AM

P: 12

For some reason I cannot edit the first post anymore. I just wanted to add that xlinearity in f is not required, also because if compromises the positivity. I needed it for other things, but I realized it's not working that way. Deriving the equations of motion shouldn't depend on that anyway.



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