Quantum Field Theory - Charged Field

Katie3
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The question is as follows;

'A free quantum field theory is related to an infinite number of quantum mechanical harmonic oscillators as unit mass particles on springs with spring constants k, where k takes all values. Now imagine the following scenario: these particles also have non-zero charges, and I suspend the system in a constant electric field Eo. Ignoring the effects of any outgoing radiations due to the oscillations of charged particles, show that this system of harmonic oscillators is equivalent to a massive scalar field theory with a classical source term int(d^4x J*(phi)). Where J is the source and phi is the scalar field. Find the connection between Eo and J.

I've been working on this all day and I feel like I've gotten no where. It *should* be an easy question. If someone could point me in the right direction, that would be great!
 
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Particles only have local charges, not global. Global charge is always net zero.
 

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