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In physics, an operator is almost always either a square matrix or a linear mapping between two function spaces (defined on, say, $\mathbb R^n$). Operators serve as observables and as time evolution operators in Quantum Mechanics. This tag will most often find valid use in quantum mechanics; don't use this tag just because your equations contain "everyday operations" like $\times$, $+$!

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Confusion about integral form of interaction potential in Dyson series

I'm reading through the Hamiltonian formulation of perturbation theory and am having trouble understanding how the integral form of the interaction potential is derived. Consider a scalar field $\phi$ …
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