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I find the whole notation here a bit confusing since we are talking about momenta modes while using real space notation (maybe I'm the only one...). To clarify what is going on we can switch to momentum space instead. Consider the quadratic term in the exponential: \begin{align} \int d^4x \phi ^2 & = \int d^4x \int d^4k d^4k' \phi _k \phi ^\ast _{ k ' ...

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I'm not being precise, but morally: Imagine you were integrating out all modes above a frequency $b\Lambda$. Consider $\omega < b\Lambda < 3 \omega$. A mode $\phi$ with frequency $\omega$ when cubed, will have some part of it as mode of frequency $3 \omega$, since: $\sin (3x) = 3 \sin (x) - 4 \sin^3 (x)$. (Easier to see that ${(e^{i \omega t})}^3 = ... 3 The Hydrogen atom can be solved exactly in the path integral formalism, see for example Fortschritte der Physik 30 401. There is no conceptual problem to compute a probability amplitude for a system with bound-states in this formalism. For example, if the system is initially in a state$\psi(x)$localized around$A=0$, the probability amplitude$a(B)$to ... 4 First, the classical and semiclassical adjectives are not quite synonyma. "Semiclassical" means a treatment of a quantum system whose part is described classically, and another part quantum mechanically. Fields may be classical, particle positions' inside the fields quantum mechanical; metric field may be classical and other matter fields are quantum ... 1 There are results that are mathematically rigorous concerning the semiclassical limit of quantum theories. It is in fact an ongoing and interesting theme of research in mathematical physics. However you need to be rather well versed in analysis to understand the results. The bibliography is quite huge, but I would like to mention the following (some quite ... 1 At the classical level, the global gauge invariance leads via Noether's theorem to electric charge conservation, cf. e.g. this Phys.SE post. The Ward-Takahashi identity (WTI) can roughly speaking be thought of as a quantum version of this. In particular, we stress that the WTI is intimately tied to electric charge conservation. OP's observation that ... 3 We can write the Fourier transform of$\langle 0|\mathcal{T}A_{\nu}(x)\psi(x_1)\bar\psi(x_2)|0\rangle$as $$S(p) D_{\nu\alpha}(q) \ e\,\Gamma^{\alpha}(p,q,p+q)S(p+q)$$ where$S(p)$is the full fermion propagator,$D_{\nu\alpha}(q)$is the full photon propagator,$\Gamma^{\alpha}(p,q,p+q)\$ is the proper vertex function, and an overall momentum conservation ...

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First of all, let me say the following: If anyone (perhaps you, @V.Moretti?) would be able to provide a more mathematically oriented perspective on this question, I think that would make a valuable complement to this answer, which can be characterized as pragmatic (or handwavey, depending who you'd ask!), rather than deep. That being said, I will now answer ...

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