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Comments to the question (v4): First of all, note that a general mathematically rigorous definition of functional integrals is a well-known open problem in mathematics. One route is to try to construct the functional integral as an appropriate continuum limit of a lattice model over a discretized space-time $M$. If we for simplicity write the phase space ...

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Kallen-Lehmann representation originates from the decomposition of $\hat{\phi}(x)|\Omega\rangle$ into the Hamiltonian eigenstates $|\lambda_{\mathbf{p}}\rangle$. If $|\langle\Omega|\hat{\phi}(x)|\lambda_{\mathbf{p}}\rangle=0$ that state doesn't give contribution into the propagator spectral representation. In the free theory $\hat{\phi}(x)|0\rangle$ is ...

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What you just did was to find a condition for attractive power-law forces to have stable orbits where stable means they remain bounded when perturbed around the circular orbit. You got the correct result. The Bertrand's Theorem though says something different: the only forces whose bounded orbits imply closed orbits are the Hooke's law and the attractive ...

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