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1) Are we able to perform perturbative analysis and use diagrammatic expansion, Green function etc. – all these field-theoretical stuff [for bosons]? In general, the field-theoretic methods (at finite or zero temperature) can be applied to both bosons and fermions with slight differences which originate from the Fermi-Dirac and Bose-Einstein ...


It's true that "the perturbative series is valid only when the perturbed state is qualitatively similar to the unperturbed state". Generally perturbation theory is acceptable when the coupling is weak, in which case the coupling can be treated as a small perturbation of the free field theory at all energies (for example Yukawa theory and $\phi^{4}$ theory. ...


1) I should note that most perturbative expansions that are of interest in physics are not formally convergent (and more often than not, not Borel-resummable either). 2) There are many examples of useful perturbative calculations for bosons. The oldest example (probably) in Many-Body physics is the calculation of the energy per particle of the weakly ...


For c) You should think about what it means that an orbit is closed. It means that after some time $t$ the particle will return to it's original position. For this to happen $m_1\tau_{osc}$ and $m_2\tau_{orb}$ have to be equal for some integer values of $m_1$ and $m_2$. This can only happen in your example if $\sqrt(n+2)$ is rational.


The initial state does not need to be one of eigenstates of the hamiltonian, it could be superposition. Therefore time evolution will change it. I don't think your first assumption is correct.


TL;DR In general, no. A longer but possibly irrelevant discussion follows. Consulting the classic review RevModPhys.58.323 by Rammer and Smith, the quantities you are considering are defined as (Eq. 2.5): $$G^{<}(\boldsymbol x_1,t_1,\boldsymbol x_{1'},t_{1'})=\mp i\langle \psi^\dagger_{\mathcal H}(\boldsymbol x_1,t_1) \psi_{\mathcal H}(\boldsymbol ...


At the risk of being too obvious, let me first point out: any state that changes with time is not an eigenstate of the Hamiltonian. So if you are describing a system that is at equilibrium for all time, then you may indeed assume that system A is in an eigenstate of the many-body Hamiltonian, but for any other situation (including anything in the real ...

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