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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 ...

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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. ...

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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 ...

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To begin with, The Amplituhedron formalism only works for a specific theory, N=4 SYM in the planar limit (only planar Feynmann diagrams are considered). Because of supersymmetry, you can classify scattering processes with two parameters: $n$ and $k$. n is the number of particles involved, and k is, roughly speaking, the number of spin flips in the process. ...

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The confusion lies with the definition of the transition probability. The transition amplitude between an $H_0$ eigenstate $H_0 \left|m\right> = E_m \left|m\right>$ and another eigenstate $H_0 \left|n\right> = E_n \left|n\right>$ due to a perturbation $V$ after a time $t$ is given by \begin{align} \left< n \right| U(t) \left| m \right> ...

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We'll give some details based on Andrew's comments. We are trying to solve $$\frac{ \partial \Delta \rho(t) }{ \partial t } = -i\mathcal L_0 \Delta \rho(t) -\{A, \rho_0(t) \} \, \mathcal F(t). \qquad (1)$$ Homogeneous solution To develop some familiarity with the Liouville operator, let us first consider the special case of $A = 0$. Then we have a ...

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It seems like your answer sidestepped the whole question. When you do the same for $\hat p$ you'll find that its derivative depends on $\hat x$, and on $\lambda$. But these coupled equations can then be solved as a second order equation for the terms individually, which should be what you are looking for.

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Basically, you do need to treat it as a perturbation and no correction is necessary... Calculating the commutator, $$\left[\hat{x}, \hat{H}\right] = \left[ \hat{x}, \frac{\hat{p}^2}{2m} - \frac{k}{2}\hat{x}^2 + \frac{\lambda}{4}\hat{x}^4 \right]$$ but as ...

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Canonical transformation A canonical transformations of the Hamiltonian is given by $$H' = e^{S} H e^{-S},$$ with $S$ anti-hermitian. The idea is to eliminate certain terms in the Hamiltonian by making a basis change. We trade old degrees of freedom by new ones in the hope that the new Hamiltonian becomes easier to solve. The ...

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As it stands the effective Hamiltonian contains the same information as the original one. But note that, given the condition $λV+[S,H₀]=0$, $S$ is proportional to $λ$ so the remaining terms are of order $λ{^2}$ and higher. Thus truncation by omitting these terms gives a result correct to order $λ$. I suppose this is the motivation behind the procedure. Does ...

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The idea is that, roughly speaking, the block moves down the gradient of the slope. Because the slope changes its direction of motion, it pushes the block left and right in roughly equal measure, and because it has a short period, the block never moves with significant horizontal velocity relative to the fixed axes. Thus, the horizontal speed of the block ...

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