Classical and quantum effective theories

A generalization of the action principle of classical mechanics, motivated by the closed time path scheme of quantum field theory, is presented to deal with initial condition problems and dissipative forces. The similarities of the classical and the quantum cases are underlined. In particular, effective interactions that describe classical dissipative forces represent the system-environment entanglement. The relation between the traditional effective theories and their closed time path extension is briefly discussed, and a few qualitative examples are mentioned.

Figure 1

An $\mathcal{O}(g_1^2{\lambda }^4)$ graph contributing to $⟨0|T[\varphi (x)\varphi (y)]|0⟩$. The circle represents the initial density matrix, $\rho (t_i)=|0_p⟩⟨0_p|$, the continuous and the dashed lines stand for the $\varphi$ and the $\psi$ propagators, respectively. The part left (right) from the circle belong to $U$ ($U^{†}$). One has $t=t_f$ at the left and right ends of the graph where the lines reaching this time from $U$ and $U^{†}$ are joined.

Figure 2

$\mathcal{O}({\lambda }^4)$ graphs that contribute to the two $\varphi$ particle nonforward scattering probability with $n_r$ real and $n_v$ virtual $\psi$ particles with $(n_r,n_v)$ given by (a) (0, 2); (b), (c) (2, 1); (d), (e), (f) (4, 0). (a) Exclusive scattering; (b), (c) inclusive scattering; and (d), (e), (f) represent the contributions of relative states (98) with $n=2$ and $n=4$, respectively.

Figure 3

Leading order graphs, contributing to the expectation value (108) with ${\sigma }^{\prime }=+$. (a)  Factorizable state; (b) entangled state contributions.