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A system's density matrix can be written in the form you cite if and only if it is a classical mixture of factorisable pure states. A factorisable pure state is, of course, one that can be written as a tensor product $\psi_A\otimes\psi_B$, where $\psi_A$ and $\psi_B$ are pure states in subsystems $A$ and $B$, respectively. Correlations between measurements ...


Consider the case of a free scalar field, governed by the usual Lagrangian, $$\mathcal{L} = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi - \frac{1}{2}m^2 \phi^2$$ The propagator, or equivalently Green's function for the theory is a function which can be though of as a response when we use a delta function as an input in the equations of motion, i.e. ...


Right, one is only supposed to put the sources $J=0$ to zero after the very last $J$-differentiation has been performed. Figuratively speaking, short of writing out the calculation in full detail: Some of the $J$s downstairs can "couple" to the $J$s upstairs in the exponential.


Basically the Green Function can be put in terms of eigenfunctions (or eigenmodes) like so: $$ G(x,x')=\sum_{\text{relevant modes}}u^{*}(x')u(x) $$ in some cases the sum turns to integral. One of the basic premises of Sturm-Liouville theorem (I hope I spelled it correctly), is that given a Linear operator $\hat{L}$, and an equation: $$ \hat{L}y(x)=f(x) $$ ...

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