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LSZ reduction vs adiabatic hypothesis in perburbative calculation of interacting fields

As far as I know, there are two ways of constructing the computational rules in perturbative field theory. The first one (in Mandl and Shaw's QFT book) is to pretend in and out states as free ...
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Transition amplitudes by functional methods in QFT

I am following section 9.2 in Peskin and Schroeder in which the Feynman rules are derived for scalar fields. They define (in eqn (9.14), page 282) the transition amplitude from $\vert\phi_a\rangle$ ...
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What is the sense of introducing generating functional to the summands of expansion of S-matrix?

Let's have generating functional $Z(J)$: $$Z(J) = \langle 0|\hat {T}e^{i \int d^{4}x (L_{Int}(\varphi (x)) + J(x) \varphi (x))}|0 \rangle , \qquad (1)$$ where $J(x)$ is the functional argument ...
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Let $n_F(\omega) = \large \frac{1}{e^{\beta (\omega)} + 1}$ be the Fermi function. A fermionic reservoir correlation function is given by: $$C_{12}(t) = \int_{-\infty}^{+\infty} d\omega~ ... 1answer 210 views Applying theorem of residues to a fermionic reservoir correlation function in order to solve the integral in the CF and obtain a summation Applying theorem of residues to a fermionic reservoir correlation function in order to solve the integral in the correlation function and obtain a summation. 1answer 134 views Two point function for massless boson in 2 dimension Is it true that two point function for massless boson theory in 2 dimension is a constant? That is to say it is independent of the distance between the two points? 2answers 148 views \langle B|A \rangle expressed in terms of the Partition Function Say you have an electron departing from point A and reaching poing B after a time t. According to some helping friend, the Partition Function for that electron going from point A to B can be written ... 1answer 159 views Significance of Poles of Correlation Function in QFT In QFT, specifically in scattering processes, what is the physical significance of the poles / residues of the N-point correlation function? And why? 1answer 126 views Product of VEVs vs. VEV of product How can we prove the following cluster decomposition formula$$\langle \phi_1 \phi_2 \rangle ~=~ \langle \phi_1 \rangle \langle \phi_2 \rangle,$$where brackets denote vacuum expectation value (VEV) ... 1answer 482 views Contact Term and Schwinger Term In field theory, when 4-divergences of time-ordered Green's functions are computed, there are extra terms known as 'Schwinger terms'. When deriving the quantum equations of motion for time-ordered ... 2answers 259 views Correlation, Time Ordering, and Observables In general, the product of two Hermitian operators \phi will not be Hermitian, unless the two operators commute. Question: is X = T \phi(t_1) \phi(t_2) Hermitian? It doesn't seem to be if T ... 1answer 355 views Why do disconnected diagrams not contribute to the S matrix? I've read somewhere that disconnected diagrams do not contribute to the S-matrix. I don't see why this is the case. I do know why vacuum bubbles do not contribute: given a generating functional for a ... 2answers 362 views Can scattering amplitudes be simplified with 1PI diagrams? I have been teaching myself quantum field theory, and need a little help connecting different pieces together. Specifically, I'm rather unsure how to tie in renormalization, functional methods, and ... 2answers 351 views How do I define time-ordering for Wightman functions? This is a follow-up question to What are Wightman fields/functions Ok, so based on my reading, the field operators of a theory are understood to be operator-valued distributions, that is, to be ... 1answer 715 views What are Wightman fields/functions Simple question: What are Wightman fields? What are Wightman functions? What are their uses? For example can I use them in operator product expansions? How about in scattering theory? 1answer 758 views Correlation function which has branch cut in momentum space When correlation function has branch cut in momentum space, how to find correlation in coordinate space? For example$$ \tilde {G}(\omega) = \frac{2i}{\omega+(\omega^2-\nu^2)^{1/2}}$$How to get the ... 0answers 752 views How to prove Wick's Theorem (Zee's eq. I.2 (16)) via Gaussian integration? I'm working through Zee's QFT in a Nutshell but there's an integral [I.2 (16)] I couldn't quite derive. The problem is to find$$\langle x_i x_j ... x_k x_l\rangle=\frac{\int ... \int dx_1 ... dx_n ...
In their book Condensed Matter Field Theory, Altland and Simons often use the following formula for calculating thermal expectation values of exponentials of a real field $\theta$:  \langle ...