29 votes

Scattering, Perturbation and asymptotic states in LSZ reduction formula

The first question we have to ask is: what is a one particle state in an interacting theory? It is reasonable to require that they are states that are both momentum eigenstates and energy eigenstates. ...
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24 votes
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Quantum Field Theory in position space instead of momentum space?

The most important reasons we use momentum space Feynman rules are: In position space, the Feynman rules generate convolutions of propagators. Because of the convolution theorem, the momentum space ...
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12 votes
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What actually means to compute things at tree level?

Suppose you want to compute a correlation say in Euclidean signature $$ \frac{1}{Z}\int D\phi\ \prod_i \phi(x_i)\ \exp\left(-\frac{1}{\hbar}S(\phi)\right) $$ with $$ S(\phi)=\frac{1}{2}(\phi,A\phi)+g\...
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10 votes

Allowed Field Re-definitions in QFT

After the field redefinition $$ \phi \rightarrow \phi + \frac{\partial_{\nu}\partial^{\nu}}{v^2}\phi\tag{1} $$ there is an apparent violation of unitarity because the propagator decays too fast in ...
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9 votes

Scattering, Perturbation and asymptotic states in LSZ reduction formula

Doubt 1: You cannot simply put $t=\pm\infty$ since all formulas become meaningless unless the limit is carefully done. The proof of the LSZ theorem by Haag and Ruelle shows that one needs the ...
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9 votes
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Is an interacting QFT Hilbert space a physical particles Fock space?

Haag's theorem says that the Hilbert space on which interacting relativistic quantum fields can be defined cannot be a standard Fock space. The finite time dynamics happens in this space, hence not in ...
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9 votes
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What does it mean for QFT to be unitary?

Maybe this is a boring answer but unitarity of a QFT means states at different times can be related to each other by \begin{equation} \left | \Psi(t_2) \right > = U \left | \Psi(t_1) \right > \...
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  • 5,184
8 votes
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The use of $a^\dagger(\mathbf{k}) = -i \int d^3x e^{ikx}\stackrel{\leftrightarrow}{\partial}_0 \phi(x)$ in the derivation of the LSZ-formula

OP asks good questions. Let us try to sketch the logic of the LSZ reduction formula. In the Heisenberg picture, a free real field $\hat{\varphi}(x)$ has a Fourier expansion $$ \hat{\varphi}(x)~=~\...
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8 votes
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The reasoning of the definition of $S$-matrix

As lurscher already said correctly, the idea is to choose an initial and a final time that is (in time) away from the actual time-dependent interaction in order to make sure that the interaction does ...
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7 votes

Why do we not have spin greater than 2?

There is a fabulous explanation in Schwartz QFT and the standard model, p153. The absence of massless particles with spin > 2 is a consequence of little group invariance and charge conservation. ...
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  • 1,147
7 votes
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Why this loop carries an integral if there is no undetermined momenta?

The momentum conservation at the vertex is NOT $p_1-p_2-k=0$ but rather $p_1-p_2-k+k=0$ which leaves $k$ completely free. The vertex has two half-edges carrying the momentum $k$ but in opposite ...
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7 votes

Spin-$J$ Amplitude $A_J(s,t) = - \frac{g^2(-s)^J}{t-M^2}$?

The intuitive proof is straightforward. The group $SU(2)$ is simple, and so any representation is contained in the tensor product of sufficiently many copies of the fundamental (à la Young). More ...
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7 votes
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The interpretation of the quantum field

(This answer is written from the perspective of lattice QFT, where everything is mathematically well-defined. Lattice QFT is not fundamental, of course, but most of the QFTs we use in physics are not ...
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7 votes

Weinberg, Effective Field Theories

Unitarity says that the sum of the probabilities for all possible out states (for a given in state) must be 1. Since the probability is the square of an amplitude, if any individual amplitude has a ...
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6 votes
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Green's Functions from Gell-Mann and Low Theorem

$\newcommand{\ket}[1]{\left|#1\right\rangle}$ $\newcommand{\bra}[1]{\left\langle#1\right|}$ $\newcommand{\braket}[1]{\left\langle#1\right\rangle}$ After reading from Yeh's notes on Advanced Condensed ...
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  • 748
6 votes

Relation between scattering matrix and an effective Hamiltonian

It's been some time since this question has been asked, but allow me to post a new answer. Its piece of my thesis, where I've put the derivation (since its so hard to find). I hope it'll help anyone, ...
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6 votes
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Why does the action have to be hermitian?

The classical action (of particles or fields) has to be real, because this means a real classical Lagrangian. This is needed because (canonical) momenta are obtained (for instance for a particle) from ...
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  • 17k
6 votes

What is meant by open-string tachyon scattering amplitude?

Veneziano amplitude is a 4 tachyon amplitude in bosonic open string theory. Two tachyons are ingoing and two are outgoing. From the stringy point of view, tachyons are present in both closed and open ...
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  • 4,390
6 votes

What actually means to compute things at tree level?

In principle in QFT you want to calculate the interaction picture unitary time evolution operator. Then you can use the operator to evolve quantum states just like you would in normal quantum ...
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  • 1,018
6 votes

How does a perturbation theory make sense in quantum field theory?

A sketch of the philosophy: Step one: Introduce a regulator. Everything is finite. Step two: Set up the perturbative expansion. This is a purely algebraic step, where there is no notion of "this is ...
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6 votes

In Feynman functional integrals why do we integrate the action over all time?

Short answer (assuming proper time-ordering of $x$ and $y$): $$ \langle \Omega| \phi(x) \phi(y) | \Omega\rangle = \frac{ \langle 0 | U(-\infty, x^0) \phi_I(x) U(x^0, y^0) \phi_I(y)...
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  • 1,332
6 votes
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Optical theorem in QFT

You might be assuming the matrix element $T_{ii}$ to be real. If so, then $$ \lvert S_{ii} \rvert^2 = 1 + \lvert T_{ii} \rvert^2 > 1 $$ Without such an assumption, $$ \begin{align*} \lvert S_{ii} ...
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  • 535
6 votes
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Significance of LSZ reduction formula

The premise of the question is that you either use LSZ formula or "directly" compute $\langle f | Te^{-i\int d^4xH_I} |i\rangle$. Implying that these are two unrelated approaches. While the LSZ ...
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6 votes
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Crossing Symmetry for Particles with Spin

General concepts Crossing symmetry is basically the CPT theorem applied in the context of the LSZ formula, using microcausality to re-order the field operators. The role of the CPT theorem is to ...
6 votes
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Why amplitudes are rational functions?

After the OP explained in the comments what exactly they're looking for, I will attempt an answer. There are a few separate facts that need explanation: Tree amplitudes are rational functions of ...
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  • 686
6 votes

QFT: relation between Cutkosky's cutting rules and the optical theorem

The case is somewhat the opposite. The cutting rules are far more general and can be used to prove the optical theorem at each order in perturbation theory. Furthermore, the cutting rules mean that ...
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  • 7,947
5 votes
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How do we calculate the S-matrix using non-perturbative QFT?

In general, for computing the scattering cross-section it is sufficient to use the pole approximation of the corresponding Green function in Heisenberg representation (details can be found in Weinberg'...
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  • 8,549
5 votes
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What are the differences (if any) between the Dyson's series definition and the "in/out" definition of the $S$-matrix

The in and out states are defined as solutions to the Lippmann-Schwinger equation, with the appropriate boundary condition prescribed by the choice of the contour ($\pm \varepsilon$). $$ | \psi^{(\pm)}...
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  • 5,375
5 votes

Quantum Field Theory in position space instead of momentum space?

I may add that the expressions for propagators $G(x,x^{\prime})\propto \int \frac{\mathrm{d}^D k}{k^2+m^2} \mathrm{e}^{-\mathrm{i}k (x-x^{\prime})}$ are quite cumbersome in the position space, and ...
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5 votes
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Question regarding the solution of Schrödinger equation for finite potential well and quantum barrier

It is two different situations of the TISE$^1$: A bound state has $E<0$ and the wave function $$ \psi(x)~=~Ae^{-\kappa |x|} , \qquad \kappa~:=~\frac{\sqrt{-2mE}}{\hbar}~>~0, \tag{1}$$ decreases ...
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