# Questions tagged [s-matrix-theory]

The S-matrix (scattering matrix) relates the initial state and the final state of a physical system undergoing a scattering process in quantum mechanics and quantum field theory. It is the unitary matrix connecting asymptotic particle states in the Hilbert space of physical states (scattering channels).

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### Ghosts use in the calculus of the scattering matrix

When you perform the calculus of $S$ matrix in, for instance, QCD maybe you need to add ghosts in external legs or internal loops. E.g.: $q\bar{q} \rightarrow gg$ needs 2 ghosts diagrams with ghosts ...
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### Deriving the homogeneous scaling of scattering amplitudes

I would like to derive the homogeneous scaling of scattering amplitudes $M(t\lambda,t^{-1}\tilde{\lambda})=t^{-2h}M(\lambda,\tilde{\lambda})$ in the following very general and first principled manner, ...
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### Calculating $S$-matrix in string theory

To calculate string $S$-matrix, we mainly use Faddeev-Popov gauge fixing method, as in chapter 6 of Polchinsky's book 《string theory》. But in section 6.2, 'tree level amplitude', I didn't find ...
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### I really wonder about the time derivative of creation and annihilation operators in the derivation of LSZ

On p. 71 below eq. (6.12) in Schwartz book, they assume that $$\lim_{t \to \pm\infty}\partial_0 a_p(t)=0.\tag{1}$$ But I thought that this is just an assumption. So we have to construct the ...
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### How to replace $T$-product with retarded commutator in LSZ formula?

I am reading Itzykson and Zuber's Quantum Field Theory book, and am unable to understand a step that is made on page 246: Here, they consider the elastic scattering of particle $A$ off particle $B$: ...
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### LSZ reduction derivation Srednicki [duplicate]

In the derivation of the LSZ reduction formula equation (5.21) Srednicki claims that in case of an interaction term in the Lagrangian density $a^{\dagger}(\textbf{k})$ will no longer be time dependent....
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### Why bound states in QFT have higher mass than single particle states?

In standard textbooks in QFT while discussing e.g. the Kallen-Lehmann formula (see e.g. Section 7.1 in the Peskin-Schroeder book) it is always assumed that bound states of two or more particles have ...
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### Interactions other than yukawa interaction which results in pair annihilation

Can boson annihilates to an electron positron pair can happen in all type of interactions which contains some number of dirac fields coupled with some number of bosonic fields? I asked this question ...
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### An example of Wick's theorem from David Tong's lecture notes

I am studying David Tong's lecture notes, But I got stuck at this point. As an example of Wick's theorem,he gives nucleon scattering. I am unable to get the final answer, I tried a lot by expanding ...
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### How to do the classification of Yangian invariant?

From Nima's paper:Scattering Amplitudes and the Positive Grassmannian arxiv:1212.5605 page91, we can see that there is a complete classification for $k=2$ Yangian inariants. But I have two questions ...
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### Independence of $S$-matrix in QED of the gauge choice

The Feynman rules in QED often use different expressions for the free photon propagator (e.g. Feynman gauge, Landau gauge, and others). Is there a textbook on the subject which explicitly checks ...
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### Why do we need to embed particles into fields?

In QFT we have the so-called embeding of particles into fields. This is discussed at full generality in Weinberg's book, chapter 5. In summary what one does is: From Wigner's classification, for each ...
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### Action of Moller operators on stable one particle states in QFT

In Weinberg's book "The QFT", vol. I, it is claimed that for theories with stable single particle states the $S$-matrix maps each such state to itself (see Section 4.3, p. 179). I am wondering if the ...
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### Differential cross-section derivation from S-matrix

I am trying to derive the usual expression for the differential scattering cross section: $\frac{d\sigma}{d\Omega} = \frac{q_f}{q_i}|f(\vec q_f,\vec q_i)|^2.$ I am familiar with the derivation which ...
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### Off-shell QFT from on-shell amplitudes

Suppose that you're given a non-perturbative $S$-matrix that corresponds to some Wightmanian QFT. By this I mean that you're given a Hilbert space and a unitary operator $S$ that acts on the Hilbert ...
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### Why would the in/out'' states asymptotically approach the free Hamiltonian eigenstates?

The ''in/out'' states of the S-matrix in QFT are defined such that at late times they are approach superpositions of direct products of eigenstates of the $\textit{free}$ Hamiltonian \...
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### Why isn't it a problem that the first term in the perturbative scattering series yields infinity (or one)?

In QFT we usually want to calculate objects of the form $\langle f|\hat S|i\rangle$ which yields the probability amplitude for the process $i \to f$. We can expand the scattering operator $\hat S$ in ...
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### S-matrix elements for Nucleon-Pion decay

I want to compute nucleon-pion decay rates. I am a bit confused how I can compute the S-matrix. Let's say we have a Nucleon Pion scattering and I want to compute their corresponding S matrix: \begin{...
As I understood, Feynman diagrams are nothing more than pictures for the transition amplitudes (up to some orders). For this we introduce a interaction vacuum state $|\Omega\rangle$ then we are able ...