What is S matrix: Definition and 12 Discussions

In physics, the S-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).
More formally, in the context of QFT, the S-matrix is defined as the unitary matrix connecting sets of asymptotically free particle states (the in-states and the out-states) in the Hilbert space of physical states. A multi-particle state is said to be free (non-interacting) if it transforms under Lorentz transformations as a tensor product, or direct product in physics parlance, of one-particle states as prescribed by equation (1) below. Asymptotically free then means that the state has this appearance in either the distant past or the distant future.
While the S-matrix may be defined for any background (spacetime) that is asymptotically solvable and has no event horizons, it has a simple form in the case of the Minkowski space. In this special case, the Hilbert space is a space of irreducible unitary representations of the inhomogeneous Lorentz group (the Poincaré group); the S-matrix is the evolution operator between

t
=

{\displaystyle t=-\infty }
(the distant past), and

t
=
+

{\displaystyle t=+\infty }
(the distant future). It is defined only in the limit of zero energy density (or infinite particle separation distance).
It can be shown that if a quantum field theory in Minkowski space has a mass gap, the state in the asymptotic past and in the asymptotic future are both described by Fock spaces.

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1. A Simple S matrix example in Coleman's lectures on QFT

In Coleman's QFT lectures, I'm confused by equation 7.57. To give the background, Coleman is trying to calculate the scattering matrix (S matrix) for a situation in which the Hamiltonian is given by $$H=H_{0}+f\left(t,T,\Delta\right)H_{I}\left(t\right)$$ where ##H_{0}## is the free Hamiltonian...
2. A Cluster Decomposition.Vanishing of the connected part of the S matrix.

Im following Weinberg's QFT volume I and I am tying to show that the following equation vanishes at large spatial distance of the possible particle clusters (pg 181 eq 4.3.8): S_{x_1'x_2'... , x_1 x_2}^C = \int d^3p_1' d^3p_2'...d^3p_1d^3p_2...S_{p_1'p_2'... , p_1 p_2}^C \times e^{i p_1' ...
3. A Quantum amplitude for a particle falling into a black hole

Here we consider a black hole formed by gravitational collapse classically. We also consider a scalar massless Klein-Gordon field propagating on this background. To quantize the field we expand it in appropriate modes. The three sets of modes required are: The incoming modes, appropriate for...
4. I Further S matrix clarifications

Hello! I attached a SS of the part of my book that I am confused about. So there they write the initial and final states in term of creation and annihilation operators, acting on the (not free) vacuum i.e. ##|\Omega>##. So first thing, the value of the creation (annihilation) operators at...
5. I Why do we assume particles are free at infinity in the S matrix theory?

Hello! I am reading about the S matrix, and I see that one of the assumption that the derivations are based on is the fact that interacting particles are free at ##t=\pm \infty## and I am not sure I understand why. One of the given examples is the ##\phi^4## theory which contains an interaction...
6. A Use of the Optical Theorem and Regge trajectories

Cutkosky rule states that: $$2Im \big(A_{ab}\big)=(2\pi)^4\sum_c \delta\Big(\sum_c p^{\mu}_{c}-\sum_a p^{\mu}_{a}\Big)|A_{cb}|^2\hspace{0.5cm} (1)$$ putting ##a=b=p## in Cutkosky rule we deduce the Optical Theorem for ##pp## scattering: 2Im \big(A_{pp}\big)=(2\pi)^4\sum_c \delta\Big(\sum_c...
7. A Pp and pBARp scattering amplitudes

Is A_pp(s,t)=A_pBARp(t,s) true based on crossing symmetry? Consider pp and pBARp elastic colissions (p + p -> p + p and p + BAR(p) -> p + BAR(p)). The scattering amplitudes are related by crossing in the following way: 1) A_pp(s,t)=A_pBARp(u,t) \simeq A_pBARp(-s-t,t) (energy large compared to...
8. A Dyson's Formula from Tong's lecture notes

I am studying quantum field theory from [David Tong's lecture notes][1] and I am stuck at a particular place. In Page 52., under the heading *3.1.1 Dyson's Formula*, Tong introduces an unitary operator U(t, t_0) = T \exp(-i\int_{t_0}^{t}H_I(t') dt') He then introduces the usual definition of...
9. S matrix Unitarity Proof, pg 298 Peskin Schroeder

I have a question regarding a derivation in Peskin and Schroeder's QFT book. On page 298, he is discussing a method for defining a gauge invariant S matrix. He does this by defining projection operators ##P_0## that project general particle states into gauge invariant states, and then defining...
10. S matrix and decaying particles

Hi All, The S-matrix is defined as the inner product of the in- and out-states, as in Eq. (3.2.1) in Weinberg's QFT vol 1: S_{βα}=(Ψ−β,Ψ+α) \Psi_{±α} are the eigenstates of the full Hamiltonian with a non-zero interaction term. Can \alpha describes a neutron ? Since it is not stable, it is...
11. Some questions about perturbative expansion of S matrix

Hi, Recently I was confronted with some difficulties in understanding the perturbative expansion of S matrix . The conventional treatment is expansing it in the interaction picture,which have to first transform Lagrangian to Hamiltonian and then replace the original field operator by...
12. Some questions on the Dyson expansion of the S matrix

I have some questions regarding: S = \sum_{n=0}^\infty\ S^n = \sum_{n=0}^\infty \frac{i^n}{n!} \idotsint \ {d^4x_1}\ {d^4x_2} \dots \ d^4x_n \ T (H_I(x_1) \ H_I(x_2) \dots \ H_I(x_n) ) 1) What is n? How do you pick n given some interaction? ( I think it might be the order in...