What is S-matrix: Definition and 37 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


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


{\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. J

    A QFT S-matrix explanations are incomprehensible

    The first look at a scattering process is something like this: We define an initial state |\textrm{in}\rangle = \int dp_1dp_2 f_{\textrm{in,1}}(p_1) f_{\textrm{in,2}}(p_2) a_{p_1}^{\dagger} a_{p_2}^{\dagger} |0\rangle Here f_{\textrm{in,1}} and f_{\textrm{in,2}} are wavefunctions that define...
  2. UnreliableObserver

    A Asymptotic states in the Heisenberg and Schrödinger pictures

    In scattering theory, the quantity of interest is the amplitude for the system—initially prepared as a collection of (approximate) momentum eigenstates—to evolve into some other collection of momentum eigenstates. For example, for ##m\to n## scattering, the amplitude we're interested in is...
  3. gremory

    A S-Matrix in Quantum Field Theory

    Hello, i need help with the S-matrix. From what i understand, with the S-matrix i would be able to compute the scattering amplitude of some processes, is that correct? If so, how would i be able to do that if i have some field ##\phi(x,t)## in hands? Is that possible?
  4. tomdodd4598

    I Fermi's Golden Rule and the S-matrix

    Hey there, This question was asked elsewhere, but I wasn't really satisfied with the answer. When I learned about Fermi's golden rule, ##{ \Gamma }_{ if }=2\pi { \left| \left< { f }|{ \delta V }|{ i } \right> \right| }^{ 2 }\rho \left( { E }_{ f } \right)##, it was derived from first order...
  5. G

    A S-matrix expansion and Wick's theorem

    My question arises when we expand the S-matrix using Wick's theorem, there we need to compute time-ordered products, but is not the same to compute $$T\{\bar{\psi}(x_1)\gamma^\mu \psi(x_1) A_\mu(x_1)\}$$ or $$T\{:\bar{\psi}(x_1)\gamma^\mu \psi(x_1) A_\mu(x_1):\}$$ While the second one is simply...
  6. A

    A Is field theory needed in Regge-theory?

    I have been reading some papers from G.F. Chew and S. C. Frautschi and they do not even bother to introduce the concept of "Field" when they describe hadron interactions. My impression is that they do not need to because interactions seem to be described by single Regge-trajectories. However...
  7. J

    Radar Cross section Vs Reflection Coefficient

    I am working on HFSS and designing Antennas for LOW RCS , in all previous works they calculate the S11 parameter for the unit cell then make an array form this unit cell and calculate the RCS , my question is what is the relation between S11 and RCS and why we calculate S11 for the unit...
  8. L

    Computing the scattering amplitude from the S-matrix

    Homework Statement Consider two real scalar fields \phi,\psi with masses m and \mu respectively interacting via the Hamiltonian \mathcal{H}_{\mathrm{int}}(x)=\dfrac{\lambda}{4}\phi^2(x)\psi^2(x). Using the definition of the S-matrix and Wick's contraction find the O(\lambda) contribution to...
  9. F

    I Higher order terms in perturbation theory (QFT)

    I'm fairly new to QFT and I'm currently trying to understand perturbation theory on this context. As I understand it, when one does a perturbative expansion of the S-matrix and subsequently calculates the transition amplitude between two asymptotic states, each order in the perturbative...
  10. M

    A Tree-level unitarity constraints in Two-Higgs Doublet Model

    Hi, I'm looking at the unitarity constraints for the Two-Higgs Doublet Model and I'm trying to follow what they do in the attached article, which can also be found here: https://arxiv.org/pdf/hep-ph/0312374v1.pdf. However I do not know how to get the scattering matrices in eq. (7). They say...
  11. S

    A Calculation of S-matrix elements in quantum field theory

    Consider the following extract taken from page 60 of Matthew Schwartz's 'Introduction to Quantum Field Theory':We usually calculate ##S##-matrix elements perturbatively. In a free theory, where there are no interactions, the ##S##-matrix is simply the identity matrix ##\mathbb{1}##. We can...
  12. L

    Why are disconnected diagrams dropped from S-Matrix

    I am talking about perturbative quantum field theory. When calculating elements of correlation functions(which then I use to calculate S-matrix elements) one always comes up with connected and disconnected diagrams. These disconnected diagrams are usually dropped from the calculation and I...
  13. FrancescoS

    S-matrix element for forward scattering and amputed green fu

    I'm studying dispersion relations applied as alternative method to perturbation theory from Weinberg's book (Vol.1) Let's consider the forward scattering in the lab frame of a massless boson of any spin on an arbitrary target ##\alpha## of mass ##m_\alpha>0## and ##\vec{p}_\alpha = 0##...
  14. T

    What is the significance of S-Matrix in Quantum Field Theory?

    Can someone please explain Heisenberg's S-Matrix theory in simple terms. Minimals maths please. Please be as expansive as possible.
  15. T

    How did S-Matrix theory affect the birth of String Theory?

    Can somebody please explain to me how the S-Matrix of Heisenberg and others helped influence the birth of String Theory. I keep hearing people say that String Theory came as a result of S-Matrix but I don't see the connection. Please keep maths to a minimal. Thanks.
  16. H

    Driven Quantum harmonic oscillator by way of the S-Matrix

    Homework Statement We have the lagragian L = \frac{m}{2} \dot{x}^2 - \frac{m \omega x^2}{2} + f(t) x(t) where f(t) = f_0 for 0 \le t \le T 0 otherwise. The only diagram that survives in the s -matrix expansion when calculating <0|S|0> is D = \int dt dt' f(t)f(t') <0|T x(t)x(t')|0>...
  17. G

    S-matrix structure of the lightest particle

    If you have a particle that is the lightest particle, then it cannot decay into other particles. As a consequence of this, its T-matrix amplitude on-shell should be zero, since the S-matrix is S=1+iT, and the amplitude for the particle to be found with the same quantum numbers and momentum is 1...
  18. G

    S-Matrix definition and terminology

    A lot of textbooks give the definition of an S-matrix element as: \langle \beta_{out}| \alpha_{in}\rangle = \langle \beta_{in}| S| \alpha_{in} \rangle=\langle \beta_{out}| S| \alpha_{out} \rangle=S_{\beta \alpha} and that S|\alpha_{out} \rangle =|\alpha_{in} \rangle I don't...
  19. G

    Does time reversal symmetry imply hermicity of s-matrix?

    If the probability for a state α prepared initially to be in a state β at a later time is given by: S_{\beta \alpha} S_{\beta \alpha}^* and for a state β prepared intitially to become a state α is: S_{ \alpha \beta} S_{ \alpha \beta}^* then in order for the two to be equal (by...
  20. D

    Understanding S-matrix elements in QFT

    Hi, I was wondering if I could test my understanding on the S-matrix and its role in evolving initial states of systems to final states (after some scattering process has occurred). Would it be correct to say the following: Given a system in an initial state \vert i \rangle, the final...
  21. alemsalem

    Exact S-matrix for potential scattering

    are there examples in non-relativistic scattering off a central potential where the S-matrix can be calculated exactly? Thanks!
  22. N

    Why can we apply the symmetries of S-Matrix to part of Feynman diagram

    How can we demonstrate that the symmetries of S-Matrix can be applyed to parts of Feynman diagrams?The S-Matrix is the sum of infinite diagrams,why we know each or part of each diagram has the same symmetries as the symmetries of S-Matrix?
  23. K

    What is the Role of S-Matrix in String Theory?

    Hi there! S-matrix is Path Integral with Vertex Operators inserted. I know how to compute Shapiro-Virasoro amplitude. So I don't have problems with calculations but with understanding. In this calculations formalism of 2-dimensional CFT is used. But there is no S-matrix in CFT, only...
  24. G

    Probability Greater than 1 in λφ^4 Theory

    Take a \lambda \phi^4 theory. To first order in λ, the 2x2 scattering amplitude is: iM=-iλ So the amplitude <f|S|i> is then <f|(1+iM)|i>=<f|i>+iM<f|i>. Letting f=i, the probability is greater than 1! It is equal to the norm |1+iM| which is sqrt[1^2+λ^2]. How is it that two particles in the...
  25. G

    Interaction picture and s-matrix

    Consider the S-matrix: <f|U(t,-t)|i> When going into the interaction picture, this becomes: <f_I|U_I(t,-t)|i_I> where the propagator is the interaction picture propagator, and the states are interaction states. Can you say that: <f_I|U_I(t,-t)|i_I>= <f|U_I(t,-t)|i> ? It...
  26. C

    Clarification of statement on how to evaluate S-matrix elements.

    Hi. I'm trying to understand a statement from the book Field Quantization on the evaluation of the S-matrix elements for a certain initial and final state in QED. The author states, if one is evalutating a matrix element on the form \langle k_1' \lambda_1', \ldots, \bar p_1' \bar s_1'...
  27. E

    Is the S-matrix in Weinberg's book always unitary?

    Hi, I've read a lot of posts about how Weinberg describes the S-matrix invariance in his book, but none of theme answered my questions. At page 116, sec 3.3 - "Lorentz Invariance" of Quantum theory of fields vol.1 Weinberg says: "Since the operator U(\Lambda, a) is unitary we may write...
  28. G

    Is the S-matrix at high energies affected by time dilation?

    The S-matrix can be written as the sum of the Feynman diagrams, divided by a factor of 1/sqrt[E] for each particle, where E is the particle's energy. Does this mean at large energies, the probability amplitude to scatter is unlikely? But how can such a statement be made when no physics is...
  29. D

    S-matrix in Weinberg's book

    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_{\beta\alpha}=(\Psi_\beta^-,\Psi_\alpha^+) When talking about the Lorentz invariance of S-matrix, the Lorentz transformation induced unitary operator U(\Lambda,a)...
  30. K

    Confusion (5) from Weinberg's QFT.(invariance of S-matrix)

    At the beginning of section 3.3, he says lorentz invariance of S-matrix means the same unitary operator acts on both in and out states. I feel a bit blur about this since he doesn't give any concrete example. Say the eletron-positron annihilation process, we have 1 electron, 1 positron, 0 photon...
  31. N

    Why do they only consider S-matrix in QFT?

    Please teach me this: Why do they only consider S-matrix(or Green functions) in Quantum Field Theory?How about other physics observations as considering in normal Quantum Mechanics,e.g angular momentum...of particles?If we consider the other physical observations,how could we represent them in...
  32. N

    Why is unitarity important in relativistic scattering processes?

    Please teach me this: Why scattering matrix in relativistic collision still must be unitary?Because in relativistic regime,the probability is not conservable. Thank you very much in advanced.
  33. M

    S-matrix formula from Lippmann-schwinger eqn

    Hi all, In chapter 3.2 of Weinberg's QFT text he asserts that one can derive the expression S_{\beta\alpha}=\delta(\beta-\alpha)-2\pi i\delta(E_\alpha -E_\beta)T_{\beta\alpha} for the S-matrix in terms of the matrix elements T^{\pm}_{\beta \alpha}=(\Phi_\beta , V\Psi^{\pm}_\alpha), where I'm...
  34. N

    S-Matrix in non-relativistic limit

    I have a problem in deriving the formula : \left \langle p^\prime | iT | p \right\rangle = -2\pi i\widetilde{ V}(q)\delta(E_{p^\prime}-E_{p}) which is in Peskin's QFT book. How to derive it? Is there anything to do with \psi =...
  35. I

    Difference between s-matrix and interactive field propagator?

    In an interactive field theory we can compute the amplitude of a particle propagating from y to x by evaluating perturbatively expressions of the form <GS|o(x)o(y)|GS> where GS stands for ground state and o are the field operators. This can be extended to higher number of operators for more...
  36. A

    Cluster decomposition of the S-matrix

    Hi, I am reading through Weinberg's "Quantum Theory of Fields" (vol. 1) and I am somewhat confused about the signs in the cluster decomposition of the S-matrix. Specifically, referring to eq. 4.3.2, let's say the term coming from the partition \alpha \to \alpha_1\alpha_2,\beta \to...
  37. T

    Does anyone know any books, websites, lecture notes etc on S-matrix?

    Does anyone know any books, websites, lecture notes etc that discuss on the poles of the S matrix being physically, the bound states?