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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244 views

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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313 views

Optical theorem applied to forward scattering of a single particle

I'm slightly confused. Write the S-matrix as $$ S = 1 + i T $$ Unitarity implies $$ T - T^\dagger = i T^\dagger T $$ In scattering from $|i\rangle$ to $|f\rangle$, $$ T_{f,i} - T^\dagger_{f,i} = i \...
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Optical theorem in $\phi^4$: which poles contribute to discontinuity in Feynman amplitude?

Section 7.3 ("The Optical Theorem") in Peskin and Schroeder's QFT text contains a leading order verification of the optical theorem in $\phi^4$ theory by calculating the (discontinuity across the ...
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991 views

Definition of the S-matrix

When I think about scattering process, I reach to slightly another definition to the S-matrix. because I understand my reasoning I hope someone could refine it to a correct one so that I can have a ...
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102 views

The LSZ formula in Peskin and Schroeder

I'm working on the Eq.(7.57) in Peskin(page 236). So I try to verify it with LSZ formula. According to Eq (7.42) So $\mathcal{M}(p \rightarrow p)=-Z M^{2}\left(p^{2}\right)$ In this I have two ...
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Questions from Srednicki's Introduction to Interacting Field Theory using the LSZ Formula

I have been reading through the chapter on the LSZ Reduction Formula from Srednicki's Quantum Field Theory, and I have a few questions about which I'm sort of confused. The questions are referenced ...
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142 views

Clarification of Path Integral formulation

I am reading from Schwarz book on QFT the Path Integral chapter and I am confused about something. I attached a SS of that part. So we have $$<\Phi_{j+1}|e^{-i\delta H(t_j)}|\Phi_{j}>=N \exp(i\...
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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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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

I noticed that in Srednicki's derivation of the LSZ-formula the expression (chapter 5) for the creation (and also later for the annihilation) operator by the field operator: $$a^\dagger(\mathbf{k}) =...
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How do the renormalization factors disappear from the computation recipe of the S-matrix in Peskin & Schroeder (p. 229 eq. (7.45) & p.324)?

In the following I limit my considerations to 4-point diagrams. After the introduction of renormalized field operator (in renormalized perturbation theory) $$\phi_r= (\sqrt{Z})^{-1} \phi\tag{10.15}...
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301 views

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 \begin{equation} \...
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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{...
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237 views

Transition amplitude for QED+QFD+QCD interactions

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 ...
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How does the interacting vacuum $|\Omega \rangle$ enter the theory?

To calculate scattering amplitudes, we consider $$ A(i\to f) = \langle{f | \hat S|i} \rangle = \langle{f |\mathrm{e}^{ -\frac{i}{\hbar} \int_{-\infty}^{\infty} dt' H_{\mathrm{i}}(t')} |i}\rangle$$ $...
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How to prove the equivalence of two different definitions of $S$-operator?

I read there are two definitions about $S$-operator: The first one (e.g (8.49) in Greiner's Field Quantization) is: $$S_{fi}\equiv \langle \Psi_p^{-}| \Psi_k^{+}\rangle$$ where $|\Psi_p^{-}\rangle$ ...
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299 views

Scattering, 4 point correlator, #distinct Feynman diagrams

In order to compute the scattering probability that two particles of type 1 (associated to $\phi_1(x)$) which come from the far past with the momenta $p_1$ and $p_2$, to scatter and evolve into two ...
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S-Matrix Generating Functional (Problem 4.1 in Weinberg)

I'm currently working through Weinberg's QFT book, but I'm somewhat stuck at problem 4.1, which states: Define generating functionals for the S-matrix and its connected part: \begin{equation} F[\...
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What should be the scattering matrix to transparent device?

The scattering matrix for a 2 port diode is given as $$S=\begin{pmatrix} 0 & 0 \\ 1 & 0 \\ \end{pmatrix}$$ Suppose I want to make this device transparent, that any mode just passes through ...
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Explicit form of S-matrix on the line

Consider the Hamiltonian $H$ on functions on the line with \begin{eqnarray} H=H_0+V,\\ H_0=-\frac{1}{2m}\frac{d^2}{dx^2} \end{eqnarray} where $V$ is a potential vanishing outside of a bounded interval....
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Is there any relationship between S-matrix elements and the path integral?

Reading Peskin&Schroeder I've made the following curious observation: Comparing S-matrix elements to the definition of the path-integral they look remarkably similar: $$_{out}\langle \mathbf{p}...
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Proton-proton and proton-antiproton elastic scattering symmetry

Is $A_{pp}(s,t)=A_{p\bar p}(t,s)$ true based on crossing symmetry? Consider $pp$ and $p\bar p$ elastic colissions ($p + p \rightarrow p + p$ and $p + \bar p \rightarrow p + \bar p$). The scattering ...
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What justifies the use of asymptotic momentum state?

The LSZ scattering approach starts with initial and final asymptotic momentum states. But we know that $\langle k' | k\rangle = \delta^3(k'-k)$, which means that it is not a properly normalizable ...
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Spin-$J$ Amplitude $A_J(s,t) = - \frac{g^2(-s)^J}{t-M^2}$?

In GSW equation (1.1.2) they define the scattering amplitude for a spin $J$ particle at high energies as $$A_J(s,t) = - \frac{g^2(-s)^J}{t-M^2}$$ mentioning it is an asymptotic approximation to a ...
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Scattering, Perturbation and asymptotic states in LSZ reduction formula

I was following Schwarz's book on quantum field theory. There he defines the asymptotic momentum eigenstates $|i\rangle\equiv |k_1 k_2\rangle$ and $|f\rangle\equiv |k_3 k_4\rangle$ in the S-matrix ...
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Renormalization condition: why must be the residue of the propagator be 1

In on-shell (OS) scheme, one of the renormalization conditions is that the propagator, say, a scalar theory $$\frac{1}{p^2+m^2-\Sigma(p^2)-i\epsilon}$$ must have a unit residue at the pole of ...
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Explanation of why poles in the S-matrix corresponding to particle production/bound states?

I've frequently heard the statement that the only singularities of the S-matrix in QFT correspond to things like the existence of bound states and the possibility of multi-particle production. I'm ...
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Deriving transition amplitude with $S$-matrix

Here is the part that is bothering me: Yeah, already here? So, my question: In the first line we have int picture states at time zero and in the second line we have limit of time evolv operator with ...
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Doubt in Weinberg's book on QFT

In chapter 3 of his book on QFT (volume 1), while discussing the symmetries of the S-matrix, Weinberg makes the following statement For any proper orthochronous Lorentz transformation $x\rightarrow ...
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$S$-matrix and in and out states

So, I have a short one. When observing scattering, we say that the amplitude for transition from one interacting state to some other interacting state same as this amplitude for free hamiltonian ...
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The kinematic region for the operator product expansion

In Ch.18 of the textbook An Introduction to Quantum Field Theory by Peskin and Schroeder, on P.613 the operator product expansion (OPE) is introduced $$\mathcal{O}_1(x)\mathcal{O}_2(0)\to \sum_n C_{...
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Can I add a classical potential to S-matrix

I'm a junior learner of QFT, and I wonder if I can add a classical approximated potential (like Coulomb potential) to a total interaction \begin{equation}V=V_{\mathrm{Coulomb}}+V_{\mathrm{internal}} \...
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Significance of LSZ reduction formula

LSZ reduction formula relates the S-matrix element and the time-ordered correlation function, in a complicated equation. However, since $$S=T e^{-i\int d^4x H_I}$$ where $H_I$ is the interaction ...
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How is “the collapse postulate is also present in QFT, only hidden inside the LSZ formula?”

Background So I am reading the following here (Blog: Not Even Wrong, Blog post: Not So Spooky Action at a Distance, Commenter: vmarko) "The collapse postulate is also present in QFT, only hidden ...
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How to properly make sense of the $\mathcal{S}$-matrix as a correlator on a sphere?

In the book "Lectures on the Infrared Structure of Gravity and Gauge Theories" by Andrew Strominger, the author discusses in Chapter 3 the idea of "The $\mathcal{S}$-matrix as a Celestial Correlator". ...
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Contradiction between aymptotically free particles in QFT and unlocalization

When studying different interactions in any QFT, one always assumes that the IN and OUT states are asymptotically free particles with definite momenta. For example, one assumes that an electron and a ...
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(Coleman's lecture note) scattering in QFT

I am currently reading Coleman's lecture note on QFT.(https://arxiv.org/abs/1110.5013) I have several questions regarding the scattering theory. Let $\phi$ be a real scalar field, and consider the ...

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