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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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172 views

Is the converse of Weinberg's statement on the cluster decomposition principle true?

In Weinberg's "The Quantum Theory of Fields, Vol. 1", Section 4.4, page 182, the author says: We now ask, what sort of Hamiltonian will yield an $S$-matrix that satisfies the cluster decomposition ...
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1answer
99 views

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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70 views

Self-energy that does not obey sum rule

Analytically, I calculated a self-energy $\Sigma(\omega)$, for which I verified that 1) $\text{Im}\big[\Sigma(\omega)\big] \leq 0$ for all $\omega$ and specifically $\text{Im}\big[\Sigma(0)\big] = 0$,...
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159 views

What is the physical interpretation of the automorphism on bounded operators induced by an S matrix?

In a QFT, the S-matrix $S$ is a unitary operator, that fixes the vacuum and commutes with the unitary operators implementing the action of the Poincare group on an appropriate Hilbert space $H$. ...
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318 views

Weinberg's S-matrix and split into free and interacting Hamiltonian

TL;DR: How can states of an interacting QFT asymptotically follow the trajectories governed by the free Hamiltonian, when, say, the free and interacting groundstates are different, and the states look ...
4
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1answer
91 views

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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49 views

Is it true that the lowest order non-zero contribution for an S matrix element has always to be convergent?

I was talking with a colleague he told me that if there is no non-zero tree level diagram contributing, then the one loop contribution cannot be divergent. I replied to this saying that this indeed is ...
4
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415 views

Polology in Functional Integration

Completeness of Hilbert space (on-shell states) is a very powerful concept in canonical quantization, for example, to study the nonperturbative characteristics of the S-matrix, like polology (pole and ...
4
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284 views

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[\...
4
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154 views

How to check the unitarity of the theory by having field equation?

Let's have some field equation of some field corresponding to particles with mass $m$ and spin $s$. How to check the unitarity of the theory? May I do it without getting $S$-matrix? May the scalar ...
3
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85 views

Scalar electrodynamics “seagull” vertex factor

By expanding the covariant derivative of the Scalar QED lagrangian one gets the following term, sometimes called "seagull" vertex. $$\mathcal{L}_{seagull} = -q^2A_\mu A ^\mu \phi^\dagger \phi$$ Most ...
3
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1answer
72 views

Scattering matrix symmetries and standard model

I am not able to get around the following question (if it make sense): Suppose I can derive the scattering matrix S for any particle scattering process. Suppose that the standard model is actually ...
3
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68 views

How do the renormalization factors disappear from the computation recipe of the S-matrix in Peskin & Schroeder (p. 229 (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) $\phi_r= (\sqrt{Z})^{-1} \phi$ in eq. (...
3
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100 views

Use of Cutkosky rule, the Optical Theorem and Regge trajectories in pp scattering total cross-section calculation

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 ...
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88 views

Number theory to estimate lower bound of S-Matrix?

I recently worked on the following idea: https://math.stackexchange.com/questions/2325724/eigenvalue-of-an-euler-product-type-operator Edit: And realised it was more probable to be used in Quantum ...
3
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99 views

How can LSZ formalism deal with “glancing blows”?

My recent answer to the question Scattering, Perturbation and asymptotic states in LSZ reduction formula got me thinking again about wave packets and the LSZ reduction formula. In my answer, I claim ...
3
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153 views

LSZ Reduction Formula and Wavepackets in Peskin

In Section 7.2 of Peskin and Schroeder, the LSZ Reduction formula is discussed. I can follow the discussion leading up to the equation after (7.41) [at the bottom of Pg. 225]: $$\sum_\lambda \int\frac{...
3
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1answer
184 views

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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62 views

Møller Opeartors and dressing relations: what picture?

Møller operators can be defined as (Urban, 2013;pg70): \[ \newcommand{\ket}[1]{\left|#1\right>} \newcommand{\braket}[2]{\left<#1|#2\right>} \Omega_{\pm}=\underset{t\rightarrow \mp 0}{\...
3
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1answer
207 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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164 views

What picture is the $S$-matrix defined in?

I am looking into the definition of the $S$-matrix, and have found two different cases. Firstly I have seen it derived that (see here, link to Google books p110): $$ S=U_S(\infty,-\infty)$$ But more ...
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200 views

Unitarity of the S-matrix and Feynman Diagrams

There are several questions on the unitarity of the S matrix, but unfortunately non of them answers directly the following question. The S matrix is unitary and that can be proven by the fact that ...
3
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333 views

How exactly analyticity of S-matrix comes from causality principle?

Recently I've read that analyticity of S-matrix ($S(k)$, where $k$ corresponds to momentum, may be analytically extended into complex values of momentum) comes from causality principle. How to prove ...
3
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252 views

Perturbation theory : quadratic external field

I'm trying to derive the explicit form of S-matrix of an interaction Hamiltonian $$H' = \frac{1}{2} \lambda \left[ \int d^3 x \rho({\vec x}) \phi({\vec x}, t)\right]^2\tag{1}$$ Even though the ...
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236 views

Møller scattering: twisted?

I am studying the Møller scattering, but I don't know how to get the twisted diagram from the S-matrix. Has anybody a good explanation?
3
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88 views

Do unoriented strings possess asymptotic states?

In QFT based particle theory, SU(N)-colored particles are not really present as asymptotic states, then raising some problems to build a S-matrix or other more axiomatic approaches to the theory. ...
3
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583 views

Validity of Cutkosky cutting rules for fermions

It is rather obvious for me that the generalized optical theorem (see e.g. Peskin&Schroeder) must hold for S-matrix elements for fermions as it is directly related to the unitarity of the S-matrix....
2
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1answer
42 views

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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33 views

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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67 views

Does somebody know where to find original paper of Lehmann, Symanzik and Zimmerman translated in English?

does anybody know where to find original paper about LSZ reduction translated in english? unfortunantely, Ive found only original German article. H. Lehmann, K. Symanzik, and W. Zimmerman, "Zur ...
2
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1answer
100 views

LSZ reduction derivation Srednicki

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....
2
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73 views

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 ...
2
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1answer
133 views

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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63 views

Contribution from $u$-channel and $t$-channel processes in OPE analysis for deep inelastic scattering

In Ch.18 of the textbook An Introduction to Quantum Field Theory by Peskin and Schroeder, on P.633 the moment sum rules for the deep inelastic form factors are discussed $$\int_0^1 dx x^{n-1}f_f^+(x,...
2
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1answer
233 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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214 views

Kallen-Lehmann representation derivation

I'm trying to understand the derivation of the Kallen-Lehmann representation given in Peskin & Schroeder (pages 211-214). I would really appreciate if anyone on here could answer a few questions I ...
2
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1answer
72 views

Clarifications on the assumptions made for QFT interactions

I am reading about scattering and S-matrix in the context of quantum field theory and although I understand the math and the physical interpretation of the final results, I am confused about some ...
2
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73 views

What is $\gamma$$\rho$ mixing?

In a previous question, I asked about the apparent universality behaviour in hadron elastic scattering. I was particularly shocked to see that even $\gamma p$ showed that universal behaviour with ...
2
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1answer
61 views

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 ...
2
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163 views

LSZ fermions (Srednicki's book)

In Srednicki's book it is stated that the LSZ formula for fermions holds only if the interacting field $\psi(x)$ is normalized to satisfy $$\langle p,s|\psi(x)|0\rangle = v(p,s)\ e^{ipx}$$ a condition ...
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128 views

Physical meaning of $\langle 0|S(+\infty,-\infty)|0 \rangle$

when I am reading text book Introduction to Many body physics Piers Coleman 2nd editor, Equation (5.26). I got confused about the meaning of $\langle0|S(+\infty,-\infty)|0\rangle$. first problem $|...
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238 views

Classical analogue of the theorem of equivalence of the S-matrix

In quantum field theory there is a statement called the equivalence theorem of the S-matrix. S-matrix is invariant under reparametrization of the field. Is there in classical mechanics, the analogous ...
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1answer
220 views

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 $|\Psi_p^{-}\rangle$ ...
2
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396 views

Calculation of $b \to s~ l^+ l^-$ penguin diagram

I'd like to calculate the matrix element amplitude for $b \to s~ l^+ l^-$ penguin diagram mediated by Z boson or the photon , like : These calculations are made of course from many time ago, so if ...
2
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1answer
271 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 ...
2
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499 views

Foundation of path Integral formulation of QFT, derivation and meaning of LSZ formulas

I'm currently studying path integral in quantum field theory. I am comfortable with path integrals, and also path integral formulation of QM, but I was asking if there is a self consistent coherent ...
2
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195 views

S-matrix element for forward scattering and amputed green function

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 ...
2
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0answers
28 views

Main differences between elastodynamic and light scattering when using S-matrix to find bound states

What are the main differences (top 5 if question is too broad), for using the S-matrix to find bound states, between elastodynamic and light scattering? (if it facilitates a higher quality question/...
2
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0answers
285 views

Probability and the propagator

Due to the Wiki article, "...In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to ...
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79 views

One question about renormalization

The idea of renormalization of "naked" perturbation theory is in principal possibility of addition counterterms which reduce infinity when calculating matrix elements. But I have met such concepts as ...