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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Why we need Hermitian conjugate amplitude rather than complex conjugate amplitude for calculating cross section?

I am trying to understand some concepts related to scattering in QED, so I would phrase my question in similar context. After calculating the scattering amplitude $\mathcal{M}$ for a process, we take ...
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Confusion on wave packet and creation operator in Mark Srednicki's book

In Mark Srednicki's QFT book, section $5$, he mentions following things: $a^{\dagger}(k)$ creats a particle with momentum $k$ and is given by \begin{equation} a^{\dagger}(k)=-i\int d^3x [e^{ikx}\...
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$qg \rightarrow qg$: Spinor Helicity Formalism

I am calculating the cross-section for quark gluon to quark gluon scattering in the spinor helicity formalism. This process has contributions from the Mandelstam channels $s$, $t$ and $u$. Using the ...
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Unitarity and amplitudes

In Bootstrap and Amplitudes: A Hike in the Landscape of Quantum Field Theory there are few statements about analytical structure of amplitudes. I want to understand statement: In a local theory of ...
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Why amplitudes are rational functions?

In Bootstrap and Amplitudes: A Hike in the Landscape of Quantum Field Theory there are few statements about analytical structure of amplitudes. I want to understand statement: Tree amplitudes must ...
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Why does sending $T\rightarrow \infty(1-i\epsilon)$ in the slightly imaginary direction cause the $n=0$ term to decay slower?

This is in reference to equation 4.27 in Peskin and Schroeder. To derive a formula for the interacting vacuum in terms of the free vacuum we evolve the free vacuum in time with the full Hamiltonian ...
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How do you decide if a process has $s$ or $t$ channel Feynman diagram?

Without working with Lagrangian, how can one explain if we are dealing with $s$ or $t$ channel diagrams? For example, for $\rm e^+e^-\to\gamma\gamma$, I thought $s$-channel diagram, but the solutions ...
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S-matrix for 2-port network

Consider an impedance $Z$ on a waveguide of impedance $Z_0$. I add a load $Z_L$ at the end of the line but I don't think it should enter in the calculation ? I call $1$ the left port, $2$ the right ...
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Time-independent source and quantum field theory

Can anyone explain the fundamental reason of why time-independent sources cannot emit or absorb energy. Does it have to do with time-translation symmetry and Noether's theorem? I was studying the ...
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For which values of lambda the euclidean two-point function $(p^2 +m^2)^{-\lambda}$ is reflection positive

The case $\lambda=1$ is well known free field kernel. What about $\lambda$ in between 0 and 1 ?? ... for $\lambda>1$ I have a proof that the kernel is not reflection positive , ...
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Does Chew's bootstrap idea accept the possibility that there may be infinitely many possible laws of physics?

Physicist Geoffrey Chew proposed the concept of bootstrap (related to S-matrix theory) where he denied that fundamental laws of nature existed at all, as it is indicated in a writing in his memory ...
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Scattering amplitudes and LSZ formula for off-shell renormalization scheme

TLDR: The question: Does it make sense to calculate scattering amplitudes using an off-shell renormalization scheme? I expand a bit by using a theory of a single self interacting massive scalar. I ...
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Why the Feynman diagram with loops attached to external legs is irrelevant to the $T$-matrix?

Hello friends I was stumbled when I learnt the scattering theory from textbook titled "Quantum Field Theory for the Gifted Amateur", which has related the scattering probability to the ...
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Is S-matrix theory a subset of modern quantum field theory?

As a graduate student of physics in the early 2000's, our particle physics classes started with quantum field theory, since QCD had long been established as a good model of the nuclear strong force. ...
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LSZ reduction formula (Schwartz)

I am currently reading Schwartz's book on QFT and the Standard Model and I'm stuck on the beginning of the proof he gives for the LSZ reduction formula. Let the initial and final states of a ...
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$S$-matrix elements in path integral approach

How to calculate $S$-matrix elements of quantum electrodynamics using path integral formalism?
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Doubt regarding interacting and free field in Schwartz

Pg no 85. Schwartz has mentioned that "The interaction picture fields are just what we had been calling (and will continue to call) the free fields". Later when he calculates the vacuum ...
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Expansion of relativistic scattering matrix

In the paper I am going through it says that the scattering amplitude $M$ (i assume this is essentially the S-matrix) can be split up into angular momentum components (I assume this means partial wave ...
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LSZ derivation and contact terms [closed]

In the derivation of the LSZ reduction formula we write asymptotic states using ladder operators at $\pm \infty$. For example if we consider for a real scalar field $\phi$ then we have the formula $\...
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Fermion LSZ reduction / Greens function

In Matthew D.Schwartz book, 'Quantum Field Theory and the Standard Model' page 227/8 he talks about Feynman diagrams in QED, specifically $e^{-}e^{-} \rightarrow e^{-}e^{-}$ u and t channel tree level ...
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Is the rate of the reaction $a+b\rightarrow c\rightarrow e+f$ the same as $e+f\rightarrow c\rightarrow a+b$

In particle physics, is the rate of the reaction $a+b\rightarrow c\rightarrow e+f$ the same as the one of $e+f\rightarrow c\rightarrow a+b$ ? If so, then what does prevent that the final states goes ...
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Black Hole S-Matrix

I am reading arXiv:2006.03606 where through Eq. (1.1) they say that the transition amplitude for collapse of matter from initial state $\Psi_{i}$ into a black hole and eventually evaporation of black ...
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Is the $S$-matrix a scalar operator?

The scattering $S$ operator which is defined to be the operator corresponding to $S$ matrix should be rotational invariance, does this imply $S$ operator is a scalar operator?
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Spinor Helicity Formalism: Reference Spinors $q$ in Compton Scattering

My question is rather straight forward, but the setup in order to pose the question is a little lengthy; please bear with me! I am trying to calculate the average over initial states and sum over ...
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Unitarity implies branch cuts in $s > 4m^2$ in the $S$-matrix

Why does unitarity imply a branch cut in the $S$-matrix after $s > 4m^2$ where $s$ is the Mandlestam variable and $m$ is the mass of the particle? Assume identical particle scattering.
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What is particle under potential in quantum field theory?

Let's say we solve the Schrodinger equation with infinite well. We can quantize the field by the resonance state and can get the annihilation and creation operator. So we will get the sort of particle ...
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Confusion among channel $u$ and channel $t$ in exclusive reactions

In the this paper (https://arxiv.org/abs/1203.4392), on page 2 it is said that Since DVCS amplitude is symmetric under $s \leftrightarrow u$ channel crossing, the CFFs and $^{S}C^i_{\cdots}$ ...
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QFT and coordinate transformations

I was going through this lecture, where Leonard Susskind explains that these two Feynman diagrams are considered to be equivalent in QFT: One diagram explains the same phenomena in one possible way ...
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Crossing Symmetry between tree-level diagrams of $e^+ e^- \rightarrow \mu^+ \mu^-$ and $e^- \mu^- \rightarrow e^- \mu^-$

According to Peskin and Schroeder (P&S)'s book, on pp. 156-157, the two processes $e^+ e^- \rightarrow \mu^+ \mu^-$ and $e^- \mu^- \rightarrow e^- \mu^-$ are connected via $s \leftrightarrow t$ ...
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Emergence of Dual Resonance model from QCD

This question is about getting to understand better what the Dual Resonance (DR) model actually managed to successfully predict or model from the nuclear interactions, in the context of understanding ...
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Poles of a transmission coefficient

I stumbled on the question I can't quite grasp: What is the meaning of poles for transmission probability $T(E)$? $$ T(E) = \left( 1+\frac{1}{4}\frac{V_0^2}{E (E+V_0)} \sin^2 \left(\frac{2 a}{\hbar }\...
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Peskin&Schroeder (4.76) - Evaluation of non-trivial part of amplitudes and Unitarity of S-matrix

(1) On Peskin&Schroeder's book (P&S), page 105, it is written that Assuming that we are not interested in the trivial case of forward scattering where no interaction take place, we can drop ...
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Physical interpretation of propagator pole

In QFT the propagator is divergent for the on-shell momentum of the particles. When e.g. calculating the amplitude for a box loop, the propagator diverges for on-shell particles running in the box-...
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Usefulness of $| {\rm in}\rangle$ and $| {\rm out}\rangle$ states in S-matrix description of QFT

I am currently reading Niklas Beisert's lecture notes on QFT, Chapter 10, on the scattering matrix $S$.$^1$ My main confusion lies in the construction of $\vert \rm in \rangle$ and $\vert \rm out \...
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Perturbative expansion of the S-matrix in QFT

Beginner to QFT - While Taylor expanding the exponential term of the S-matrix, why is the 2nd order term (quadratic time integration term) written with two different dummy variables for time? (...
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Analytic continuation of scattering phase shift

In scattering theory, the resolvent, $S$-matrix, $T$-matrix, etc. are formally defined for any complex energy, $z$, even if most practical calculations consider just the positive real energy axis, $z=...
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Asymptotic states in strongly-coupled QFTs

In QFT the asymptotic states play a very important role, as they are the basis on which we decompose our in-states and out-states when we calculate correlation functions. However, I have not been ...
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$u$-channel in Feynman diagrams

I have two questions regarding the $u$-channel in Feynman diagrams. $\textbf{Question 1:}$ Suppose I have a $\gamma\gamma\to\bar{\nu}_{\mu}\nu_{\mu}.$ One of the diagrams will look as one of the ...
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Why are the renormalisation constants the same in LSZ and renormalisation?

To keep it simple I'll phrase everything in terms of scalar fields. We seem to have three constants called $Z$: When we do LSZ reduction we say, as $t\rightarrow-\infty$ then $\phi\rightarrow \sqrt{...
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Why can we shift the field $\phi$, so that $\langle \Omega | \phi(x) | \Omega \rangle = 0$?

Problem Introduction In different derivations of the LSZ reduction formula the author makes a shift of the field $\phi(x)$ $$ \phi'(x) = \phi(x) - \langle \Omega | \phi(x) | \Omega \rangle, $$ and ...
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Is tree-level QCD on-shell constructible, with BCFW?

Is tree-level QCD on-shell constructible, with BCFW? Pure yang mills is on-shell constructible, what is one add into massless fermions?
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Deriving Unitarity of $S$-matrix in 1D Quantum Mechanics

So, I was studying about scattering across a one-dimensional unknown potential ( pretty elementary Quantum Mechanics) and how if we know the $S$-matrix of such a system, we can deduce an awful lot of ...
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Why does the S matrix always contain a factor of $(2\pi)^4?$

In quantum field theory, one usually defines the scattering amplitude as $$S-1=(2\pi)^4\delta(p_{out}-p_{in})M_{Scattering Amplitude}$$ Where S is the S matrix element for any scattering process. It's ...
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Do all unitary-preserving regulators necessarily turn real loop integrals into pure imaginary numbers?

The optical theorem, which results from the unitarity of the $S$-matrix, relates the imaginary part of the forward scattering amplitude to the total cross section. When using this theorem in practice, ...
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Perturbative proof of unitarity of $S$-matrix in QED

In any standard textbook on QFT I know it is claimed that the $S$-matrix in QED is a unitary operator. I have never seen any proof of it. This should be compared with the analogous property of $S$-...
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Dependence of $S$-matrix on a coordinate system in QFT

The $S$-matrix is defined as follows (see e.g. Section 3.2 in Weinberg's "The quantum theory of fields"): $$S=\lim\exp(iH_0\tau)\exp(-iH(\tau-\tau_0))\exp(-iH_0\tau_0),$$ where the limit is taken when ...
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Which vacuum do I use for the path-integral?

In Weinberg, vol. 1, Section 9.2, Weinberg defines the in and out vacua as states with no particles (9.2.4): $$a_{\rm in}|{\rm VAC,in}\rangle=0$$ $$a_{\rm out}|{\rm VAC,out}\rangle=0$$ He does this ...
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What is algebraic structure, operation or relation explains relationship between the 2 diagonal terms of the 2 density matrices?

Quantum decoherence therefore prescinds from the observer and from the measurement process in a certain way preceding it and simulating the collapse of the wave function. In particular, "...
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Relation between in-states and free states in adiabatic approximation

I've been reading the recent book "Quantum Field Theory Lectures of Sidney Coleman", which I find great. I am, however, confused about one passage relating in-states and free states in the adiabatic ...
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On interacting QFTs with two masses $m$ and $M>2m$

hep-th/0412302v2 is an interesting paper by Shvedov about rigorous semiclassical covariant QFT. Shvedov talks on p.27 and p.30 about ''well-known'' properties of a putative interacting QFT with two ...

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