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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QFT: relation between Cutkosky's cutting rules and the optical theorem
I'm self-studying QFT and am trying to see the relation between Cutkosky rules and the optical theorem, which are presented together as consequences of unitarity in almost every elementary ...
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A short question on Ryder's proof of LSZ formula
I am reading Ryder's derivation of LSZ formula and I do not follow one intermediate step. It first solves the inhomogeneous Klein Gordon equation using Green's function. The result with retard Green's ...
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Where do the traces come from in Casimir's trick?
I am following the derivation for electron-muon scattering amplitude in Griffiths textbook and got to the section where they use Casimir's trick. I can't see where the traces come from. Equation 7.123 ...
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What is the role of wave packets in LSZ formulae?
When deriving LSZ formulae, we assume asymptotic particles’ creation/annihilation operators as:
$$a_\text{g,in/out}\ \ (\mathbf{p})\equiv \int d^3k \ g(\mathbf{k}) a_\text{in/out}(\mathbf{k}), \ \text{...
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Energy Renormalization and Vacuum Diagrams
I have been reading the lecture notes of Coleman's course on QFT. When developing scattering theory with the use of a cutoff function, he mentions that, in order to ensure that the free vacuum ...
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Transition amplitude, $\langle f|i\rangle$ or $\langle f|S|i\rangle$?
I have seen the transition amplitude was written in different ways, but I don't understand.
For example, in Mark Srednicki's textbook section $10,11$, it is written as $\langle f|i\rangle$. But in ...
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LSZ reduction formula for free fields
I'm reading the section on the LSZ reduction formula in Schwartz's QFT book and he talks about the action of free fields in the formula. Specifically he says (sec. 6.1.1, p. 73):
The LSZ reduction ...
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How and why the state of free particle in quantum physics is represented by plane wave packet? [closed]
In Quantum Mechanics (Cohen Tannoudji) Topic: "Asymptotic Form Of Stationary Scattering States"
It is written that for large negative values of $t$, the incident particle is free and it's ...
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Electron-molecule scattering: what is the physical meaning of the $S$, $K$ and $T$ matrices?
In electron-molecule scattering, I am told that the $S$, $K$, $T$ matrices are fundamental in order to gain insight into some physical scattering of electrons from molecules. The equations are defined ...
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Physical interpretations of Scattering Matrix $S$, Transition matrix $T$ and amplitude $M$
In QFT we define the scattering matrix from the scattering amplitude as
$$
S_{fi} = \lim\limits_{t\rightarrow \infty}\lim\limits_{t_0\rightarrow -\infty }\left\langle f\left|U(t,t_0)\right|i \right\...
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Degeneracy of energy eigenstates when $E > V$
Reading my text book on quantum physics, I found the following statement:
Let's suppose we have a short-scale force, so we have a potential energy such that
$$
V(x) = V_- \quad \text{if} \quad x \ll 0 ...
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$q^2=4m^2$ as the threshold for creation of a real electron-positron pair
On page 252, Peskin & Schroeder remark that the branch cut of the quantity
$$\widehat \Pi_2(q^2) \equiv \Pi_2(q^2)-\Pi_2(0) = -\frac{2\alpha}{\pi}\int_0^1dx\,x(1-x)\log\left(\frac{m^2}{m^2-x(1-x)q^...
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S-matrix expansion for the $\phi^4$ theory and the interaction picture
My question is about the perturbative expansion of the S-matrix using Dyson's expansion.
Let the Lagrangian density of the $\phi^4$ theory be
\begin{equation}
\mathcal{L} = \frac{1}{2}\left[\partial_\...
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In and out states, QFT [duplicate]
In s matrix calculation we use in and out states. I have few doubts about in and out states.
In which representation do we discuss about in and out states (Heisenberg picture or interaction picture)?
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Localised wavepacket (basic question from Srednicki chap. 5)
I'm currently working through Srednicki and I am confused by a couple of lines in chap. 5 (the LSZ reduction formula). He wants a wavepacket localised around $\mathbf{k = k}_1$ and $\mathbf{x} = 0$ at ...
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Trouble deriving expression for differential scattering cross section from S-matrix
I am following the derivation of the scattering cross-section from Peskin and Schroeder textbook. On page 105, we get an expression for the differential cross-section:
$$d\sigma = \left(\prod_f \frac{...
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Why are resonances poles on the unphysical sheet?
When studying the s-matrix in QFT I’ve seen it said that resonances correspond to poles on the unphysical sheet. Why can’t they be poles on the physical sheet?
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Do we include momentum conservation factors in amplitudes?
Let $\phi$ be a real scalar field and
$$\mathcal L = \frac{1}{2}(\partial_\mu \phi)^2 + \frac{1}{2}m^2\phi^2 - \frac{\lambda}{4!}\phi^4.$$
Given the standard Feynman diagrams and Feynman rules for $\...
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Scalar Electrodynamics or QED?
What's the difference between doing calculations in scalar electrodynamics and QED? I've seen that some people use scalar electrodynamics for Compton scattering and someone else using QED to solve for ...
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On which propagator does the field self-contraction loop go on this Feynman diagram?
This question relates to page 111 in Peskin and Schroeder.
I am trying to do the derivation of the 2-particle to 2-particle Feynman diagrams in $\phi^4$ theory by hand, following Peskin and Schroeder. ...
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$\phi^4$-theory, S-matrix Feynman diagram to first order from Peskin and Schroeder
This relates to page 111 in Peskin and Schroeder.
We have the $\phi^4$ S-matrix for a 2-particle to 2-particle scattering reaction:
$$-i\frac{\lambda}{4!}\int d^4x \langle p_1p_2|\mathcal T\left(\phi(...
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Why is there a negative sign in the time evolution operator when defining in/out states (Peskin/Schroeder)
This relates to Peskin & Schroeder's QFT book, equation 4.70 on page 104.
To define in and out states we take our initial state and evolve it far into the past, and do the same for our final state....
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Has it been proven that causality means the S-matrix is analytic?
In books like 'the analytic S-matrix' they give justification that causality implies analyticity however they also said it hasn't been explicitly proven.
This book was written a while ago, has it been ...
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What does QFT say about non-linear processes?
In QFT one can use the S matrix theory. We have a IN free system in the far past. It interacts in a black box in the present and there is a free OUT system in the far future.
We have OUT = S IN with ...
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Prove that the scattering operator is unitary [duplicate]
Let $H_0$ be some initial time-independent hamiltonian, and let $V$ be a scattering potential, such that the hamiltonian of a scattering process is:
$$H=H_0+V$$
Define the quantum states $|\psi_i\...
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Why do we demand that symmetries commute with $S$-matrix?
I am working in the context of Wightman Quantum Field Theory, and under a symmetry group I understand a group together with a unitary projective representation on the projective Hilbert space, unitary ...
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Satisfying the equation of motion of the time evolution operator
In Cohen-Tannoudji's Atom-Photon Interactions, he gives the integral form of the time evolution operator in the Schrodinger representation as
\begin{equation}
U(t_{f},t_{I}) = U_{0}(t_{f},t_{i}) + \...
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Existence of the S-matrix in AQFT
I am reading the book "An introduction to Symmetry and Supersymmetry in Quantum field theory" by Lopuszanski, and I have some problems understanding his argumentation about the existence of ...
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What is the continuum function in $R$-matrix basis?
In the book $R$-Matrix Theory of Atomic Collisions it is defined the $R$-matrix basis
\begin{aligned}
\psi_{k}^{\Gamma}\left(\mathbf{X}_{N+1}\right)=& \mathcal{A} \sum_{i=1}^{n} \sum_{j=1}^{n_{c}}...
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Is Compton Scattering the “Abelian limit” of $qg \rightarrow qg$?
I have calculated the average over initial and sum over final states of the squared amplitude for both Compton scattering $e^-\gamma \rightarrow e^-\gamma$ (QED) and quark-gluon scattering $qg \...
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Rescaling/renormalisation of the $n$-point function in $\phi^4$-theory by an unique $Z$?
In the chapter 12.2 of Peskin & Schroeder they introduce the rescaled renormalised $n$-point function respectively Green's function:
$$\langle \Omega|T\phi(x_1)\phi(x_2)\ldots \phi(x_n)|\Omega\...
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On the creation of wave packets with particular properties in quantum field theory
At the start of chapter 5 of Mark Srednicki's lecture notes on quantum field theory we define an operator that creates a particle that is "localised in momentum space near $\mathbf {k_1}$, and ...
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Any good references on the analytic structure of scattering amplitudes?
In papers they often say things about the analytic structure of S matrices - things like resonances are poles on the unphysical sheet, particle channels cause a square root branch cut etc.
I've seen ...
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Describing small, NRQM systems purely in terms of photons
Is there a canonical way to describe an open, non-relativistic quantum system with density matrix $\rho(t)$ entirely in terms of the light that it emits and absorbs (and vice versa?) Or is it possible ...
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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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Amplitude of quark$+$antiquark $\rightarrow$ ghost$+$antighost in QCD
Since the BRST charge operator commutes with the Hamiltonian of QCD, a physical state such as $q+\bar q$ should not be allowed to evolve into an unphysical one like $\chi+\bar\chi$, where these two ...
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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:
$a^{\dagger}({\bf k})$ creates 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 ...
