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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Particle Creation by a Source
I am currently self-studying Quantum Field Theory and am using the textbook Introduction to Quantum Field Theory by Peskin and Schroeder. Currently I am in chapter 4, and am doing the first problem in ...
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Why are the poles of the propagator the masses of bound states and not the masses of unbound states as well?
For a real scalar field $\phi$, the Kallén-Lehmann representation of a two-particle propagator in a translation invariant theory in QFT is:
$G(x_1,x_2) = \sum_{m, \; other \; quantum \; numbers} \int \...
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Contractions of probability amplitude in quantum field theory
Given a Lagrangian $$\mathcal{L}=\frac12(\partial_\mu \phi(x))^2+\frac12m^2 \phi(x)^2-\frac{\lambda}{4!} \phi(x)^4$$
The probability amplitude in each order is given by: $$A=(2\pi)^3 (2E_q)^{1/2} (...
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Proof that different integral prescriptions lead to different scattering matrices?
So I have read that if you calculate feynman diagrams with an $i\epsilon$ prescription, that is with propagators of the form $\frac{i}{p^2-m^2+i\epsilon}$ then you get the standard $\mathcal{iM}$ ...
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In and out states in QFT
I have a few questions about how in and out states are defined in QFT. I was reading Schwartz's book but I found some confusing stuff about some of the things he does.
I'm going to work with a real ...
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LSZ formula and connected Green functions
My question is relatively simple. In the LSZ formalism, it is said that S-matrix elements correspond to on-shell limits of Green's functions. On the other hand, what people usually do is that they ...
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Scattering from finite square well and the transmission coefficient
Suppose we have a typical finite square well where $\lim_{x \to \pm\infty} V(x)=0$ and $V(x)=-V_0$ for all $x\in[-a,a]$ where $V_0>0$. The finite square well admits both bounds state solutions (...
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Dyson's formula $\phi^4$ theory
I have some difficulties when calculating the amplitude of a S-matrix using Wick's theorem. The evolution of the $U$-matrix is
\begin{align}
U(t, t_0) = T \exp(-i\int_{t_0}^{t}H_I(t') dt')=1 - i\int_{...
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“In”- and “out”-states in scattering theory
In scattering theory the Hamiltonian $H$ can be written as the sum
$H = H_0 + V$
where $H_0$ is the Hamiltonian of free particles and $V$ shall contain the interaction between particles. We can find ...
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Classical action as $S$-matrix generating functional at the tree-level
I have seen repeated statements that the $S$-matrix generating functional at the tree-level is the same as the action calculated on the classical solutions of the equation of motion corresponding to ...
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Connected part of $S$-matrix generating functional
I am currently studying an article by A.Jevicki et. al. (https://doi.org/10.1103/PhysRevD.37.1485) and I am a little confused. They say that the generating functional of the $S$-matrix is related to ...
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Polarization sum for spin 1 massive particles, for the perpendicular component of the field
I have a massive vector field $V_{\mu}$ appearing in an operator only with its perpendicular. So the operator looks something like this
$\bar{f}_1\gamma^{\mu}{V}_{\mu}^{\perp}f_2$
(the $\gamma$ matrix ...
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On contractions and S-Matrix in $\phi^4$ scalar theory
If you have a self-interacting Lagrangian for a scalar field theory:
$$L= L_0 + L_I = \frac{1}{2} (\partial_\mu\phi)^2 - \frac{1}{2} m^2\phi^2- \frac{g}{4!}\phi^4$$
where $g$ is the coupling constant, ...
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S-Matrix of Modular Tensor Category and physical S-matrix
Let $\mathscr{C}$ be a Modular Tensor Category (MTC). Then it has a finite set of simple objects $\{X_i\}$. Moreover, we define the S-matrix as $$S_{ij} = \text{Tr}(B_{X_i,X_j}^{-1}B_{X_j,X_i})$$
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How to interpret in and out states in scattering process for $\phi^4$ theory [closed]
What can be considered as in and out states when we calculate the scattering amplitude for $\phi^4$ theory? Also, whats the role of adiabatic evolution here?
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On the S-Matrix and correlation functions
I'm learning about interactions in QFT. My primary source has been the book "Quantum Field Theory For The Gifted Amateur" by Tom Lancaster. From there I've learned that in QFT one is ...
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Reference request for computing gluon scattering amplitudes in $\mathcal{N} = 4$ SYM in the planar limit
Could someone please provide me some reference(s), preferably that I can find online for free (arXiv for example or other), which would explain in some detail how to calculate gluon scattering ...
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Why does S-matrix have poles?
Why does unitarity require amplitudes to have singularities? I can see this by constructing amplitudes for specific processes, but what is the rationale behind the general rule? For concreteness, ...
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Interference term in QED Bha-bha scattering in terms of Mandelstam variables
In quantum field theory, I'm trying to calculate the invariant amplitude for the QED process of Bha-Bha scattering $e^+e^- \rightarrow e^+e^-$ and I found that two diagrams contribute at the second ...
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Confusion on $S$-Matrix and scattering amplitude in Matthew Schwartz's QFT textbook
In section 6.1, eqs(6.4) and eq(6.5), the $|i\rangle$ and $|f\rangle$ are defined as
$$|i\rangle=\sqrt{2\omega_1}\sqrt{2\omega_2} a_{p_1}^{\dagger}(-\infty)a_{p_2}^{\dagger}(-\infty)|\Omega\rangle,\...
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LSZ formula V.S. Long-range force
This site may already have a similar question, but I haven't been able to get a convincing answer, so let me ask you again.
My question is “ Why does LSZ formula apply to theories that include long-...
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Connected diagrams and keldysh
I have seen that:
In the ground state ($T = 0$) formulation of the Green’s function written in terms of operators
in the interaction picture, the Green’s function reads:
$$G(r,t;r',t') = -i\frac{\...
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Is $|i\rangle=\sqrt{2\omega_1}\sqrt{2\omega_2}a^\dagger_{p_1}(-\infty)a^\dagger_{p_2}(-\infty)|\Omega\rangle$ a momentum eigenstate?
Define an asymptotic state in the far past as $$|i\rangle=\sqrt{2\omega_1}\sqrt{2\omega_2}a^\dagger_{{\vec p}_1}(-\infty)a^\dagger_{{\vec p}_2}(-\infty)|\Omega\rangle$$ where $|\Omega\rangle$ is the ...
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QFT: relation between Cutkosky's cutting rules and the optical theorem
I'm self-studying QFT and am trying to see the relation between Cutkosky rules and the optical theorem, which are presented together as consequences of unitarity in almost every elementary ...
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A short question on Ryder's proof of LSZ formula
I am reading Ryder's derivation of LSZ formula and I do not follow one intermediate step. It first solves the inhomogeneous Klein Gordon equation using Green's function. The result with retard Green's ...
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Where do the traces come from in Casimir's trick?
I am following the derivation for electron-muon scattering amplitude in Griffiths textbook and got to the section where they use Casimir's trick. I can't see where the traces come from. Equation 7.123 ...
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What is the role of wave packets in LSZ formulae?
When deriving LSZ formulae, we assume asymptotic particles’ creation/annihilation operators as:
$$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}) + \...
