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 do we use pole mass in LSZ formula?

I am reading "Quantum Field theory and the Standard Model" by Schwartz. It derives LSZ formula in chapter 6, \begin{equation} \langle f|S|i\rangle =\left[i\int d^4x_1e^{-ip_1x_1}(\Box + m^2)\...
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Calculating a four-point Green function using Wick's theorem (problem 12.1 in Mandl & Shaw)

In problem 12.1 in Quantum Field Theory, Mandl & Shaw the aim is to calculate the four point green function $$ G^{\mu\nu}(x,y,z,w) = \frac{\langle 0 | T\big(A^{\mu}A^{\nu}\psi(z)\bar{\psi}(w)S\big)...
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Contraction with external legs in $S$-matrix

If we consider following $S-$matrix element:$$\left\langle\mathbf{p}_1 \mathbf{k}_2|T\{\phi(x_1) \phi(x_2)\}| 0\right\rangle_0 $$ where $\phi$ denote Klein-Gordon field, and apply the convention in ...
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How is Lorentz invariance of $S$-matrix related to vanishing of Hamiltonian density commutator at spacelike separations?

In Section 5.1 of the book, 'Quantum Theory of fields Vol-1' by Steven Weinberg, he says that if the Hamiltonian density commutes with itself at spacelike separation then the $S$-Matrix satisfies ...
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Energy and momentum conservation using Dirac delta function

I found in many text of QED dealing with scattering, the scattering matrix $S_{fi} \propto$ $\delta^4(p_f -p_i)$. They say that the $\delta$ function ensures the conservation of momentum and energy. ...
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A question for "Scattering amplitudes for all masses and any spins" : What is the exact two-component expression of the high spin wave function?

Now I'm studying the spinor helicity formalism with several liturature. I could understand roughly how to calculate the $n$-point on-shell amplitudes for massless particles very efficiently and ...
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Computing $\langle 0|S |0\rangle$ in $\phi^4$ theory [closed]

$\newcommand{\bra}[1]{\langle #1|}$ $\newcommand{\ket}[1]{|#1\rangle}$ I have been reading David Tong's QFT notes. As part of an exercise, I am asked to examine $\bra{0} S \ket{0}$ to order $\lambda^2$...
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QFT- Computing matrix elements when momentum is NOT conserved

In QFT, when computing cross sections one calculates the $T$ matrix elements (probability amplitudes that the interaction Hamiltonian takes us from the initial state $|\mathbf{k}_A,\mathbf{k}_B\big>...
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How to study the calculation of scattering amplitudes in particle physics?

I am looking for literature on the topic of calculating scattering amplitudes for particle physics processes, but this is a very big subject. I have studied introductory particle physics before, as ...
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Different forms of the LSZ reduction formula

I'm studying Chapter 7 section 2 of Peskin and Schroeder on the LSZ reduction formula, on page 227 they write the LSZ reduction formula $$\tag{7.42}\prod_1^n \int d^4x_i e^{ip_i\cdot x_i}\prod_1^m\int ...
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Visual explanation for $|\psi(-\infty)\rangle^{\text{in}}= \lim_{t\rightarrow -\infty} e^{iH_0 t} e^{-iHt}|\psi\rangle $

I'm reading section 7.4, Scattering and the $S$-matrix of Quantum Field Theory: Lectures of Sidney Coleman. It says For a scattering of particles in potential, we have a very simple formula for the S-...
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Scalar derivative couplings: Effects on S-matrix and Feynman Rules

In Schwartz's field theory book ch. 7.4.2 he claims that interaction Lagrangians like $${\cal L}_{\rm int} = \lambda \phi_1(\partial_{\mu}\phi_2)(\partial_{\mu}\phi_3)\tag{7.101}$$ lead to the Feynman ...
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Scattering in Shallow Bound States - Suppression of Matrix Element

In Weinberg's Lectures on Quantum Mechanics, given the S-matrix $$S_{\beta \alpha}=\delta(\beta-\alpha)-2 \pi i \delta\left(E_\beta-E_\alpha\right) T_{\beta \alpha}\tag{8.8.4}$$ where $$T_{\beta \...
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Optical theorem and singularities branch cut in Quantum Field Theory

On Peskin and Schroeder's QFT book, page 233, the book is trying to deal Feynman diagram's relation with $\text{Im}M$, where $M$ is the scattering amplitude. First, the book is trying to calculate ...
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Derivation in LSZ Reduction Formula

In deriving the LSZ formula, a crucial step is to show $$\langle|a_{p}^{\dagger}|\rangle=-i\int dx^0 \int \mathrm{d}^{3} x \partial_{0}\langle | e^{-i p\cdot x} \overleftrightarrow{\partial_{0}} \phi(...
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(Srednicki) How to obtain the normalization condition for Dirac field?

I'm reading through srednicki's qft and I met a problem. In its section 41, after he make an assumption that the creation operators of free field theory would work comparably in the interacting theory ...
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Resolution of the Identity in Quantum Field Theory

In Peskin and Schroder's QFT book, on page 212, eq.7.2, they use the completeness relation in a derivation involving the two-point correlation function: $$ \mathbf{1}=|\Omega\rangle\langle\Omega|+\...
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Weinberg, Effective Field Theories

Weinberg in his QFT Volume 1 points out in Chapter 12, section 12.3, near Fig. 12.4 (Is Renormalizability necessary?) that for expansions in EFTs in powers of $k/M$, where $k$ is the energy scale of ...
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Mass in the LSZ reduction formula

I'm reading wikipedia's page on the LSZ reduction formula https://en.wikipedia.org/wiki/LSZ_reduction_formula For the scalar LSZ reduction formula, after performaing a Fourier transform on the $n$-...
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Yang-Mills optical theorem computation

In Peskin&Schroeder, chapter 16.3 (page 516) we compute in the context of the optical theorem the quantity $$ \frac{1}{2}\left[\left(i\mathcal{M}^{\mu\nu}\epsilon^{-*}_\mu\epsilon^{+*}_\nu\right)\...
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Imaginary part of IR divergence in gravitational S-matrix

I was recently studying Weinberg's computation of the IR divergence in QED and gravity from virtual soft exchanges (https://doi.org/10.1103/PhysRev.140.B516). He does the computation where the matter ...
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Conditions for potential for scattering theory?

From the book, Scattering Theory: The Quantum Theory of Non-Relativistic Collisions the author does not derive the conditions but mentions in scattering theory we have a potential $V$ satisfies: (the ...
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Why do we get away with these assumptions of scattering theory?

The first questionable assumption I can think of is: The integration of the Hamiltonian density in the Dyson series is done over an infinite spacetime volume. Why do we get away with this assumption? ...
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Low energy theory of any S-matrix is a QFT

I have seen similar claims to the following: Any (sufficiently nice) S-matrix at low energy can always be described by an effective Quantum Field Theory. I would like to understand the extent of the ...
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Why are virtual photons never observed, if QFT predicts them as a possible outcome of experiments?

Consider the Dyson series of the $S$-matrix of Quantum Electrodynamics. The second term in this series predicts non-zero probability amplitudes of observing photons that are off-shell. For an incoming ...
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Polarization structure of photon scattering

I'm having trouble understanding the polarization structure of 2-2 photon scattering, but I'm pretty sure that my doubt lies in the photon polarization concept, independently of what scattering I'm ...
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Momentum non-conservation while on-shell condition is satisfied

There are two particles $\it{N}$ and $\pi$ with masses $m_N$ and $m_\pi$ associated with Hermitian scalar fields $\phi_N$ and $\phi_\pi$. The matrix element for the process $N\rightarrow N'\pi$ is $$...
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Polology of Feynman amplitudes, Section 10.2, Weinberg

In Weinberg's QFT Volume 1 section 10.2, we basically find that we have a pole when we have an intermediate particle state; the momentum of the intermediate state is on-shell. I am having trouble ...
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Correct dimensions of a single particle state?

Good day to everyone, this is my first time asking a question on the site so I hope the formalism isn't too sloppy. During my studies in QFT it was often remarked that given a single particle ...
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Scattering matrix ($S$-matrix), looking for a detailed guide to calculate it

I cannot find a clear guide for finding a scattering matrix (aka $S$-matrix) in electrodynamics, from the very first steps to the final step. What is the procedure to follow to obtain a scattering ...
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If the scattering amplitudes are Lorentz scalars, why is S-matrix Lorentz covariant?

All observers should agree on the probabilities: $\mathcal{P}(\mathcal{R}_1 \rightarrow \mathcal{R}_2)$ in an inertial frame $\mathcal{O}$ = $\mathcal{P}(\mathcal{R}_1' \rightarrow \mathcal{R}_2')$ ...
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Are there redundancy in Feynman diagrams for a particular scattering, in terms of rotation of the diagram?

In the lecture (see after 00:11:40) Leonard Susskind discusses a particular instance of indication of redundancy in the Feynman diagrams of scattering of $\pi$ mesons via $\rho$ mesons. He continues ...
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Form of the optical theorem in non-Abelian theory

I am studying chapter 16.3 from Peskin & Schroeder and I am trying to follow through the argument where we include contributions from ghosts to satisfy the Ward identity in non-Abelian gauge ...
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LSZ reduction for Bhabha scattering

This may be a silly question. As far as I know, the LSZ reduction formula is an equation to relate S-matrix elements and the expectation value of the time-ordered field operator. This approach allows ...
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Confused about the scattering Operator in LSZ reduction formula

In Greiner's Field quantization book, Chapter 9 on the LSZ reduction formalism, he states $$S_{fi}=\langle q_1,...,q_m;\text{out}| p_1,...,p_m;\text{in}\rangle\tag{9.10}$$ where $S$ is the scattering ...
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Asymptotic States, Propagator and Commutation Relations

Following Fradkin's discussion in the book QFT Integrated Approach, the commutation relation for asymptotic states satisfies $$\left\langle 0\left|\left[\phi(x), \phi\left(x^{\prime}\right)\right]\...
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The reasoning of the definition of $S$-matrix

The definition of the $S$-matrix is given by $$S=\lim_{t_{f}\rightarrow\infty}\lim_{t_{i}\rightarrow-\infty}U(t_{f},t_{i}).$$ Where $U(t_{f},t_{i})$ is the evolution operator, given by the $$U(t_{f},...
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The relation between full Green's function and S-matrix

I'm learning Green's function in condensed matter. The full Green's function is defined as $$G(k_2,t_2;k_1,t_1) = \langle\Omega |T a_{k_1}(t_1)a_{k_2}^{\dagger}(t_2) |\Omega \rangle $$ The $\Omega$ is ...
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$S$-Matrix and Relation to Phase Shifts

Given some potential $V(x)$, we can describe the amplitude of incoming and outgoing waves through the scattering matrix $S$ whereby $$\begin{pmatrix} B \\ F \end{pmatrix}= \begin{pmatrix} S_{11} & ...
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Symmetry implies Ward identity

I am thinking about symmetries and that their "quantum" consequences are Ward identities of the form $$<\beta|[Q,S]|\alpha>=0,$$ where $Q$ is the conserved charge associated with the ...
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$A(\phi\phi^*\gamma \gamma)$ with Spinor Helicity Formalism

I am calculating $A_4(\phi\phi^*\gamma\gamma)$ with the spinor helicity formalism (Exercise 2.16). I am following the conventions defined in Elvang's notes Such computation requires three diagrams, ...
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Is there a physical interpretation of the connection between the scattering matrix and bound states?

The square integrability condition of a scattering wavefunction can be written for imaginary wavenumber $k = -\mathrm{i}\kappa$ as $$\int_0^\infty \mathrm{d}r\left|(-1)^l \mathrm{e}^{-\kappa r} - S_l(-...
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$S$-matrix commutation with Hamiltonian

I know from scattering theory that $S$-matrix and the free Hamiltonian $H_{0}$ commute due to energy conservation of incident and outgoing asymptotic states, but can the $S$-matrix and $H = H_{0} + V$,...
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Why does a pole in the Green function correspond to a bound state?

Consider the many-body (zero temperature) fermion Green function $$ G(a,b;t)=-i\theta(t)\langle\psi_a(t)\psi_b^\dagger\rangle $$ Where I'm restricting $t>0$ for causality and that the free ...
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Wave function of a real scalar field in interacting quantum field theory

In interacting real scalar field theory, if I intuitively define the "wave function" of a state as $$\Psi(x)\equiv\langle\Omega|\hat{\phi}(x)|\Psi\rangle.$$ Does this wave function satisfy ...
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State time-evolution in the Interaction picture

What is the Schrödinger-like equation $$i\frac{d}{dt}|\psi(t)\rangle_I=V_I|\psi(t)\rangle_I$$ telling us for the behavior of the interaction picture state vectors, $|\psi(t)\rangle_I$, at infinity/...
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LSZ reduction formula relation

LSZ formula gives a relation between the scattering amplitudes and correlators as $$\langle f |i \rangle = (-i)^{m+n}\int \Pi_{i=1}^m d^4x_, e^{ik_i'x_i (\Box_{x_i} -m^2)}\Pi_{j=1}^n d^4x_, e^{ik_j ...
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More general propagator of a real scalar field

I have some Lagrangian containing a real scalar field $\phi$ with mass $m$. Let $A \in \mathbb{R}$ be some constant. The Lagrangian takes the form: \begin{equation} \mathcal{L} = -\frac{A}{2} (\...
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Lorentz invariance, locality and unitarity of the S-matrix

I am reading these notes on cosmological bootstrap by Baumann and found the following statement, which I am trying to understand / see where it comes from [page 2 of the linked notes]: "[...] S-...
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Why does the LSZ reduction formula only give the connected part of the $S$ matrix?

As an example, using the LSZ reduction formula, the $S$ matrix element for $2\rightarrow 2$ scattering is found in Peskin and Schroeder to be $$\langle \boldsymbol{p}_1 \boldsymbol{p}_2\rvert S \lvert ...
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