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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How do we measure the gluon scattering?

In QFT, we often calculate the 'Feynman amplitude' for gluon scattering. As we know, Gluon is gauge boson mediating the strong force between quarks. By color confinement, Quarks are never seen in our ...
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Expectation value of number operators in interactive vacuum

Non-interactive vacuum (say, a simple non-interacting scalar field) gives zero as the expectation value of number operator. With added interaction (say, phi-4 theory), the new ground state vacuum is ...
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Understanding from $S$-Matrix to Feynman-Rules in scalar QFT [closed]

I am learning QFT at the moment and the process from defining the S-Matrix to deriving the feynman rules is in my opinion pretty complicated, since there are many different things to pay attention to. ...
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Calculating Diagrams with Wilson Lines to One Loop

I have a question concerning the calculation of amplitudes containing Wilson lines. I want to calculate the jet function defined by equation (3.3) from this paper. $$\mathcal{J}(\text{arguments})u_s(p)...
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Scattering matrix in time domain, causality

In this question, I consider scattering problems in one dimension. In the scattering matrix formulation in quantum mechanics, the scattering outgoing (out) waves can be written as, $$\psi^{(out)}(E)=\...
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Are amplitudes for inverse processes related to each other?

The (generalized) optical theorem is presented in the book of Peskin and Schroeder (An introduction to Quantum Field Theory - chapter 7-Radiative Corrections:Some Formal Developments) as follows $-i (\...
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What is meant by a dispersion relation when complex integration in the Cutkosky rules' derivation or the computation of form factors $F(q^2)$ is used?

In the context of calculations of loop diagrams with Cutkosky-rules very often the concept of dispersion relation is mentioned. For instance this technique is used in Vol. 4 of Landau-Lifshitz for the ...
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Work of LSZ reduction formula

I want to know the mechanism of the LSZ reduction formula. The left side will have $\langle f|S|i\rangle$ and the right side has Fourier transform of $(\Box+m^2)$ times multiplication of Heisenberg ...
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LSZ reduction formula for scalar field

I am using Schwartz QFT and the LSZ reduction formula at pp 70. The scalar field was written as $$ \phi(x)=\phi(\vec{x}, t)=\int \frac{d^3 p}{(2 \pi)^3} \frac{1}{\sqrt{2 \omega_p}}\left[a_p(t) e^{-i p ...
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Calculating location of a complex pole of a Scattering Matrix

I am asked to calculate a pole in the lower complex momentum plane of an element from the Scattering Matrix for the potential $ V(x) = V_{0}(δ(x-1)+δ(x+1)$ . The potential parity even so it easier to ...
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Simplify calculation for matrix elements in quantum electrodynamics

So I am learning quantum field theory. At the moment I have a look at the interactions between electrons/positrons and photons, which is quantum electrodynamics. I want to calculate matrix elements of ...
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Why can we use the scattering matrix formula for decay rate?

The derivation of the scattering matrix $S_{\alpha\beta}$ requires the states to exist asymptoitcaly, why can we use it for a decay rate where clearly the decaying particles does not exist ...
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Scattering ${\cal M}$- and $S$-matrix

I am reading QFT book, like Introduction to QFT by Peskin and Schroeder, I would like to know conceptually what is the difference between $S$-matrix and invariant matrix element ${\cal M}$ in ...
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Constructing the Density Matrix from the $S$-Matrix

The $S$-matrix (scattering matrix) in Peskin & Schroeder's Introduction to QFT is given as $$ \langle p_1 p_2 \cdots | k_1 \cdots k_{n}\rangle_{in} $$ where $| k_1 \cdots k_{n} \rangle_{in} $ ...
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Finding the interaction vertices

Given a Lagrange density $$\mathcal{L} = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi - \frac{m^2}{2}\phi^2 - \frac{\lambda_3}{3!}\phi^3 - \frac{\lambda_4}{4!}\phi^4$$ where $\phi$ is a scalar field,...
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Expanding ladder operators in terms of field operator

In the LSZ reduction formula for calculating the S-matrix of real scalar interacting fields, one of the crucial steps in the derivation is to write the annihilation operator as $$a(\vec{k})=i\int d^3\...
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Proof of boost generator $K_0$ commute with $S$-matix in Weinberg QFT 1

In Weinberg QFT Vol.1, Weinberg defines a boost operator K when there exists an interaction $V$ as $$\textbf{K}=\textbf{K}_0+\textbf{W}, \tag{3.3.20}$$ where $\textbf{W}$ is expected as a correction ...
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How to obtain (interacting) time-ordered correlation functions from the S-matrix - reverse of the LSZ formula?

The LSZ formula shows how to obtain the S-matrix elements from the time-ordered correlation functions of the interacting fields. I wonder if there is a reverse formula; that is, can we find the ...
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Resistance of obstacles in one-dimensional quantum scattering

The transfer matrix in one-dimensional quantum mechanics, fulfills the property \begin{align} M_{12}=M_2M_1\,. \end{align} If we consider two obstacles, from the above matrix multiplication, it is ...
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Optical theorem for Feynman diagrams

I'm studying section 7.3 of Peskin and Schroeder. In the middle of page 232, the book says: For our present purposes, let us define $M$ by the Feynman rules for perturbation theory. This allows us to ...
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Optical theorem Peskin and Schroeder

I'm trying to understandd the optical theorem of peskin nad schroeder $$\tag{7.50} \text{Im} M(k_1,k_2\rightarrow k_1,k_2)=2E_{cm}p_{cm}\sigma_{tot}(k_1,k_2\rightarrow\text{anything})$$ which Peskin ...
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Are all amplitudes evaluated on-shell?

In [1] the authors state that for a free scalar theory (eq 7) under field redefinition $\phi\rightarrow a_1 \phi^2 + a_2 \phi^3 + ...$: "Once evaluated on-shell, the n-point tree-level ...
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Question about asymptotic assumption in LSZ reduction formula derivation

I have a silly question in derivation of LSZ reduction formular, I can go directly with the derivation until I found a assumption that I can't convince myself. In the book Quantum Field Theory and the ...
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Deriving the optical theorem, order-by-order unitarity

I am reading from Matthew Schwartz' QFT textbook. In Chapter 24.1, beginning on page 453, the optical theorem is derived from the principle of unitarity. Recalling that the $S$-matrix is given as the ...
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LSZ reduction formula in Peskin and Schroeder

In derivation of the LSZ reduction formula in Peskin and Schroeder, on page 227, the book says Let us analyze the relation between the diagrammatic expansion of the scalar field four-point function ...
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Interaction Picture Ladder Operators vs Free Field Operators

Consider the case of an interacting scalar field theory with bare mass $m_0$. After having derived the LSZ reduction formula, all that is left is to compute the time ordered products of the Heisenberg ...
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Is it possible that the scattering matrix is the minus identity matrix $S=-\mathbb{I}$?

We know that if the scattering matrix is an identity matrix ($S=+\mathbb{I}$), it means that transmission is zero and there is full reflection. My question: Is it possible that the scattering matrix ...
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Peskin & Schroeder LSZ formula missing in- and out states

In Peskin and Schroeder the LSZ-formula is given as below where the states in the $S$-matrix element are fully interacting Heisenberg states. $$\begin{array}{l}\prod_{1}^{n} \int d^{4} x_{i} e^{i p_{i}...
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Weinberg, off the mass shell Feynman diagrams

In section 6.4 of Weinberg QFT, the book says on page 286: It is important to also consider Feynman diagrams "off the mass shell", for which the external line energies like the energies ...
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Does the orientation of an anistropic potential impact the Scattering-Matrix?

I was curious if an anistropic potential's orintation depended on a scattering matrix. Is it feasible to create a dependence if not?
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LSZ reduction formula vs Dyson's expansion

In quantum field theory, we have use perturbation series to compute the $S$-matrix elements. For example: $$S=1+\sum_{i=1}^\infty\frac{(-i/\hbar)^n}{n!}\int_{-\infty}^\infty...\int_{-\infty}^\infty T[...
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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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