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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The definition of the path integral

I still have big conceptual questions about the path integral. According to (24.6) of the book "QFT for the gifted amateur" from Lancaster & Blundell the path integral is equal to $$Z =\...
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Non-relativistic Quantum Mechanical Scattering Theory Textbook Recommendations

I am looking for textbooks that cover non-relativistic quantum mechanical scattering theory. An ideal text would not "brush mathematical details under the rug" and would also make contact ...
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Diverging Scattering Amplitudes and Transmission/Reflection Coefficients

I am currently studying scattering theory from Sakurai and Griffiths and I have noticed that for the 1D Dirac potential, the transmission and reflection coefficients diverge when the energy ...
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Scattering Matrix and the Lippmann-Schwinger equation in QM

I am currently studying scattering theory from the Sakurai's quantum mechanics. I have previously studied this subject from Griffith's quantum mechanics. In the latter textbook, scattering matrices ...
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Scattering matrix and transfer matrix relationship in the case of arbitrary matrix dimensions

Finding the relationship between scattering and transfer matrix elements is trivial in the case of 2 by 2 matrices when there are two inputs and two outputs. However, how should I approach the task of ...
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Why does $S$-matrix theory end up being a covariant formalism when it is not obvious that it is?

A principle of QFT that is frequently invoked, repeated, and potentially subject to rigorous verification is that the theory in question must exhibit Lorentz covariance and be invariant under the ...
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Justification of discarding the backward wave in step potential scattering

I'm following Shankar's treatment of 1D scattering in Principles of Quantum Mechanics (Page 167 to Page 172). In general, the eigenstates of the single-step potential $$V(x)=\begin{cases} 0 & \...
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An S-matrix description of the photoelectric effect?

As is well known, the photoelectric effect is an experimental phenomenon that had enormous historical importance for the emergence of the concept of photons and quantum mechanics itself. As is well ...
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Discontinuity of the scattering amplitude and optical theorem

The generalized optical theorem is given by: \begin{equation}\label{eq:optical_theorem} M(i\to f) - M^*(f\to i) = i \sum_X \int d\Pi_X (2\pi)^4 \delta^4(p_i-p_X)M(i\to X)M^*(f\to X).\tag{Box 24.1} ...
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How is dimensionality of $S$ preserved term by term in a perturbative expansion?

In a schematic notation, the scattering matrix element $$\langle p_{out}|S|p_{in}\rangle := 1 + i (2 \pi)^4 \delta^4(p_{in} -p_{out}) M$$ between an incoming state with momentum $|p_{in}\rangle$ and ...
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Deriving a contradiction from the LSZ condition

I'm reading the LSZ reduction formula in the wikipedia: https://en.wikipedia.org/wiki/LSZ_reduction_formula To make the argument simple, let $$\mathcal{L}=\frac{1}{2}(\partial \varphi)^2 - \frac{1}{2}...
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Reason to consider only compact world-sheets in string theory

Generally speaking, the "sum over world-sheets" in string theory involves summing over all possible topologies of compact, orientable and connected, as Polchinski says in page $100$ of his ...
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Quantization of a massless scalar

Let $t$:time, $r$:distance, and $u=t-r$. Since any massless particle should propagate along u=const. , we need to change the asymptotic infinity of a massless scalar from time infinity to null ...
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Schrodinger's picture and Heisenberg's picture in finding interaction ground state and two-point correlator

In section 4.2 of An Introduction to Quantum Field Theory by M.E.Peskin and others, it derives interaction ground state by observing the time evolution of ground state in free field theory (pg.86), ...
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$T$ Matrix elements in Scattering Theory (Sakurai 2nd edition)

I am currently unable to see how the $T$ Matrix elements discussed in 6.1 of Sakurai's Modern Quantum mechanics 2$^\mathtt{nd}$ edition can be expressed as they are in equation (6.1.26) (see below). ...
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Explict Form of Ground State in Interacting Field Theory

In An Introduction to Quantum Field Theory by Peskin and Schroeder chapter 4, it has discussed about the ground state $|\Omega\rangle$ (where $|0\rangle$ is the ground state in free field theory) in ...
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Derivation of Peskin & Schroeder eq. (4.29)

Background material: These are the parts that I can follow. Previously Peskin & Schroeder have derived already the expression of the interaction ground state $|\Omega\rangle$ in terms of the free ...
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LSZ reduction formula and connected Feynman diagrams in Peskin & Schroeder [duplicate]

I don't understand why in the LSZ reduction formula I need to consider only connected Feynman diagrams when I compute scattering amplitudes. From what I read in Peskin & Schroeder it seems that ...
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LSZ theorem for trivial scattering

The $1\to1$ scattering amplitude is trivial and is given by (take massless scalars for simplicity) $$ \tag{1} \langle O(\vec{p}) O^\dagger(\vec{p}\,')\rangle = (2 | \vec{p}\,|) (2\pi)^{D-1} \delta^{(...
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QED Feynman graph Coordinate space doubt

So I know that there are the Feynman rules to transform mathematical equation into graphs but to me it's not too much clear when I should draw the graph vertically or horizontally i.e. how I determine ...
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Conservation of angular momentum in LSZ reduction formula

I recently solved a problem involving calculating an LSZ reduction formula for the decay of a polarized photon into two pions. Specifically, I wrote an expression for the matrix element $\langle p_+,...
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Angular momentum and the $S$-matrix

I have been curious about the status of angular momentum in the context of the $S$-matrix and scattering amplitudes. In particular, if we pass to a classical scattering problem and imagine scattering ...
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How to compose scattering matrices?

Imagine I have a scattering region (denoted as sample). Scattering matrix and transfer matrix gives the same information about scattering. The scattering matrix tells us how incoming modes are ...
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Some calculation in Mahan book, p73 [closed]

On page 73 of Mahan, Many-particle physics, 3rd edition, one finds $$ _0\langle|S(-\infty,0) = e^{-iL}_0\langle|S(\infty,-\infty)S(-\infty,0). $$ I'm wondering why this is true, as in the previous ...
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Calculate first-order term of the $S$-matrix for the $\phi^{4}$ theory [closed]

Before I ask a question, I will start with a small introduction. I want to evaluate the $S$-matrix order-by-order in an expansion in small $\lambda$ for a $2 \rightarrow 2$ scattering in $\phi^{4}$ ...
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Confusion regarding the $S$-matrix in Quantum Field Theory

In his Harvard lectures on QFT, Sidney Coleman defines the $S$-matrix as, $$ S \equiv U_{I}(\infty, -\infty) $$ Where $U_{I}(-\infty, \infty)$ is the time evolution operator in the interaction picture....
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Derive equations for the $S$-matrix composition in case of $2n$ by $2n$ full $S$-matrices [closed]

I am studying computational physics, but for me it seems that this question should be asked in physics section. i need to derive equations for the $S$-matrix composition in case of $2n$ by $2n$ full $...
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Sidney Coleman's Lectures Notes on QFT: Question regarding incoming states and free states

In Sidney Coleman's Lecture Notes on Quantum Field Theory, under section 7.4, we have the following, For a scattering of particles in a potential, we have a very simple formula for the S-matrix. We ...
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General interpretation of the poles of the propagator

I am somewhat familiar with the fact that the poles of the Feynman propagator in QFT give the momentum of particle states. I'm also familiar with the KL spectral representation in that context (See ...
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How to apply multiple Klein-Gordon operators to products of propagators?

I have the 4-point correlation function for a scalar free field $$ \langle{0} | T \phi_1 \phi_2 \phi_3 \phi_4 | 0 \rangle = -\left[ \Delta_F(x_1-x_2) \Delta_F(x_3-x_4) + \Delta_F(x_1-x_3) \Delta_F(x_2-...
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$S$-matrix in Dirac picture

Let's define the interaction Hamiltonian as $$\hat{H}(t) = \hat{H}_{\text{S}}+\hat{V}_{\text{S}}(t)\tag{1}$$ Where $\hat{V}_{\text{S}}\in \mathcal{L}(\mathcal{H})$ represents time-dependent ...
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Question about In-States in S.Weinberg Lectures On Quantum Mechanics

At the beginning of chapter 7 Potential Scattering of Lectures on Quantum Mechanics by S. Weinberg, he has introduced the in states (in Heisenberg picture) $\Psi^{\text{in}}_{\pmb{k}}$, which ...
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How to recognize Feynman diagrams from the $S$-matrix expansion?

I'm studying scattering processes in QED and one usually have to compute first of all the Scattering matrix $$\hat{S}=T\biggl (\exp\{-i\int d^{4}x:\bar{\psi}(x)\gamma_{\mu}\hat{A}^{\mu}(x)\hat{\psi}(x)...
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Step Potential versus Free particle [duplicate]

In the step potential \begin{equation} V= \begin{cases} 0 &, \text{x<0}\\ V_0 &, \text{x>0} \end{cases} \end{equation} for the scattering states$(E>V_0)$, the states on the right and ...
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Scalar particle Compton scattering using relativistic Lagrangian formulation of electromagnetism

We know that parallel to scalar QED, a common formalism that describes a massive particle coupled to electromagnetism is through a relativistic worldline formalism, which writes $$\mathcal{S}=\int\ ds\...
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What is a particle in the context of QFT with interactions?

This is a crossposting of the same question from mathoverflow: https://mathoverflow.net/q/454768/ It seems that this question was not received well there, claiming that this question is not ...
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What physical processes other than scattering are accounted for by QFT? How do they fit into the general formalism?

For background, I'm primarily a mathematics student, studying geometric Langlands and related areas. I've recently been trying to catch up on the vast amount of physics knowledge I'm lacking, but I've ...
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How do we interpret disconnected diagrams in scattering theory?

It is apparent that disconnected diagram contributes additional delta functions to the corresponding matrix element. For example, we consider the scalar $\phi^3$ theory and the following $2\...
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Why do the eigenvalues of the 4-momentum operator organize themselves into hyperboloids?

Specifically I'm asking for the motivation behind figure 7.1 in page 213 of the QFT textbook by Peskin and Schroeder. In that section they just consider eigenstates of the 4-momentum operator $P^\mu=(...
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How can we prove that Compton scattering has two equivalent terms in the $S$-matrix expansion?

Consider the Compton scattering $$e^{-}(p,s)+\gamma(k,\lambda)\rightarrow \gamma(k',\lambda')+e^{-}(p',s')$$ To calculate the process' amplitude one has to compute the matrix element $$S_{fi}=<f|\...
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CPT invariance and Soft Theorems

I am reading the paper IR Dynamics and Entanglement Entropy, written by Toumbas and Tomaras and I have a question on using the CPT invariance of the QED $S$-matrix elements in order to derive the ...
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Understanding the potential step for a particle in 1D

In an exercise, I consider a particle moving from $x=-\infty$ towards a potential step, where $V(x)=0$ for $x\leq 0$ and $V(x)=V_0$ for $x>0$. If we consider the case of $0<E<V_0$, we have; $$...
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$S$-matrix from LSZ

Considering $2 \rightarrow 2$ scattering in $\phi^4$, this loop diagram gives a contribution of $$\int{dx_{1}dx_{2}dy_{1}dy_{2}dk_{1}dk_2 dp_1 dp_2 dq_1 dq_2 e^{-ik_1 x_1}e^{-ik_2 x_2}e^{ip_1 y_1}e^{...
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How diagrams with loop and several propagators contribute to $S$-matrix element?

I studied Feynman rules with Schwartz textbook and what caught my eye was diagrams such as second and third on this picture (diagrams to the second order of $g$ for $\mathcal{L} = \frac{g}{3!}\phi^3$ ...
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Redefinition of fields and interpretation of the particle content

Suppose I have some Lagrangian $\mathcal L_1$ involving multiple fields $\phi_i$ with interactions. I can reparametrize the Lagrangian in terms of new fields $\psi_i$ by inserting some ...
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Elastic phase shifts in relativistic QFT

Are there any methods known for computing the elastic phase shifts in a relativistic QFT such as $(\lambda/4!)\phi^4$, which ensure that unitarity is satisfied?. The naive approach of simply ...
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Proof that asymptotic particle states are free

In quantum field theory, It’s often said that the interacting annihilation operator (defined by the Klein Gordon inner product between the interacting field and a plane wave) behaves like the free ...
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Simplify a vertex in the on-shell form

I am calculating with a vertex connecting a pion, delta and a nucleon. In general, the vertex is calculated as $$ \Gamma_{\pi N \Delta, a}^{\mu} \sim \gamma^{\mu\nu\rho} (p_\pi)_{\nu}(p_\Delta)_\rho ...
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Is the $S$-Matrix analytic in Planck constant?

Taking the scattering amplitude as a function of $\hbar$, is such function necessarily analytic in this variable. Suppose I'm concerned with Relativistic Quantum Field Theory. In QED, the tree level ...
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Reference request: Non-relativistic scattering in second quantization

There are at least two possible ways to go about computing the amplitude for $2\to2$ scattering of indistinguishable particles in non-relativistic quantum mechanics. The first is the method we all ...

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