Skip to main content

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).

Filter by
Sorted by
Tagged with
3 votes
1 answer
57 views

Independence of $S$-matrix of $\xi$-gauge in QED

On page 298 in Peskin and Schroeder, the authors attempt to argue that the $S$-matrix should be independent of the $\xi$-gauge in QED. However, I don't understand their argument, in particular the ...
User3141's user avatar
  • 863
0 votes
1 answer
52 views

Square of the Feynman amplitude for $a +b\to c+d$ and its reverse

In quantum field theory, if a process $a +b\to c+d$ is allowed by a certain interaction Lagrangian (hermitian), the reverse process, $c+d\to a+b$, must also be allowed (as far as I understand) by the ...
Solidification's user avatar
2 votes
2 answers
86 views

Question on 1D Scattering Resonances

I'm reading Henley and Garcia's Subatomic Physics. To introduce the concept of resonances they use a 1D square well scattering example. Resonances are where the transmission coefficient goes to one. ...
CGS's user avatar
  • 2,540
2 votes
0 answers
60 views

Asymptotic states and physical states in QFT scattering theory

Context In the scattering theory of QFT, one may impose the asymptotic conditions on the field: \begin{align} \lim_{t\to\pm\infty} \langle \alpha | \hat{\phi}(t,\mathbf{x}) | \beta \rangle = \sqrt{Z} \...
Steven Chang's user avatar
0 votes
0 answers
21 views

On the symmetry of changing the sign of helicity of incoming and outgoing particles in the invariant matrix element

Let $\Psi_\Lambda^{\{\mu\}}\propto U_\Lambda^{\{\mu\}}$ and $\psi_\lambda^{\{\nu\}}\propto u_\lambda^{\{\nu\}}$ be spinors of spin $s$ fermions where $s \geq 1/2$ with respective helicites $\Lambda$ ...
infinitezero's user avatar
  • 1,312
3 votes
0 answers
49 views

Field strength renormalization for fermions

Following section 7.1 and 7.2 in Peskin and Schroeder (P&S), I've tried to consider what the derivation of the LSZ formula looks like for (spin $1/2$) fermions (in the text, they explicitly ...
User3141's user avatar
  • 863
0 votes
0 answers
60 views

How can I calculate the cross-section of a $N+\pi \rightarrow N + \pi$?

In the same theme as my previous question, I have the diffusion process $$N+\pi \rightarrow N + \pi$$ where the Lagrangian for this theory is $$L = \partial^\mu\psi\partial_\mu\psi^* - M²\psi\psi^*-\...
LittleBlue's user avatar
0 votes
0 answers
54 views

Independence of $S$-matrix in QED of a gauge of EM field

Due to existence of several ways to fix a gauge of an EM field in QED, there are several ways to quantize it. That leads to non-uniqueness of photon propagator and hence to non-uniqueness of integrals ...
2 votes
0 answers
39 views

A problem in Weinberg QFT Vol.1 Chapter 3

This is related to problem 5 of Weinberg's QFT vol.1 Chapter 3. The standing wave states $\Psi_\alpha^0$ are defined by a modified version of Lippmann-Schwinger equation, \begin{equation} \Psi_\...
Damo's user avatar
  • 56
2 votes
1 answer
78 views

Fermi theory cutoff from unitarity bound

Tree-level cross sections for processes described by Fermi theory behave like $\sigma $ $\sim$ $G_{F}^2 \cdot s$, where $G_{F}$ is the Fermi constant and $\sqrt s$ is the energy entering in the ...
onibaku's user avatar
  • 31
2 votes
1 answer
64 views

Field redefinitions in the Higgs mechanism

Consider the Higg's mechanism for a simple $U(1)$ theory. Leaving aside the lagrangian which consists of a kinetic term for the gauge field, a covariant derivative term and the potential term for the ...
Nakshatra Gangopadhay's user avatar
2 votes
1 answer
98 views

Feynman diagrams in string theory

I am beginning to study string theory, I have a beginner level doubt: If we consider a Feynman torus diagram in string theory, it is a worldsheet. What does it represent? Does it actually mean that in ...
SX849's user avatar
  • 306
3 votes
0 answers
52 views

Existence of eigenstates in the context of continuous energies in the Lippmann-Schwinger equation

In the book QFT by Schwartz, in section 4.1 "Lippmann-Schwinger equation", he says that: If we write Hamiltonian as $H=H_0+V$ and the energies are continuous, and we have eigenstate of $H_0$...
Gao Minghao's user avatar
2 votes
0 answers
65 views

Calculating LSZ reduction for higher order in fields terms

Consider a theory with only a single massless scalar field $\phi(x)$ and a current $J^\mu(x)$ which can be polynomially expanded as fields and their derivatives and spacetime \begin{align} J^\mu(x) = ...
Mmmao 's user avatar
  • 78
2 votes
0 answers
55 views

Does Sakurai's definition of $S$-matrix assume a particular type of scattering?

I am using Sakurai's Modern Quantum Mechanics 3ed. In chapter 6, Sakurai defines the $T$-matrix via the equation $$\langle \vec{k}' \lvert U_I(t, t_0) \lvert \vec{k} \rangle = \delta_{k'k} - \frac{i}{\...
Silly Goose's user avatar
  • 2,676
2 votes
3 answers
112 views

How do vacuum bubbles "dress" terms in the $S$-matrix numerator?

I am self-studying QFT using the book "A modern introduction to quantum field theory" by Maggiore. On page 124-125 he's doing the calculation in the interaction picture for a process with ...
Andrea's user avatar
  • 603
0 votes
1 answer
124 views

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 =\...
Frederic Thomas's user avatar
0 votes
0 answers
23 views

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 ...
1 vote
1 answer
37 views

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 ...
StackUser's user avatar
  • 199
0 votes
1 answer
122 views

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 ...
StackUser's user avatar
  • 199
0 votes
1 answer
112 views

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 ...
Karen's user avatar
  • 1
2 votes
1 answer
80 views

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 ...
Davius's user avatar
  • 1,640
0 votes
2 answers
63 views

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 & \...
Jason Chen's user avatar
2 votes
0 answers
51 views

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 ...
Davius's user avatar
  • 1,640
1 vote
0 answers
100 views

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} ...
Andrea's user avatar
  • 53
2 votes
1 answer
88 views

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 ...
Albert's user avatar
  • 307
3 votes
0 answers
64 views

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}...
Sung Kan's user avatar
2 votes
0 answers
77 views

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 ...
Генивалдо's user avatar
1 vote
1 answer
74 views

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 ...
gerogero's user avatar
2 votes
1 answer
114 views

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), ...
Ting-Kai Hsu's user avatar
5 votes
1 answer
181 views

$T$ 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). ...
elscan's user avatar
  • 138
3 votes
1 answer
121 views

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 ...
Ting-Kai Hsu's user avatar
4 votes
2 answers
296 views

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 ...
Rescy_'s user avatar
  • 838
3 votes
0 answers
151 views

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 ...
Alex's user avatar
  • 357
2 votes
0 answers
77 views

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^{(...
stringynonsense's user avatar
0 votes
0 answers
52 views

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 ...
Lip's user avatar
  • 41
2 votes
0 answers
56 views

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_+,...
user1394273's user avatar
1 vote
0 answers
187 views

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 ...
Panopticon's user avatar
1 vote
1 answer
240 views

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 ...
Galilean's user avatar
  • 988
-3 votes
1 answer
91 views

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 ...
user2820579's user avatar
1 vote
1 answer
180 views

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 $\phi^{4}$ ...
Jochem4T's user avatar
  • 237
2 votes
1 answer
159 views

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....
ShKol's user avatar
  • 322
1 vote
0 answers
33 views

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 $...
Andrew's user avatar
  • 23
3 votes
1 answer
183 views

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 ...
ShKol's user avatar
  • 322
3 votes
1 answer
325 views

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 ...
P. C. Spaniel's user avatar
1 vote
0 answers
47 views

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-...
SrJaimito's user avatar
  • 601
1 vote
1 answer
112 views

$S$-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 ...
user avatar
0 votes
0 answers
51 views

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 ...
Ting-Kai Hsu's user avatar
1 vote
0 answers
65 views

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)...
Filippo's user avatar
  • 475
0 votes
1 answer
75 views

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 ...
EM_1's user avatar
  • 860

1
2 3 4 5
12