All Questions
Tagged with quantum-field-theory s-matrix-theory
463
questions
0
votes
1
answer
52
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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 ...
- 11.8k
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} \...
- 108
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votes
0
answers
21
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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 $\Lambda$ ...
- 1,312
3
votes
0
answers
49
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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 ...
- 863
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0
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60
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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^*-\...
- 11
0
votes
0
answers
54
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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
1
answer
65
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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 ...
- 1,791
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 ...
- 306
3
votes
0
answers
53
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$...
- 57
2
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0
answers
65
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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) = ...
- 78
2
votes
3
answers
112
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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 ...
- 613
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 =\...
- 9,778
2
votes
1
answer
80
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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 ...
- 1,640
1
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0
answers
101
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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}
...
- 53
2
votes
1
answer
88
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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 ...
- 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 $$\mathcal{L}=\frac{1}{2}(\partial \varphi)^2 - \frac{1}{2}...
- 61
1
vote
1
answer
74
views
Quantization of a massless scalar
Let $t$:time, $r$:distance, and $u=t-r$.
Since any massless particle should propagate along u=const. , we need to change the asymptotic infinity of a massless scalar from time infinity to null ...
- 11
2
votes
1
answer
114
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Schrodinger's picture and Heisenberg's picture in finding interaction ground state and two-point correlator
In section 4.2 of An Introduction to Quantum Field Theory by M.E.Peskin and others, it derives interaction ground state by observing the time evolution of ground state in free field theory (pg.86), ...
- 529
3
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1
answer
121
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Explict Form of Ground State in Interacting Field Theory
In An Introduction to Quantum Field Theory by Peskin and Schroeder chapter 4, it has discussed about the ground state $|\Omega\rangle$ (where $|0\rangle$ is the ground state in free field theory) in ...
- 529
4
votes
2
answers
296
views
Derivation of Peskin & Schroeder eq. (4.29)
Background material:
These are the parts that I can follow.
Previously Peskin & Schroeder have derived already the expression of the interaction ground state $|\Omega\rangle$ in terms of the free ...
- 838
3
votes
0
answers
151
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LSZ reduction formula and connected Feynman diagrams in Peskin & Schroeder [duplicate]
I don't understand why in the LSZ reduction formula I need to consider only connected Feynman diagrams when I compute scattering amplitudes. From what I read in Peskin & Schroeder it seems that ...
- 357
2
votes
0
answers
77
views
LSZ theorem for trivial scattering
The $1\to1$ scattering amplitude is trivial and is given by (take massless scalars for simplicity)
$$
\tag{1}
\langle O(\vec{p}) O^\dagger(\vec{p}\,')\rangle = (2 | \vec{p}\,|) (2\pi)^{D-1} \delta^{(...
- 181
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_+,...
- 31
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 ...
- 858
-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 ...
- 793
1
vote
1
answer
181
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 $\phi^{4}$ ...
- 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....
- 322
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 ...
- 322
3
votes
1
answer
326
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 ...
- 4,827
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
$$
\langle{0} | T \phi_1 \phi_2 \phi_3 \phi_4 | 0 \rangle = -\left[ \Delta_F(x_1-x_2) \Delta_F(x_3-x_4) + \Delta_F(x_1-x_3) \Delta_F(x_2-...
- 601
1
vote
1
answer
112
views
$S$-matrix in Dirac picture
Let's define the interaction Hamiltonian as
$$\hat{H}(t) = \hat{H}_{\text{S}}+\hat{V}_{\text{S}}(t)\tag{1}$$
Where $\hat{V}_{\text{S}}\in \mathcal{L}(\mathcal{H})$ represents time-dependent ...
user375448
1
vote
0
answers
75
views
Scalar particle Compton scattering using relativistic Lagrangian formulation of electromagnetism
We know that parallel to scalar QED, a common formalism that describes a massive particle coupled to electromagnetism is through a relativistic worldline formalism, which writes
$$\mathcal{S}=\int\ ds\...
- 76
0
votes
1
answer
104
views
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 ...
- 121
17
votes
2
answers
1k
views
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 ...
- 171
1
vote
1
answer
112
views
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\...
- 619
2
votes
1
answer
127
views
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=(...
- 151
1
vote
1
answer
110
views
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
$$S_{fi}=<f|\...
- 475
2
votes
1
answer
113
views
CPT invariance and Soft Theorems
I am reading the paper IR Dynamics and Entanglement Entropy, written by Toumbas and Tomaras and I have a question on using the CPT invariance of the QED $S$-matrix elements in order to derive the ...
- 3,992
1
vote
2
answers
221
views
$S$-matrix from LSZ
Considering $2 \rightarrow 2$ scattering in $\phi^4$, this loop diagram gives a contribution of
$$\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^{...
user310742
0
votes
0
answers
60
views
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 $\mathcal{L} = \frac{g}{3!}\phi^3$ ...
4
votes
0
answers
70
views
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 ...
- 153
3
votes
2
answers
363
views
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 ...
user310742
0
votes
1
answer
62
views
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
$$ \Gamma_{\pi N \Delta, a}^{\mu} \sim \gamma^{\mu\nu\rho} (p_\pi)_{\nu}(p_\Delta)_\rho ...
- 55
0
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0
answers
129
views
Is the $S$-Matrix analytic in Planck constant?
Taking the scattering amplitude as a function of $\hbar$, is such function necessarily analytic in this variable.
Suppose I'm concerned with Relativistic Quantum Field Theory.
In QED, the tree level ...
- 1,268
2
votes
0
answers
60
views
How to perform the limit of infinite time in the LSZ approach?
I am computing the scattering matrix using the LSZ reduction formula in a semiclassical limit. The result that I am getting has the following form:
$$
S = \lim_{t_i \to - \infty} \lim_{t_f \to \infty} ...
- 81
0
votes
1
answer
149
views
$S$-matrix from interacting picture
I’ve been reading a lot about the interaction picture, and I’m trying to string the ideas behind it together.
Essentially, the goal is to calculate something like $<f(\infty)|i(-\infty)>$. We ...
user310742
3
votes
1
answer
445
views
Asymptotic states in the Heisenberg and Schrödinger pictures
One can show that, in the interacting theory, the operators that create single-particle energy-momentum eigenstates from the vacuum are
\begin{align}
(a_p^{\pm\infty})^\dagger=\lim_{t\to\pm\infty}(...
- 1,060
1
vote
0
answers
89
views
LSZ Reduction Formula (Weinberg Derivation)
In section 10.3 of Weinberg's Volume 1 in deriving LSZ reduction Formula, the author says,
We also define a 'truncated' matrix element $M_l$ by
$$\int d^4 x_2 \cdots e^{-q_2x_2} <\textbf q \sigma| ...
- 96
4
votes
2
answers
215
views
What are wave packets for (probabilities from the $S$-matrix)?
I am looking into the section of the book by Peskin and Schroeder in which they connect the $S$-matrix to probabilities.
They start by considering the in state, which is a two-particle state
\begin{...
- 705
2
votes
1
answer
132
views
On "waiting" and Coleman's derivation of LSZ
I'm going through the derivation LSZ in Coleman's QFT notes. The math is perfectly clear (or at least I don't mind being handwavy about convergence issues), and I'm happy with the idea that \begin{...
- 215