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).
253
questions
1
vote
1answer
83 views
(Coleman's lecture note) scattering in QFT
I am currently reading Coleman's lecture note on QFT.(https://arxiv.org/abs/1110.5013) I have several questions regarding the scattering theory. Let $\phi$ be a real scalar field, and consider the ...
1
vote
1answer
72 views
I really wonder about the time derivative of creation and annihilation operators in the derivation of LSZ
In Schwartz book, they assume that
$$\lim_{t \to \pm\infty}\partial_0 a_p(t)=0.\tag{1}$$
But I thought that is just assumption. so we have to construct the mathematical description. I found the Gell-...
0
votes
0answers
38 views
Off-shell vs half off-shell vs fully off-shell $T$-matrix
I know what are on-shell particles, but I want to know what are off-shell, and half off-shell, and fully off-shell states? and how we decide to consider one of these states in evaluating $T$-Matrix?
2
votes
1answer
32 views
Questions about scattering matrix theory of non-free particles
Hi，I have a problem for scattering matrix theory.
Currently, the book I've read is about collision between free particles.
What if collision between non-free particles? For example, in lattice, only ...
2
votes
1answer
102 views
What is the intuitive reason why matter and antimatter should be highly reactive?
Common knowledge has it that when an amount of matter and an amount of antimatter come anywhere near each other, they annihilate, leaving nothing but "pure energy".
In more technical terms, maybe we ...
0
votes
0answers
10 views
Necessary and suficient conditions for Scattering to be Elastic
Pretty straightforward: what basic assumptions must we make in constructing a Scattering Theory, in Quantum Mechanics, in order for it to conserve energy of the incident particle (i.e. to be elastic)?
...
3
votes
0answers
52 views
Scalar electrodynamics “seagull” vertex factor
By expanding the covariant derivative of the Scalar QED lagrangian one gets the following term, sometimes called "seagull" vertex.
$$\mathcal{L}_{seagull} = -q^2A_\mu A ^\mu \phi^\dagger \phi$$
Most ...
1
vote
1answer
42 views
Condition for finitely many bound states in one dimension
This came up in the context of the inverse scattering transform for the KdV equation. My primary reference, a set of lecture notes on integrable systems by Maciej Dunajski, makes the claim that the ...
2
votes
1answer
89 views
Energy Interpretation of Quantum Effective Action From Weinberg's “The Quantum Theory of Fields”
In section 16.3 of Weinberg, he attempts to prove that the effective potential energy $V(\phi)$ is equal to the minimum energy density of a state with field expectation value $\phi$. I am confused ...
0
votes
0answers
25 views
Eigenstates of the scattering matrix?
Consider a single-particle non-relativistic problem.
Consider a 3D spherically symmetric potential.
What are the eigenstates of the $S$-matrix? Are they spherically symmetric?
And what are the ...
1
vote
0answers
50 views
Scattering amplitudes vs correlators
What are the practical differences between correlators and scattering amplitudes in quantum field theory?
On a very practical level: scattering amplitudes describe the evolution of an IN state into ...
1
vote
1answer
53 views
Itzykson and Zuber: Reduced $T$-matrix
Could someone help me understand the reduced $T$-matrix mentioned in Itzykson and Zuber, eq.
$$\langle{f}| T|p_1p_2\rangle=(2\pi)^4\delta^4(P_f-p_1-p_2)\langle f|\mathcal{T}|p_1p_2\rangle. \tag{5-7}$...
2
votes
0answers
43 views
Does somebody know where to find original paper of Lehmann, Symanzik and Zimmerman translated in English?
does anybody know where to find original paper about LSZ reduction translated in english? unfortunantely, Ive found only original German article.
H. Lehmann, K. Symanzik, and W. Zimmerman, "Zur ...
1
vote
0answers
71 views
A Question about In/Out States in Quantum Field Theory
When I was reading the lecture notes Advanced Quantum Field Theory by Jorge Crispim Romao, I accidentally found the following thing that I don't understand.
On page 56, section 2.2, the author ...
1
vote
0answers
56 views
S-matrix branch cuts properties
I'm trying to formally understand some non-perturbative results in scattering theory but the material available on the topic are not too friendly, so there are some very simple and essential facts I ...
2
votes
1answer
83 views
Asymptotic LSZ reduction formula (Peskin & Schroeder)
Peskin & Schroeder, An Introduction to Quantum Field Theory, write at page 224
$$\int d^{4} x e^{i p \cdot x}\left\langle\Omega\left|T\left\{\phi(x) \phi\left(z_{1}\right) \cdots\right\}\right| ...
1
vote
0answers
85 views
Applying Schwinger-Dyson equations within the LSZ formula
My problem will be formulated in terms of $\phi^3$ theory, and I would appreciate answers within the framework of $\phi^3$ or another scalar field theory. This question is to help me understand what ...
2
votes
1answer
76 views
LSZ reduction derivation Srednicki
In the derivation of the LSZ reduction formula equation (5.21) Srednicki claims that in case of an interaction term in the Lagrangian density $a^{\dagger}(\textbf{k})$ will no longer be time dependent....
2
votes
0answers
64 views
Ghosts use in the calculus of the scattering matrix
When you perform the calculus of $S$ matrix in, for instance, QCD maybe you need to add ghosts in external legs or internal loops. E.g.: $q\bar{q} \rightarrow gg$ needs 2 ghosts diagrams with ghosts ...
8
votes
1answer
153 views
Why should there be one-particle states in an interacting quantum field theory?
I'm a mathematician trying to learn quantum field theory. This question has two parts: first, I want to double check that I'm thinking about the surrounding issues correctly, after that I'll ask my ...
2
votes
1answer
91 views
Optical theorem in $\phi^4$: which poles contribute to discontinuity in Feynman amplitude?
Section 7.3 ("The Optical Theorem") in Peskin and Schroeder's QFT text contains a leading order verification of the optical theorem in $\phi^4$ theory by calculating the (discontinuity across the ...
3
votes
1answer
72 views
Weinberg's Coleman-Mandula theorem proof sufficient condition for isomorphism?
In Weinberg's QFT Volume 3 book on Supersymmetry, he presents his own proof of the Coleman-Mandula theorem. As part of the proof, he proves that the only possible internal symmetry generators must ...
0
votes
1answer
65 views
What people mean by “state evolving with the interacting/free theory”?
This is a quite basic question but I confess it is something I didn't get up to this point.
When defining the Moller operators and hence the $\cal{S}$-matrix one usually considers "states $\Psi$ ...
0
votes
1answer
43 views
Size of transversal momenta in Multi-Regge kinematics?
Considering a scattering process in which $2$ incoming particles annihilate and produce $n-2$ other particles, one can consider the particle momenta $p_i^\mu$ (with $i=1,2,3,...,n$) to be in so called ...
1
vote
0answers
50 views
LSZ reduction formula for massive vector bosons
What is the precise form of the LSZ reduction formula for massive vector bosons?
The LSZ formula for scalar bosons, fermions, and photons is given e.g. in the textbook "Quantum field theory" by ...
1
vote
0answers
54 views
Calculating vacuum expectation value of graviton field
I've been reading a section in this thesis (pp 25-27) which reviews Duff's paper (pp 6-7 in particular) in which he calculates the tree-level vacuum expectation value (vev) of the graviton field (upon ...
1
vote
0answers
81 views
Calculate the scattering matrix of a 3-port lumped element network [closed]
I can get S11 and S12 by terminating port 2 and 3 with Z0 and Zr respectively. But when I try to calculate S13 it seems less straight forward. The following is the network, taken from Gao, J. The ...
0
votes
0answers
9 views
Potential from scattering matrix [duplicate]
Given a scattering matrix, is there a procedure to find the potential from the scattering matrix?
I think there should be a way as the scattering matrix holds the information of the boundary ...
2
votes
0answers
52 views
Contribution from $u$-channel and $t$-channel processes in OPE analysis for deep inelastic scattering
In Ch.18 of the textbook An Introduction to Quantum Field Theory by Peskin and Schroeder, on P.633 the moment sum rules for the deep inelastic form factors are discussed
$$\int_0^1 dx x^{n-1}f_f^+(x,...
2
votes
1answer
104 views
How to understand complex masses of unstable particles? The conceptual problem of calculating decay rate
If a particle has a complex mass, $p^2-m^2=0$ leads to $p^μ \notin \mathbb R^4$. What does it mean?
When you want to calculate S-matrix elements of decay process $\langle p_f,\ldots\mid p_i\rangle$, ...
1
vote
1answer
240 views
Mass dimension of an $n$-particle scattering amplitude in 4D
For the 4-dimensional case, and using the cross-section formula, how can we show that the mass dimensions of an $n$-particle amplitude must be $$[A_n] = 4-n~?\tag{2.99}$$
My understanding is that the ...
6
votes
1answer
96 views
QFT Why do in and out states have a non-trivial overlap?
Im trying to follow chapter 4 about interacting fields in Peskin and Schröder. They define the S matrix by
$_{out}<p_1 p_2 | k_a k_b>_{in} = <p_1 p_2 | S | k_a k_b>$,
where $S = \lim_{T\...
1
vote
0answers
67 views
Meson scattering amplitude in the linear sigma model
I am trying to calculate scattering amplitudes with linear sigma model Lagrangian, given as
$$\mathcal L= \frac{1}{2}(\partial_{\mu}\sigma)^2+\frac{1}{2}(\partial_{\mu}\vec{\pi})^2-\mathcal U(\sigma,\...
1
vote
0answers
69 views
A puzzle about Green's function and S-matrix
From the discussion in some posts example1, example2, we know that the S-matrix is the residue of the corresponding Green's function. On the other hand, S-matix is a physical observable in QFT, but ...
2
votes
1answer
308 views
Why first-order Born Approximation doesn't satisfy optical theorem?
First-order Born Approximation in Quantum Mechanics states that scattering amplitude is a Fourier transform of potential:
$$
f(\theta) = \int d^3 r^{\prime} e^{-i (\bf k - k_i)r^{\prime}} V(r^{\prime}...
6
votes
0answers
57 views
Self-energy that does not obey sum rule
Analytically, I calculated a self-energy $\Sigma(\omega)$, for which I verified that
1) $\text{Im}\big[\Sigma(\omega)\big] \leq 0$ for all $\omega$ and specifically $\text{Im}\big[\Sigma(0)\big] = 0$,...
4
votes
0answers
74 views
The kinematic region for the operator product expansion
In Ch.18 of the textbook An Introduction to Quantum Field Theory by Peskin and Schroeder, on P.613 the operator product expansion (OPE) is introduced
$$\mathcal{O}_1(x)\mathcal{O}_2(0)\to \sum_n C_{...
1
vote
0answers
52 views
Why should vacuum energy be zero for LSZ formalism?
Can anyone explain why vacuum energy must be zero if we are to use LSZ formalism?
1
vote
0answers
86 views
Is it possible to define Feynman diagrams in curved space-time?
I have a very simple question:
"Is it possible to talk about Amplitudes and Feynman diagrams assuming a different background than the usual Minkowski one?
Let's assume for example that the background ...
2
votes
1answer
67 views
Why can we not define asymptotic states in CFTs?
I have known that we can't define asymptotic states in CFTs, because we can't use Fock spaces to describe CFTs. But is that right and why? I want to know some details about it.
3
votes
1answer
64 views
Scattering matrix symmetries and standard model
I am not able to get around the following question (if it make sense):
Suppose I can derive the scattering matrix S for any particle scattering process.
Suppose that the standard model is actually ...
1
vote
2answers
100 views
Can a black hole ever be the output of a Feynman diagram in momentum space?
For a Feynman diagram representing a collision, particles come in from infinity with certain momenta, collide and then go off to infinite with other momenta. At collision vertices we integrate over ...
2
votes
1answer
72 views
How does one prove the channel independent inequality satisfied by the product of the three Mandelstam variables?
How does one prove the following equation (67.5) from the BLP Quantum Electrodynamics book? The q's are the 4 momenta, and h is the sum of all four masses. Two q's written after one another in the ...
3
votes
1answer
171 views
Can interacting quantum field theory describe more than just scattering?
From my understanding we do not yet know how to make much out of interacting QFT other than scattering amplitude at asymptotic infinity. (Correct me if I misunderstand.) But path integral, in ...
0
votes
0answers
18 views
What do experiments look like wherein the in-going momentum-space wavefunctions have overlap?
The LSZ formula relies on your ability to separate the momentum-space support of your wavefunctions in order to compute amplitudes from correlation functions. Are there any experiments in which:
You ...
2
votes
1answer
198 views
Why bound states in QFT have higher mass than single particle states?
In standard textbooks in QFT while discussing e.g. the Kallen-Lehmann formula (see e.g. Section 7.1 in the Peskin-Schroeder book) it is always assumed that bound states of two or more particles have ...
2
votes
1answer
129 views
Why the vacuum expectation value?
I am reading "QFT in a Nutshell", and the beginning of the book progresses like this:
Show how $\langle q_F|e^{-iHt}|q_I\rangle=\int Dq\ e^{iS}$
Says that we are more interested in $\langle F|e^{-iHt}...
2
votes
1answer
334 views
Optical theorem in QFT
I've been working with the Optical theorem in the case in which final and initial states are equals and I have the following doubt. Let's write the scattering matrix $S$ as:
$$S = 1 + i·T \tag1$$
...
3
votes
0answers
64 views
How do the renormalization factors disappear from the computation recipe of the S-matrix in Peskin & Schroeder (p. 229 (7.45) & p.324)?
In the following I limit my considerations to 4-point diagrams.
After the introduction of renormalized field operator (in renormalized perturbation theory)
$\phi_r= (\sqrt{Z})^{-1} \phi$
in eq. (...
1
vote
1answer
91 views
Clarification of Path Integral formulation
I am reading from Schwarz book on QFT the Path Integral chapter and I am confused about something. I attached a SS of that part. So we have $$<\Phi_{j+1}|e^{-i\delta H(t_j)}|\Phi_{j}>=N \exp(i\...
