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).
2
votes
0answers
56 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
133 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
0answers
46 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
63 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
60 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
34 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
30 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
44 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
55 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
8 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
35 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
$$\int_0^1 dx x^{n-1}f_f^+(x,...
2
votes
1answer
96 views
How to understand complex masses of unstable particles? The conceptual problem of calculating decay rate
If a particle has a complex mass, $p^2-m^2=0$ leads to $p^μ \notin \mathbb R^4$. What does it mean?
When you want to calculate S-matrix elements of decay process $\langle p_f,\ldots\mid p_i\rangle$, ...
1
vote
1answer
103 views
Mass dimension of an $n$-particle scattering amplitude in 4D
For the 4-dimensional case, and using the cross-section formula, how can we show that the mass dimensions of an $n$-particle amplitude must be $$[A_n] = 4-n~?\tag{2.99}$$
My understanding is that the ...
6
votes
1answer
83 views
QFT Why do in and out states have a non-trivial overlap?
Im trying to follow chapter 4 about interacting fields in Peskin and Schröder. They define the S matrix by
$_{out}<p_1 p_2 | k_a k_b>_{in} = <p_1 p_2 | S | k_a k_b>$,
where $S = \lim_{T\...
1
vote
0answers
60 views
Meson scattering amplitude in the linear sigma model
I am trying to calculate scattering amplitudes with linear sigma model Lagrangian, given as
$$\mathcal L= \frac{1}{2}(\partial_{\mu}\sigma)^2+\frac{1}{2}(\partial_{\mu}\vec{\pi})^2-\mathcal U(\sigma,\...
1
vote
0answers
49 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
170 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:
$$
f(\theta) = \int d^3 r^{\prime} e^{-i (\bf k - k_i)r^{\prime}} V(r^{\prime}...
5
votes
0answers
51 views
Self-energy that does not obey sum rule
Analytically, I calculated a self-energy $\Sigma(\omega)$, for which I verified that
1) $\text{Im}\big[\Sigma(\omega)\big] \leq 0$ for all $\omega$ and specifically $\text{Im}\big[\Sigma(0)\big] = 0$,...
4
votes
0answers
61 views
The kinematic region for the operator product expansion
In Ch.18 of the textbook An Introduction to Quantum Field Theory by Peskin and Schroeder, on P.613 the operator product expansion (OPE) is introduced
$$\mathcal{O}_1(x)\mathcal{O}_2(0)\to \sum_n C_{...
1
vote
0answers
50 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
66 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
54 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
57 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
82 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
155 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
17 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
99 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
219 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
54 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
87 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 $$<\Phi_{j+1}|e^{-i\delta H(t_j)}|\Phi_{j}>=N \exp(i\...
1
vote
0answers
125 views
Kallen-Lehmann representation derivation
I'm trying to understand the derivation of the Kallen-Lehmann representation given in Peskin & Schroeder (pages 211-214). I would really appreciate if anyone on here could answer a few questions I ...
1
vote
0answers
153 views
S-matrix and Green's function
I'm considering one paper about electron recombination and there is an expression for S-matrix that confuses me
$${S_{fi}} = i\mathop {\lim }\limits_{t' \to \infty \atop t \to - \infty } \left\langle ...
2
votes
1answer
66 views
Clarifications on the assumptions made for QFT interactions
I am reading about scattering and S-matrix in the context of quantum field theory and although I understand the math and the physical interpretation of the final results, I am confused about some ...
2
votes
0answers
67 views
What is $\gamma$$\rho$ mixing?
In a previous question, I asked about the apparent universality behaviour in hadron elastic scattering. I was particularly shocked to see that even $\gamma p$ showed that universal behaviour with ...
2
votes
1answer
70 views
Do Reggeons-Pomerons-Odderons offer an Universal picture of hadron interactions?
As far as I know, the total cross-sections of the following hadron interactions are well described by a single Reggeon trajectory and a single Pomeron (soft Pomeron) trajectory. It seems to work for ...
3
votes
0answers
77 views
Use of Cutkosky rule, the Optical Theorem and Regge trajectories in pp scattering total cross-section calculation
Cutkosky rule states that:
$$2Im \big(A_{ab}\big)=(2\pi)^4\sum_c \delta\Big(\sum_c p^{\mu}_{c}-\sum_a p^{\mu}_{a}\Big)|A_{cb}|^2\hspace{0.5cm} (1)$$
putting $a=b=p$ in Cutkosky rule we deduce the ...
0
votes
1answer
83 views
Delta function from poles of Green's function
In quantum mechanical scattering theory, we often use Green's functions which contain poles. For example, in Schroedinger quantum mechanics the free Green's function is given by
$$
G_0(\vec{p}) = \...
1
vote
1answer
179 views
Optical theorem applied to forward scattering of a single particle
I'm slightly confused.
Write the S-matrix as
$$ S = 1 + i T $$
Unitarity implies
$$ T - T^\dagger = i T^\dagger T $$
In scattering from $|i\rangle$ to $|f\rangle$,
$$ T_{f,i} - T^\dagger_{f,i} = i \...
2
votes
1answer
235 views
$pp$ and $p\bar p$ scattering energy scaling exponents and 3d directed percolation model critical exponents similarity/equality, why?
$pp$ and $p\bar p$ scattering can be approximately described (in the Regge limit, that is, when $s \gg m \gt |t|$) by the exchange of Reggeons defined by the following Regge trajectory (low $s$):
$$\...
1
vote
1answer
141 views
Proton-proton and proton-antiproton elastic scattering symmetry
Is $A_{pp}(s,t)=A_{p\bar p}(t,s)$ true based on crossing symmetry?
Consider $pp$ and $p\bar p$ elastic colissions ($p + p \rightarrow p + p$ and $p + \bar p \rightarrow p + \bar p$). The scattering ...
12
votes
0answers
112 views
Is the converse of Weinberg's statement on the cluster decomposition principle true?
In Weinberg's "The Quantum Theory of Fields, Vol. 1", Section 4.4, page 182, the author says:
We now ask, what sort of Hamiltonian will yield an $S$-matrix that satisfies the cluster decomposition ...
14
votes
2answers
243 views
Why do we need to embed particles into fields?
In QFT we have the so-called embeding of particles into fields. This is discussed at full generality in Weinberg's book, chapter 5. In summary what one does is:
From Wigner's classification, for each ...
0
votes
0answers
30 views
Why can we use time-dependent perturbations when evaluating the S-matrix?
Suppose we have Hamiltonian $H_0 + V$. When working in the interaction picture we may derive the evolution operator of $|\psi_I(0)\rangle$ which is given by $$S(t,t_0) = T\left[\exp \left( -i \int_{...
3
votes
0answers
81 views
Number theory to estimate lower bound of S-Matrix?
I recently worked on the following idea:
https://math.stackexchange.com/questions/2325724/eigenvalue-of-an-euler-product-type-operator
Edit: And realised it was more probable to be used in Quantum ...
1
vote
0answers
103 views
What exactly are we doing when we “invent” Feynman Diagrams?
So, I am trying to derive the Feynman rules for Yukawa theory (following the section in Peskin). Specifically, for the process 2 fermions $\rightarrow$ 2 fermions. To second order, I then have that ...
0
votes
1answer
119 views
Time Evolution of Asymptotic Free States in QFT
In equation (4.70) of Peskin, he states that
$$_{out}\langle \mathbf{p_1, p_2, \cdots} | \mathbf{k_A,k_B}\rangle_{in} = \lim_{T\rightarrow \infty}\langle \mathbf{p_1, p_2, \cdots} | e^{-iH(2T)} |\...
3
votes
0answers
62 views
How can LSZ formalism deal with “glancing blows”?
My recent answer to the question Scattering, Perturbation and asymptotic states in LSZ reduction formula got me thinking again about wave packets and the LSZ reduction formula.
In my answer, I claim ...
3
votes
1answer
71 views
Scattering amplitude with a change in basis of fields
Suppose I know the Feynman rules for the scattering process $\pi^j \pi^k \rightarrow \pi^l \pi^m$ where $j,k,l,m$ can be $1, 2$ or $3$. Define the charged pion fields as $\pi^\pm=\frac{1}{\sqrt{2}}(\...
