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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Explanation of why poles in the S-matrix corresponding to particle production/bound states?
I've frequently heard the statement that the only singularities of the S-matrix in QFT correspond to things like the existence of bound states and the possibility of multi-particle production. I'm ...
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Deriving transition amplitude with $S$-matrix
Here is the part that is bothering me:
Yeah, already here? So, my question: In the first line we have int picture states at time zero and in the second line we have limit of time evolv operator with ...
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Doubt in Weinberg's book on QFT
In chapter 3 of his book on QFT (volume 1), while discussing the symmetries of the S-matrix, Weinberg makes the following statement
For any proper orthochronous Lorentz transformation $x\rightarrow ...
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$S$-matrix and in and out states
So, I have a short one. When observing scattering, we say that the amplitude for transition from one interacting state to some other interacting state same as this amplitude for free hamiltonian ...
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Can I add a classical potential to S-matrix
I'm a junior learner of QFT, and I wonder if I can add a classical approximated potential (like Coulomb potential) to a total interaction \begin{equation}V=V_{\mathrm{Coulomb}}+V_{\mathrm{internal}}
\...
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Significance of LSZ reduction formula
LSZ reduction formula relates the S-matrix element and the time-ordered correlation function, in a complicated equation. However, since
$$S=T e^{-i\int d^4x H_I}$$
where $H_I$ is the interaction ...
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How is “the collapse postulate is also present in QFT, only hidden inside the LSZ formula?”
Background
So I am reading the following here (Blog: Not Even Wrong, Blog post: Not So Spooky Action at a Distance, Commenter: vmarko)
"The collapse postulate is also present in QFT, only hidden ...
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How to properly make sense of the $\mathcal{S}$-matrix as a correlator on a sphere?
In the book "Lectures on the Infrared Structure of Gravity and Gauge Theories" by Andrew Strominger, the author discusses in Chapter 3 the idea of "The $\mathcal{S}$-matrix as a Celestial Correlator".
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Schwinger paper about charge in QED
I would like to find the paper by Schwinger where he discusses analytic properties of physical values in the limit $e\rightarrow 0$ and physical meaning of $e^2$ sign. I know only that this paper was ...
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Contradiction between aymptotically free particles in QFT and unlocalization
When studying different interactions in any QFT, one always assumes that the IN and OUT states are asymptotically free particles with definite momenta. For example, one assumes that an electron and a ...
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(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 ...
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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-...
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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?
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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 ...
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1answer
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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 ...
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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)?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
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Itzykson and Zuber: Reduced $T$-matrix
Could someone help me understand the reduced $T$-matrix mentioned in Itzykson and Zuber, eq.
$$\langle{f}| T|p_1p_2\rangle=(2\pi)^4\delta^4(P_f-p_1-p_2)\langle f|\mathcal{T}|p_1p_2\rangle. \tag{5-7}$...
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Does somebody know where to find original paper of Lehmann, Symanzik and Zimmerman translated in English?
does anybody know where to find original paper about LSZ reduction translated in english? unfortunantely, Ive found only original German article.
H. Lehmann, K. Symanzik, and W. Zimmerman, "Zur ...
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A Question about In/Out States in Quantum Field Theory
When I was reading the lecture notes Advanced Quantum Field Theory by Jorge Crispim Romao, I accidentally found the following thing that I don't understand.
On page 56, section 2.2, the author ...
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S-matrix branch cuts properties
I'm trying to formally understand some non-perturbative results in scattering theory but the material available on the topic are not too friendly, so there are some very simple and essential facts I ...
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Asymptotic LSZ reduction formula (Peskin & Schroeder)
Peskin & Schroeder, An Introduction to Quantum Field Theory, write at page 224
$$\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| ...
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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 ...
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1answer
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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....
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\Psi$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
$$\int_0^1 dx x^{n-1}f_f^+(x,...
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How to understand complex masses of unstable particles? The conceptual problem of calculating decay rate
If a particle has a complex mass, $p^2-m^2=0$ leads to $p^μ \notin \mathbb R^4$. What does it mean?
When you want to calculate S-matrix elements of decay process $\langle p_f,\ldots\mid p_i\rangle$, ...
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Mass dimension of an $n$-particle scattering amplitude in 4D
For the 4-dimensional case, and using the cross-section formula, how can we show that the mass dimensions of an $n$-particle amplitude must be $$[A_n] = 4-n~?\tag{2.99}$$
My understanding is that the ...
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QFT Why do in and out states have a non-trivial overlap?
Im trying to follow chapter 4 about interacting fields in Peskin and Schröder. They define the S matrix by
$_{out}<p_1 p_2 | k_a k_b>_{in} = <p_1 p_2 | S | k_a k_b>$,
where $S = \lim_{T\...
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Meson scattering amplitude in the linear sigma model
I am trying to calculate scattering amplitudes with linear sigma model Lagrangian, given as
$$\mathcal L= \frac{1}{2}(\partial_{\mu}\sigma)^2+\frac{1}{2}(\partial_{\mu}\vec{\pi})^2-\mathcal U(\sigma,\...
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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 ...
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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:
$$
f(\theta) = \int d^3 r^{\prime} e^{-i (\bf k - k_i)r^{\prime}} V(r^{\prime}...
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Self-energy that does not obey sum rule
Analytically, I calculated a self-energy $\Sigma(\omega)$, for which I verified that
1) $\text{Im}\big[\Sigma(\omega)\big] \leq 0$ for all $\omega$ and specifically $\text{Im}\big[\Sigma(0)\big] = 0$,...
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The kinematic region for the operator product expansion
In Ch.18 of the textbook An Introduction to Quantum Field Theory by Peskin and Schroeder, on P.613 the operator product expansion (OPE) is introduced
$$\mathcal{O}_1(x)\mathcal{O}_2(0)\to \sum_n C_{...
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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?
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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 ...
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1answer
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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.
