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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59 views

Physical interpretation of propagator pole

In QFT the propagator is divergent for the on-shell momentum of the particles. When e.g. calculating the amplitude for a box loop, the propagator diverges for on-shell particles running in the box-...
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Usefulness of $| {\rm in}\rangle$ and $| {\rm out}\rangle$ states in S-matrix description of QFT

I am currently reading Niklas Beisert's lecture notes on QFT, Chapter 10, on the scattering matrix $S$.$^1$ My main confusion lies in the construction of $\vert \rm in \rangle$ and $\vert \rm out \...
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47 views

Perturbative expansion of the S-matrix in QFT

Beginner to QFT - While Taylor expanding the exponential term of the S-matrix, why is the 2nd order term (quadratic time integration term) written with two different dummy variables for time? (...
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Analytic continuation of scattering phase shift

In scattering theory, the resolvent, $S$-matrix, $T$-matrix, etc. are formally defined for any complex energy, $z$, even if most practical calculations consider just the positive real energy axis, $z=...
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Asymptotic states in strongly-coupled QFTs

In QFT the asymptotic states play a very important role, as they are the basis on which we decompose our in-states and out-states when we calculate correlation functions. However, I have not been ...
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$u$-channel in Feynman diagrams

I have two questions regarding the $u$-channel in Feynman diagrams. $\textbf{Question 1:}$ Suppose I have a $\gamma\gamma\to\bar{\nu}_{\mu}\nu_{\mu}.$ One of the diagrams will look as one of the ...
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Why are the renormalisation constants the same in LSZ and renormalisation?

To keep it simple I'll phrase everything in terms of scalar fields. We seem to have three constants called $Z$: When we do LSZ reduction we say, as $t\rightarrow-\infty$ then $\phi\rightarrow \sqrt{...
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Why can we shift the field $\phi$, so that $\langle \Omega | \phi(x) | \Omega \rangle = 0$?

Problem Introduction In different derivations of the LSZ reduction formula the author makes a shift of the field $\phi(x)$ $$ \phi'(x) = \phi(x) - \langle \Omega | \phi(x) | \Omega \rangle, $$ and ...
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Is tree-level QCD on-shell constructible, with BCFW?

Is tree-level QCD on-shell constructible, with BCFW? Pure yang mills is on-shell constructible, what is one add into massless fermions?
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Deriving Unitarity of $S$-matrix in 1D Quantum Mechanics

So, I was studying about scattering across a one-dimensional unknown potential ( pretty elementary Quantum Mechanics) and how if we know the $S$-matrix of such a system, we can deduce an awful lot of ...
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Why does the S matrix always contain a factor of $(2\pi)^4?$

In quantum field theory, one usually defines the scattering amplitude as $$S-1=(2\pi)^4\delta(p_{out}-p_{in})M_{Scattering Amplitude}$$ Where S is the S matrix element for any scattering process. It's ...
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Do all unitary-preserving regulators necessarily turn real loop integrals into pure imaginary numbers?

The optical theorem, which results from the unitarity of the $S$-matrix, relates the imaginary part of the forward scattering amplitude to the total cross section. When using this theorem in practice, ...
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Perturbative proof of unitarity of $S$-matrix in QED

In any standard textbook on QFT I know it is claimed that the $S$-matrix in QED is a unitary operator. I have never seen any proof of it. This should be compared with the analogous property of $S$-...
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Dependence of $S$-matrix on a coordinate system in QFT

The $S$-matrix is defined as follows (see e.g. Section 3.2 in Weinberg's "The quantum theory of fields"): $$S=\lim\exp(iH_0\tau)\exp(-iH(\tau-\tau_0))\exp(-iH_0\tau_0),$$ where the limit is taken when ...
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Which vacuum do I use for the path-integral?

In Weinberg, vol. 1, Section 9.2, Weinberg defines the in and out vacua as states with no particles (9.2.4): $$a_{\rm in}|{\rm VAC,in}\rangle=0$$ $$a_{\rm out}|{\rm VAC,out}\rangle=0$$ He does this ...
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What is algebraic structure, operation or relation explains relationship between the 2 diagonal terms of the 2 density matrices?

Quantum decoherence therefore prescinds from the observer and from the measurement process in a certain way preceding it and simulating the collapse of the wave function. In particular, "...
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Relation between in-states and free states in adiabatic approximation

I've been reading the recent book "Quantum Field Theory Lectures of Sidney Coleman", which I find great. I am, however, confused about one passage relating in-states and free states in the adiabatic ...
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On interacting QFTs with two masses $m$ and $M>2m$

hep-th/0412302v2 is an interesting paper by Shvedov about rigorous semiclassical covariant QFT. Shvedov talks on p.27 and p.30 about ''well-known'' properties of a putative interacting QFT with two ...
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Sign of residues of S-matrix elements for bound states

I am currently studying S-matrices in 2D QFT and came across a statement I do not fully understand. If one parametrizes 2-particle S-matrix elements with relative rapidity $\theta$, crossing-symmetry ...
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Deriving the homogeneous scaling of scattering amplitudes

I would like to derive the homogeneous scaling of scattering amplitudes $M(t\lambda,t^{-1}\tilde{\lambda})=t^{-2h}M(\lambda,\tilde{\lambda})$ in the following very general and first principled manner, ...
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The LSZ formula in Peskin and Schroeder

I'm working on the Eq.(7.57) in Peskin(page 236). So I try to verify it with LSZ formula. According to Eq (7.42) So $\mathcal{M}(p \rightarrow p)=-Z M^{2}\left(p^{2}\right)$ In this I have two ...
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Interactions other than yukawa interaction which results in pair annihilation

Can boson annihilates to an electron positron pair can happen in all type of interactions which contains some number of dirac fields coupled with some number of bosonic fields? I asked this question ...
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The use of $a^\dagger(\mathbf{k}) = -i \int d^3x e^{ikx}\stackrel{\leftrightarrow}{\partial}_0 \phi(x)$ in the derivation of the LSZ-formula

I noticed that in Srednicki's derivation of the LSZ-formula the expression (chapter 5) for the creation (and also later for the annihilation) operator by the field operator: $$a^\dagger(\mathbf{k}) =...
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How to do the classification of Yangian invariant?

From Nima's paper:Scattering Amplitudes and the Positive Grassmannian arxiv:1212.5605 page91, we can see that there is a complete classification for $k=2$ Yangian inariants. But I have two questions ...
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An example of Wick's theorem from David Tong's lecture notes

I am studying David Tong's lecture notes, But I got stuck at this point. As an example of Wick's theorem,he gives nucleon scattering. I am unable to get the final answer, I tried a lot by expanding ...
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Independence of $S$-matrix in QED of the gauge choice

The Feynman rules in QED often use different expressions for the free photon propagator (e.g. Feynman gauge, Landau gauge, and others). Is there a textbook on the subject which explicitly checks ...
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Action of Moller operators on stable one particle states in QFT

In Weinberg's book "The QFT", vol. I, it is claimed that for theories with stable single particle states the $S$-matrix maps each such state to itself (see Section 4.3, p. 179). I am wondering if the ...
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Differential cross-section derivation from S-matrix

I am trying to derive the usual expression for the differential scattering cross section: $\frac{d\sigma}{d\Omega} = \frac{q_f}{q_i}|f(\vec q_f,\vec q_i)|^2.$ I am familiar with the derivation which ...
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Off-shell QFT from on-shell amplitudes

Suppose that you're given a non-perturbative $S$-matrix that corresponds to some Wightmanian QFT. By this I mean that you're given a Hilbert space and a unitary operator $S$ that acts on the Hilbert ...
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S-matrix elements for Nucleon-Pion decay

I want to compute nucleon-pion decay rates. I am a bit confused how I can compute the S-matrix. Let's say we have a Nucleon Pion scattering and I want to compute their corresponding S matrix: \begin{...
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How does the interacting vacuum $|\Omega \rangle$ enter the theory?

To calculate scattering amplitudes, we consider $$ A(i\to f) = \langle{f | \hat S|i} \rangle = \langle{f |\mathrm{e}^{ -\frac{i}{\hbar} \int_{-\infty}^{\infty} dt' H_{\mathrm{i}}(t')} |i}\rangle$$ $...
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What should be the scattering matrix to transparent device?

The scattering matrix for a 2 port diode is given as $$S=\begin{pmatrix} 0 & 0 \\ 1 & 0 \\ \end{pmatrix}$$ Suppose I want to make this device transparent, that any mode just passes through ...
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Explicit form of S-matrix on the line

Consider the Hamiltonian $H$ on functions on the line with \begin{eqnarray} H=H_0+V,\\ H_0=-\frac{1}{2m}\frac{d^2}{dx^2} \end{eqnarray} where $V$ is a potential vanishing outside of a bounded interval....
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Is there any relationship between S-matrix elements and the path integral?

Reading Peskin&Schroeder I've made the following curious observation: Comparing S-matrix elements to the definition of the path-integral they look remarkably similar: $$_{out}\langle \mathbf{p}...
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Why isn't it a problem that the first term in the perturbative scattering series yields infinity (or one)?

In QFT we usually want to calculate objects of the form $\langle f|\hat S|i\rangle$ which yields the probability amplitude for the process $i \to f$. We can expand the scattering operator $\hat S$ in ...
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Spin-$J$ Amplitude $A_J(s,t) = - \frac{g^2(-s)^J}{t-M^2}$?

In GSW equation (1.1.2) they define the scattering amplitude for a spin $J$ particle at high energies as $$A_J(s,t) = - \frac{g^2(-s)^J}{t-M^2}$$ mentioning it is an asymptotic approximation to a ...
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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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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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1answer
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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

On p. 71 below eq. (6.12) in Schwartz book, they assume that $$\lim_{t \to \pm\infty}\partial_0 a_p(t)=0.\tag{1}$$ But I thought that this is just an assumption. So we have to construct the ...
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58 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?
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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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