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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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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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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.
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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}...
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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: $$S = 1 + i·T \tag1$$ ...
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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. (...
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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\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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}) = \...
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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 \...
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$pp$ 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 $s$): $$\...
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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 ...
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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 ...
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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 ...
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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_{...
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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 ...
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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 ...
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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)} |\...
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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 ...
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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}}(\...
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LSZ reduction formula derivation

I am following the derivation of the LSZ reduction formula on Weigand's notes. We arrive at eq. 2.64, schematically \begin{equation} \tag{1} \label{LSZ} \langle p_n, \text{out}|q_r, \text{in}\...
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Is an interacting QFT Hilbert space a physical particles Fock space?

There are "Lectures on Quantum Field Theory" by P.A.M. Dirac, in which he claims that QFT state space is not a separable Hilbert space. Also, I have seen some research papers (in axiomatic QFT), which ...
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Green function for the partial wave solution vs regular solution

In the book "The Quantum Theory of Nonrelativistic Collision" from John R. Taylor, the Lippmann-Schwinger equantion for the partial-wave is given as: $$\psi_{l,p}^+=j_l(pr)+\int_0^\infty\,dr'G_{l,p}^{...
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LSZ fermions (Srednicki's book)

In Srednicki's book it is stated that the LSZ formula for fermions holds only if the interacting field $\psi(x)$ is normalized to satisfy $$\langle p,s|\psi(x)|0\rangle = v(p,s)\ e^{ipx}$$ a condition ...
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Why are resonances poles in the S Matrix? A question from Shankar's quantum book

We choose to analyze the problem through the S matrix. Near a resonance we have $$ s_l(k) = e^{2i\delta_l} = ... = \frac{1+i\tan\delta_l}{1-i\tan\delta_l} = \frac{E-E_0-i\Gamma/2}{E-E_0+i\Gamma/2} \...
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Is there an analogue of the LSZ reduction formula in quantum mechanics?

In quantum field theory the LSZ reduction formula gives us a method of calculating S-matrix elements. In order to understand better scattering in QFT, I will study scattering in non-relativistic ...
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Relation between “amplitudes program” and S-matrix theory?

I've been hearing about recent progress in amplitudes, which, as I understand, uses unitarity, locality, and Lorentz invariance to find scattering amplitudes (I often hear buzz words like BCFW ...
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Show $S$-operator is unitary

In an exercise, we are supposed to show that the scattering matrix on the right of $$S_1(E)= \begin{pmatrix}t_1 & r_1' \\ r_1& t_1'\end{pmatrix}\delta(E_f-E_i)$$ is unitary. We are explicitly ...
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Jost function in Scattering Theory

I have a doubt about the behaviour of the Jost function $f_l(p)$. The chapter 12 of the book of Scattering Theory of John R. Taylor shows that the eigenvalues of the $\hat{S}$ matrix may be given as ...