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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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1answer
91 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_{...
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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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108 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 ...
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1answer
78 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.
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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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2answers
110 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 ...
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1answer
75 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 ...
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179 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 ...
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1answer
230 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 ...
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1answer
148 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}...
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459 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: $$S = 1 + i·T \tag1$$ ...
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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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1answer
106 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\...
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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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323 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 ...
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1answer
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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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1answer
75 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 ...
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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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1answer
277 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 \...
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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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1answer
151 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 ...
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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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109 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 ...
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1answer
151 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)} |\...
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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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1answer
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What justifies the use of asymptotic momentum state?

The LSZ scattering approach starts with initial and final asymptotic momentum states. But we know that $\langle k' | k\rangle = \delta^3(k'-k)$, which means that it is not a properly normalizable ...
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1answer
73 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}}(\...
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320 views

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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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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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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135 views

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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1answer
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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 ...
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1answer
110 views

Interpreting operator insertions in the S-Matrix

At several points throughout Weinberg QFT Volume I, Weinberg claims that the sum of all diagrams which have in states $\alpha$ and out states $\beta$ and one off-shell photon at position $x$ is given ...
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LSZ Reduction Formula and Wavepackets in Peskin

In Section 7.2 of Peskin and Schroeder, the LSZ Reduction formula is discussed. I can follow the discussion leading up to the equation after (7.41) [at the bottom of Pg. 225]: $$\sum_\lambda \int\frac{...
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1answer
183 views

Why would the ``in/out'' states asymptotically approach the free Hamiltonian eigenstates?

The ''in/out'' states of the S-matrix in QFT are defined such that at late times they are approach superpositions of direct products of eigenstates of the $\textit{free}$ Hamiltonian \begin{equation} \...
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1answer
401 views

Question regarding the solution of Schrödinger equation for finite potential well and quantum barrier

When solving the Schrödinger equation for finite potential well, the solution outside of the well is $$\psi _{1}=Fe^{{-\alpha x}}+Ge^{{\alpha x}}\,\!$$ and $$\psi _{3}=He^{{-\alpha x}}+Ie^{{\alpha ...
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Quantum symmetries: $S$ or $Z$?

Let $I$ be the action of some QFT (gauge-fixed and including all the necessary counter-terms); $S$ the associated scattering-matrix; and $Z$ the partition function (in the form of, say, a path ...
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1answer
100 views

Why do particle resonances lead to peaks in the cross section?

Since bound states lead to poles in Green functions, I wonder if this is the reason for peaks in the cross section. From a QFT point of view, the infinitesimal cross section $\text d\sigma/\text d\...
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1answer
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Srednicki LSZ reduction formula motivation once more

I have seen this question asked before, but have not really found a satisfying answer, or maybe more exactly, I have not seen an answer that I can really understand. I refer to section 5 of his book,...
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1answer
188 views

Conservation of momentum in scattering process

On page 60 of Quantum Field theory and the standard model from Schwartz, he talks about scattering process with the $S$ matrix. He says: "Since the S-matrix should vanish unless the initial and ...
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1answer
220 views

Calculating $S$-matrix in string theory

To calculate string $S$-matrix, we mainly use Faddeev-Popov gauge fixing method, as in chapter 6 of Polchinsky's book 《string theory》. But in section 6.2, 'tree level amplitude', I didn't find ...
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Weinberg's S-matrix and split into free and interacting Hamiltonian

TL;DR: How can states of an interacting QFT asymptotically follow the trajectories governed by the free Hamiltonian, when, say, the free and interacting groundstates are different, and the states look ...
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128 views

Physical meaning of $\langle 0|S(+\infty,-\infty)|0 \rangle$

when I am reading text book Introduction to Many body physics Piers Coleman 2nd editor, Equation (5.26). I got confused about the meaning of $\langle0|S(+\infty,-\infty)|0\rangle$. first problem $|...
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364 views

In Feynman functional integrals why do we integrate the action over all time?

Say the definition of a propagator in quantum field theory is: $$G_F(x,y)=\int \phi(x)\phi(y) e^{i S[\phi] } D\phi$$ where $S$ is the action. Why do we integrate the Lagrangian density from $t=-\...