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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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3
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1answer
453 views

Scattering matrix elements for potential $q(x)$

Suppose I have S.E. with potential $q(x)$, which is known, how do I compute the scattering or S-matrix with respect to this potential? I've tried searching for non-trivial (i.e constant potential ...
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1answer
343 views

Why is the S-Matrix element essentially the residue of the Green function (LSZ formula)?

On Wikipedia, quite similar to the script I am following the LSZ formula is given as $$ _{out}\left<p_1,...,p_n| q_1,...,q_m \right>_{in} =\\ \int \prod_i^m \left(\textrm{d}x^4\, i e^{-q_ix_i}(\...
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1answer
501 views

The derivation steps of the LSZ reduction formula

I am following the derivation steps of the LSZ reduction formula as done in the lecture notes by Timo Weigand the section on S matrix in the full interacting theory, here: http://www.thphys.uni-...
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1answer
202 views

Scattering amplitude calculation using first order Chiral Perturbation Theory Lagrangian

Consider the Chiral Perturbation Lagrangian to first order(quark masses set to zero): $$L = L^{(2)} = \frac{f^2}{4}tr[\partial_{\mu} U \partial^{\mu} U^{\dagger}] ,$$ where U is a $2 \times 2 $ ...
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1answer
290 views

Status of particles in interacting QFT

From my readings in QFT and answers such as this, I've read that the concept of particles and particle-number in interacting systems becomes ill-defined in QFT. Of course, in the real world, a number ...
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0answers
62 views

Møller Opeartors and dressing relations: what picture?

Møller operators can be defined as (Urban, 2013;pg70): \[ \newcommand{\ket}[1]{\left|#1\right>} \newcommand{\braket}[2]{\left<#1|#2\right>} \Omega_{\pm}=\underset{t\rightarrow \mp 0}{\...
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0answers
102 views

Field redefinitions involving derivatives

Consider the field equation $(\partial^2-m_1^2)(\partial^2-m_2^2)\phi =0 $ Now let me make the field redefinition $\psi = (\partial^2-m_2^2) \phi$ The question is, will the S-matrix be invariant ...
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1answer
284 views

Non-local field redefinition and $S$-matrix

It is known that for local field redefinitions for which the LSZ formula is valid: $$\langle 0|\phi(x)|p\rangle \neq 0$$ field redefinitions don't change the S-matrix. (See QMechanic's answer to ...
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1answer
631 views

Derivation of the S-matrix/Dyson's formula from David Tong's lecture notes

I am studying quantum field theory from David Tong's lecture notes and I am stuck at a particular place. In Page 52., under the heading 3.1.1 Dyson's Formula, Tong introduces an unitary operator $U(...
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1answer
49 views

How are the linear differential operator equations are solved in perturbation theory?

I have a problem in solving the first order linear differential equation: where Uo and Ho are [n x n] matrices and |i> is a [n x 1] column vector. The author states that by considering a complete ...
3
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1answer
344 views

Contact terms in Dyson-Schwinger equation can be ignored?

According to this text here http://www.physics.indiana.edu/~dermisek/QFT_09/qft-II-4-4p.pdf contact terms do not affect the scattering amplitude. But These contact Terms are there; the question is: ...
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1answer
154 views

How is scattering possible?

Bjorken and Drell's book shows that the 'in' and 'out' states are eigenstates of the full interacting theory. If this is true, then how is scattering possible if both in and out states are eigenstates ...
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3answers
449 views

How does a perturbation theory make sense in quantum field theory?

The idea of a perturbation series in powers of a coupling $\alpha\ll1$ (for example, the fine structure constant in QED) make sense if the contribution of $(n+1)^{th}$ term in the series is smaller ...
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1answer
135 views

Why this loop carries an integral if there is no undetermined momenta?

Consider the following Feynman diagram: I've read that it will have associated with it one integral over the loop. The issue is, in Schwartz book the Feynman rules for momentum space are: Internal ...
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3answers
2k views

What actually means to compute things at tree level?

In his QFT book, Matthew Schwartz first talks about tree level as follows: We will begin by going through carefully some of the predictions that the theory gets right without infinities. These are ...
4
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2answers
484 views

S-matrix derived directly in terms of the interaction picture

Consider a quantum mechanical system with Hamiltonian $$H=H_0+H_{\text{int}}.$$ Consider $H_0$ to be time-independent, so that its associated time-evolution operator is $U_0(t,t_0)=e^{-i(t-t_0)H_0}$....
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1answer
324 views

How to use Wick's theorem to compute this matrix element?

I wanted to see how to use Wick's theorem in practice (I know with Feynman diagrams it is better, but here I want to do this with Wick's theorem only), so I considered computing the matrix element for ...
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1answer
207 views

One Particle State in Interacting QFT (Eqs. 4.88 4.89 in Peskin & Schroeder)

How to derive equation 4.88 in section 4.6, page 108, of Peskin & Schroeder? $$\left|k_{1}k_{2}\right\rangle\propto\lim_{T\rightarrow+\infty(1-i\epsilon)}e^{-iHT}\left|k_{1}k_{2}\right\rangle_{0}....
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1answer
274 views

Vacuum bubbles and LSZ reduction

Let me preface this by saying that I don't have an issue with this: $$ \langle\Omega|T\phi_H\cdots\phi_H|\Omega\rangle = \frac{\langle 0|T\phi_I\cdots\phi_IS|0\rangle}{\langle 0|S|0 \rangle}, $$ ...
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1answer
664 views

How to tell whether a Feynman diagram is $t$-channel or $s$-channel by looking?

By looking at a diagram, how does one tell whether it represents a $s$-channel process or a $t$-channel process i.e., without finding the amplitude? I'm familiar with Mandelstam variables but I've ...
3
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1answer
1k views

LSZ Reduction Formula

In Section 3.7.2 of Tong's QFT notes the LSZ reduction formula is briefly discussed. Essentially, this tells us that for S-matrix elements we can use the same momentum space Feynman rules as for ...
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1answer
415 views

Computing S-Matrix Elements from Feynman Diagrams

In Peskin and Schroeder (PS), the Feynman rules for calculating correlation functions are first presented. Only terms involving all field contractions need to be considered. In Section 4.6, this is ...
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0answers
84 views

Finding a reference or a proof for scattering matrix of a dihedral corner reflector and Bragg surface?

I'm reading the book Polarimetric Radar Imaging: From Basics to Applications, on chapter 6 it is said that: The s-matrix for a dihedral corner reflector has the form: $$S=\begin{bmatrix}e^{2j\...
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0answers
49 views

Is it true that the lowest order non-zero contribution for an S matrix element has always to be convergent?

I was talking with a colleague he told me that if there is no non-zero tree level diagram contributing, then the one loop contribution cannot be divergent. I replied to this saying that this indeed is ...
3
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1answer
207 views

Transition amplitude for QED+QFD+QCD interactions

As I understood, Feynman diagrams are nothing more than pictures for the transition amplitudes (up to some orders). For this we introduce a interaction vacuum state $|\Omega\rangle$ then we are able ...
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0answers
238 views

Classical analogue of the theorem of equivalence of the S-matrix

In quantum field theory there is a statement called the equivalence theorem of the S-matrix. S-matrix is invariant under reparametrization of the field. Is there in classical mechanics, the analogous ...
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1answer
294 views

Gauge transformations at infinity

Consider the following paragraph taken from page 15 of Thomas Hartman's lecture notes on Quantum Gravity: In an ordinary quantum field theory without gravity, in flat spacetime, there two types of ...
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84 views

Double poles in S-matrx

Are double (or higher) poles forbidden in general in an S-matrix? If so, why? If not, under which conditions can they appear, and what would be the interpretation?
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1answer
220 views

How to prove the equivalence of two different definitions of $S$-operator?

I read there are two definitions about $S$-operator: The first one (e.g (8.49) in Greiner's Field Quantization) is: $$S_{fi}\equiv \langle \Psi_p^{-}| \Psi_k^{+}\rangle$$ where $|\Psi_p^{-}\rangle$ ...
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1answer
61 views

How can we justify $\psi(\textbf{r})\to\psi_{in}(\textbf{r})+\psi_{sc}(\textbf{r})$ at $|\textbf{r}|\to \infty$ but not at finite $|\textbf{r}|$?

In quantum scattering theory, the outgoing wave at $|\textbf{r}|\to\infty$, scattered from a localized potential, can be written as $$\psi(\textbf{r})\to\psi_{in}(\textbf{r})+\psi_{sc}(\textbf{r})$$ ...
3
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1answer
238 views

Confusion over assumptions made in the LSZ reduction formula

I've been reading through a derivation of the LSZ reduction formula (http://www2.ph.ed.ac.uk/~egardi/MQFT_2013/, lecture 2, pages 2-3) and I'm slightly confused about the arguments made about the ...
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1answer
57 views

On the identification of asymptotically free in and out states for a certain interaction Lagrangian

Consider the Lagrangian $ L= L_D + L_{KG} +L_{int}$ where the first term is the Dirac Lagrangian, the second is the Klein-Gordon and the interaction term is $L_{int} = g \bar \psi \tau \cdot \phi \psi$...
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1answer
292 views

Is Feynman diagram all about S-matrix?

This is what I am concluding after reading texts about Feynman diagram: that Feynman diagram is all about finding S-matrix or scattering amplitudes. Is there anything more to Feynman diagram?
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1answer
470 views

Allowed Field Re-definitions in QFT

I am trying to understand which field redefinitions are allowed in a QFT. The textbooks I have read appear to treat this topic flippantly. I assume that one cannot arbitrarily manipulate the ...
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0answers
397 views

Calculation of $b \to s~ l^+ l^-$ penguin diagram

I'd like to calculate the matrix element amplitude for $b \to s~ l^+ l^-$ penguin diagram mediated by Z boson or the photon , like : These calculations are made of course from many time ago, so if ...
0
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1answer
552 views

S-matrix and time-evolution operator

On page 108 of Peskin Shroeder. If the formula $$ |\mathbf{P_\cal{A}}\mathbf{P_\cal{B}} \rangle \propto \lim_{T\to \infty(1-i\epsilon)} e^{-iHT}\, | \mathbf{P_\cal{A}}\mathbf{P_\cal{B}} \rangle_0 \...
2
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1answer
272 views

Scattering, 4 point correlator, #distinct Feynman diagrams

In order to compute the scattering probability that two particles of type 1 (associated to $\phi_1(x)$) which come from the far past with the momenta $p_1$ and $p_2$, to scatter and evolve into two ...
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0answers
161 views

Orthogonality of matrix elements, trace techniques

I am currently reading Thomson's "Modern particle physics", and I have trouble understanding a concept that he somewhat gives for granted. Using Feynman rules, I am able to compute the matrix element ...
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2answers
2k views

Quantum Field Theory in position space instead of momentum space?

What are the reasons why we usually treat Quantum Field Theory in momentum space instead of position space? Are the computations (e.g. of Feynman diagrams) generally easier and are there other ...
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0answers
506 views

Position space Feynman rules to momentum space

For example consider $2\to2$ scattering in scalar $\phi^4$ theory. When the in/out coming positions are fixed, it is easy to calculate in terms of contractions. When we fix the incoming / outgoing ...
3
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1answer
124 views

Why does the $\mathbb{1}$ in $S = \mathbb{1} + i \mathcal{M}$ correspond to no scattering?

I am a beginner in quantum field theory and I am learning from the lecture notes by David Tong. On Page 58, he gives an example of two nucleons scattering and says that we are only interested in $\...
3
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1answer
281 views

Bare mass to physical mass in the limit of vanishing interaction as $t\rightarrow \pm\infty$

In the Quantum Field Theory by Itzykson and Zuber (page 202), they assume that the coupling terms in the Equation of motion (of an interacting theory) vanishes smoothly as $t\rightarrow \pm \infty$. ...
3
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1answer
174 views

What are the asymptotic momentum eigenstates? Dressed quanta or quanta of free theory?

Suppose I consider an interacting theory, say QED (with electrons and photons). Let, free electrons, I mean the quanta of the free Dirac Lagrangian. The dressed electron differs from the free electron ...
6
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1answer
609 views

Positivity of residues and unitarity in scattering amplitudes

I am reading "Superstring Theory" by Green, Schwarz, Witten. In the introduction, about the Veneziano amplitude (below eq. 1.1.16/17), they say that The residues of poles must be positive in a ...
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0answers
499 views

Foundation of path Integral formulation of QFT, derivation and meaning of LSZ formulas

I'm currently studying path integral in quantum field theory. I am comfortable with path integrals, and also path integral formulation of QM, but I was asking if there is a self consistent coherent ...
3
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1answer
343 views

About asymptotic field assumption in QFT

I'm studying QFT and in a trouble about asymptotic assumption. It states every Heisenberg field converges to free field (asymptotic field) if one takes a limit of $x_0\rightarrow \pm \infty$. $$ \phi(...
6
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1answer
180 views

S-Matrix Interpretation and Predictions

How does one distinguish between the second-loop contribution of a known particle, and the first-loop contribution of a more massive-and as yet undiscovered-particle in the S-matrix and/or ...
6
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1answer
225 views

Mass of the asymptotic fields: physical or bare?

If I understand it correct, then the physical mass $m$ of a particle, is the mass in presence of the interaction (i.e., the mass of the dressed particle) where as the bare mass $m_0$ is the mass in ...
3
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0answers
164 views

What picture is the $S$-matrix defined in?

I am looking into the definition of the $S$-matrix, and have found two different cases. Firstly I have seen it derived that (see here, link to Google books p110): $$ S=U_S(\infty,-\infty)$$ But more ...
5
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1answer
258 views

Feynman diagram representation of variational derivative of S-matrix

For quite some time I am struggling to understand section 6.4 in Weinberg volume 1. He observes there that if interaction hamiltonian density is extended by coupling to c-number fields $\epsilon$, $$ \...