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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$S$-Matrix and Relation to Phase Shifts

Given some potential $V(x)$, we can describe the amplitude of incoming and outgoing waves through the scattering matrix $S$ whereby $$\begin{pmatrix} B \\ F \end{pmatrix}= \begin{pmatrix} S_{11} & ...
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Symmetry implies Ward identity

I am thinking about symmetries and that their "quantum" consequences are Ward identities of the form $$<\beta|[Q,S]|\alpha>=0,$$ where $Q$ is the conserved charge associated with the ...
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$A(\phi\phi^*\gamma \gamma)$ with Spinor Helicity Formalism

I am calculating $A_4(\phi\phi^*\gamma\gamma)$ with the spinor helicity formalism (Exercise 2.16). I am following the conventions defined in Elvang's notes Such computation requires three diagrams, ...
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Is there a physical interpretation of the connection between the scattering matrix and bound states?

The square integrability condition of a scattering wavefunction can be written for imaginary wavenumber $k = -\mathrm{i}\kappa$ as $$\int_0^\infty \mathrm{d}r\left|(-1)^l \mathrm{e}^{-\kappa r} - S_l(-...
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$S$-matrix commutation with Hamiltonian

I know from scattering theory that $S$-matrix and the free Hamiltonian $H_{0}$ commute due to energy conservation of incident and outgoing asymptotic states, but can the $S$-matrix and $H = H_{0} + V$,...
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Why does a pole in the Green function correspond to a bound state?

Consider the many-body (zero temperature) fermion Green function $$ G(a,b;t)=-i\theta(t)\langle\psi_a(t)\psi_b^\dagger\rangle $$ Where I'm restricting $t>0$ for causality and that the free ...
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Wave function of a real scalar field in interacting quantum field theory

In interacting real scalar field theory, if I intuitively define the "wave function" of a state as $$\Psi(x)\equiv\langle\Omega|\hat{\phi}(x)|\Psi\rangle.$$ Does this wave function satisfy ...
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State time-evolution in the Interaction picture

What is the Schrödinger-like equation $$i\frac{d}{dt}|\psi(t)\rangle_I=V_I|\psi(t)\rangle_I$$ telling us for the behavior of the interaction picture state vectors, $|\psi(t)\rangle_I$, at infinity/...
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LSZ reduction formula relation

LSZ formula gives a relation between the scattering amplitudes and correlators as $$\langle f |i \rangle = (-i)^{m+n}\int \Pi_{i=1}^m d^4x_, e^{ik_i'x_i (\Box_{x_i} -m^2)}\Pi_{j=1}^n d^4x_, e^{ik_j ...
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More general propagator of a real scalar field

I have some Lagrangian containing a real scalar field $\phi$ with mass $m$. Let $A \in \mathbb{R}$ be some constant. The Lagrangian takes the form: \begin{equation} \mathcal{L} = -\frac{A}{2} (\...
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Lorentz invariance, locality and unitarity of the S-matrix

I am reading these notes on cosmological bootstrap by Baumann and found the following statement, which I am trying to understand / see where it comes from [page 2 of the linked notes]: "[...] S-...
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Why does the LSZ reduction formula only give the connected part of the $S$ matrix?

As an example, using the LSZ reduction formula, the $S$ matrix element for $2\rightarrow 2$ scattering is found in Peskin and Schroeder to be $$\langle \boldsymbol{p}_1 \boldsymbol{p}_2\rvert S \lvert ...
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Physical intuition behind the optical theorem

I am taking a course of QM (as part of my maths studies) and we saw the optical theorem in scattering theory. Namely, we have $$\sigma = \frac{4\pi}{k}\Im f(\vec{q}=0),$$ where $\sigma$ is the total ...
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Schwartz book: Heisenberg equation of motion for quantum fields (equation 7.28)

In his book "Quantum field theory and the standard model", section 7.2 "Hamiltonian derivation" (of the Feynman rules), Schwartz states that the equations of motion $i\partial_t\...
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LSZ formula for initial and final one particle states

The LSZ formula for a real scalar field $\varphi$ is (Srednicki 5.24) $$ \left<f|i\right>=i^{n+n'}\int d^4x_1e^{ik_1x_1}(-\partial_1^2+m^2)...\\ \quad d^4x'_1e^{ik'_1x'_1}(-\partial_{1'}^2+m^2).....
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Proof of unitarity of gauge-invariant S-matrix in Peskin and Schroeder

I'm reading chapter 9.4 "Quantization of the electromagnetic field" of Peskin's and Schroeder's book. When proving the unitarity of the gauge-invariant S-matrix, a trick is used. $$ SS^\...
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(Coleman's notes) Constructing in and out states

In Coleman's QFT (section November 13 in the notes, section 14.1 in the textbook compiled by Chen, Derbes, et al) we associate to any nice, normalizable one-meson state $| f \rangle$ the operator \...
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Should the $S$-matrix always analytic in coupling constant?

If we use Dyson series, the $S$-matrix is always an analytic function of the coupling constant. However, if that is the case, how can non-perturbative effects arise in QFT? My question is, should the $...
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On-shell propagator to an off-shell propagator

I am learning the Ward-Takahashi Identity part of Peskin and Schroeder's textbook of quantum field theory. In the prove process, it involves a diagram 7.66. Then it says that I can understand ...
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Is renormalization needed if non-perturbative approach of $S$-matrix works?

Assume there is a way to construct $S$-matrix in a non-perturbative way, is renormalization still needed in computing $S$-matrix? Since when I encounter regularization and renormalization, they are ...
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PCAC - Ward Identity for non-conserved currents - Derivative and $T$-Order Commutation

I'm currently studying Goldberger-Treiman relation from the book by S. Coleman (Aspects of Symmetry, chapter 2) in which, working in the framework of a not better precised "weak interaction ...
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Why "in" and "out" states $\Psi^\mp_\alpha$ are eigenstates of total Hamiltonian $H$?

"in" and "out" states, $\Psi^\pm$, with reference to Weinberg Vol. 1 pages 109 and 110 could be defined by $$\Psi_\alpha^\pm = \Omega(\mp \infty)\Phi_\alpha\tag{3.1.13}$$ where $$\...
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Finding relation between matrix $S$ and matrix $M$ for wave propagation

we have the same Scattering matrix concept in RF as in quantum physics however, I couldnt derive an expression for the $S$ matrix using the $M$ matrix elements and vice-versa. How can I derive eq 1.13 ...
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Can the $S$-matrix always be decomposed as $S = 1 + iT$?

The LSZ formula for a scalar field $\phi$ with $n$ out-states and $r$ in-states is $$ \langle p_1,\dots,p_n\vert S \vert q_1,\dots, q_r \rangle = \left(\mathrm{i}Z^{-1/2}\right)^{n+r}\prod_i (-p_i^2 + ...
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LSZ formula applied to two-point correlation function

I was trying to find the scattering amplitude using the LSZ formula for a trivial process i.e. applying it to the two-point correlation function, but I kept getting 0 as the answer. I'm not sure ...
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Infinite time limit in two-point correlation function

I am reading the derivation of the two-point correlation function in Peskin and Schroeder (section 4.2). I don't understand the infinite time limit that is taken between eq. (4.26) and (4.27). They ...
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Why does a gluon soft factor depend only on the adjacent legs?

The gluon soft factor can be obtained by taking the soft gluon limit in the $n$-point MHV amplitude $$A(1,2,\cdots, n)=\frac{\langle i,j\rangle^4}{\langle 1,2\rangle\cdots\langle n, 1\rangle},$$ to be ...
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QED Soft Theorems and Momentum Conserving $\delta$ function

The sub-leading soft theorem (or Low theorem) states that the radiative Feynman amplitude is associated to the non-radiative Feynman amplitude in the following way $$\mathcal{M}_{\text{rad}}(\omega_k,\...
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Is the potential matrix-element of the Schrödinger equation in 2D polar coordinates always real?

Assume we want to solve the Schrödinger equation in 2D polar coordinates for an anisotropic potential (in order to get K-matrix). using this separation wavefunction $$ \Psi(r,\phi) = \sum_{j,m} \frac{\...
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Mass counterterms in Yukawa-type meson-nucleon theory (Coleman's notes)

On page 102 of his QFT notes (https://arxiv.org/abs/1110.5013) Coleman uses the following reasoning for calculating the counterterms to the meson and nucleon masses in what he calls "quantum ...
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Branch Cut of Wavepacket - LSZ Reducation Formula Peskin

Peskin & Schroeder, An Introduction to Quantum Field, page 224-225, formula 7.41 \begin{multline}\sum_{\lambda} \int \frac{d^{3} k}{2(2\pi)^3 E_{\mathbf{k}}(\lambda)}\Phi (\mathbf{k}) \frac{i}{p^{...
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QFT's bound states references

At a graduate level, QFT courses teach very well how to perform perturbative calculations using LSZ or even the background field method. Plenty of books are suggested to go into the details of this ...
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QED vertex as 2 equivalent matrix elements

Consider just right handed fields $e_R,\bar{e}_R$ making up the standard electromagnetic current $A_\mu \bar{e}_R\gamma^\mu e_R$. Consider the matrix elements $$\langle 0|J^\mu|e\bar{e}\rangle, \...
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How are annihilation/creation operators used to reach an external state of $|0 \rangle$ in an $S$-matrix?

I'm trying to understand how to compute the $S$-matrix element for $\phi \phi \to \phi \phi$. In "Peskin and Schroeder's Ch. 4.6". I'm lead to believe that, in $\phi^4$ theory, $$ S = \...
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Why do we need to normalise states in quantum field theory?

In QM its obvious that we need to normalise quantum states since their inner product squared represents a probability. This normalization leads to physical states in QM being represented by 'rays' of ...
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Confusion with $S$-matrix of Bhabha scattering

Here are the two Feynman diagrams that corresponds to $e^- + e^+ \rightarrow e^- + e^+$ processes: electron positron scattering and electron positron annihilation respectively. I can clearly see the ...
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Momentum-space Feynman rules (Matthew D. Schwartz, Quantum field Theory and the Standard Model)

I'm reading the Matthew D. Schwartz, Quantum field Theory and the Standard Model, section 7.3 Momentum-space Feynman rules. I think that I'm beginner in quantum field theory and so please understand ...
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Is the matrix element symmetric in a cross-section computation?

Let's consider the cross-section of $A+B\rightarrow C+D$. This involves the usage of an "matrix element" : $M$ for the amplitude of the reaction. Is the element matrix of $A+B\rightarrow C+D$...
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$S$-matrix element for real photon production

In this book (Thermal field theory by Bellac) on page 109 the $S$-matrix element for the transition from an initial state to a final state plus photon $(i)\to(f,\gamma)$ is given: $$S_{fi}^{(\lambda)}(...
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Interacting Picture in QFT

I'm having trouble understanding how the interaction picture describes scattering. In quantum theory, the probability amplitude for a system in state $|i(t_i) \rangle$ to be measured in state $|f(t_f) ...
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Analytic continuation of Feynman amplitudes seems ill-defined

I was reading Peskin & Schroeder's book on Quantum Field Theory and on chapter 7, "The optical theorem for Feynman diagrams" (page 232) they extend analytically the Feynman amplitude $i \...
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Why is this the definition of the scattering matrix and reflection/transmission coefficients?

I've been having trouble with the choice commonly used to name the elements of the scattering matrix in quantum mechanics. Let's say we are dealing with a simple potential barrier in 1D, such that if ...
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Crossing symmetry for mediators

The principle of crossing symmetry holds for all interactions. Consider the reaction: $𝐴+𝐵→𝐶+𝐷$ If one of the particles is swapped between the right and left sides of the reaction, it turns into ...
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How to derive Collinear amplitude proportional to Born amplitude

In the collinear limit, the squared matrix element factorises into (for partons 4 and 5 going collinear) \begin{eqnarray} \overline{\sum}|M_3(1+2 \to 3+4+5)|^2 \approx \overline{\sum}|M_2(1+2 \to 3+4')...
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Is there time reversal symmetry for delta potentials?

I was looking at problem $4.6$ of Gasiorowicz's Quantum Physics, where he asks to prove that the scattering matrix of a potential of the form $$V(x)=\frac{\hbar^2}{2m}\frac{\lambda}{a}\delta(x-b)$$ is ...
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Meaning of Scattering Amplitude in QFT

I've been reading a textbook on QFT, and learned that we can calculate the probability amplitude that a system in state $|i\rangle$ will "collapse" into state $|f\rangle$ after some amount ...
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Feynman diagrams in configuration or momentum space

Some QFT textbooks (for example Mandl-Shaw at page 113, but also Greiner and others) refer to Feynman diagrams in coordinate space or momentum space. What is exactly the difference between the two ...
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Confusion about in and out states, interacting Hilbert space etc, referring to Weinberg QFT

There are many posts related to this issue on this site, but I have found none that answer my specific questions about this matter. I review my understanding of Weinbergs approach. There are probably ...
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Adiabatic turn-on of free multi-particle states

Consider a second-quantized operator $\mathcal{H}_{full}$ describing some interacting QFT, whose action is known on a set of Fock states $\{\mathcal{|F\rangle}\}$, which, in turn, are the eigenstates ...
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Am I free to add a contact term to Feynman diagram calculations?

In a model with 3 particles $\psi$, $\phi$, and $\gamma$, suppose we have three diagrams and subsequently three amplitudes $\mathcal{M}^\mathrm{s}_\mu$, $\mathcal{M}^\mathrm{t}_\mu$ and $\mathcal{M}^\...
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