Questions tagged [propagator]

propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum.

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28 views

Scattering Greens function exactly at energy of bound state

I have a small bit of confusion about the expansion I am seeing in literature for the Greens function in time independent scattering theory. For example here is an excerpt from Scattering Theory of ...
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Use of Cauchy's integral formula in the derivation of the Feynman propagator

In deriving the Feynman propagator in Timo Weigand's 2014 QFT2 notes, at the top of page 37, (equation 1.170), we use Cauchy's integral formula: $$g(z_0)=\frac{1}{2\pi i}\oint_{C_1}\frac{g(z)}{z-z_0}\...
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Question about the Lehmann-Kallen representation in Srednicki [duplicate]

So when deriving the LK form of the exact propagator he says (chapter 13): $$ \langle 0| \phi(x)\phi(y)|0\rangle=\int d\tilde k e^{ik(x-y)}+\int_{4m^2}^\infty ds\rho\int d\tilde ke^{ik(x-y)}. \tag{13....
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Why commutator of positive and negative parts of scalar field is equal to the Feynman propagator?

Peskin & Schroeder state that the contraction of two fields, defined as the commutator: $$ [\phi^+(x),\phi^-(y)]\qquad \text{assuming}\ x^0>y^0$$ is equal to the Feynman propagator $D_F(x-y)$. ...
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Quark propagator

I was reading about Large N QCD, and specifically, T Hooft Double line notation, when I stumbled across the quark propagator- $$\langle \psi^a (x) \overline{\psi^b}(y) \rangle = \delta^{ab} S(x-y) $$ ...
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Path integral formulation for Green's functions

In the first place, I am struggling when trying to derive the path integral formulation of the Green function for non-interacting particles $$G_{ij}(\tau)=-\frac{1}{Z}\int D(\bar{\psi},\psi) \psi_i(\...
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1answer
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How to use principal value in propagator definition?

The propagator for a scalar particle can be written as $$ \frac{1}{x + i\epsilon} = {\rm PV}\left( \frac{1}{x} \right) - i\pi\delta(x), \quad x = p^2 - m^2, \tag{1} $$ where $p, m$ are the momentum ...
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Simple relation of Dirac matrices in Dirac propagator

There is this equivalence between representations for the Dirac propagator: $$\frac{i}{\not{p}-m}=\frac{i(\not p+m)}{p^2-m^2}.$$ The only way i could think this to be true is if $\gamma^{\mu}p_{\mu}\...
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Propagator for Free Particle in Energy Basis

I'm continuing this question because the answer to this is not helping me. As the OP said the propagator in the problem should be given in energy basis : $$U(t)=\sum_{\alpha=\pm}\int_0^\infty |E,\...
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Propagator from a spacetime point to itself and physical interpretation of vacuum

I am following a lecture note on the QFT. But am a little confused about some parts related to the vacuum bubbles. We define the Feynman propagator, $D_{F}(x-y)$, as giving the amplitude for a ...
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Integral of gauge field bulk to boundary propagator in AdS

I'm studying from the book "Introduction to the AdS/CFT Correspondance" by Horatiu Nastase. In page 190, he defines the gauge field bulk-to-boundary propagator in Euclidean $AdS_{d+1}$ given ...
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Self-loops in Chern-Simons theory

Consider the dumbbell graph decorated with propagator $P$ one the edges and with integration variables $x$ and $y$ on the vertices. We associate to it the following integral: $$ I = \int_{x,y} P(x,x)P(...
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1answer
61 views

What is going on with this generating functional? (QFT)

I am reading Peskin and Schroeder's chapter on functional methods and they compute the following correlation function: \begin{equation*} \begin{split} \langle 0| T\phi_1\phi_2\phi_1\phi_3 |0\rangle ...
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Diffusion and dispersion relation

I'm looking at some dispersion relations for some complex systems and realised I actually don't have a clear understanding of what physics I can get from a dispersion relation from equations that ...
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Is the Green function of electromagnetism a scalar or a tensor?

When I check classical electromagnetism books Maxwell equations \begin{equation} \Box A^\nu (x)=\frac{4\pi}{c}j^\nu (x) \end{equation} can be solved using a scalar Green function $G(x,x')$ \begin{...
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1answer
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Trying to prove that a propagator only depends on the difference of coordinates if our system has translational symmetry

As the title says, I'm trying to prove that if a system is translation invariant, then its propagator depends only on the difference of coordinates. That is to say $$K(\boldsymbol{x},\boldsymbol{x}';t ...
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Calculating the Nambu-Gorkov propagator (given its inverse) for quarks

I am working through this paper on the calculation of a Nambu-Gorkov propagator, could someone explain how did we reach the next step right after equation (16)?
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1answer
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SDE with drift multiplied by telegraph like random process

Let the stochastic process $\{X_t\}$ be defined by the following SDE (Ito's convention for discretization) $dX_t=\frac{1}{p}S_tg(X_t)dt+\sqrt{2}dW_t$ where $W_t$ is a standard Wiener process, $g: \...
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Expanding the quantum mechanical propagator in terms of the (non-degenerate) eigenvalues of the Hamiltonian

Could anyone please help me with this derivation? I am struggling to see how the Propagator Can be expanded out into the form This is a non-degenerate two-level system. Any help would be greatly ...
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A Oscillatory integral in light-cone coordinates

I am trying to evaluate an integral in light-cone coordinates Where light-cone coordinates in 1+1D are defined by $x^+=\frac{x^0+x^1}{\sqrt 2}$ and $x^-=\frac{x^0-x^1}{\sqrt 2}$. The integral that I ...
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Is the free Dirac fermion propagator divergent in 2+1 dimensions?

I think it diverges since we have $$g(p)\sim\int dp p^2\frac{1}{p}\sim p^2.$$ So do we need to regularize even a free theory?
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How does Tx Power affect the Free Space Path Loss model?

On this website one can use the FSPL model to calculate the Path Loss. This is typically interpreted online as the theoretical signal strength (in dB) that one may receive (e.g. from some WiFi). ...
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Massless scalar propagator in Euclidean space and Green's equation

In this paper (Erickson et al, 2000), the authors claim in eq. (46) that the Green's equation corresponding to a bosonic propagator $\Delta(x)$ in $2\omega$ dimensions is: $$ - \partial^2 \Delta(x) = \...
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Schrödinger equation and propagator

I took a quantum mechanics course many years ago and I'm currently revising it and learning some more advanced subjects of it. Now, I'm trying to understand the use of propagator to find the wave ...
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Integral for 2-point function in $d$ dimension

When I was reading Joao Penedones's TASI Lecture on AdS/CFT, it says in (104): $$ \lim_{z\rightarrow 0}\int \frac{d^dx}{z^{d-1}}\frac{z^\Delta}{(z^2+(x-y_1)^2)^\Delta}\partial \frac{z^\Delta}{(z^2+(x-...
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1answer
67 views

Why do some of the Feynman propagators for a complex scalar field vanish?

I am learning QFT. Earlier we showed that a complex field can be decomposed like so: $$\begin{align*} \phi &= \int \frac{d^3k}{(2\pi)^3}\frac{1}{\sqrt{2\omega_k}}\big(a(\mathbf{k})e^{-ikx}+b^\...
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Ordering ambiguity in the Feynman propagators obtained using Wick's theorem

Applying Wick's theorem to a string of four field operators, $\phi_a\equiv\phi(x_a)$: $$T(\phi_1\phi_2\phi_3\phi_4)=\{...\}, \tag{1}$$ we obtain several terms, three of which are fully contracted ...
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1answer
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Why is it necessary to wrap our contour around the branch cut at $+ im$ in the spacelike Klein-Gordon propagator? (P&S)

This question is in reference to eq. (2.52) on the bottom of page 27 in Peskin and Schroeder. To evaluate the Klein-Gordon field propagator along a spacelike interval we wrap the contour around the ...
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1answer
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Difficulty with Kallen-Lehmann spectral functions

The following is for $D=4$. The correlators at a fixed point are power laws of the form $x^{-2\Delta}$, where $\Delta$ is the scaling dimension. Suppose I wish to find the nature of the spectrum at ...
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Branch cut singularity in photon propagator at one loop

I'm going through Peskin & Schroeder chapter 7, and I am finding difficult to understand how the branch cut singularity appears in the one loop correction to the photon propagator. Essentially, ...
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1answer
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Why is $\langle 0| \phi (x) \phi (y) |0 \rangle = \int \frac{d^3p}{(2\pi)^2} \frac{1}{2E_p} e^{-i p (x-y)}$ ? (Peskin and Schroeder equation 2.50)

More specifically, starting from $\langle 0| \phi (x) \phi (y) |0 \rangle$ I have arrived at the expression: $D(x-y)=\langle 0| \phi (x) \phi (y) |0 \rangle= {\Large\int \int} \frac{d^3p \cdot d^3q}{(...
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Renormalization at different scales in $\phi^4$ theory

Starting from the bare Lagrangian-density $\mathcal{L} = \frac{1}{2}(\partial^2 - m_0^2) \phi_0 - \frac{\lambda_0}{4!}\phi_0^4$ one introduces the renormalized field and parameters as $\phi_0 = \sqrt{...
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How to show the (Klein-Gordon) retarded propagator satisfies its equation of motion?

The retarded propagator for a massless scalar field is $$ G_R(t,\mathbf{x} ;t',\mathbf{x}' ) = \frac{ \Theta(t-t') \delta\big( - (t-t')^2 + |\mathbf{x} - \mathbf{x}'|^2 \big)}{2\pi} \tag{1} $$ which ...
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Why are the propagators in old-fashioned QED oblique, while in modern QED they are horizontal (or vertical)?

In old-fashioned Quantum Electrodynamics, one can find diagrams such as these (probably Stückelberg was the first to use this notation, a kind of predecessor of Feynman diagrams): In modern QED this ...
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51 views

Vanishing terms in calculating a simple propagator in QFT

In David Tong's lecture notes on quantum field theory, at the top of page 38, we calculate the amplitude for a particle to propagate from $y$ to $x$: $$\begin{align}\langle0|\phi(x)\phi(y)|0\rangle&...
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2answers
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Retarded vs Feynman Klein-Gordon Propagators

Although I follow all the manipulations -- Green's functions, choice of contour/i$\epsilon$ prescription, etc -- I seem to be struggling with too many trees. The forest remains blurry. In ...
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How can the connected Greens 2pt function be summed as a geometric series of the self-energy if the self-energy contains divergent terms?

In Ryder's book on QFT page 341 we can see $$\begin{align} D_{\mu\nu}'=D_{\mu\nu}-D_{\mu\alpha}\big(k^\alpha k^\beta-g^{\alpha\beta}k^2\big)\Pi(k^2)D_{\beta\nu} \end{align}$$ and hence putting $D_{\mu\...
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Expressing Feynman propagator with absolute value instead of Heaviside step function?

I recently saw an interesting paper: https://arxiv.org/abs/1912.13435, where the author wrote the Feynman propagator of free massless scalar field as $$ D_F(x-x') = -i\int\limits_{-\infty}^\infty \...
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1answer
36 views

Propagator in normal modes

I started with the Hamiltonian of coupled oscillators in a circular lattice(with $m=\hbar=1$ and $x_{a+N}=x_{a}$) $$H=\frac{1}{2}\sum_{a=0}^{N-1}\left[p_a^2+\omega^2 x_a^2+\Omega^2\left(x_a-a_{a+1}\...
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Discrepancy in two-point correlator integral of Harmonic Oscillator

The two point correlator of Quantum H.O. of natural frequency $\omega$, calculated using path integrals, is $$C_{2}=D\left(t_{2}-t_{1}\right) \propto \int \frac{d w^{\prime}}{2 \pi} \frac{e^{-i w^{\...
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Closed loop in scalar quantum field or QED

I want to know how to write the propagation amplitude of a 2 point closed loop in scalar quantum field or QED (i think it should be even possible as a scattering process in RQM). As an example, in ...
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1answer
166 views

What is the string equivalent of the Feynman Propagator?

The Feynman propagator for a massless point particle is proportional to: $$\Delta(x-y;t_1-t_2)=\frac{1}{|x-y|^2-(t_1-t_2)^2}$$ which is, formerly, the Fourier transform of $\dfrac{1}{|k|^2-E^2}$. For ...
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1answer
93 views

Generator of QED in path integral approach

Consider the interacting field Lagrangian density of the real KG field \begin{equation}\mathscr{L}=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{m^2}{2}\phi^2-\frac{\lambda}{4!}\phi^4 \end{...
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1answer
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How to organize this strong coupling perturbation theory?

Consider a 2d scalar field theory with quartic interaction $$S[\phi]=\int d^2x \left((\nabla\phi)^2+m^2(\phi^2+g\phi^4)\right)$$ I want to compute the partition function $$ Z[m,g]=\int\mathcal{D}\phi\,...
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Is there a non-degenerate propagator for scalar fields?

I am trying to work out a functional propagator $K[\phi_{in},\phi_{out};t]$ for a scalar field for a free Klein Gordon field. It must satisfy the Hamiltonian equation: $$\left(i\frac{\partial}{\...
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1answer
32 views

Exact solution for propagator with general quadratic potential?

Given the Hamiltonian: $$H = \frac{1}{2m}p^2 + x_i M^{ij} x_j.$$ Is there an exact non-relativistic propagator for this, where $M$ is a general symmetric matrix? Similar to the harmonic oscillator ...
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Understanding ground state expectation values of time ordered products of operators

Working through chapters 6 through 8 in Srednicki's "quantum field theory", I am having some trouble conceptually understanding what is happening when we take time ordered products of ...
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2answers
71 views

Infinite correlation functions in free field theory

In a free scalar field theory, Wick's theorem guarantees that $\langle \hat\phi(x)\rangle = 0$ and $\langle \hat\phi(x)^2\rangle = \infty$. Given that $\hat \phi(x)$ creates a particle at $x$, these ...
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80 views

Generating Functional for Complex Scalar Theory

The generating functional for a free complex scalar field theory is given by: $$W[J,J^*]=\int D\phi D\phi^* \exp (i \int_{}^{} d⁴ x [(\partial_{\mu}\phi)^*(\partial^{\mu}\phi) -m^2\phi^*\phi + J^*\...
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Taylor Expansion of Feynman Propagator with regulator masses and gamma matrices

I am currently trying to understand a really old paper of Jackiw and Coleman: "Why dilatation generators do not generate dilatations". There, at some point they arrive at the following ...

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