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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Feynman propagator from Hadamard propagator
The Feynman propagator is defined as
$$i G_F = \theta(t-t')G^+ + \theta(t'-t)G^-. \tag{1}$$
Using
$$G^{(1)} = G^+ + G^-,$$
$$G_R = -\theta(t-t')G, $$
$$G_A = \theta(t'-t)G, $$
$$\bar{G} = \frac{1}{2}(...
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Finding scalar propagators in QFT for specific spacetime dimension $d$ and mass $m$
I need to understand how in practice one finds propagators for given $d$ and $m$ in quantum field theory. I can write down the theory provided for it but I don't know how to use it.
We will compute ...
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How to show a propagator of massive spin-1 field is the Green function to its equation of motion?
During my QFT course my teacher said the propagator of the massive spin-1 field is the green function of the equation of motion derived from the Proca Lagrangian, which is shown in the next two ...
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Non-vanishing amplitude outside light cone doesn't violate causality? [duplicate]
I am following Peskin & Schroeder's QFT book. And on equation 2.51, we get an expression for the free Klein-Gordon propagator for timelike intervals $x^0-y^0=t$, $x-y=0$:
$$D(x-y) \sim e^{-imt}\...
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Is the scalar-field Feynman propagator at the origin ($x=0$) equal to 1?
I was reading about Feynman rules for scalar field in $\phi^4$ theory in section 4.6, pages 113-114 of Peskin & Schroeder, and, calculating amplitudes for processes, the authors show that Feynman ...
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Wick Rotation vs Sokhotski-Plemeli Method to compute internal loop of Feynman correlators
When computing loop integrals in QFT, one often encounters integrals of the form
$$\int_{-\infty}^\infty\frac{dp^4}{(2\pi)^4}\frac{-i}{p^2+m^2-i\epsilon},$$ where we are in Minkowski space with metric ...
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Wick contraction between two scalar fields
I have a short question about Wick contraction.
It is given that $$\phi\left(x\right) = \phi^{+}\left(x\right) + \phi^{-}\left(x\right)\tag{1}$$ where:
$$\phi^{+}\left(x\right) = \int \frac{d^3p}{\...
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CDW correlation function for 1D Dirac fermion in condensed matter
I am following Shankar's lecture notes on bosonization, specifically the theory of left-/right-moving fields for a low-energy 1D fermionic chain. For now, I ignore the Heisenberg time dependence of ...
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Laplace Green function on $R \times S^3$
On flat Euclidean $R^4$ the Laplace operator has the Green function $G(x,y) = \frac{1}{4\pi^2(x-y)^2}$, i.e.
$$-\Delta G(x,y) = \delta^4(x-y).$$
What would be the corresponding Green function on $R \...
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Particle density and current in terms of Green function
Consider a non-relativistic free-fermion system. I am wondering how to calculate observables like average particle density and average current in terms of momentum-space Green functions. I know that ...
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Calculating gauge propagator in minimally coupled, non-relativistic fermion system
For context, I am trying to derive Eq. 4.1 of $T_c$ superconductors">this paper. Consider the action
$$S[\psi^\dagger, \psi, a] = -\int d\tau \int d^2r \sum_\sigma \psi^\dagger (D_0-\mu_F-\frac{1}{...
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Exact results for gluon propagator diagrams in QCD
I am looking for analytic results for gluon propagator Feynman graphs. Ideally as a power series in the dimensional regularization parameter $\epsilon$ and more ideally involving some of the non-...
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Fradkin renormalization group propagator integral
I am studying Fradkin's lecture notes on advanced field theory, and in the section on Wilson RG he has a list of integrals of the propagators of the fast fields. One of such integrals is:
$$
\int d^D ...
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Proof that $-\partial^2 G(x, y) = \delta(x-y)$ for free field propagator
I recently realized that there is a slightly pedantic issue when one normally proves that the equations of motion acting on the free field propagator gives a delta function which I have become ...
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Cancelling one-loop divergences in non-linear sigma model expansion term
In the appendix A of this paper by Braaten et al., the authors try to compute the divergences of two integrals that come from an expansion of an action $I$ in $\langle e^{iI} \rangle$, via dimensional ...
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General interpretation of the poles of the propagator
I am somewhat familiar with the fact that the poles of the Feynman propagator in QFT give the momentum of particle states. I'm also familiar with the KL spectral representation in that context (See ...
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Inverse of an operator [closed]
I want to understand how to find the Inverse of an operator.
I know it involves the use of Green's function but I can't seem to figure out how.
Here is the actual problem:
On page 302 of Peskin&...
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Klein-Gordon and Green's Function
I want to prove the following relation:
$$(\partial_x^2 + m^2)\langle0|T\phi(x)\phi(x_1)|0\rangle = -i\delta^{(4)}(x - x_1).$$
My Approach: Consider LHS
$$(\partial_x^2 + m^2)\langle0|T\phi(x)\phi(x_1)...
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How to apply multiple Klein-Gordon operators to products of propagators?
I have the 4-point correlation function for a scalar free field
$$
\langle{0} | T \phi_1 \phi_2 \phi_3 \phi_4 | 0 \rangle = -\left[ \Delta_F(x_1-x_2) \Delta_F(x_3-x_4) + \Delta_F(x_1-x_3) \Delta_F(x_2-...
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Fermion Propagator
Will the fermion propagator change if instead of deriving it from the Lagrangian
$$\mathcal{L}=i\bar{\Psi}\gamma^{\mu}\partial_{\mu}\Psi
-m\bar{\Psi}\Psi\tag{1}$$
I derive it from
$$\mathcal{L}'=\frac{...
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Curved spacetime generalization of Bethe-Salpeter equation
I am interested in the problem of bound states in QFT in curved spacetime. I was wondering if the generalization of the Bethe-Salpeter equation is as simple as replacing the Green’s functions in the ...
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How to compute the Feynman propagator for the Proca field?
I was repeating each step of the exercise 6.4 of the Greiner's book "Field quantization" when I discovered that there is a passage which I can't reproduce, the calculations are lengthy and ...
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Schwinger proper-time representation of Feynman propagator in Coordinate space
I'm working on a QCDSR paper which calculates mass of $B$ meson in the present of external magnetic field.
the author works with Schwinger proper-time representation of Feynman propagator in momentum ...
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Confusion Regarding the Propagator [duplicate]
To my understanding, the expression $$G^+=\theta(t_f-t_i)\langle x_f|\mathcal{\hat U}(t_f,t_i)|x_i\rangle$$ represents the probability amplitude that a particle starting at position $x_i$ at time $t_i$...
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Normalization of the harmonic oscillator propagator
The propagator of a quantum system is defined by $$\mathcal{K}(t,x;\,t_{0},x_{0})\,\equiv\,\left\langle x\right|\hat{U}(t,\,t_{0})\left|x_{0}\right\rangle.$$ In this notation, the unitarity demands ...
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How is van Kortryk's identity used? [closed]
The Wikipedia article on propagators mentions
that the harmonic oscillator propagator can be derived
from the free particle
propagator using van Kortryk's identity.
This is followed by the identity ...
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Why must the propagator exponent be imaginary?
In response to asmaier's question, qmechanic showed why the propagator must be $\exp(cS)$. That made perfect sense. But can it also be shown that $c$ is imaginary? I believe it follows from ...
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How is the free particle propagator derived? [closed]
The free particle propagator is a well known function,
for example, see Wikipedia.
However, I cannot find a source that explains how to derive
the free particle propagator.
Please explain how the free ...
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Momentum Space Propagator from Path Integral Formulation of “Polyakov-style” action for a massive relativistic point particle
I have the derived the following expression for the propagator of a “Polyakov-style” action for a massive relativistic point particle:
\begin{equation*}
G(X_2 - X_1) = \mathcal{N}'\int_{0}^{\infty}...
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Propagator for a massless scalar field in $d$-dimensional spacetime [closed]
I'm trying to show that for a free massless scalar field, the 2-point correlation function in $d$-dimensional spacetime has the following form:
$$<\phi(x)\phi(y)> = \int \frac{d^d{p}}{(2\pi)^d}\...
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How do we interpret the second-order differential operator in the QFT path integral?
For the free scalar field theory, the path integral has a differential operator term in the exponent,
$$
Z[J] = \int \mathcal{D}\phi \, \exp\left( i \left[ -\frac{1}{2} \int d^d x \, \phi(x) A \phi(x) ...
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Description about cancellation of bubble diagrams while computing correlation function by M. Schwartz
I'm trying to understand M.Schwartz's description on his own QFT & SM book, which is about cancellation of disconnected diagrams so called bubbles when we compute two point correlation function ...
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Problematic Factor of 2 in Klein-Gordon Propagator Derivation
I want to derive the Klein-Gordon Green function equation
$$(\Box_b + m^2) D_F(x_b - x_a) = - i \delta^4(x_b - x_a)$$
by using the same steps taken when fixing the 'exact' Green function of the non-...
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What's the quantum operator for the diffusion Green's function?
I am trying to understand the following equation about the diffusion Green's function ("Principles of condensed matter physics" by Paul Chaikin, chapter 7.4 Diffusion)
$$n(x,t)=\int G(x-x',t-...
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Pauli-Villars regularization and self-energy
In the calculation of the electron self-energy in QED (one-loop level), there is a UV and IR divergent integral that needs to be regularized. A common choice for the regularization is the Pauli-...
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Peskin and Schroeder Pg. 228: Expanding resummed propagator around the physical pole
I am having difficulty wrapping my head around this particular statement and equation (7.44) on pg. 228 of P&S,
I think I understand that "~" sign here is trying to denote at under $p^0 ...
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Diquark propagators in color superconductivity
I’m studying color superconductivity referring to “The Phases of Quantum Chromodynamics From Confinement to Extreme Environments” by John B. Kogut and Mikhail A. Stephanov (link).
In chapter 9, the ...
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Retarded Green's function for Fermions vs Bosons
Normally, retarded non-interacting Green's function for Fermions is:
$$
G_0^R(k,E) = \frac{1}{E-\epsilon_k+i\eta}
$$
But recently, I read a few research articles (for example, equation (46) of [1]), ...
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Is QFT linear with respect to superposition of multi-particle states?
I saw other posts such as this one but I don't think it's quite the same question, or even if it is, the answer employs the operator formalism and I'm not sure I follow it. I'm wondering, if you have ...
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Propagator and Ward identity in the $R_\xi$ gauge
The full gauge propagator in the $R_\xi$ gauge is
$$D_{\mu\nu} = \frac{i}{k^2+i\epsilon}\left(-g_{\mu\nu}+\frac{1-\xi}{k^2}k_\mu k_\nu\right).\tag{1}$$
Now if we take $\xi=0$, we get the Lorenz gauge, ...
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Why does it make sense to do Dyson Resummation with first-order 1PI-diagrams?
This is somewhat related to This question.
The way I understand renormalisation (e.g of the mass) is that we consider the loop-corrections order by order in the coupling constant. If we then consider ...
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Is the propagator the same as the matrix elements of the time evolution operator?
So Sakurai in their QM book defines the propagator in wave mechanics as:
$$K(x'',t;x',t_0)=\sum_{a'}\langle x''\vert a'\rangle \langle a'\vert x'\rangle \exp\left[\dfrac{-iE_{a'}(t-t_0)}{\hbar}\right]....
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Propagator Terms for Different Types of Particle [duplicate]
It is possible in the correct gauge to write the first term of the Yang-Mills Lagrangian as $$\text{tr} ( -\frac{1}{2} \partial^{\mu} A^{\nu} \partial_{\mu} A_{\nu}),$$ where $A_{\mu}$ is a traceless ...
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Renormalization of the photon propagator at loop-level
I am trying to understand the photon propagator renormalization procedure, followed in M. Srednicki's book Quantum Field Theory. Specifically, I am reading Chapter 62, titled "Loop Corrections in ...
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How do we know there will be enough counterterms to renormalize a theory?
I am wondering if it is always certain that there are enough counterterms to renormalize a renormalizable (e.g. non-negative mass dimension of coupling constant) theory. Through some methods such as ...
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Transition amplitude between field configurations from the path integral
In the path integral formulation of QFT, we should in principle be able to calculate the transition amplitude from a classical field configuration $\phi_{in}(x,t=0)$ to $\phi_{out}(x,t=T)$ using the ...
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Diverging integral in massive fermionic field correlator
I'd like to understand the concept of the 2-particle quantum correlator for massive fermions with mass $m>0$ in 1 spatial dimension: $$C(x,y)=\langle 0|\psi(x)\psi^{\dagger}(y)|0\rangle=\int_{-\...
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Feynman Technique for Greater Green's Function in Many-Body Quantum Theory
In many body quantum theory, Feynman diagrams are commonly used to calculate green's function. My question is: does the diagrammatic method works for all kinds of green's function? For casual green's ...
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Generalities about propagators and its application to scalar chromodynamics
In an exercise for a course on quantum field theory, I am given the following Lagrangian:
$$
\mathcal{L} = -\frac{1}{2} G_{\mu\nu}^a G^{a\mu\nu} + 2 (D_\mu \phi^\dagger)^a(D^\mu \phi)^a - 2 m^2\phi^{\...
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Properties of analytic continuation of two point/ Wightman function
In this paper, the author considers Wightman functions calculated on an accelerating detector for a massless scalar field, namely
$$G_+^R = {}_M \langle 0 | \phi(x) \phi^{\dagger}(x') | 0 \rangle_M$$
$...
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