Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
0answers
23 views

How does one calculate Fourier transform of Feynman propagator?

I am struggling with calculating the following integral on Sredinicki: How did he get the second line of (10.6)? That is, how did he calculate the Fourier transform of Feynman propagator?
0
votes
2answers
68 views

Confusion about functional derivative in path integral

If we act a functional derivative $$\frac{\delta}{\delta J(z)}$$On the expression$$\int\int d^4x d^4y \space J(x)\Delta(x-y)J(y)$$ where $\Delta(x-y)$ is Feynman propagator. What one should get is ...
1
vote
0answers
53 views

When can you simplify the W boson propagator

I have seen in several sources that the propagator of the $W$ boson is: $$\frac{- i \left( g^{\mu\nu} - \frac{P^\mu P^\nu}{m_W^2} \right)}{p^2 - m_W^2} $$ But then in some calculations (usually ...
0
votes
0answers
57 views

How to grasp the limits of these two integrals? [duplicate]

I find some difficulty in understanding the limits of the two integral below (on Page 27 of Peskin & Schroeder's Quantum Field Theory): $$D(x-y)=\frac{1}{4\pi^2}\int_m^\infty d E \sqrt{E^2-m^2}e^{...
0
votes
0answers
43 views

Why is the imaginary part of the Breit-Wigner propagator given by the total decay width?

The optical theorem links the imaginary part of the forward scattering amplitude to the total decay width of a particle: $\mathrm{Im}\,M_{i\to i} = m\Gamma_{tot}$. Here $\Gamma_{tot} = \frac{1}{2m} \...
2
votes
1answer
62 views

Are powers of the harmonic oscillator semiclassically exact?

The Duistermaat-Heckman theorem, although too complex for me to completely grasp, states that under some conditions, the partition function for a special class of Hamiltonians is semiclassically exact....
4
votes
1answer
90 views

Wick rotation vs. Feynman $i\varepsilon$-prescription

The generating functional $Z[J]$ of some scalar field theory is \begin{equation} Z[J(t,\vec{x})]=\int \mathcal{D}\phi e^{i\int (\mathcal{L}+J\phi)d^4x} \end{equation} This integral is not well ...
0
votes
0answers
28 views

Propagators in interaction with derivatives

Given a Lagrangian density containing an interaction with derivates, it's easy how to guess the Feynman rules for vertexes. However i was wondering about propagators: in S-matrix expansion it's ...
2
votes
0answers
92 views

Calcluating the photon propagator with gauge fixing parameter

I'm trying to calculate the photon propagator via the functional integral, with lagrangian (plus source) $L = -\frac{1}{4}F^{ab}F_{ab} - \frac{\lambda}{2}\left(\partial^aA_a\right)^2 + J^aA_a $ ...
1
vote
0answers
10 views

Propagators for polarised photons

Photons can be thought to come in two types. Depending on which way they are spinning. The propagator for a photon (sum of both helices) in a certain gauge is $\frac{\eta^{\mu\nu}}{|k|^2}$. I read ...
0
votes
0answers
40 views

Are vacuum-fluctuations a consequence of causality?

I'n new to QFT, and recently lerned about the propagator of a free scalar field theory in Minkowski-space, which according to our lecture notes looks like $$G(p, q) = \frac{1}{q^\mu q_\mu + M^2} \...
1
vote
1answer
53 views

Propagator for W boson

I've found in different literature that some write the propagator for the W boson as $\frac{g_{\mu\nu}-\frac{k_\mu k_\nu}{M^2_W}}{k^2-M^2_W+iM_W\Gamma_W}$ and others like $\frac{g_{\mu\nu}-\frac{k_\mu ...
2
votes
0answers
33 views

Is one allowed to split path integrals in the Feynman-Vernon Influence theory

In QFT the propagator $J(t,t_0,x_f,x_i) = \langle x_f | U(t,t_0) | x_i \rangle$ fulfills the property $$ J(t,t_0,x_f,x_i) = \int_{-\infty}^{\infty}dx' J(t,t',x_f,x')J(t',t_0,x',x_i) $$ and can be ...
0
votes
0answers
43 views

Geometric series of two point function and self energy

This question is related to this question Proof of geometric series two-point function. Suppose we have a graph $A$ with a symmetry factor $s_1$. According to Srednicki (chapter 9, eq. (9.13)) for a ...
2
votes
1answer
79 views

Proof of geometric series two-point function

In deriving the expression for the exact propagator $$G_c^{(2)}(x_1,x_2)=[p^2-m^2+\Pi(p)]^{-1}$$ for $\phi^4$ theory all books that i know use the following argument: $$G_c^{(2)}(x_1,x_2)=G_0^{(2)}...
1
vote
1answer
118 views

Definition of one particle irreducible diagrams

Text books often defines One-Particle Irreducible diagram (1PI diagram) as a connected diagram which does not fall into two pieces if you cut one internal line. Is this internal line the full ...
2
votes
1answer
104 views

How many counterterms does QED have?

I have read the statement that QED has four counterterms to cancel divergences. However, I have learnt that there are only three counterterms (vertex, electron propagator, photon propagator), which is ...
1
vote
1answer
69 views

Proof of 1-particle irreducible (1PI) diagrams

If we split the effective action into $$Γ[Φ] =\frac{1}2ΦiG_0^{-1}Φ + Γ^{int} [Φ]$$ we can show that the full propagator is given by $$G= i[iG − Σ]^{-1}$$ With $$Σ=-Γ_{ΦΦ}^{int} [Φ]$$ Here $Γ_{...
2
votes
0answers
64 views

Conformal transformation of a vertex operator before normal ordering

Let us consider a free scalar boson $\varphi(z,\bar{z})$ on the complex plane and assume the following two-point correlation function \begin{eqnarray} \langle\varphi(z,\bar{z})\varphi(w,\bar{w})\...
3
votes
1answer
117 views

Resonances in QFT

I am a bit confused about resonances in QFT. I am reading Schwarz's QFT book and as far as I understand, if in a reaction the mass of the particle acting as a propagator is bigger than the sum of the ...
0
votes
0answers
22 views

Loop-loop propagator in bosonic string theory

I have been searching for information on how to compute (or at least partially compute) the propagator of a string given an initial and a final loop in bosonic string theory. I have found several ...
0
votes
0answers
11 views

Propagator for two spins with time-dependent field?

I'm looking for a closed-form solution for a problem that SEEMS to me to be so simple and basic it HAS to have a solution - but I can't find it. I was wondering if the great minds of this august ...
0
votes
1answer
68 views

Dirac propagator causality

I was studying the Dirac propagator and came across an excelent article which includes all the derivation, and interestingly we can conclude that the anticommutator is zero for space-like intervals. ...
1
vote
1answer
32 views

How does the Lorenz gauge eliminate the scalar component of the vector field?

Wikipedia states that by using the Lorenz gauge, $\partial_\mu A^\mu=0$, we eliminate the scalar part (spin-0) of the vector potential that previously had spin-1 and spin-0 components${}^1$. However,...
3
votes
2answers
132 views

What is the meaning of propagator in the context of lattice theory?

Say in $1+1D$ free fermion theory, it is easy to calculate the propagator in the (effective) field theory to be $$\langle \psi^\dagger(z)\psi(z')\rangle = \frac{1}{2\pi}\frac{1}{z-z'}$$ (in the ...
0
votes
1answer
71 views

Resonance propagator properties

(This is part of a problem from Schwarz book on QFT). 1. Show that a propagator only has an imaginary part if it goes on-shell. Explicitly, show that $$Im(M)=-\pi\delta(p^2-m^2)$$ when $$iM=\frac{i}{p^...
1
vote
1answer
57 views

Correction to the fermion propagator

Given the Lagrangian $$\mathscr{L}=\bar{\psi}\left(i\partial\!\!\!/-m\right)\psi +\frac{1}{2}\left(\partial\phi\right)^2- \frac{1}{2}M^2\phi^2 - g\bar{\psi}\psi\phi^2,$$ calculate the propagator ...
3
votes
1answer
111 views

How do you write the Wightman function $\langle\phi(t_1)\phi(t_2)\rangle$ for a massive scalar field in position space?

For a free real scalar field $\phi(t,\mathbf{x})$, we define the Wightman function as: $$ W(t_1,t_2) \equiv \langle 0 | \phi(t_1,\mathbf{x}_1) \phi(t_2,\mathbf{x}_2) | 0 \rangle $$ I'm suppressing the ...
0
votes
1answer
56 views

Propagator in Wave Mechanics Laplace-Fourier transform

In my Modern quantum mechanics, J. J. Sakurai p.119-120, when considering the integral of the propagator $K$ in whole space, he gets: $$G(t)= \int d^3 x' K(\textbf{x'},t;\textbf{x'},0) = \sum_a \exp \...
1
vote
1answer
58 views

Correction to the scalar propagator - derivative coupling

Given the scalar field Lagrangian $$\mathscr{L}=\frac{1}{2}e^{-\lambda\phi}\partial_\mu\phi\partial^\mu\phi,$$ evaluate the order $\lambda^2$ correction to the propagator. At that order in $\...
3
votes
1answer
86 views

Wick rotation of the propagator in quantum mechanics

I am told that making the substitution $t\to-i\tau$, or a 'Wick rotation', can be used to study the propagator in imaginary time, making some problems easier. For example, this source proposes that we ...
3
votes
1answer
132 views

Quantum Harmonic Oscillator propagator in Sakurai

In Sakurai the derivation of the propagator leads to the expression $$u_n(x)\exp{\left(\frac{-iE_nt}{\hbar}\right)} = \left(\frac{1}{2^{n/2}\sqrt{n!}}\right) \left(\frac{m\omega}{\pi\hbar}\right)^{1/...
0
votes
1answer
85 views

Propagator in $\phi^3$ Theory in $d=0$ Spacetime Dimensions

I am working through some problems in Srednicki's book in QFT (page 70). In one of the problems we work with the following integral: $$Z[g,J]=\frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{\infty}dx \; \exp \...
3
votes
3answers
194 views

Peskin & Schroeder: Free particle propagation

In Peskin & Schroeder Ch. 2, p. 14, in proving that the NRQM propagation amplitude for a free particle is nonzero everywhere, they move from \begin{equation} U(t)~=~ \frac{1}{(2\pi)^3} \int d^3p \...
0
votes
1answer
193 views

Derivation of propagator for free particle

On Sakurai page 127 he gives the formula $$ (1)~~~~~\langle x_n,t_n|x_{n-1},t_{n-1}\rangle = \left[\frac{1}{w(\Delta t)}\right] \exp\left[\frac{im(x_n-x_{n-1})^2}{2\hbar\Delta t} \right]$$ Noting ...
1
vote
1answer
248 views

Causal propagator and Feynman propagator

I have some questions about the Green’s function of the Klein-Gordon operator and the Feynman propagator. The first is about retarded Green’s function: \begin{eqnarray} \int_{-\infty}^\infty\frac{d^...
0
votes
0answers
203 views

Propagator solution to Klein-Gordon equation

We know that the Klein-Gordon operator is given by $(\partial^2+m^2)=(\partial_\mu\partial^{\mu}+m^2)$, which is used to describe the evolution of relativistic free particles. How can we show that ...
1
vote
0answers
56 views

What effect does multiplying $\mathscr{L}$ by $-1$ have on the propagator?

I am following along Ashok Das' development of Thermofield dynamics in his book Finite Temperature Field Theory. Here you have two real scalar fields $\phi_1$ and $\phi_2$ with Lagrangian density $$ \...
3
votes
1answer
96 views

What is the reason for the extra minus sign in $(\Box_x - m^2)G_\mathrm{F}(x,y) = - \delta^{(4)}(x-y)$?

The Feynman propagator is given by the expectation value of two time-ordered (scalar) field operators (evaluated in the vacuum): $$ G_\mathrm{F}(x,y) \equiv \langle 0 | \mathcal{T}\big( \hat{\phi}(x) \...
2
votes
0answers
46 views

If you knew the full interacting 2-point correlation function, do you know everything?

I'll specify to the case of a real scalar field to be concrete. If you have some interacting theory with Lagrangian $$ \mathscr{L}[\phi] = -\tfrac{1}{2} (\partial_{\mu} \phi)(\partial^{\mu} \phi) - \...
1
vote
1answer
298 views

QFT in A Nutshell Zee I.3.1

In Zee's solution for (I.3.1) on p. 483 at the back of the book (2003 edition) he writes: $$D(x)=-\frac{1}{2(2\pi)^2 r}\int_0^\infty \frac{dk~k}{\sqrt{k^2 +m^2}}(e^{ikr}-e^{-ikr})=-\frac{1}{8\pi^2 r}\...
2
votes
0answers
76 views

How can the propagator be written in the below integral form?

I’am finding it difficult to understand as to how the delta function is written as a product of many delta functions with the integral
0
votes
0answers
64 views

Lippmann-Schwinger Equation in Dirac theory

Consider a scattering process of some particle which is described by a Dirac Equation. We use the Lippmann-Schwinger Equation for the total scattering wave function in representation-free form up to ...
2
votes
1answer
345 views

Feynman rules from Lagrangian

I know there are already questions about determining the Feynman rules given a particular Lagrangian, but this question is more about finding out whether I understand correctly how to do this. I'm ...
3
votes
1answer
292 views

Correlators for space-like photon bouncing

Consider the diagram below of two detectors-emitters exchanging photons: Thin lines are null-paths between detectors. Detectors are also synchronized in such a way that measurements outside the blue ...
6
votes
1answer
220 views

Path integral in even number of spatial dimensions: does it exist?

The path integral formulation of Quantum Mechanics is related to Huygens principle, as stated by Feynman in his seminal article [1] and widely commented since then. However Huygens principle does not ...
0
votes
2answers
232 views

Propagator in momentum basis from first principles

I have a propagator for a free particle, $$ G(x,t;y,t_0) = \sqrt{\frac{-im}{2\pi \hbar(t-t_0)}} \exp\bigg(\frac{im(x-y)^2}{2\hbar(t-t_0)}\bigg). $$ one of the exercises in the quantum section of my ...
0
votes
2answers
218 views

Show that Propagator satisfies Schrödinger equation

I want to show that $$K=K(x,x',t-t')=\sum_{\beta}\exp\left[\frac{-iE_{\beta}}{\hbar}(t-t')\right]$$ satisfies the Schrödinger equation $$ H|\psi\rangle = i\hbar\partial_t|\psi\rangle$$ with respect ...
3
votes
2answers
73 views

Multiplying Distributions in finite-temperature Keldysh/Thermo-field field theory

In the real-time finite temperature formalisms (Schwinger-Keldysh or Thermo-field), the free propagators are often defined with terms like: $$ \mathrm{Dirac\ Delta}\ \times \ \mathrm{Thermal\ ...
0
votes
0answers
48 views

Derive vector meson propagator? [duplicate]

Starting with $$ D_{\nu\lambda}\left[ -\big( k^2-m^2 \big)g^{\mu\nu}+k^\mu k^\nu \right]=\delta^\mu_\lambda~~, $$ what are the steps to find the massive vector meson propagator $$ D_{\nu\lambda}=...