All Questions
46
questions
0
votes
1
answer
121
views
Fourier transform of the Gaussian action for the real scalar bosonic field
In my current homework, we have to get familiar with quadratic theory in order to reach $\phi^4$-theory. So the starting point is
$$Z = \int Dx e^{-S[\phi]}$$
with the action for the real scalar ...
- 1
1
vote
0
answers
67
views
Derivation of massive photon propagator
I'm trying to derive the massive photon propagator using the path integral formalism for a theory with
$$
\mathcal{L} = -\dfrac{1}{4} F_{\mu\nu} F^{\mu\nu} + \dfrac{1}{2} m^2 A_\mu A^\nu, \text{with } ...
- 1,611
1
vote
1
answer
215
views
Path Integral Formalism and Two-Point Function
I'm given an action,
$$A[\vec{S}] = \frac{\Theta}{2} \int_{-\infty}^{\infty} dt \left(\frac{d\vec{S}}{dt}\right)^2, \tag{1}$$
with $$\vec{S}^2 = 1.\tag{2}$$ I'm asked to calculate the two point ...
- 11
1
vote
1
answer
137
views
Peskin and Schroeder eqn 9.14 [closed]
I am not familiar with functional integral, and in the text like $$\int D\phi D\pi \exp [i\int^T_od^4x(\pi\dot{\phi}-\frac{1}{2}\pi^2-\frac{1}{2}(\nabla \phi)^2-V(\phi))].\tag{9.14}$$ I try to compile ...
- 326
1
vote
0
answers
44
views
Feynman parametrization with three denominators [duplicate]
I'm triyng compute some integral loops for my thesis and I need use Feynman's parametrization with three denominators, I've used the general formula like:
\begin{equation}
\frac{1}{A_1A_2...A_n}=(n-1)!...
0
votes
0
answers
87
views
Field shift in Generating functional for the Dirac field
On P&S's QFT page 302, eq.(9.73) defined the generating functional for the Dirac field.
$$Z[\bar{\eta}, \eta]=\int \mathcal{D} \bar{\psi} \mathcal{D} \psi \exp \left[i \int d^4 x[\bar{\psi}(i \not ...
- 1,421
1
vote
0
answers
152
views
Explicit calculation of the effective action for $\phi^4$ by a Legendre-transformation
Let's say my generating functional for the connected moments is given by
\begin{align}
W[J]&=\underbrace { -\frac { 1 }{ 2N } \ln { ( } Na) }_{ { ring } } +\underbrace { \frac { 1 }{ N^{ 2 } } \...
- 498
3
votes
1
answer
200
views
Deriving a scattering amplitude of a loop diagram using path integral formulation
I am following Zee's QFT in a nutshell and his excerise I.7.2 states to derive
$$\frac{1}{2}(-i\lambda)^2 \int \frac{d^4k}{(2 \pi)^4}\frac{i}{k^2 -m^2 +i\epsilon}\frac{i}{(k_1+k_2-k)^2 -m^2 +i\epsilon}...
- 31
2
votes
2
answers
2k
views
How to perform a Gaussian functional integral?
I'm completely beginner to the quantum field theory and try to learn the basics of functional integrals. However, I could not understand clearly. Could someone please explain the idea with the help of ...
- 83
1
vote
1
answer
67
views
Disappearing symmetry in gaussian functional determinant
I have the following integral
$$I=\int D\varphi \; e^{-\int d^4p d^4p' \left[ -\frac{1}{2}\varphi(p) g(p) \delta(p+p') \varphi(p') \right]}.\tag{1}$$
This is the continuum limit of a gaussian matrix ...
- 861
4
votes
0
answers
188
views
Feynman Rules from Generating Functional
For the following Lagrangian:
$$\mathcal{L}= \overline \psi \left(i \gamma^{\mu}D_{\mu} - m \right)\psi -\frac{1}{2}\left(F_{\mu\nu}\right)^2,$$
I'm trying to find the Feynman rules. I know that the ...
- 1,213
1
vote
0
answers
352
views
Complete the square for the generating functional of the Dirac field
Quote Peskin page 302 the Dirac generating function was
$$Z[\bar \eta ,\eta ]=\int D\bar\psi D\psi\exp[i\int dx^4 (\bar\psi (i\gamma^\mu\partial_\mu -m )\psi+\bar\eta \psi+\bar\psi \eta)]$$
could be ...
- 3,256
3
votes
1
answer
305
views
How do I show that the $n$-point correlator $\left\langle\phi(x_1)\phi(x_2)...\phi(x_n)\right\rangle$ is equal to this expression?
Given the Euclidean action \begin{equation}
S_E(\phi) = \int d^d x \frac{1}{2}\big(\nabla\phi\cdot\nabla\phi + m^2\phi^2\big)\end{equation} and the partition function \begin{equation}\mathcal{Z} = \...
1
vote
0
answers
79
views
How to justify $\int D\phi\exp[-\frac{1}{2} \int d^4 x'\int d^4 x\phi(x')M(x',x)\phi(x)]$
It's related to a homework and exercise. The homework was more complicated, but I needed this to figure out the convention that was used.
Consider the integral
$$\int D\phi\exp[-\frac{1}{2} \int d^4 x'...
- 3,256
2
votes
0
answers
455
views
Feynman diagram for cubic and quartic interaction
I have a question about the scalar quantum field theory and potential $V(\phi) = \alpha \frac{\phi^3}{3!} + \beta \frac{\phi^4}{4!}$. I want to find some of the possible connected one- and two-loop ...
- 86