All Questions
Tagged with quantum-field-theory homework-and-exercises
94
questions
65
votes
2
answers
19k
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How do I construct the $SU(2)$ representation of the Lorentz Group using $SU(2)\times SU(2)\sim SO(3,1)$?
This question is based on problem II.3.1 in Anthony Zee's book Quantum Field Theory in a Nutshell
Show, by explicit calculation, that $(1/2,1/2)$ is the Lorentz Vector.
I see that the generators of ...
- 1,505
14
votes
2
answers
5k
views
Position operator in QFT
My Professor in QFT did a move which I cannot follow:
Given the state $$\hat\phi|0\rangle = \int \frac{d^3p}{(2\pi)^3 2 E_p} a^\dagger_p e^{- i p_\mu x^\mu}|0\rangle,$$ he wanted to show that this ...
- 263
1
vote
4
answers
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Is this useful identity valid only under the integral sign?
Studying dimensional reugularization one often encounters the following identity:
$$
\int d^d q\, \, q^\mu q^\nu f(q^2) = \frac{1}{d}g^{\mu\nu}\int d^d q\,\, q^2 f(q^2)
$$
often justified by some ...
- 505
24
votes
3
answers
7k
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Problem understanding the symmetry factor in a Feynman diagram
I am trying to understand a $1/2$ in the symmetry factor of the "cactus" diagram that appears in the bottom of page 92 In Peskin's book. This is the diagram in question (notice that we are ...
- 6,067
12
votes
1
answer
5k
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Bessel function representation of spacelike KG propagator
Preliminaries: In their QFT text, Peskin and Schroeder give the KG propagator (eq. 2.50)
$$
D(x-y)\equiv\langle0|\phi(x)\phi(y)|0\rangle = \int\frac{d^3p}{(2\pi)^3}\frac{1}{2\omega_\vec{p}}e^{-ip\...
- 695
8
votes
1
answer
1k
views
Schrodinger Wave Functional (quantum fields) - Solving Functional Gaussian Integrals
Okay, So i'm doing some research that involves the Schrodinger representation in quantum field theory. The ground state wave functional for the Klein Gordon field is a generalized gaussian in position ...
- 305
8
votes
2
answers
1k
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Simple QFT exercise
Consider a particle on the real line with:
$L=\frac{1}{2}(\partial_0q)^2 + f(q)\partial_0q$
the equation of motion is that of a free particle $\partial_0^2q=0$. In fact $\delta[f(q)\partial_0q]=0$. ...
- 353
63
votes
0
answers
4k
views
How to apply the Faddeev-Popov method to a simple integral
Some time ago I was reviewing my knowledge on QFT and I came across the question of Faddeev-Popov ghosts. At the time I was studying thеse matters, I used the book of Faddeev and Slavnov, but the ...
- 1,210
15
votes
3
answers
4k
views
Three integrals in Peskin's Textbook
Peskin's QFT textbook
1.page 14
$$\int_0 ^\infty \mathrm{d}p\ p \sin px \ e^{-it\sqrt{p^2 +m^2}}$$
when $x^2\gg t^2$, how do I apply the method of stationary phase to get the book's answer.
2....
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7
votes
3
answers
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Deriving Schrodinger equation from Klein-Gordon QFT with the definition $\psi(\textbf{x},t)\equiv \langle 0|\phi_0(\textbf{x},t)|\psi\rangle$
In the book "Quantum Field theory and the Standard Model" by Matthew Schwartz, page 23-24, the position space wavefunction is defined as
$$\psi(x)=\langle 0|\phi(x)|\psi\rangle, \tag{2.82+2.83}$$
...
- 26.8k
5
votes
2
answers
3k
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How do you prove that $L=I-V+1$ in $\lambda\phi^4$ theory?
It is known that the number of loops in $\lambda\phi^4$ theory is given by the formula
$$L=I-V+1$$
where $L$ is the number of loops, $I$ the number of internal lines and $V$ the number of vertices. ...
- 6,067
2
votes
1
answer
691
views
Recovering QM from QFT
Reading through David Tong lecture notes on QFT.
On pages 43-44, he recovers QM from QFT. See below link:
QFT notes by Tong
First the momentum and position operators are defined in terms of "...
user56963
17
votes
3
answers
7k
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How to obtain the explicit form of Green's function of the Klein-Gordon equation?
The definition of the green's function for the Klein-Gordon equation reads:
$$
(\partial_t^2-\nabla^2+m^2)G(\vec{x},t)=-\delta(t)\delta(\vec{x})
$$
According to these resources:
Green's function ...
- 6,095
8
votes
1
answer
2k
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How to derive completeness relation in quantum field theory with a Lorentz invariant measure?
$\bullet$ 1. For the one-particle states, the completeness relation is given in Peskin and Schroeder, $$(\mathbb{1})_{1-particle}=\int\frac{d^3\textbf{p}}{(2\pi)^{3}}|\textbf{p}\rangle\frac{1}{2E_\...
- 26.8k
8
votes
2
answers
3k
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Deriving photon propagator
In Peskin & Schroeder's book on page 297 in deriving the photon propagator the authors say that
$$\left(-k^2g_{\mu\nu}+(1-\frac{1}{\xi})k_\mu k_\nu\right)D^{\nu\rho}_F(k)=i\delta^\rho_\mu \tag{9....
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