All Questions
18
questions
1
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1
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127
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Homework, demonstrate a translation in QFT using the momentum operator [duplicate]
The question is to demonstrate the following relation in case of fermionic field:
$$ e^{i\vec{x_0}.\vec{P}} \psi(\vec{x}) e^{-i\vec{x_0}.\vec{P}} = \psi(\vec{x} - \vec{x_0})$$
where $\psi(\vec{x})$ is ...
- 131
0
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0
answers
87
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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.
$$Z[\bar{\eta}, \eta]=\int \mathcal{D} \bar{\psi} \mathcal{D} \psi \exp \left[i \int d^4 x[\bar{\psi}(i \not ...
- 1,421
1
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1
answer
48
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Resonance level model: Commutator
As a small part of an exercise on the resonant level model (all fermionic (field-)operators, $\Psi(\vec{x}) = \sum\limits_{\vec{k}}e^{i\vec{k}\vec{x}}c_{\vec{k}} $, $V$ is a constant, $d$ and $c$ ...
- 31
2
votes
1
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357
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Proof involving exponential of anticommuting operators
Problem:
On page 23 of the book "Quarks, gluons and lattices" by Creutz, he defines a state
$$\langle\psi|=\langle 0|e^{bFc}e^{\lambda b^\dagger G c^\dagger}$$
where $\lambda$ is a number, $...
- 7,720
-1
votes
1
answer
94
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Action of fermion fields on fermionic states
I was asked to show that if $c^\dagger_r(p) |0 \rangle = |p,r\rangle$ is a massive vector particle state with momentum $p$ and polarisation $\epsilon^\mu_r(p)$ then
$$\langle 0 \lvert A^\mu(x) \lvert ...
- 373
0
votes
1
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143
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How to prove the following identity of fermion creation and annihilation operators [closed]
Define $$M_{\theta} \equiv \exp\left[\theta \sum_s \left(d^{\dagger}(\vec{p},s)b(\vec{p},s) -b^{\dagger}(\vec{p},s)d(\vec{p},s)\right)\right],$$ where $\theta$ is a continuous real parameter. Show via ...
- 503
1
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1
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168
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Formal identity involving fermion propagator in quantum field theory
I'm studying from here: Roberto Soldati - Field Theory 2. Intermediate Quantum Field Theory (A Next-to-Basic Course for Primary Education)
I'm trying to understand and prove an equality at page 52, ...
- 505
2
votes
0
answers
133
views
Transformation of fermionic creation and annihilation operators
How do the creation and annihilation operators of Dirac fermions transform under a Lorentz transformation whose axis is not parallel with the axis of spin quantization?
- 870
0
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1
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221
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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.
...
- 1,172
1
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1
answer
272
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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 ...
- 205
4
votes
1
answer
1k
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How to derive this Matsubara sum, as presented in Wikipedia?
On the Wikipedia page for Matsubara frequencies, the following formula is presented,
$$
\sum_{i\omega_n} \frac{(i\omega_n)^2}{(i\omega_n)^2 - \xi^2} = -\frac{\xi}{2}\Big(1 - 2 N_{\text{FD}}(\xi)\Big),
...
- 2,910
2
votes
1
answer
194
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Spinor quantization: contradiction between covariant anticommutator and canonical rules?
Starting from the free lagrangian
$$\mathscr L = \bar\Psi(i\displaystyle{\not}\partial - m)\Psi$$
I compute the canonical momenta
$$\Pi =\frac{\partial \mathscr L}{\partial\dot{\Psi}}=i\Psi^\dagger ...
- 523
0
votes
1
answer
932
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Fermion anti-commutation relations
The fermion anti-commutation relations are given as $$\{\psi_{\alpha}({\bf x},t),\psi_{\beta}^{\dagger}{(\bf x'},t)\} = \delta_{\alpha,\beta} \, \delta({\bf x} - {\bf x'}).$$ I am interested in ...
- 3,856
1
vote
0
answers
790
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Fermion - Antifermion (annihilation) scattering amplitude
I'm trying to get the scattering of the diagrams described here in the "annihilation, part ii" (fermion/antifermion - scalar/scalar)
http://www.physics.umd.edu/courses/Phys624/agashe/F10/solutions/HW7....
- 536
0
votes
2
answers
193
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Explanation on anticommutation relations
Setup
Given two states: $|K\rangle=a_i^+a_j^+|\rangle$ and $|L\rangle=a_k^+a_l^+|\rangle$.
Evaluating the overlap: $\langle K|L\rangle=\langle|a_ja_ia_k^+a_l^+|\rangle$
Introducing: $a_ia_k^+=\delta_{...
- 139