All Questions
Tagged with anticommutator homework-and-exercises
19
questions
-1
votes
1
answer
24
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Deriving the equal time anti-commutator of the Dirac fields [closed]
I am trying to solve an exercise on deriving the equal-time anti-commutator of the Dirac fields. But I got stuck somewhere and couldn't get the desired result.
I would like to show that
$$
\{\psi(x), \...
- 41
1
vote
0
answers
61
views
(Anti) Commutation relation of derivative of the fermionic operator
While deriving Semiconductor Bloch equation, I stumbled upon a commutation relation that I have never seen before. It looks like,
$$[\alpha_k^{\dagger}\alpha_k, \alpha_{k'}^{\dagger}(\nabla_{k'}\...
- 1,007
0
votes
1
answer
88
views
Help with commutator algebra with fermionic operators
I am struggling to understand how the following is true for the fermionic creation/annihilation operators $a^\dagger, a$: $$[a^\dagger a, a]=-a$$
If someone could walk me through the math derivation ...
- 101
2
votes
1
answer
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
0
votes
1
answer
260
views
How to express the anti-commutator in the form of a density operator?
$
\newcommand{\ket}[1]{|{#1}\rangle}
\newcommand{\bra}[1]{\langle{#1}|}
\newcommand{\braket}[2]{\langle{#1}|{#2}\rangle}
\newcommand{\acomm}[2]{\left\{#1,#2\right\}}
$Let $\{ \ket{1} \ket{1} \ket{2} .....
- 241
0
votes
1
answer
607
views
Creation and annihilation operators for fermions from anticommutator
In a question, I was given that $a^{\dagger}a + a a^{\dagger} =1$ and asked to show what $a|n\rangle$ and $a^{\dagger}|n\rangle$ would be, given that $H|n\rangle=(a^{\dagger}a + 1/2)$.
I am getting ...
- 182
1
vote
1
answer
128
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Chiral Symmetry and Charge Algebra
I'm following Section 5.1 in Cheng and Li's Particle Physics book and I am having trouble reproducing some of the commutation relations.
The axial current is given by
$$
(J_A^a)^\mu = \bar{\psi}_\...
- 574
0
votes
1
answer
110
views
Dirac spinor and field quantization
Can we simplify $$ Σ_s Σ_r [b_p^s u^s(p)\mathrm e^{ipx} (b_q^r)^†(u^r)^†(q)\mathrm e^{-iqy} + (b_q^r)^†(u^r)^†(q)\mathrm e^{-iqy} b_p^s u^s(p)\mathrm e^{ipx}]\tag{1}$$ as $$Σ_sΣ_r[ \{b_p^s, (b_q^r)^†...
- 369
0
votes
1
answer
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
0
votes
1
answer
532
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Proving identity $\mathrm{Tr}[\gamma^{\mu}\gamma^{\nu}] = 4 \eta^{\mu\nu}$
In the lecture notes accompanying a course I'm following, it is stated that
$$\DeclareMathOperator{\Tr}{Tr} \Tr\left[\gamma^{\mu}\gamma^{\nu}\right] = 4 \eta^{\mu\nu}
$$
Yet when I try to prove this,...
- 1,438
0
votes
0
answers
223
views
Anticommutation relation for the exponential field of the bosonic field
In 1+1 dimensions, the massless KG equation has the general solution $$\phi(x,t)=\int_{-\infty}^{\infty}dp/(4\pi E_p)[a_pe^{i(px-E_pt)}+a^{\dagger}_pe^{-i(px-E_pt)}]$$ where $E_p^2=p^2$. The operator ...
- 1,669
-1
votes
1
answer
181
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Anti-commutative Hermitian operators in an infinite dimensional Hilbert space
An example of a pair of anti-commutative Hermitian operators in a finite Hilbert space is $\sigma_x$ with $\sigma_z.$
Indeed $\sigma_z\sigma_x=i\sigma_y$, whereas $\sigma_x\sigma_z=-i\sigma_y$.
My ...
- 13
5
votes
2
answers
37k
views
Properties of anticommutators [closed]
Do anticommutators of operators has simple relations like commutators.
For example:
$$[AB,C]=A[B,C]-[C,A]B.$$
But I don't find any properties on anticommutators. Do same kind of relations exists ...
- 1,450
4
votes
1
answer
7k
views
Confusion about slash notation
I am confused about the slash notation and especially taking the square of a slashed operator.
Defining $\displaystyle{\not} a \, = \, \gamma^\mu a_\mu$ we have $\,\,$ $\displaystyle{\not} a \...
- 2,347
0
votes
1
answer
932
views
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