All Questions
Tagged with anticommutator clifford-algebra
8
questions
0
votes
1
answer
79
views
What is the motivation for the Self-Dual Canonical Anticommutation Relation (CAR) algebra?
What exactly is the motivation for the use of the Self-Dual Canonical Anticommutation Relation (CAR) algebra in the context of infinite lattice systems? Why not remain with the CAR algebra, as both ...
- 71
1
vote
1
answer
270
views
(Anti)commutation relations for higher-dimensional anti-symmetrized Gamma matrices
Suppose $a,b,c...=0,....,D-1$ are Lorentz indices of $SO(1,D-1)$ tangent space and consider $D$-dimensional Clifford algebra defined by the usual anticommutation relation
$$\{\Gamma^a,\Gamma^b\}=2\eta^...
- 123
1
vote
0
answers
117
views
Is there a bosonic representation of Clifford algebra in (1,3) spacetime?
By a suitable combination of Dirac's $\gamma_\mu$ matrices one can define creation and destruction operators satisfying fermionic anticommutators. Is there a similar result for bosons in the context ...
5
votes
2
answers
5k
views
Is there an identity for anti-commutator $\{ A B, \, C D \}$ in terms of commutators $[\, , \,]$ only?
I'm looking for an identity that could express the anti-commutator
$$\tag{1}
\{ A B , \, C D \} \equiv A B C D + C D A B
$$
expressed as a combination of commutators only: $[A,\, C]$, $[A, \, D]$, etc....
- 7,592
2
votes
1
answer
374
views
Matrix representation of the CAR for the fermionic degrees of freedom
The canonical anticommutation relations (CAR) for a fermionic degree of freedom can be written as follows:
$$ a^2 = \left( a^{\dagger} \right) ^2 = 0, $$
$$ a a^{\dagger} + a^{\dagger} a = 1. $$
...
- 16.2k
2
votes
1
answer
3k
views
Anticommutator of spin-1 matrices
We know that in the spin-1/2 representation the anticommutation relation of the Pauli matrices is $\{\sigma_{a},\sigma_{b}\}=2\delta_{ab}I$. Does a similar relation hold for the spin-1 representation?
0
votes
1
answer
532
views
Proving identity $\mathrm{Tr}[\gamma^{\mu}\gamma^{\nu}] = 4 \eta^{\mu\nu}$
In the lecture notes accompanying a course I'm following, it is stated that
$$\DeclareMathOperator{\Tr}{Tr} \Tr\left[\gamma^{\mu}\gamma^{\nu}\right] = 4 \eta^{\mu\nu}
$$
Yet when I try to prove this,...
- 1,438
6
votes
4
answers
13k
views
Proof of the Anti-Commutation Relation for Gamma Matrices from Dirac Equation
My textbook on QFT says that the Dirac equation can be used to show the following relation:
$$\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}$$
I have searched around and unable to find how to prove this ...
- 1,410