Dirac matrices, or gamma matrices, are a set of matrices with specific anticommutation relations that generate a matrix representation of the Clifford algebra which acts on spinors, fundamental to the Dirac equation describing spin-1/2 charged particles.
Questions tagged [dirac-matrices]
595 questions
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votes
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Dimension of Dirac $\gamma$ matrices
While studying the Dirac equation, I came across this enigmatic passage on p. 551 in From Classical to Quantum Mechanics by G. Esposito, G. Marmo, G. Sudarshan regarding the $\gamma$ matrices:
$$\tag{16.1.2} (\gamma^0)^2 = I , (\gamma^j)^2 = -I \…
31
votes
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Lorentz transformation of Gamma matrices $\gamma^{\mu}$
From my understanding, gamma matrices transforms under Lorentz transformation $\Lambda$ as
\begin{equation}
\gamma^{\mu} \rightarrow S[\Lambda]\gamma^{\mu}S[\Lambda]^{-1} = \Lambda^{\mu}_{\nu}\gamma^{\nu}
\end{equation}
Where $S[\Lambda]$ is the…
user113988
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Spinors, Spacetime and Clifford algebra
I'm looking to understand the intrinsic connection that Clifford algebra allows one to make between spin space and spacetime. For a while now I've trying to wrap my head around how the Clifford algebra fits into this story, with the members of my…
Jack Hughes
- 659
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votes
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Commutator of Dirac gamma matrices
Quick question...For some reason I'm having trouble finding an identity or discussion for the commutator of the gamma matrices at the moment...i.e $$\gamma^u\gamma^v-\gamma^v \gamma^u$$ but I am not finding this anywhere. I have an idea of what it…
IntuitivePhysics
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spinor vs vector indices of Dirac gamma matrices
I am struggling to understand the nature of the components of the Dirac matrices.
If we view the four components of a Dirac spinor as $\psi^a$ with $a$ being a 'spinor' index, then if a gamma matrix acts on this to give another spinor, then it's…
Coconut
- 223
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votes
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Why are usually 4x4 gamma matrices used?
As far as I understand gamma matrices are a representation of the Dirac algebra and there is a representation of the Lorentz group that can be expressed as
$$S^{\mu \nu} = \frac{1}{4} \left[ \gamma^\mu, \gamma^\nu \right]$$
Usually the…
Wolpertinger
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Can one show that ${\gamma^5}^\dagger = \gamma^5$ directly from the anticommutation relations?
Is it possible to show that ${\gamma^5}^\dagger = \gamma^5$, where
$$ \gamma^5 := i\gamma^0 \gamma^1 \gamma^2 \gamma^3,$$
using only the anticommutation relations between the $\gamma$ matrices,
$$…
PPR
- 2,234
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votes
1 answer
Confusion about gamma matrices in Euclidean spacetime
I have encountered a number of sources with differing definitions of the transition from Minkowski spacetime to Euclidean spacetime.
I'd like some clarification as to how to go from Minkowski to Euclidean spacetime; in particular, how the Dirac…
user4580791
- 321
11
votes
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How to show that $\bar\psi\gamma^\mu\psi$ of a Dirac spinor $\psi$ transforms as a vector?
This is part 2 of exercise II.1.1 of Zee's QFT in a Nutshell (here's part 1).
This is what I have got:
\begin{align}
\bar\psi\gamma^\lambda\psi \mapsto \bar\psi^{\,\prime}\gamma^\lambda\psi^{\,\prime} & =…
Bass
- 1,527
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Do gamma matrices form a basis?
Do the four gamma matrices form a basis for the set of matrices $GL(4,\mathbb{C})$? I was actually trying to evaluate a term like $\gamma^0 M^\dagger \gamma^0$ in a representation independent way, where $M, M^\dagger$ are $4\times 4$ matrices.
SRS
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votes
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How to find a particular representation for the gamma matrices?
I asked this question as a subquestion in another thread, but got the answer below and thought it deserved a thread of its own.
Two well-known representation of the gamma matrices are the Weyl and Dirac-Pauli reps. The Weyl rep is often used when…
Physics_maths
- 5,717
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votes
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Dimension of gamma matrices in dimensional regularization
When performing loop integrals in theories containing Dirac fermions, one almost always confronts terms of the form
$$\text{Tr}\left[\gamma^{\mu_1}\cdots\gamma^{\mu_n}\right].$$
For instance, in $d$ dimensions, we could compute the simple trace…
Bob Knighton
- 8,762
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votes
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Can we make the Dirac representation a gauge theory?
I'm looking for comments and references about an idea : gauging the Dirac representation of the Dirac matrices. What kind of field interaction would it give ?
Specifically, the Dirac equation is defined as this (free field, to begin with)…
Cham
- 8,015
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votes
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How does Schur's Lemma mean that the Dirac representation is reducible?
In chapter 3 of Peskin and Schroeder, when they're talking about "Dirac Matrices and Dirac Field Bilinears," they introduce $\gamma^{5}$ and give some properties of it. One of the properties is $[\gamma^{5},S^{\mu\nu}]=0$. Then they say that this…
Akorl
- 118
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votes
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Why the lowest order of matrices in Dirac equation are 4x4 matrices?
Why the lowest order of matrices in Dirac equation (Relativistic Quantums) are 4x4 matrices (and can not be 2x2 matrices)?
How to prove it?
fronthem
- 527