Linked Questions

11
votes
2answers
1k views

Why are usually 4x4 gamma matrices used? [duplicate]

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^\...
8
votes
2answers
2k views

Why the lowest order of matrices in Dirac equation are 4x4 matrices? [duplicate]

Why the lowest order of matrices in Dirac equation (Relativistic Quantums) are 4x4 matrices (and can not be 2x2 matrices)? How to prove it?
0
votes
0answers
138 views

Clifford Algebra in 3D [duplicate]

Why the gamma matrices are taken 2 by 2 (Pauli matrices) in 3 dimensional Clifford Algebra. As in 4D Clifford Algebra the matrices are 4 by 4, in 3D Algebra why are they not 3 by 3 matrices? The ...
7
votes
2answers
905 views

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 $[\...
8
votes
2answers
430 views

Are there projective representations of the Lorentz Group NOT coming from a Clifford algebra?

Let $\mathrm{SO}(1,d-1)_{+}$ be the restricted Lorentz Group in $d$ dimensions. Are there projective irreducible representations of this group that do not descend from a representation of $\mathrm{C}\...
3
votes
1answer
1k views

Matrix order in Dirac equations

The trace of matrix is always sum of its eigen values , which can be seen if $\hat{U}$ transforms the matrix $\alpha_i$ into it's diagonal form . $$ \begin{pmatrix} A_1 & 0 & \cdots & 0 \...
4
votes
1answer
851 views

Spinors in 2+1 dimensions

I am trying to understand representations of the Poincare/Lorentz group, and in particular spinors, in 2+1 dimensions. I know some of the math, but I'm not sure about the physical interpretation of it ...
1
vote
2answers
848 views

Dirac spinors in 2+1 dimensions

In 3+1 dimensions, Dirac spinors have four complex components. In 2+1 dimensions, the representation of the Clifford algebra by $\sigma^3$ and $-i\sigma^3\sigma^i$, with $i\in\{1,2\}$ is 2-dimensional,...
0
votes
1answer
521 views

Dimension of gamma matrices in higher dimensional Dirac equations

Reading about Dirac's equation in higher dimensional space-times I have read that the gamma matrices are $2^{[D/2]}\times{}2^{[D/2]}$. So, if we have $D=11$, for example, how is this formula supposed ...
3
votes
1answer
268 views

Gamma matrices in (2+1)

I am sure that is very well-known question and see on this site several similar questions but I would like to specify the answer 1) I know that in $(2+1)$-dimensions one can construct $\gamma$-...
2
votes
1answer
434 views

Representations of the Dirac algebra, hermitian adjoint and traces

Strictly speaking this is a math question, but since the Dirac algebra is much more important in physics than in math I thought I'd have a better chance of getting an answer here. The Dirac algebra ...
0
votes
1answer
384 views

Dimensionality of Gamma Matrices

If I express the Dirac equation in the form of $$i\hbar \frac{\partial}{\partial t} \psi_a(x) = \left(-i\hbar c(\alpha^j)_{ab}\partial _j + mc^2(\beta)_{ab}\right)\psi_b(x),$$ with the constraints $...
2
votes
2answers
290 views

Is there a bi-4-vector representation of the Dirac gamma matrices and the spinor?

I learned recently that if you have the Dirac spinor represented in the Weyl (chiral) basis $\Psi = \begin{pmatrix} \psi_L \\ \psi_R \end{pmatrix}$, then given a Lorentz Transformation $\Lambda = exp[\...
7
votes
1answer
238 views

Why complexify in order to construct Dirac representation?

Suppose we have a theory is covariant under the Spin group Spin(2n-1; 1). We consider the real vector space $V = R^{2n-1,1}$, which naturally comes with a Lorentzian inner product. On this vector ...
2
votes
1answer
134 views

How to understand spinors in 1+1 spacetime?

I am struggling to understand spinors in 1+1 spacetime. I know in this case the Clifford algebra is realized by two by two matrices so the spinors have two components. Then what do we mean by spin or ...

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