Questions tagged [dirac-matrices]

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26
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4answers
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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: $$\...
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 \...
11
votes
2answers
2k 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^\...
4
votes
2answers
308 views

What is the relationship between the Lorentz group and the $CL(1,3)$ algebra?

In my classes the dirac equation is always presented as the "square root" of the Klein Gordon equation, then from this you can demand certain properties from the Matrices (anticommutation relations, ...
3
votes
2answers
2k views

Construction of Pauli Matrices

How can we construct the Pauli matrices starting from $$\sigma_i=\begin{bmatrix} a & b\\ c& d \end{bmatrix}$$ by using the conditions $$\sigma^2_i=1,$$$$\left [ \sigma_x,\sigma_y \right ]=2i\...
1
vote
0answers
249 views

What is $\langle G_{\mu\nu}\rangle\langle G_{\mu\nu}\rangle$ for the Dirac gamma matrices?

Given the following 16 matrix multiplications of the Dirac gamma matrices \begin{align} G_{\mu\nu} = \dfrac{1}{2} \begin{pmatrix} I && \gamma_{0} && i\gamma_{123} && i\gamma_{...
21
votes
3answers
6k views

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^{\...
6
votes
1answer
518 views

What is the role of the spacetime algebra?

For Minkowski space $M^4=\mathbb{R}^{1,3}$ the Clifford algebra $Cl_{1,3}$ (Dirac algebra) with $\{\gamma^\mu, \gamma^\nu \}=2 g^{\mu \nu}$ is sometimes called "spacetime algebra". What is its ...
9
votes
2answers
2k views

Do Dirac Gamma Matrices act like Tensors?

Do Dirac Gamma Matrices act like Tensors? Is it true that $$ \gamma_\mu = \eta_{\mu\nu}\gamma^\nu~? $$ Also what about $\sigma_{\mu\nu}$, where $\sigma_{\mu\nu}$ is defined to be: \begin{align*} \...
9
votes
2answers
4k views

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, $$ \...
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?
1
vote
2answers
144 views

Why in the relativistic quantum mechanics $ \gamma_4$ name is not used instead of $ \gamma_5$?

I have seen in the in the Dirac equation $$\gamma_0,\gamma_1,\gamma_2,\gamma_3.$$ Then I have seen the definition of a new matrix $$\gamma_5=i\gamma_0\gamma_1\gamma_2\gamma_3.$$ Now my question is why ...
0
votes
1answer
610 views

Gordon Identity confusion

For the Gordon identity $$2m \bar{u}_{s'}(\textbf{p}')\gamma^{\mu}u_{s}(\textbf{p}) = \bar{u}_{s'}(\textbf{p}')[(p'+p)^{\mu} -2iS^{\mu\nu} (p'-p)_{\nu}]u_{s}(\textbf{p}) $$ If I plug in $\mu$=5, ...
0
votes
2answers
362 views

Need help with solution of the Dirac equation

$$\left(\vec\sigma \cdot \vec{p} \right)^2=\left(\vec\sigma \cdot \vec{p}\right) \left(\vec\sigma \cdot \vec{p} \right)=\vec{p} \cdot \vec{p}+\mathrm{i}\left(\vec\sigma \cdot \left[ \vec{p} \times \...
11
votes
3answers
3k views

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, ...
5
votes
2answers
1k views

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 ...
4
votes
1answer
4k views

How can pseudospin be a vector? (Graphene)

In graphene science, I don't understand how one interprets pseudospin as a vector. I thought 'pseudospin' was the vector of Pauli matrices. So how can it be a vector that one can plot for example in ...
5
votes
2answers
686 views

Choice of Dirac gamma matrix representation and definition of adjoint spinor

Is the definition of the adjoint spinor $\bar{\psi}=\psi^\dagger \gamma^0$ forcing a particular choice of representation of the Dirac matrices (or a subset of the possible choices)? More precisely, I ...
6
votes
1answer
992 views

Hermitian properties of Dirac operator

I am trying to understand the Hermiticity of the (massless) Dirac operator in both (flat) Minkowski space and Euclidean space. Let us define the Dirac operator as $D\!\!\!/=\gamma^\mu D_\mu$, where $...
10
votes
4answers
16k views

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 ...
6
votes
1answer
737 views

How to show that $\bar\psi\psi$ of a Dirac spinor $\psi$ transforms as a scalar?

I would like to show that for a Dirac spinor $\psi$, the scalar product $\bar\psi\psi$ transforms as a scalar under a Lorentz transformation $\Lambda$, where $\bar\psi = \psi^\dagger\gamma^0$. This is ...
6
votes
4answers
897 views

Representation under which Pauli matrices transform

In Peskin and Schroeder's Quantum Field Theory, there is an identity of Pauli matrices which is connected to the Fierz identity, (equation 3.77) $$(\sigma^{\mu})_{\alpha\beta}(\sigma_\mu)_{\gamma\...
4
votes
3answers
459 views

Why do we need matrices in the Dirac equation?

Consider the following equation: \begin{equation} \nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2} = \left(A \partial_x + B \partial_y + C \partial_z + \frac{i}{c}D \partial_t\right)\left(A \...
2
votes
0answers
135 views

Do Lorentz rotations transform the Gamma matrices $\gamma_a$?

Do local Lorentz rotations (see below definition) actually transform the Dirac Gamma matrices? If so, how can they collude with coordinate transformations to make the Gamma matrices $\gamma_a$ ...
4
votes
2answers
1k views

Has the relative sign in the Dirac equation any meaning?

I know there are many questions about this topic and also various answers, but it's never stated explicitly, why there is a certain sign before the mass term in the Dirac Lagrangian. I'm also confused,...
3
votes
1answer
403 views

Identities of Pauli matrices in two-component spinor formalism

I'm reading the review by H. K. Dreiner, H. E. Haber and S. P. Martin (arXiv:0812.1594) about the two-component spinor formalism. There are some identities and notational conventions which lead to ...
6
votes
1answer
225 views

Time reversal for Dirac particles in 2+1 or 6+1 (mod 8) spacetime dimensions

It is my understanding that time reversal invariance for Dirac fermions is usually (in 3+1 dimensions at least) implemented by an antiunitary operator ${\mathfrak T}$ that acts on the Dirac field ...
6
votes
2answers
2k views

Hermitian adjoint of 4-gradient in Dirac equation

I'm having issues deriving the Dirac adjoint equation, $$\overline{\psi}(i\gamma^{\mu}\partial_{\mu}+m)=0.\tag{1}$$ I started by taking the Hermitian adjoint of all components of the original Dirac ...
2
votes
0answers
284 views

Are Lifshitz and Berestetskii right in this case?

In the Quantum electrodynamics book (look at the problem) its authors Lifshitz and Berestetskei claim that operator of charge conjugation $\hat {C} = -\alpha_{2}$ in Majorana basis transforms as $\hat ...
3
votes
1answer
453 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 $\gamma$-...
0
votes
3answers
547 views

Is spinor the sum of scalar, vector, bi-vector, pseudo-vector, and pseudo-scalar?

Is spinor $\psi$ actually the sum of scalar, vector, bi-vector, ..., pseudo-scalar? Before talking about spinors, we have to differentiate two kinds of spacetime, demonstrated with the example of ...