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Questions tagged [dirac-matrices]

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Converting two component product to four component notation

Consider the product of two left Weyl spinors in the notation commonly found in supersymmetry, \begin{equation} \chi ^\alpha\eta_\alpha = \chi ^\alpha \epsilon _{ \alpha \beta } \eta ^\beta \end{...
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Time reversal operator symmetry of dirac lagrangian

I want to prove time reversal symmetry of Dirac Lagrangian, I have some problems with calculations. I start with \begin{eqnarray} T\psi T = U \psi \end{eqnarray} \begin{eqnarray} T\bar{\psi } T = ...
358 views

Motivation for spinors

After it was found that the gamma matrices couldn't be Pauli matrices and only had to be larger and even, why was their need to define a new algebraic object (i.e a Dirac spinor)? Why couldn't a ...
3k views

Parity on gamma matrices

I want to understand clearly why $P \gamma^{\mu} P = \gamma^{\mu}$, where $P$ is the parity operator. This result follow for example from pag. 66 of Peskin-Schroeder. The parity operator acts ...
349 views

Simplifying a seemingly simple gamma matrix identity

When studying from the book by Wise and Manohar, Heavy Quark Physics (pg 102), I came across a seemingly simple identity that I am not able to prove. It's likely an easy problem but I can't for the ...
156 views

$\gamma^5$ factor in Quantum Field Theory

I have a problem with interpretation of $\gamma^5$ factor in the interaction Hamiltonian. I know that $\frac{1\pm\gamma^5}{2}$ is a helicity projection and it requires helicity conservation in ...
644 views

Proof of equivalence of different representations of the $\gamma$-matrices in the Dirac equation

This question concerns the Dirac equation and the $4\times4$ $\gamma$-matrices. The task is to prove that a similarity transformation of the standard $\gamma$-matrix conserves the commutation relation ...
515 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 ...
235 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_{...
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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*} \...
136 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 ...
1k views

What is unitarily similar matrices?

In one of tasks I met the concept of unitarily similar matrices: in particular, I need to prove that sets $\gamma_{\mu}, -\gamma_{\mu}$ (Dirac gamma matrices) are unitarily similar. I don't know what ...
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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 . $$\begin{pmatrix} A_1 & 0 & \cdots & 0 \... 0answers 232 views Are Lifshitz and Berestetskii right in this case? In the Quantum electrodynamics book (look at the problem) its authors Lifshitz and Berestetskei claim that operator of charge conjugation \hat {C} = -\alpha_{2} in Majorana basis transforms as \hat ... 2answers 2k views Why the lowest order of matrices in Dirac equation are 4x4 matrices? [duplicate] Why the lowest order of matrices in Dirac equation (Relativistic Quantums) are 4x4 matrices (and can not be 2x2 matrices)? How to prove it? 1answer 1k views Closed formula for product of gamma matrices I was asking myself if there is a closed formula for the following product of gamma matrices:$$\gamma_\mu\gamma_\nu \gamma_5.$$I would like to express this matrix in terms of the basis$${\...
The spinor inner product in particle physics is given by $\overline{\psi} \psi = \psi^{\dagger} \gamma_0 \psi$, where I take the convention that the zeroth gamma matrix is hermitian while the rest ...