# Questions tagged [dirac-matrices]

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### Traces in 't Hooft-Veltman scheme

I'm currently looking at the 't Hooft-Veltman regularization scheme and I'm a bit confused on how exactly one calculates traces in this scheme. As far as I understand one has to divide the $D$-...
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### Four-momentum matrix dot product sigma matrix

In the QFT book of Peskin and Schroeder, they introduce the notation: \begin{align} \sigma^{\mu}=(I,\sigma^{i})\\ \bar{\sigma^{\mu}}=(I,-\sigma^{i}). \end{align} On page 46 (Eq.(3.50)), They take the ...
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### Clarifying Griffiths syntax when solving Feynman Diagrams

In Griffiths textbook "Introduction to Elementary Particles", he details how to use Feynman diagrams and walks through some examples of solving some essential particle interactions. The ...
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### Deriving the Generalized Fierz Transformation from Schroeder's Textbook

I am self studying QFT from the textbook An Introduction of Quantum Field Theory and the corresponding solutions from Zhong-Zhi Xianyu. The generalized Fierz Transformation is derived in problem 3.6. ...
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### (Non-)Hermiticity of Dirac operator

I have a Dirac operator given by \begin{equation} D\!\!\!/[A, A^{5}]=\gamma^\mu D_\mu=\gamma^\mu (\partial_{\mu} - {\rm i} A_{\mu} - {\rm i} \gamma_{5} A_{\mu}^{5}), \end{equation} where $A_{\mu}$ ...
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### Respresentation Dirac group and Lie algebra

I am reading Peskin and Schroeder's book on QFT and have some difficulties with representation groups. Let's start with the Lorentz group since it is easier. let $\Lambda$ be a Lorentz transformation, ...
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### Gamma identities computations

I got my BsC in Physics many years ago (18yrs) but at those times QFT was not covered during the career, only in Ph.D. courses. Now I'm studying it at my own during my free time using several books on ...
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### Trace of two Dirac matrices in 4 dimensions

I want to show that tr($\gamma^\mu \gamma^\nu$) = 4$\eta^{\mu \nu}$. I know that {$\gamma^\mu , \gamma^\nu$} = 2$\eta^{\mu\nu}I_4$ and tr($\gamma^\mu \gamma^\nu$) = tr($\gamma^\nu \gamma^\mu$), so tr(...
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### Accounts on the solutions of the Dirac equation

Consider the Dirac equation $(i\gamma^{\mu}\partial_{\mu}-m)\psi = 0$. As it is well known, there are different representations for the matrices $\gamma^{\mu}$, $\mu = 0,1,2,3$, the most famous ones ...
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### Does $\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}\mathbb1$ determine the hermiticity of the gamma matrices?

If I remember correctly, the derivation of the Dirac equation requires that $\gamma^0$ is Hermitian while $\gamma^i$ for $i=1,2,3$ is anti-Hermitian. This is clearly true for the standard Dirac ...
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### Current divergence

How do I calculate the current divergence $$\partial_\mu A_{ud}^\mu = \partial_\mu\left (\overline{u} \gamma^\mu \gamma^5 d(x)\right )?$$ I don't understand exactly how to manipulate gamma matrices.
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### Expression of $\not{p}$ in Dirac equation

In scattering amplitudes, page 9, equation (2.6), (2.7), $\not{p}$ (in the Dirac equation (2.4)) is as follows: \begin{align} \not{p} = \left( \begin{matrix} 0 & p_{a\dot{b}} \\ p^{\dot{a}b} & ...
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### Proof that $\gamma_{5}^2=1$

I need to prove that $\gamma_{5}^2=1$, and in order to do this I wrote: \begin{equation} (\gamma_{5})^2=\gamma^{5}\gamma_{5}=\left(-\frac{i}{4!}\epsilon^{\mu\nu\rho\sigma}\gamma_{\mu}\gamma_{\nu}\...
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### Dirac spinor in the chiral basis

In the chiral basis, the gamma matrices take the form $$\gamma^0=\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}, \quad \gamma^j=\begin{bmatrix}0 & -\sigma^j \\ \sigma^j & 0\end{bmatrix}$$...
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### How to take trace over group and Dirac indices?

I'm currently reading Pokorski's book "Gauge Field Theories" and in Chapter 13 he discusses, among other things, Fujikawa's method of deriving the chiral current (see page 488 and the ...
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### How to invert this matrix?

Is there a smarter method for finding the determinant and inverse of the following matrix, without using the brute force procedure? (When I say brute force, it is to write the matrix with each term ...
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### The tensor product in the Hamiltonian of graphene

I have the Hamiltonian of pristine graphene \begin{equation} H=v_{F}.\boldsymbol{\gamma}.\boldsymbol{p} \end{equation} with $\boldsymbol{p}=(p_{x},p_{y})$ is the momentum operator, $v_{F}$ is the ...
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### Proof of $\mathrm{tr}(\gamma^{5}\gamma^\mu\gamma^\nu)=0$

Using $\gamma^{5}\gamma^\mu=-\gamma^\mu\gamma^{5}$ and $\mathrm{tr}(AB)=\mathrm{tr}(BA)$ I obtain \begin{equation}\tag{1} T_{\mu\nu}:=\mathrm{tr}(\gamma^{5}\gamma^\mu\gamma^\nu)=-\mathrm{tr}(\gamma^\...
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### Property of Dirac operator in spinor frame bundle

The Dirac operator is defined by $$d=g(e_\alpha,e_\beta) \gamma(e_\beta)\nabla_{e_\alpha} \tag1$$ Here $\nabla$ is the spin covariante derivative $e_\alpha$ basis of the tangent space and $\gamma$ the ...
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### Derivation of Dirac matrices in spherical polar coordinates

How to derive Dirac $\gamma$ matrices in spherical polar coordinates?
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### How these two approaches to spinors in curved spacetimes relate?

Regarding spinors in curved spacetimes I have seem basically two approaches. In a set of lecture notes by a Physicist at my department he works with spinors in a curved spacetime $(M,g)$ by picking a ...
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### Why are the 4-vector and bispinor representation of the Lorentz algebra in particular so related?

When learning about the Dirac equation, there are several indications that the fundamental (4-vector) representation and the bispinor representations are connected in some way. To give an example, the ...