Questions tagged [dirac-matrices]

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30 views

Why $(\vec{\sigma}\cdot\vec{p}) (\vec{\sigma}\cdot\vec{p}) = (\vec{\sigma}\cdot\vec{p}) p^0$ for Pauli matrices?

I am trying to verify that the following equation is true: $$(\vec{\sigma}\cdot\vec{p}) (\vec{\sigma}\cdot\vec{p}) = (\vec{\sigma}\cdot\vec{p}) p^0$$ where $p^\mu=(p^0,\vec{p})$ is the four momentum ...
0
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1answer
41 views

Spinor matrix representation of a spacetime vector in 2+1D

Let $a^\mu$ be a spacetime vector in 2+1D: $(t,x,y)$. What would be the spinor-matrix representation ${A^\alpha}_\beta$ of the spacetime vector $a^\mu$? I have not been able to even find examples or ...
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0answers
26 views

Calculation of Parity in Quantum Field Theory

In the book "Relativistic Quantum Mechanics An Introduction To Relativistic Quantum Fields" by Luciano Maiani Omar Benhar, page 174, the picture of that page is provided below. I don't ...
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0answers
40 views

Show $[\gamma^\mu,\eta^{\nu\lambda}I]=0$ for Dirac matrices

I am trying to show the following commutation relation for the Dirac matrices $\gamma^\mu$ and the metric $\eta^{\nu\lambda}$: $$2[\gamma^\mu,\eta^{\nu\lambda}I]=0$$ where $I$ is the 4x4 identity ...
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1answer
45 views

Interchanging positions of Gell-Mann matrices with Dirac matrices, Pauli matrices

The anti-commutation relations for Gamma matrices $\big\{\gamma ^\mu , \gamma ^\nu \big\} = 2g ^{\mu \nu} $ can be used for interchanging positions of the respective matrices in a given expression, ...
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1answer
43 views

What is $D$ or $D$-with-a-slash-through-it in the Standard Model equation(s)?

In the mathematical formulation of the Standard Model, which I do not understand yet, there is a capital letter $D$ or $D$-with-a-slash-through-it that I can't find an explanation for. Flip Tanedo (a ...
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1answer
21 views

Commutation relation of gamma matrices

While going through the third chapter of Peskin's QFT book, I am stuck at the following proof: $$[\gamma^\mu, S^{\rho\sigma}] = (\mathcal{J}^{\rho\sigma})^\mu_{~\nu} \gamma^\nu,$$ where, $$S^{\mu\nu} =...
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0answers
44 views

Unitary transformation of Dirac equation

Dirac equation is given by $$(i\gamma^\mu\partial_\mu-m)\psi=0.$$ The matrices $\gamma^\mu$ satisfy the relation $$\{\gamma^\mu,\gamma^\nu\}=\gamma^\mu\gamma^n+\gamma^\nu\gamma^\mu=2g^{\mu\nu},$$ ...
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1answer
38 views

How to evaluate traces of Dirac matrices with slashed momentum in between?

I am trying to calculate the tree level scattering cross section for Moeller scattering using QED. I have got some traces which I cannot evaluate. e.g.$$Tr(\gamma^\nu \not{p_2}\gamma^\mu \gamma_\nu \...
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0answers
35 views

Time-reversal symmetry and the generalized special axes (eg: $y$) in any $D$ space dimension

In 3D space, it is common to choose the time-reversal symmetry acting on spin-1/2 doublet fermions as $$ T = i \sigma_y K = \begin{pmatrix} 0 & 1\\ -1& 0\end{pmatrix} $$ where $K$ is complex ...
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1answer
105 views

Prove $\displaystyle{\not}{\nabla}^2=\nabla_{\mu} \nabla^{\mu}-R/4$ [closed]

I am trying to show $\displaystyle{\not}{\nabla}^2=\nabla_{\mu} \nabla^{\mu}+R/4$ where $R$ is Ricci scalar. $\nabla_{\mu}$ is covariant derivative for spinor: \begin{equation} \nabla_{\mu}=\partial_{...
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0answers
45 views

Dirac Bilinear Transformation Laws (vector)

The question is to verify that the following Dirac Bilinears obey the following transformation law: $$\bar\psi'(x') \gamma^\mu \psi'(x') = \Lambda^\mu_\nu \bar\psi(x) \gamma^\nu \psi(x)$$ What I know ...
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0answers
61 views

Quantum Field Theory Identity

We are asked to show that $$\not A\not B = A\cdot B - i\sigma_{\mu\nu}A^\mu B^\nu $$ I know that $$\not A = A_\mu \gamma^\nu $$by definition, and: $$\sigma_{\mu \nu}=\frac{i}{2} [\gamma_\mu, \gamma_\...
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1answer
64 views

Evaluating the modulus squared of a spinor chain with different number of spinor and anti-spinors

I want to evaluate the interference between diagrams in a BSM model whose relevant part of the contributions are \begin{equation} \begin{split} A&=[\bar{u}_e(k_2)v_e(k_3)] [\bar{u}_e(k_1)u_\mu(p_1)...
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0answers
15 views

Gamma matrices with holomorphic indices? Exercise 9.5 (String Theory and M-Theory by Becker Becker Schwartz)

In Exercise 9.5 of String theory text by Becker, Becker and Schwartz, the holomorphic indices are used where $\gamma_7 = (1-\gamma_1)(1-\gamma_{\bar 1}) (1-\gamma_2)(1-\gamma_{\bar 2}) (1-\gamma_3)(1-...
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0answers
44 views

How to prove that helicity is conserved for massive Dirac particles

The helicity is defined as: $$h=\frac{\vec{S}.\vec{p}}{||\vec{p}||}$$ where $$S= \frac{\hbar}{2} \zeta$$ and $\zeta$ equals \begin{pmatrix} \vec{\sigma} & 0\\ 0 & \vec{\sigma} \end{pmatrix} ...
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2answers
127 views

What kind of tensor is $\psi^\dagger\psi$?

I'm trying to find how does this quantity $$\psi^\dagger\psi$$ transforms under a Lorentz transformation. Where $\psi$ is a Dirac spinor. What I've tried so far: It is known that a Dirac spinor ...
3
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1answer
62 views

Becker, Becker, Schwarz: “String Theory and M Theory” Exercise 5.3

Supersymmetric transformation: $$ \begin{align} \delta\Theta^{Aa} =& \varepsilon^{Aa}, \tag{5.3} \cr \delta X^\mu =& \bar{\varepsilon}^A\Gamma^{\mu}\Theta^A. \tag{5.4} \end{align} $$ The ...
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1answer
49 views

Dirac conjugation of a 3x3 matrix

This question might be stupid, but when I compute $\bar{B}$ in the Lagrangian, I have to multiply 3x3 $B$ matrix with 4x4 $\gamma_0$ matrix (Dirac's conjugation) which are incompatible in size. What ...
1
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1answer
39 views

Sign of pair of Dirac spinor bilinear

I don't understand the following statement: Any pair of Dirac spinors verifies $(\bar{\Psi}_1\Psi_2)^\dagger=\bar{\Psi}_2\Psi_1$ and it is valid for both commuting and anti-commuting (Grassmann-valued)...
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2answers
112 views

Product of an odd number of Dirac $\gamma$ matrices [closed]

Suppose $n$ is an odd number. Why can we write $a\llap{/}_1 a\llap{/}_2 ... a\llap{/}_n$ as $$a\llap{/}_1 a\llap{/}_2 ... a\llap{/}_n = V_\mu \gamma^\mu + A_\mu \gamma^\mu \gamma_5$$ for some $V_\mu, ...
1
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1answer
46 views

(Anti)commutation relations for higher-dimensional anti-symmetrized Gamma matrices

Suppose $a,b,c...=0,....,D-1$ are Lorentz indices of $SO(1,D-1)$ tangent space and consider $D$-dimensional Clifford algebra defined by the usual anticommutation relation $$\{\Gamma^a,\Gamma^b\}=2\eta^...
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1answer
29 views

How does following relation generates Lorentz generator?

In Schwartz book sec 10.3, Schwartz says following: The Lorentz generators when acting on Dirac spinors can be written as $$S^{\mu \nu}=\frac{i}{4}[\gamma^{\mu},\gamma^{\nu}]$$ But what I am able to ...
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1answer
104 views

Matrix “dimensional analysis” of Lagrangians in QFT

Since the important things in the QFT Lagrangian are vectors and matrices, I wanted to do a "matrix dimensional analysis" of each term. The electromagnetic Lagrangian (ignoring all constants ...
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1answer
61 views

Proving an equation built up out of Dirac-$\gamma$ matrices

Given the following Feynman Amplitude: $$\mathscr{M}=\bar{u_s} (\vec p') \Gamma u_r (\vec p) \tag 1$$ Where: $\bar u_s, u_r$ are Dirac spinors ($1\times 4$ and $4 \times 1$ matrices respectively) $\...
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0answers
55 views

Dirac matrices in 2 dimensions and $\gamma_5$

The gamma matrices obeying the Clifford algebra can be represented by Pauli matrices in (1+1) and in (1+2) dimensions. But is it possible to define a $\gamma_5$ in these dimensions? In the sense of an ...
3
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1answer
126 views

Why does the trace show up in such expressions?

I've been studying different scattering processes (from Mandl & Shaw QFT's book, chapter 8) and there's always a purely-mathematical common step I do not understand: the showing-up of the trace. ...
0
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1answer
129 views

Normalization for a free Dirac plane wave

I've recently come about the free plane wave solutions to the Dirac equation, and i'm having a hard time proving that the normalization factor $n$ is $$n=\frac{1}{\sqrt{2m(m+\omega)}}$$ Where $\omega$ ...
1
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1answer
58 views

$\gamma_5 \gamma^\sigma$ expressed with Levi-Civita tensor

We have that $\gamma_5 = -\frac{i}{4!} \epsilon^{\mu \nu \rho \sigma} \gamma_\mu \gamma_\nu \gamma_\rho \gamma_\sigma$. Using this, what approach would be suggested in showing that $\gamma_5 \gamma^\...
1
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1answer
81 views

Is this a typo in Peskin's QFT?

In ''An intro to QFT (2018)'' chapter 3, Peskin does the following: Let me introduce some notation first, let $v^s_k=\begin{pmatrix}\;\;\,\sqrt{k\cdot\sigma}\,\xi^{-s}\\-\sqrt{k\cdot\bar{\sigma}}\,\...
2
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1answer
73 views

Calculating traces for triangle diagrams with massless fermions

I am following Schwarz Quantum Field Theory textbook. In particular, I am looking at triangle diagrams with massless fermions. On pg. 623 - 624 Schwarz attempts to calculate $q_\mu^1 M_{5}^{\alpha\mu\...
0
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1answer
41 views

Expansion of trace in photon self-energy

I am studying through the photon self-energy $$ i\Pi_{\mu\nu}(q) = \int\frac{d^4 k}{(2\pi)^4}Tr\left[(-ie\gamma_\mu)\frac{i(\require{cancel}\cancel k+m)}{k^2-m^2+i\epsilon}(-ie\gamma_\nu)\frac{i(\...
0
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1answer
46 views

Is it right to think of Parity as a change of basis in Dirac's Lagrangian?

I'm trying to understand CPT symmetries in the Dirac Lagrangian but, so far, I've had more questions than answers. My naive view of CPT transformations is the following (please don't doubt to correct ...
0
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2answers
97 views

How do we know there are only 16 Dirac bilinears?

We know there are 5 types of bilinears in 4 dimensions, all of them add up to contribute with 16 independent DoF (degrees of freedom). Namely, these bilinears are known as: scalar (1DoF), pseudoscalar(...
0
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1answer
68 views

Spinor transformation matrix derivation

The spinor in the Dirac equation should transform via a 4x4 matrix $S$ that depends on the specific Lorentz boost/rotation: $\psi '(x')=S(\Lambda )\psi(x)\tag1$ Where S satisfies: $S^{-1}\gamma ^{\...
2
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0answers
156 views

An $SL(2,C)$ representation and Dirac Spinor

In PCT, spin and statistics, and all that book, the following example is given: Let $S(A)$ be a representation of $SL(2,C)$ given as : $$S(A)=\frac{1}{2}\left(a^{0} \mathbf{1}+\mathbf{a} \cdot \...
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0answers
65 views

Is there a bosonic representation of Clifford algebra in (1,3) spacetime?

By a suitable combination of Dirac's $\gamma_\mu$ matrices one can define creation and destruction operators satisfying fermionic anticommutators. Is there a similar result for bosons in the context ...
0
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1answer
61 views

How does $\{\gamma^\mu , \gamma^\nu\}= 2g^{\mu\nu}I$ imply that $\gamma^\mu$ is traceless?

How does $\{\gamma^\mu , \gamma^\nu\}= 2g^{\mu\nu}I$ imply that $\gamma^\mu$ is traceless? where I represents the identity matrix I know that $$\{\gamma^\mu , \gamma^\nu\}=\gamma^\mu\gamma^\nu + \...
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1answer
46 views

Equivalence of the Dirac representation of the Lorentz algebra and its conjugate in even dimensions - Polchinski

In Polchinski's String Theory, Appendix B.1 we look at the smallest irreps of the Clifford algebra in even-dimensional spacetimes $d=2k+2$. In (B.1.16) he defines two operators from the gamma matrices ...
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2answers
108 views

Hermitian Adjoint of Dirac Equation vs Dirac Lagrangian

I have a question about the self-adjointness of the gradient in spinor space. In the derivation of the Dirac adjoint equation, as in Hermitian adjoint of 4-gradient in Dirac equation , it has been ...
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2answers
59 views

Why did Dirac choose a linear equation in momentum for formulating a relativistic wavefunction?

The Dirac equation arose out of the need for a quantum mechanical wave equation that treats position and time on an equal basis, just as relativity expects them to be. Now the Schrodinger equation is ...
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1answer
120 views

Understanding the Dirac equation

I am reading a physics book where the Dirac equation is being introduced in the form: $$\left[c \boldsymbol{\alpha} \cdot\left(\boldsymbol{p}+\frac{e \boldsymbol{A}}{c}\right)-e \phi+\beta m c^{2}\...
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2answers
66 views

Dirac equation: covariant form and original form

In my lecture book, the Dirac equation is derived and given as the equation: $$i \hbar \gamma^{\mu} \partial_{\mu} \psi-m c \psi=0 \tag{1}$$ Where: $$\gamma^{0}=\left(\begin{array}{rr} 1 & 0 \\ 0 ...
-3
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1answer
60 views

Gamma matrices - Dirac [closed]

I tried to ask this question: Prove that $\{\gamma_\mu , \gamma_\nu\} = 0$, but I was unable to resolve it. Can someone help me?
0
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0answers
86 views

Gamma matrices and the metric

There is a step in a Dirac spinor manipulation problem I'm working on that requires me to make the following remark, $$ 2 \gamma^\sigma \eta^{\rho\mu}\mathbb{1}_4 \gamma^\nu = 2 \eta^{\rho\mu}\mathbb{...
1
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0answers
78 views

Is this an alternative Dirac Equation in curved space?

The usual covariant derivative for the Dirac equation in curved space is: $$D_\mu \psi = (\partial_\mu - {i \over 4} {\omega_\mu}^{ab} \sigma_{ab}) \psi$$ However, I think I found another ...
1
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2answers
41 views

Using the infinitesimal-angle form of the rotation operator to perform infinitesimal rotaions on spin-1/2 states

Im stuck on a homework problem where I must use the rotation operator $$\hat{R}_{e_z,d\phi}=\hat{I}-i\frac{\hat{S}_z}{\hbar}d\phi,$$ to act on $|\psi_{\theta,\phi}\rangle=\cos(\theta/2)|\uparrow_z\...
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1answer
50 views

What is the old (50's) convention on Dirac gamma matrices?

What were the standard relations for gamma matrices in the mid 50's, when 4-vectors where represented by $(x_1, x_2, x_3, ict)$? In particular the values of $\gamma^\mu\gamma^\nu$ , the definition of $...
1
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1answer
75 views

Partial derivatives in the derivation of a Dirac Spinor

As per JF132's answer to Conservation of the axial current using Dirac equations of motion, "since the gamma matrices $\gamma^\mu$ are $4\times 4$ matrices, and the conjugate Dirac spinors $\bar{\...
0
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1answer
56 views

Dirac adjoint of a gamma matrix product

I was wondering if, because for generic bounded operators, anti-distributivity applies, i.e. $$(AB)^{\dagger} = B^{\dagger}A^{\dagger},$$ the same is true of gamma matrices. I was asked to prove $$\...

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