Questions tagged [dirac-matrices]

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Particles physics - Projection matrices

I'm struggling with a question were I'm asked to simply the following expression such that my result only involves one projection matrix. $$P_L \not p \not k P_L $$ Where $$P_L = (\frac{1-\gamma_5}{2})...
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Commutator and Anti-commutator

In spinor field representation I found that for any scalar $\#$, following happens: $[\Sigma^{ij},\#\gamma^0]=0$ and $\{\Sigma^{i0},\#\gamma^0\}=0$ Is there any analogue for vector representation too? ...
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21 views

Fierz Identity calculation

While reading an article, it's said that to simplify the following Dirac structure $$\left(P_Lv_j^d\bar{v}^s_kP_R\right)_{\alpha\beta}\tag{1}\label{1}$$ where $j,k$ are color indices and $d,s$ ...
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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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71 views

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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1answer
67 views

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 ...
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33 views

What books would you recommend to understand mathematically the Dirac matter

I am a math student who got interested in the topics above also i want to learn about the Dirac matter the spinors the Einsteins GR and the Yang-Mills Maxwell Anderson Higgs theories and models and ...
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1answer
31 views

Finite spinor transformation

I am currently studying the finite spinor transformations in QFT. There is equation which i do not fully understand. Rather i don't understand the notation and what it represents: In the script, we ...
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1answer
71 views

A calculation on the relation between gamma-matrices and total antisymmetric tensor

In some papers I found the following relations: $$\left[\gamma^N,\left[\gamma^I,\gamma^J\right]\right]=4\left(\eta^{NI}\gamma^J-\eta^{NJ}\gamma^I\right)$$ $$\left\{\gamma^N,\left[\gamma^I,\gamma^J\...
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50 views

How to deal with cross-terms in squared slashed covariant derivative in QED?

I am trying to compute the square of the slashed covariant derivative (the sum of the partial derivative and vector potential contracted with the gamma matrices) in terms of the square of the standard ...
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1answer
55 views

Complex conjugate of the Dirac equation

(Following the calculations done in 'Quantum Field Theory in a Nutshell' [Second Edition] by Zee, Page 101) The Dirac equation in the presence of an electromagnetic field is given by: $$ [i \gamma^{\...
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32 views

How to decompose the spinor kinetic term of $D=10$ SYM in terms of lower dimensional parts?

The kinetic term for the spinors in $D=10$ SYM is $\lambda \gamma^\mu \partial_\mu \lambda$, where $\lambda$ is a 16 component Majorana-Weyl spinor and $\gamma^\mu$ is a 16 by 16 matrix satisfying $\{\...
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Deriving the Dirac equation & Dirac matrices

I'm learning about how the Dirac equation came about. In pg. 20 of the Lectures on Quantum Field Theory book by Ashok Das, pg.20, the author starts with the energy momentum relation $$p^\mu p_\mu=m^...
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34 views

Question about the trace in simplifying $e\!-\!e$ cross section [duplicate]

In computing a cross section for the mutual scattering of two electrons (Zee, Sec II.6), I have gotten the expression factorized so that it contains a term \begin{align*} \tau^{\mu\nu}&=\frac{1}{...
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32 views

Anti-commutator of Dirac matrices

Consider $$ \beta = \begin{pmatrix} \mathbf{1} & 0 \\ 0 & -\mathbf{1} \end{pmatrix},\quad \alpha_i = \begin{pmatrix} 0 & \mathbf{\sigma}_i \\ \mathbf{\sigma}_i&0 \end{pmatrix}.$$ The ...
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1answer
130 views

Trace of the log of a matrix

When Zee computes the Grassmann path integral in Section II.5 of his QFT book, he uses an algebraic step I can't follow. I follow this part: \begin{align} \text{Tr}\ln\!\big[\gamma^5\big(-i\gamma^\mu\...
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Why do physicists use wave functions with more than two components?

For $n$-body systems you just need a single component wavefunction. For example, for a two-body system you would need a wavefunction of 6 variables. $\psi(x_1,x_2,y_1,y_2,z_1,z_2)$ That satisfies the ...
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Simple relation of Dirac matrices in Dirac propagator

There is this equivalence between representations for the Dirac propagator: $$\frac{i}{\not{p}-m}=\frac{i(\not p+m)}{p^2-m^2}.$$ The only way i could think this to be true is if $\gamma^{\mu}p_{\mu}\...
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2answers
105 views

Notations related to identities for spinors in the rest frame

This question pertains to some notation in Zee's QFT book, Section II.2. The Dirac equation is $$ (i\gamma^\mu\partial_\mu-m)\psi(x)=0, $$ which we can write in momentum space with the Fourier ...
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33 views

Indices in Dirac's spinorial representation

The other day in class we derived Dirac's equation and talked a little about its relativistic covariance. In particular, we said that the electronic wavefunction $\psi(x)$ transforms as $\psi'(x'(x))=...
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1answer
58 views

Cyclic identity of $\gamma$-matrices in 3D and 4D

Reading the book on Supergravity from Freedman & van Proeyen I was very bewildered by the so called cyclic identity of $\gamma$-matrices of the Clifford-algebra(eq. 3.67) important in string ...
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1answer
73 views

Prove the hermiticity of $\sigma_{\mu\nu} F^{\mu\nu}$

In exercise 4.2, "Relativistic Quantum Mechanics" by Bjorken-Drell, an additional term is added to the Dirac Hamiltonian such that new equation of motion is $$\left(i\gamma_\mu{\nabla^\mu}-...
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50 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 ...
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50 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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1answer
1k 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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54 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
62 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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122 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
38 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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65 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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61 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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39 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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109 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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58 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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62 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
146 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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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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50 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
153 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 ...
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1answer
74 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
56 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 ...
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1answer
42 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
257 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, ...
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1answer
57 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
34 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
119 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
65 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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72 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 ...
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1answer
140 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. ...
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1answer
139 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 $\omega$ ...

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