Questions tagged [dirac-matrices]

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Property of Charge Conjugation Operator

In class, we have defined the Charge Conjugation Operator ($C$) such that: \begin{equation} C \left(\gamma^\mu\right)^T C^{-1} = - \gamma ^\mu , \end{equation} \begin{equation} \psi^C \equiv C\,\...
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96 views

How are the covariant Pauli matrices defined?

When doing calculations with Weyl spinors, terms like $\theta\sigma^\mu\theta^\dagger$ appear. I know that for 3+1 spacetime dimensions, $\sigma^\mu = (\textbf{1}, \sigma^i)$ with $i=1,2,3$ the usual ...
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1answer
646 views

Trace of Gamma Matrices [closed]

If I have: $Tr(\gamma^{\mu}\gamma^{\alpha}\gamma^{\nu}\gamma^{\beta}\gamma^{\rho}\gamma^{\gamma}\gamma^{\sigma}\gamma^{\delta})$ and I want to get it re-ordered like $Tr(\gamma^{\alpha}\gamma^{\...
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1answer
268 views

Trace technology with polarisation vectors

Consider $d$-dimensional gamma matrix structures. I have an expression like $$ \sum_{h_2=\pm}\text{Tr}(\not{\xi}_2\not{p}_3\bar{\not{\xi}}_2\not{p}_1), $$ where $\not p=p^\mu \eta_{\mu\nu}\gamma^\nu$ ...
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2answers
398 views

Zee's use of Kronecker Product in “QFT in a Nutshell” to represent Dirac matrices

In his book Quantum Theory in a Nutshell (2nd edition, p. 94), Zee describes the Dirac gamma matrices and lists a representation using Pauli matrices and the identity matrix. For example he writes $$ ...
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2answers
408 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,...
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2answers
609 views

Why is $\gamma^5$ used to define the projection operator?

The properties of the projection operators are defined as: $$P_+ = \frac{1}{2}(1+\gamma^5)$$ $$P_- = \frac{1}{2}(1-\gamma^5)$$ where $\gamma^5 = -i\gamma^0\gamma^1\gamma^2\gamma^3$ and their key ...
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0answers
150 views

Transpose Noether current from $U(1)$ symmetry of the free Dirac field

When I read through the notes of a particle physics script there is the following identity that I don't understand $$\left(\psi^T {\gamma^\mu}^T \bar{\psi}^T \right)^T = -\bar{\psi}\gamma^\mu \psi.$$ ...
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0answers
132 views

Lorentz invariance of the axial current

I want to determine whether $$\bar{\psi}\gamma^\mu \gamma^5 \psi \bar{\psi}\gamma_\mu \gamma^5\psi = j_5^\mu j_{5\mu},$$ where $\bar{\psi} = \psi^\dagger \gamma^0 $ and $\gamma^5 = -i\gamma^0\gamma^1\...
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0answers
36 views

What happened to $\gamma^4$? [duplicate]

Kind of a simple question, but I was looking up some things related to the Dirac equation, and I noticed something about the $\gamma$ matrices. There's $\gamma^0,\,\gamma^1,\,\gamma^2,\,\gamma^3$, ...
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2answers
309 views

Derivation of generators of Lorentz group for spinor representation

I want to prove $$S^{\mu \nu}=\frac{i}{4}[\gamma^\mu,\gamma^\nu].$$ I started from $$[\gamma^\mu,S^{\alpha\beta}]=(J^{\alpha\beta})^\mu_\nu \gamma^\nu$$ Putting the value of $(J^{\alpha\beta})^\mu_\...
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1answer
155 views

Green's function for Dirac operator on $S^4$

Let $S^4$ be a round sphere of radius 1 (with the standard Riemannian round metric), and let $D_\text{F} \equiv \gamma^\mu \nabla_\mu$ be the Dirac operator on $S^4$, acting on the usual spinors for ...
2
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1answer
1k views

Charge conjugation operator and gamma matrices

The gamma matrices are defined by their anticommutation relations, which are symmetrical in permutations of $\gamma_1, \gamma_2, \gamma_3$. Given this symmetry, why is the change conjugation operator $...
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0answers
127 views

Product of Lorentz transformation with gamma matrices

In my QFT course, we wrote the following things : $$ S(\Lambda)^{-1} \gamma^\mu S(\Lambda)=\Lambda^{\mu ~ .}_{. ~ \nu} ~ \gamma^\nu $$ So when we ""apply"" a Lorentz boost to gamma matrices, they ...
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0answers
164 views

Dimension of Representation of Majorana Fermions with Euclidean Metric?

It is possible to represent the Dirac matrices in the Majorana basis using $N= 2^{⌊d/2⌋}$-dimensional matrices, as shown here. This source uses a Minkowski metric. It would then be possible to move to ...
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1answer
447 views

About the fifth gamma matrix [closed]

How can one prove that $$\gamma^5=\frac{i}{4!}\varepsilon_{\mu\nu\alpha\beta}\gamma^{\mu}\gamma^{\nu}\gamma^{\alpha}\gamma^{\beta}$$ from the following: $$\gamma^5:=i\gamma^0\gamma^1\gamma^2\gamma^...
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1answer
266 views

Charge conjugation in arbitrary basis

Consider the matrix $C = \gamma^{0}\gamma^{2}$. It is easy to prove the relations $$C^{2}=1$$ $$C\gamma^{\mu}C = -(\gamma^{\mu})^{T}$$ in the chiral basis of the gamma matrices. Do the two ...
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1answer
77 views

Spin states in a finite potential well

i have a question concerning an electron in an attractive potential well. Let's suppose the potential function is defined as $$V = \left\{ \begin{array}{cl}0, & \mbox{for } z < 0\\ ...
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0answers
100 views

Clifford Algebra in 3D [duplicate]

Why the gamma matrices are taken 2 by 2 (Pauli matrices) in 3 dimensional Clifford Algebra. As in 4D Clifford Algebra the matrices are 4 by 4, in 3D Algebra why are they not 3 by 3 matrices? The ...
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1answer
573 views

Pauli matrices and Lorentz transformations

Consider the Weyl equations: \begin{align} i\sigma^{\mu} \partial_{\mu} \psi_{L} & = 0 \\ i\overline{\sigma}^{\mu} \partial_{\mu} \psi_{R} & = 0, \end{align} where $\sigma^{\mu} = \left ( \...
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1answer
2k views

Confusion about slash notation

I am confused about the slash notation and especially taking the square of a slashed operator. Defining $\displaystyle{\not} a \, = \, \gamma^\mu a_\mu$ we have $\,\,$ $\displaystyle{\not} a \...
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1answer
333 views

Spinor indices manipulation in the Noether's current for free fermions

I can't solve this apparent paradox; I have the free lagrangian for massive fermions $\mathscr L = i\bar\Psi\gamma^\mu\partial_\mu\Psi - m\bar\Psi\Psi$ which is invariant under the global phase ...
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1answer
821 views

Partial completeness relation for Dirac spinors

in studying trace techniques to obtain matrix elements, I came across a problem when we treat scattering of neutrinos on protons. Indeed, since those neutrinos are supposedly created in a weak decay, ...
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2answers
687 views

Solution of Dirac equation-Positive and Negative energy

For particles defined with positive energy, we use $$\phi= \begin{pmatrix} 1 \\ 0 \\ \end{pmatrix} $$ or $$ \begin{pmatrix} 0 \\ 1 \\ \end{pmatrix} $$...
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2answers
319 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 \...
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1answer
71 views

What is the implication of the existence of a non-singular matrix $S: \gamma^\lambda S=S \gamma^m u$? [closed]

$$\gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu=2g^{\mu\nu}I_4$$ Pauli's fundamental theorem states that given two sets of matrices $\gamma^\mu$ and $\gamma^\nu$ which obey the commutation rules (...
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1answer
446 views

Magnetic moment of the electron and gamma matrices

What is the relation between the magnetic moment of the electron and $\sigma^{\mu\nu}= \frac{i}{2}[\gamma^\mu,\gamma^\nu]$ ? that I'd like to answer this question: Consider the coupling: $\psi' \...
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1answer
404 views

Covariance of the Dirac equation

In the Ashok Das book "Lectures on quantum field theory" , it's written that in page 76 : therefore, the matrix $ S~ \gamma^0~ S^\dagger ~ \gamma^0$ must be proportional to the identity matrix (this ...
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1answer
60 views

The anticommutativity of $K=\gamma_4 (\vec\Sigma \cdot \vec L +\hbar)$ and $\frac{\vec\sigma\cdot \vec x}{r}$

I have been trying to follow the solution to the Dirac equation for the hydrogen atom. There is a claim from an online source that the following two operators anticommute: $$K=\gamma_4 (\vec\Sigma \...
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2answers
120 views

How to calculate the trace below?

I am currently reading Peskin's QFT book on my own. Though it introduces the Trace Technology in Section 5.1, the trace calculations in the following sections are still far from clear to me. Here is ...
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1answer
254 views

Solution of the Dirac equation by Pauli four-vector

Reading through David Tong lecture notes on QFT. On page 100, he solves the Dirac equation by Pauli four-vector. See below link: QFT notes by Tong, Chapter 4 In (4.107) he gives the solution in ...
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0answers
253 views

What is the Hermitian conjugate of the 4-vector momentum in Dirac equation?

I am quite confused about the Hermitian conjugate of the 4-vector momentum $p=(p^0, p^1, p^2, p3)$. The confusion mainly arises when deriving the Dirac adjoint and the charge current probability. (1)....
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2answers
862 views

Hermitian properties of the gamma matrices

The gamma matrices $\gamma^{\mu}$ are defined by $$\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}.$$ There exist representations of the gamma matrices such as the Dirac basis and the Weyl basis. Is it ...
4
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1answer
582 views

Normalization of Dirac bispinors

Let $u_\lambda(\vec{k})$ and $v_\lambda (\vec{k})$ be solutions of the following equations $$(\not k-m)u_\lambda(\vec{k})=0$$ $$(\not k+m)v_\lambda(\vec{k})=0$$ Suppose that $u_\lambda(\vec{k})^\...
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1answer
388 views

Representations of the Dirac algebra, hermitian adjoint and traces

Strictly speaking this is a math question, but since the Dirac algebra is much more important in physics than in math I thought I'd have a better chance of getting an answer here. The Dirac algebra ...
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2answers
738 views

Dirac spinors in 2+1 dimensions

In 3+1 dimensions, Dirac spinors have four complex components. In 2+1 dimensions, the representation of the Clifford algebra by $\sigma^3$ and $-i\sigma^3\sigma^i$, with $i\in\{1,2\}$ is 2-dimensional,...
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2answers
397 views

How to express $\gamma^{\mu} \gamma^{\nu}$ as a linear combination of {1, $\gamma^5, \gamma^{\mu}, \gamma^{u} \gamma^5, \sigma^{\mu \nu}$}?

** EDIT: I think I have completely missed the mark on asking my question. Here is another try. I do not understand what a linear combination means in this situation. My naive desire is to have an ...
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1answer
261 views

Program for Dirac Matrix Traces

I want to calculate traces of Dirac matrices with a program like Mathematica. I found the package FeynCalc or Tracer.m but they seem to be outdated. Are there any better / newer solutions to this? I ...
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1answer
314 views

Dimensionality of Gamma Matrices

If I express the Dirac equation in the form of $$i\hbar \frac{\partial}{\partial t} \psi_a(x) = \left(-i\hbar c(\alpha^j)_{ab}\partial _j + mc^2(\beta)_{ab}\right)\psi_b(x),$$ with the constraints $...
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1answer
478 views

Spinor representation and Lorentz transformation in Peskin &Schroeder

I am a newbie in group theory. In Peskin & Schroeder's QFT P.42 (3.29) it says that, since we have $$\Lambda_{\frac{1}{2}}^{-1}\gamma ^{\mu}\Lambda_{\frac{1}{2}}~=~\Lambda^{\mu}_{~~\nu }\gamma^{\...
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0answers
284 views

Spinors in dimensions greater than $4$

The Dirac equation describes the behaviour of non-interacting spin-$1/2$ fermions in a quantum-field-theoretic framework and is given by $$i\gamma^{\mu}\partial_{\mu}\psi=-m\psi,$$ where $\gamma^{\...
3
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1answer
820 views

A more general completeness relation for Dirac spinors

Assume that we have two 1/2-spin particles with four-momenta $p$ and $p'$. Particle Dirac spinors satisfy the completeness relation $$ \sum_{s=1}^2u_s(p)\overline{u}_s(p)=\not p+m $$ My goal now is to ...
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1answer
138 views

QFT, show that $(\vec{p}\cdot \vec{\sigma})^2 = |p|^2$

I am following Peskin. During the derivation of the Dirac field (boosting the solution from the rest frame), we use that $(\vec{p}\cdot \vec{\sigma})^2 = |p|^2$. Where $\vec{p}$ is momentum vector, ...
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1answer
216 views

Decomposition of gamma matrices into sigma matrices and their equvialence [closed]

Considering even dimension. From the definition of $\gamma^{(d+1)}$ (all products of gamma matrices) and its anti commutation, $\{ \gamma^\mu, \gamma^{(d+1)}\}=0$, if we choose $\gamma^{(d+1)}$ as ...
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2answers
1k 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 ...
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2answers
117 views

Transforming a QFT identity (quantisation of the Dirac field)

When (falsely) quantizing the Dirac-field, Peskin/Schroeder (Introduction to Quantum Field Theory) get with $$\psi(\vec{x})=\int\dfrac{d^3p}{(2\pi)^3}\dfrac{1}{\sqrt{2E_{\vec{p}}}}e^{i\vec{p}\cdot\...
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0answers
75 views

From $\gamma$ to $\sigma$, finding proper basis

For $d$ dimensional case, usual gamma matrices have basis $\Gamma^A = \{1, \gamma^a, \cdots \gamma^{a_1 \cdots a_d}\}$ (Let's think about even case only for simplicity, I know for odd case only up to ...
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1answer
493 views

Some general formula with trace of gamma matrices relating $\gamma^{(d+1)}$

I want to figure out the trace of gamma matrices relating with $\gamma^{(d+1)}$ for even $d$ dimensional case. First define $\gamma^{(d+1)}$ as \begin{align} \gamma^{(d+1)} = \gamma^1 \gamma^2 \...
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0answers
70 views

Time derivative of total angular momentum in Dirac equation

I would like to calculate the time derivative of the total angular momentum of a Dirac particle in an electromagnetic field $(\vec A, \phi)$: $$\vec J = \vec r \times (\vec p - \frac{q}{c} \vec A) + \...
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0answers
204 views

Basic calculus of the adjoint spinor being transformed under parity

In Modern Particle Physics (p.287) Thompson says that under the parity transformation of the adjoint spinor we have $$\bar u=u^\dagger\gamma^0\rightarrow^p (\hat Pu)^\dagger\gamma^0= u^\dagger\gamma^{...