Questions tagged [dirac-matrices]

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469 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
425 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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2answers
125 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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2answers
662 views

Representation under which Pauli matrices transform

In Peskin and Schroeder's Quantum Field Theory, there is an identity of Pauli matrices which is connected to the Fierz identity, (equation 3.77) $$(\sigma^{\mu})_{\alpha\beta}(\sigma_\mu)_{\gamma\...
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0answers
264 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
894 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
617 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
401 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
416 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
279 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
331 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
500 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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4answers
5k views

Dimension of Dirac $\gamma$ matrices

While studying the Dirac equation, I came across this enigmatic passage on p. 551 in From Classical to Quantum Mechanics by G. Esposito, G. Marmo, G. Sudarshan regarding the $\gamma$ matrices: $$\...
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288 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^{\...
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1answer
140 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
229 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 ...
4
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1answer
292 views

Why do we need tetrads/vierbeins/frame-fields to describe fermions in curved space?

I'm learning about the frame formalism and read that to couple fermions to gravity you need to go to the frame-formalism. As a motivation to learn more about frame-fields would someone sketch me why ...
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2answers
118 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
78 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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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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1answer
415 views

Why gamma-matrices are associated with tetrads Lorentz rotation?

In Zee's "QFT in nutshell" in a paragraph "Differential geometry of Riemann manifold" he states that Dirac gamma-matrices are associated with tetrads Lorentz rotation, so Dirac lagrangian in curved ...
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206 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^{...
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2answers
457 views

Can we make the Dirac representation a gauge theory?

I'm looking for comments and references about an idea : gauging the Dirac representation of the Dirac matrices. What kind of field interaction would it give ? Specifically, the Dirac equation is ...
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1answer
578 views

Why is the unitary matrix relating the gamma matrices and their complex conjugates antisymmetical?

In Messiah's Quantum Mechanics Vol. II, properties of the Dirac matrices are derived. There is so-called fundamental theorem, which states that, Let $\gamma^\mu$ and $\gamma^{'\mu}$ be two systems of ...
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1answer
223 views

Proof of two Lorentz-algebra identities

I am currently working through the QFT introduction text by Peskin and Schroeder and try to fill in two identities that I wasn't able to prove (it should be fairly simple, but my experience with this ...
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2answers
836 views

How does Schur's Lemma mean that the Dirac representation is reducible?

In chapter 3 of Peskin and Schroeder, when they're talking about "Dirac Matrices and Dirac Field Bilinears," they introduce $\gamma^{5}$ and give some properties of it. One of the properties is $[\...
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1answer
497 views

Symmetry properties of gamma matrices

While reading a paper on supersymmetry i faced the following problem. Its about the symmetry property of charge conjugation matrix in different space time dimension. The charge conjugation matrix is ...
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2answers
1k views

Why are usually 4x4 gamma matrices used? [duplicate]

As far as I understand gamma matrices are a representation of the Dirac algebra and there is a representation of the Lorentz group that can be expressed as $$S^{\mu \nu} = \frac{1}{4} \left[ \gamma^\...
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2answers
271 views

Do the matrices $S^{\mu\nu} = \frac{1}{4}[\gamma^\mu, \gamma^\nu]$ have a name?

Do the matrices $S^{\mu\nu}$ defined by $$ S^{\mu\nu} = \frac{1}{4}[\gamma^\mu, \gamma^\nu] $$ have a name ($\gamma^\mu$ are the gamma matrices)? They feel very important to me since they form a ...
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1answer
232 views

Should the trace of a product of gamma matrices depend on the convention I use?

I am trying to work out $$\text{Tr}[\gamma_5\gamma_\mu\gamma_\nu\gamma_\alpha\gamma_\beta]$$ using the same convention as J.J. Sakurai (Advanced Quantum Mechanics), what I get is $$\text{Tr}[\gamma_5\...
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1answer
189 views

Components of Dirac equation solve the Klein Gordan equation derivation

On page 90 of this set of lecture notes on quantum field theory, http://www.damtp.cam.ac.uk/user/tong/qft/four.pdf a simple derivation is given to show that each component Dirac equation solves the ...
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1answer
442 views

Invertibility of Dirac matrices

Suppose we have Dirac equation in the following form : $i\partial_t\psi = (-i\vec{\alpha}\cdot \nabla +m\beta)\psi$ and assume that the Klein-Gordon equation is satisfied, i.e. $\partial_t^2 \psi =(...
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1answer
3k views

Deriving the Spinor Completeness Relation without using a Representation

Reference: DAMTP problem set 3, question 5 but ignore the spinor solutions given. To preface, this has taken up 1 entire day and a further 2 afternoons of work so I will just list the most promising ...
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1answer
475 views

Behaviour of Dirac Bilinears

Dirac bilinears transform in the Lorentz indices as, $\bar{\psi}\psi$ scalar $\bar{\psi}\gamma^\mu\psi$ vector $\bar{\psi}\sigma^{\mu\nu}\psi$ 2nd rank (antisymmetric) tensor $\bar{\psi}\gamma^{\mu}\...
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1answer
696 views

Gamma matrices relations (Dirac Spinors: QFT) [closed]

The entry question in an exam paper: I think I have made an elementary error in the transpose somewhere invoked by a conceptual misunderstanding of how spinors behave with gamma matrices under a ...
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1answer
773 views

Hermitian properties of Dirac operator

I am trying to understand the Hermiticity of the (massless) Dirac operator in both (flat) Minkowski space and Euclidean space. Let us define the Dirac operator as $D\!\!\!/=\gamma^\mu D_\mu$, where $...
4
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1answer
113 views

Some diracology in traces

Suppose I want to evaluate the trace $p_{\alpha} q_{\beta}\text{Tr}(\gamma^{\alpha} \gamma^0 \gamma^{\beta} \gamma^0)$. Using the standard trace formula for four gamma matrices I arrive at $$p_{\...
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1answer
258 views

New “oscillator basis” of gamma matrices?

It was mentioned in http://kclpure.kcl.ac.uk/portal/files/12371620/Studentthesis-Mehmet_Akyol_2013.pdf page 28, a new concept "oscillator basis" or more precisely the author defines gamma matrices of ...
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0answers
555 views

Feynman amplitude for electron-positron annihilation and $W^{\pm}$ production

I'm working with this interaction Hamiltonian density $$ H_{int}(x) = ig\bar{\Psi}_{\nu_e}(x)\gamma^\rho P_L \Psi_e(x)V_\rho(x) + igV^\dagger_\rho(x)\bar{\Psi}_e(x)\gamma^\rho P_L \Psi_{\nu_e} $$ ...
2
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1answer
83 views

Positive free particle Dirac equation

I've been set the task of showing that: $$ \bar{\psi^{s}}\psi^{s}=2m $$ For s=0,1. Where: $$ \psi^{0,1}=\sqrt{|E|+m}\begin{pmatrix}\chi^{0,1}\\ \frac{\vec{\sigma}\cdot\vec{p}}{E+m}\chi^{0,1}\end{...
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2answers
136 views

Pseudoscalar current of Majorana fields

Consider a Majorana spinor $$ \Phi=\left(\begin{array}{c}\phi\\\phi^\dagger\end{array}\right) $$ and an pseudoscalar current $\bar\Phi\gamma^5\Phi$. This term is invariant under hermitian conjugation:...
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1answer
668 views

How to show that $\bar\psi\gamma^\mu\psi$ of a Dirac spinor $\psi$ transforms as a vector?

This is part 2 of exercise II.1.1 of Zee's QFT in a Nutshell (here's part 1). This is what I have got: \begin{align} \bar\psi\gamma^\lambda\psi \mapsto \bar\psi^{\,\prime}\gamma^\lambda\psi^{\,\...
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1answer
557 views

How to show that $\bar\psi\psi$ of a Dirac spinor $\psi$ transforms as a scalar?

I would like to show that for a Dirac spinor $\psi$, the scalar product $\bar\psi\psi$ transforms as a scalar under a Lorentz transformation $\Lambda$, where $\bar\psi = \psi^\dagger\gamma^0$. This is ...
2
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1answer
58 views

Where did the gamma matricies disappear to?

In the discussion of external bremsstrahlung, the following amplitude is used: $$M=i\bar{u}_e(k')e \left(\gamma^\nu\epsilon_\nu \left[ \frac{i \gamma^\nu(k'_\nu+\omega_\nu) + m}{(k'+\omega)^2-m^2}\...
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2answers
339 views

Local Lorentz transformations

If $\gamma^m$ denotes a tangent space gamma matrix, and $\gamma^\mu$ denotes a curved space gamma matrix, then they are related by $$\gamma^\mu(x) = \gamma^m e_{m}^{\mu}(x)$$ where $e_{m}^{\mu}(x)$ ...
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2answers
4k views

Lorentz transformation of Gamma matrices $\gamma^{\mu}$

From my understanding, gamma matrices transforms under Lorentz transformation $\Lambda$ as \begin{equation} \gamma^{\mu} \rightarrow S[\Lambda]\gamma^{\mu}S[\Lambda]^{-1} = \Lambda^{\mu}_{\nu}\gamma^{\...
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2answers
2k views

Gamma matrices and trace operator

I'm trying to show that the trace of the product of the following three Gamma (Dirac) matrices is zero, i.e. $$\text{tr}(\gamma_{\mu} \gamma_{\nu} \gamma_{5})=0 \text{.}$$ I attempted to use the fact ...
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1answer
46 views

Regarding properties of matrices involved in Dirac equation

In this document, after equation 62 on page 9, the author says that we can rewrite $\alpha^i \alpha^j \partial_i \partial_j$ as $\frac{1}{2} (\alpha^i \alpha^j + \alpha^j \alpha^i)\partial_i\...
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2answers
218 views

Why does the Dirac equation require matrices to be rotationally invariant?

Why does the Dirac equation derivation require matrices? Starting from $$i\hbar \frac{\partial \psi}{\partial t} = \left(\frac{\hbar c}{i}\alpha^k\partial _k + \beta m_0 c^2 \right) \psi =H \psi.$$ ...