Questions tagged [dirac-matrices]

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4
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108 views

How many Killing spinors exist on $S^5$?

So, I know that on $S^n$, a spinor of the form $$ \Sigma^\pm = \frac{1 \pm i\gamma^\alpha z_\alpha}{\sqrt{1+z^2}}\Sigma_0$$ where $\Sigma_0$ is a constant spinor, is a Killing spinor on $S^n$ ...
3
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0answers
111 views

Dimension of gamma matrices in dimensional regularization

When performing loop integrals in theories containing Dirac fermions, one almost always confronts terms of the form $$\text{Tr}\left[\gamma^{\mu_1}\cdots\gamma^{\mu_n}\right].$$ For instance, in $d$ ...
3
votes
1answer
106 views

What does $\Lambda^{-1}_{\frac{1}{2}}\gamma^\mu\Lambda_{\frac{1}{2}}=\Lambda^\mu_{\phantom{\mu}\nu}\gamma^\nu$ mean?

\begin{equation} \Lambda^{-1}_{\frac{1}{2}}\gamma^\mu\Lambda_{\frac{1}{2}}=\Lambda^\mu_{\phantom{\mu}\nu}\gamma^\nu \end{equation} In P&S, p. 42: Equation (3.29) says that the $\gamma$ ...
3
votes
1answer
272 views

Identities of Pauli matrices in two-component spinor formalism

I'm reading the review by H. K. Dreiner, H. E. Haber and S. P. Martin (arXiv:0812.1594) about the two-component spinor formalism. There are some identities and notational conventions which lead to ...
3
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0answers
47 views

Mapping from spinor to tetrad

I am reading the journel by Patrick l. Nash: mapping from tetrad to Dirac spinor. While reading this ,I came across the term concrete real 4*4 irreducible representation of SO(3,3). I know SO(3) is ...
3
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0answers
178 views

Question about the Dirac equation

Energy and momentum of a particle can be expressed by equation $$E^2=p_1^2c^2+p_2^2c^2+p_3^2c^2+m^2c^4\hspace{40pt}(1)$$ Equation (1) can be divided into $E$ on both sides. We obtain $$E=\frac{v_1}{c}...
3
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0answers
292 views

Transformations of gamma-matrices through Pauli matrices transformations

I have the transformation law of the Lorentz group for Pauli matrices: $$ \tag 0 (\sigma^{\mu})_{a \dot {a}}{'} = \Lambda^{\mu}_{\quad \nu} N_{a}^{\quad c}(\sigma^{\nu})_{c \dot {c}}(N^{-1})^{\dot {c}}...
2
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0answers
76 views

Almost complex structure $J_i^j$ from covariantly constant spinor $\eta$

In the context of superstring compactification on a 6-manifold which admits a covariantly conserved spinor $\eta$, which we normalize so that $\eta^{\dagger} \eta = 1$, I am trying to show that the ...
2
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0answers
78 views

Supergravity and Gamma Matrices

On page 101 of Freedman and Van Proeyen's book on Supergravity they find the propagator of the gravitino, however I'm not sure how to work through the steps in (5.30), and hints or answers would be ...
2
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0answers
55 views

How to relate $\gamma^5$ to the spacetime volume form? In regards to axial current anomaly

In playing with gamma matrices of the $\mathcal{\mathscr{C}}l_{1,3}(R)$ variety, it's not uncommon to hear allusions to $\gamma^5$ being related to the volume 4-form. To illustrate the similarities: $...
2
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0answers
63 views

Motivating the Unintuitive Properties of Spinors

In the usual treatment of (Dirac) spinors, one usually starts with "factoring" the energy-momentum relation, deducing the properties of the $\gamma$ matrices by requiring the cross terms to cancel, ...
2
votes
1answer
106 views

Dirac matrices in dimensional regularization, get correct order epsilon

Let us work in dimension $D = 4-2\epsilon$. In 4-dimension, we can write $\text{Tr}[A B]$, where $A$ and $B$ are string of gamma matrices, as $\sum_m \text{Tr}[A~\Gamma^m]\text{Tr}[B~\Gamma^m]$, ...
2
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0answers
29 views

Vertex with spinorial structure and scattering amplitude

Consider the Lagrangian $$\mathcal{L}=\bar\psi_1\left(i\partial\!\!\!/-m_1\right)\psi_1 + \bar\psi_2\left(i\partial\!\!\!/-m_2\right)\psi_2 - g\bar\psi_1\gamma_\mu\psi_1\bar\psi_2\gamma^\mu\psi_2.$$ ...
2
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0answers
223 views

Gamma matrices in higher (even) spacetime dimensions

Suppose we write the gamma matrices in this following representation: \begin{align*} \gamma^{0}=\begin{pmatrix} \,\,0 & \mathbb{1}_{2}\,\,\\ \,\,\mathbb{1}_{2} & 0\,\, \end{...
2
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0answers
360 views

Spin covariant derivative of gamma matrices?

Where can I find a general expression (on curved manifolds) in local coordinates, for the following: $$\nabla^S_{\mu}\gamma^{\nu} = ?$$ $\nabla^S_{\mu} = \partial_{\mu} + \omega^S_{\mu}$ is the spin ...
2
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0answers
291 views

Charge conjugation in chiral representation

I'm reading Maggiore's book and I got to the part of charge conjugation symmetry for Dirac spinor. I get that the definition of charge conjugation is representation-dependent, however I couldn't find ...
2
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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 ...
2
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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{...
2
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157 views

How one can write $\bar{\psi}$ in odd dimension?

I know that the Dirac equation in general dimensions has a form of $$ (i\gamma_{\mu} \nabla_\mu - m ) \psi =0 $$ and the action for that is written as $$ S = \int d^d x \bar{\psi} (i\gamma_{\mu} \...
2
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0answers
629 views

Proof of equivalence of different representations of the $\gamma$-matrices in the Dirac equation

This question concerns the Dirac equation and the $4\times4$ $\gamma$-matrices. The task is to prove that a similarity transformation of the standard $\gamma$-matrix conserves the commutation relation ...
2
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0answers
227 views

Are Lifshitz and Berestetskii right in this case?

In the Quantum electrodynamics book (look at the problem) its authors Lifshitz and Berestetskei claim that operator of charge conjugation $\hat {C} = -\alpha_{2}$ in Majorana basis transforms as $\hat ...
1
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0answers
56 views

Dirac matrices for generalized metric tensors

The Dirac matrices are defined by the relations $$\left [\gamma^{i},\gamma^{j}\right]_{+}=2\eta^{i,j}\mathbb{1}$$ where $[\cdot,\cdot]_{+}$ is the anti-commutator. What happens if I replace $\eta^{i,...
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0answers
83 views

Again on Spin Operator in Dirac Field Theory (Peskin & Schroeder)

Good morning, I've already seen that this topic has been discussed so long, but my doubts remain unchanged. At page 61 of Peskin & Schroeder, An Introduction to QFT, there is the demonstration ...
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0answers
37 views

Question about Pauli Matrices

I found the following identities about Pauli matrices from the lecture notes of Supersymmetry. $$((\sigma^{\mu})^{\alpha\dot{\alpha}})^{\ast}=(\bar{\sigma}^{\mu})^{\dot{\alpha}\alpha}$$ $$((\sigma_{\...
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78 views

My Struggle with Fierz Identity

I am following BUSSTEPP Lectures on Supersymmetry to learn SUSY. The Lagrangian of a interacting Wess-Zumino model in 4D is given by $$\mathcal{L}=-\frac{1}{2}(\partial_{\mu}S)(\partial^{\mu}S)-\...
1
vote
1answer
55 views

States of spin of a quantum mechanical particle

Assuming a spin is prepared in the positive $x$-direction ($|r\rangle$) and a measuring apparatus is oriented on the $z$ axis, does this equation apply? $$|r\rangle=\frac{1}{\sqrt{2}}|u\rangle+\frac{...
1
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0answers
124 views

Relation between Dirac spinors, quaternions, and bicomplex numbers

Superficially Dirac spinor resp. Dirac gamma matrices and quaternions and bicomplex numbers seems to be very similar objects. all can be expressed by unitary 4x4 matrices so they seem to represent ...
1
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0answers
274 views

Gamma matrices invariant under lorentz transformation

I know this has been asked before but I just can't seem to get my head around it based on the answers I've read. So the idea is that we have the gamma matrices $\gamma^{\mu}$. Now from my ...
1
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0answers
92 views

Dirac Fields and Derivatives (Am I gaining extra minus signs?)

I've given myself a severe headache jumping between East/West Coast sign conventions; I have picked up an extra minus sign and could do with a hand. I am currently using $\eta=\textrm{Diag}[-,+,+,+]$ ...
1
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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
133 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\...
1
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0answers
130 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 ...
1
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0answers
167 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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0answers
257 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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0answers
298 views

Dirac Spinors as Eigenvalues of Helicity Matrix

I am trying (unsuccessfully) to verify this relation regarding the helicity of Dirac spinors: $$ { \sigma }_{ \vec { p } }u_{ r }\left( \vec { p } \right) =\frac { \vec { \Sigma } \cdot \vec { p } ...
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0answers
231 views

What is $\langle G_{\mu\nu}\rangle\langle G_{\mu\nu}\rangle$ for the Dirac gamma matrices?

Given the following 16 matrix multiplications of the Dirac gamma matrices \begin{align} G_{\mu\nu} = \dfrac{1}{2} \begin{pmatrix} I && \gamma_{0} && i\gamma_{123} && i\gamma_{...
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35 views

As a returning adult learner from engineering and math background, what recommendations to get the basics on QM?

I'm building up the foundations to work on quantum mechanics. I am comfortable with the wave function and the PDE aspect of the field, but my mathematical background, though it covers linear algebra, ...
0
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0answers
37 views

Gamma traceless

I read this Under what conditions is a vector-spinor gamma trace free. And also read many papers about higher spin, but no one explains why irreducible spinor is gamma traceless spinor? Can anyone ...
0
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0answers
45 views

Relativistic quantum field theory

Let $\psi(x)$ be solution of Dirac equation $$ (\gamma^\mu\Pi_\mu-mc) \psi(x)=0 $$ where $\Pi_\mu=i\partial_\mu-eA_\mu$ is momentum operator in present electromagnetic field . We consider tow ...
0
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0answers
43 views

Spinor transformations as representations of $\mathrm{SL}(2, \mathbb{C})$

Background In the Weyl representation of the Dirac $\gamma$-matrices, the spinor transformations $S=e^{\frac{1}{2} \omega_{\alpha \beta} \Sigma^{\alpha \beta}} \in \,\mathrm{G}_{\mathrm{L}} \leq \...
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0answers
38 views

Spin for Dirac fermions proof $\Sigma_{i}=\frac{1}{2} \gamma_{5} \gamma_{0} \gamma_{i}$

The spin for Dirac fermions is defined as: $$\Sigma_{i} \equiv \frac{1}{4}\epsilon_{ijk}\sigma_{jk}$$ Without using an explicit representation for the matrices, I would like to show that the spin ...
0
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0answers
40 views

Sakurai Quantum Mechanics problems

I just began studying QM on Sakurai's "Modern Quantum Mechanics" and just finished chapter 1. I am now approaching the exercises. On exercise 2 there is a notation I can't understand: "A 2x2 square ...
0
votes
1answer
40 views

Commutation of alpha dirac matrix

I want to calculate the commutation of $[\hat{x},\vec{\alpha}\;\vec{p}]$. This boils down to $$[\hat{x},\vec{\alpha}\;\vec{p}] = i\hbar\hat{\alpha_x}+\left[\hat{x},\hat{\alpha_x}\right]\hat{p_x} +\...
0
votes
1answer
134 views

From relativistic equation to find Dirac matrices

Is this possible and then how? $$((\gamma \otimes \mathbf\sigma)\bullet\mathbf p)(\gamma^\prime\otimes\mathbf 1_2) = \gamma\gamma^\prime\otimes\sigma \bullet \mathbf p $$ where $\gamma$ and $\gamma^\...
0
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0answers
25 views

Charge conjugation opearation in odd dimension

In my problem, I use the following set of $\gamma$-matrices (in (2+1) spacetime): $$\gamma^0=\sigma^1;\quad \gamma^{1}=i\sigma^2;\quad \gamma^{2}=i\sigma^{3},$$ where $\sigma^{(i)}$ are usual Pauli ...
0
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0answers
113 views

Does anyone know how to symmetrize $\gamma$-matrices?

I'm trying to construct the SO(5, 5) $\gamma$-matrices which are real and symmetric. Recently, I have 6 symmetric and 4 antisymmetric $\gamma$-matrices ($6_S + 4_A$ representation). How can I ...
0
votes
0answers
25 views

density of state function for semi Dirac materials

Respected members I have a problem in finding density of state for semi Dirac system (linear dispersion relation in one direction and parabolic in other direction). $$E(k)=\pm\sqrt{{(\hslash^2k_x^...
0
votes
1answer
109 views

Why do the $\gamma$ matrices behave like vectors (tensors)?

In the study of Quantum Field Theory and Group Theory for the spinor representation of $SO$ groups, we know the following correspondence: $\chi C\psi$ scalar $\chi C\gamma^\mu\psi$ vector $\chi C\...
0
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0answers
22 views

Zero momentum in Non relativistic Quantum Mechanics and about Dirac matrices

In relativistic quantum mechanics, we can solve the Dirac's equation with an added condition that the momentum of the particle is $0$. However, such independence isn't provided by the Schrodinger's ...