Questions tagged [dirac-matrices]

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4
votes
0answers
186 views

Confusion about gamma matrices in Euclidean spacetime

I have encountered a number of sources with differing definitions of the transition from Minkowski spacetime to Euclidean spacetime. I'd like some clarification as to how to go from Minkowski to ...
4
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126 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
151 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
347 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 $\sum_m \text{Tr}[A~\Gamma^m]\text{Tr}[B~\Gamma^m]$, ...
3
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0answers
55 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
votes
1answer
555 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 ...
3
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0answers
311 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}}...
3
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1answer
244 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}...
2
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0answers
146 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 \...
2
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0answers
65 views

Dirac matrix algebra from bosonic creation/anihilation operators?

Using some generic fermionic creation/anihilation operators $a_i$ and $a_i^{\dagger}$ ($i, j = 1, 2, 3, \dots, N$) such that \begin{align} \{ a_i, \, a_j^{\dagger} \} &= \delta_{ij}, \tag{1} \\[...
2
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0answers
134 views

Do Lorentz rotations transform the Gamma matrices $\gamma_a$?

Do local Lorentz rotations (see below definition) actually transform the Dirac Gamma matrices? If so, how can they collude with coordinate transformations to make the Gamma matrices $\gamma_a$ ...
2
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0answers
98 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
79 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
69 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
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0answers
33 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
277 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
556 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
418 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
82 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
572 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
89 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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0answers
171 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
703 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
283 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
61 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 ...
1
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0answers
77 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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0answers
73 views

Generalization of Dirac matrices

Is it possible to have a set of 16-dimensional matrices $\gamma_{\mu}^{a}$ such that $$\{\gamma_{\mu}^{a},\gamma_{\nu}^{b}\} = 2\delta^{ab}\eta_{\mu\nu}$$ where $\eta_{\mu\nu}$ is the Minkowski metric ...
1
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0answers
48 views

Index manipulation of Dirac matrices

In several places I see that Dirac matrix indexes are treated as usual 4-vector indexes that can be changed with the metric tensor, for example $$\gamma_\mu=g_{\mu\nu} \gamma^\nu. $$ Why is it true?
1
vote
1answer
223 views

Question about the Dirac adjoint and Feynman slash notation

I was trying to prove the identity $\overline{\displaystyle{\not}{a}\displaystyle{\not}{b}\dots \displaystyle{\not}{p}} = \displaystyle{\not}{p}\dots \displaystyle{\not}{b}\displaystyle{\not}{a}$. On ...
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0answers
55 views

Gamma Matrices as nonstandard numbers, and Grassman Numbers

I'm in the process of exploring the Dirac equation and its forms and consequences, and as such have just been initiated into the theory of spinors and their accompanying formalism. One of the things ...
1
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0answers
70 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 ...
1
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0answers
102 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,...
1
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0answers
101 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 ...
1
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0answers
147 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
47 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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0answers
155 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
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1answer
57 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{...
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0answers
221 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 ...
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0answers
102 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
167 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
172 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
168 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
178 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 ...
1
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0answers
340 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
272 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^{...
1
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0answers
328 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 } ...
1
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0answers
249 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_{...
0
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49 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 ...
0
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0answers
24 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 ...
0
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0answers
67 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{...