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Questions tagged [spinors]

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Transformation of Vectors

let $\Psi \in V$ be a vector and we have the action of a lorentz transformation on the object $\sigma_2 \Psi $. And $\sigma_2 \Psi $ is then in V as well. V is "Weyl or Dirac space". And the lorentz ...
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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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Fierz identities and Weinberg operator

I've been told that $$ (\bar{L_i^c}\widetilde{\phi}^{\ *})(\widetilde{\phi}^\dagger L_k) = -\frac{1}{2}(\bar{\widetilde{L}}_i \vec{\sigma}L_k)(\widetilde{\phi}^\dagger \vec{\sigma} \phi) \tag1 $$ by ...
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Generally Covariant Dirac equation: The spin connection

Wikipedia, an answer on stackexchange and a few papers in the Arxiv I've found all have different definitions of the spin connection found in the Dirac equation. Can anyone please tell me what the ...
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Periodic Killing spinor on $S^1$?

In Cartesian coordinates on $R^2$ we have a constant two component Killing spinor $\epsilon_0$. If we use polar coordinates $x = R \cos t$ $y = R \sin t$ we have the vielbein $e^R = dR$ $e^t = ...
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Lorentz transformation of a Weyl Spinor?

A left handed Weyl Spinor belongs to the $(\frac{1}{2},0)$ representation of the Lorentz group. So given the Spinor, the unitary representation of the Lorentz transformation should look like $\exp{iA\...
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Dirac equation in 1+1D spacetime compared to “standard” 3+1D Dirac equation

In the past couple of weeks I've been studying the Dirac equation and its solutions. During a discussion with a tutor it was pointed out to me that one could formulate something similar to the Dirac ...
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A question about Lorentz transformations in spinor representation

For $$\Lambda^{\mu}{}_{\nu}= \frac{1}{2} Tr(\bar{\sigma}^{\mu} S \sigma_{\nu}S^{\dagger}) $$ We need to prove that $$\Lambda (S)= \Lambda (-S)$$ Am I naive to say that by adding $-S$, $S^{\dagger}...
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Sources to learn about Killing spinors

What are some good sources to learn about Killing spinors from? I am currently learning about Killing vectors and how they are the generators of a Lie algebra that corresponds to the isometry group ...
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Trying to prove the Wess Zumino invariance under a SUSY transformation

I have the Lagrangian density $$L=-\partial_\mu \phi^\star \partial^\mu \phi - \bar{\chi}_R \gamma^\mu \partial_\mu \chi_L - \bar{\chi}_L\gamma^\mu\partial_\mu \chi_R.$$ where $\epsilon$ is the ...
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1answer
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Infinitesimal parameter of Lorentz transformation

I'm working through the SUSY lecture notes by Lambert, and he does something which seems strange to me during the calculation of the Wess Zumino model. He says the spinor $\psi$ has the ...
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Spinor representation

I am trying to study Special unitary group of order 2 and some textbooks mention objects transform under special unitary group are called Spinors. then How can we represent a spinor using matrix?
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Vanishing of a solution of Dirac equation

Let $\psi(x,t)$ be a solution of the free Dirac equation. Assume that $$\psi(\vec x,0)=\delta^{(3)}(\vec x) u,$$ where u is a fixed spinor. (In other words $\psi(\vec x,0)$ is assumed to be supported ...
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1answer
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How does one find the parity trasformation matrix of spinors for non-free field theory?

In many QFT textbook, for example, the book of Srednicki, they use free field theory to derive the transformation matrix of the Spinors: $$P^{-1}\Psi(x)P=D(P)\Psi(P^{-1}x)$$ Then we have a relation: ...
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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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Unphysical degrees of freedom for the Weyl spinor?

I am attempting to solve the Weyl equation: $$\bar\sigma^{\mu}\partial_{\mu}\phi=0$$ Where $\bar\sigma^{\mu}=(-1,\vec{\sigma})$ in my convention, and $\phi$ is a two component Weyl spinor. I consider ...
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I want to know the conformal weights of spinors in 2D

I want to know the conformal weights(or dimensions) of left/right-moving fermions in 2D, ${\cal N}=(2,2)$ superconformal theory. More specifically, what is the left/right-moving conformal dimension ($...
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Naive question about particles and spinor fields [duplicate]

What is the difference between the "real" particle electron (for instance) and the spinor field of electron? I mean, which means that the electron have been described by a spinor field?My question is ...
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How does the spin connection affect the dynamics of a fermion in curved space?

Consider a massless right-handed Majorana fermion in curved spacetime. Without any other fields present, the Lagrangian density is (I believe) the following: $$ \mathcal{L}_{\psi} = \sqrt{g}i\bar{\...
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1answer
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Identity Involving Grassmann Variables and Pauli Matrices

I am trying to prove the following identity: $$\theta\sigma^{\mu}\bar{\theta}\theta\sigma^{\nu}\bar{\theta}=\frac{1}{2}g^{\mu\nu}\theta\theta\bar{\theta}\bar{\theta}$$ Where $\theta$ and $\bar{\...
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Quantization of the massless neutrino field

If a massless neutrino or anti-neutrino is considered (in the whole post I consider neutrinos res. anti-neutrinos as mass-less), it is described by the Weyl-equation: $$\overline{\sigma}^{\mu}\...
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Identity involving Majorana spinors and Pauli matrices

How to prove that: $$(\sigma^\mu \bar{\xi}_2)_\alpha \partial_\mu (\xi_1 \psi)=-(\sigma^\mu\bar{\xi}_2)_\beta \xi_{1\alpha}\partial_\mu\psi^\beta-(\xi_1\sigma^\mu\bar{\xi}_2)\partial_\mu \psi_\alpha\...
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Why can the spin operator be written as a product of fermions?

I was studying the Hubbard model, where we define the spin operator $\vec{S} = \frac{1}{2} c^\dagger \vec{\sigma} c$, where the creation and annihilation operators are both vectors of the form $c^\...
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Four-brackets (Hodges, Momentum Twistors)

I use the reference from Andrew Hodges, available at https://arxiv.org/abs/0905.1473. I am having trouble understanding his use of the four-bracket. I refer to equation 6 and equation 9, where he ...
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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)-\...
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Is there any Classical Mechanics system which needs to be described by a spinor?

We need an ordinary number (scalar) to describe a harmonic oscillator, and a vector to describe, for example, a pendulum. Is there any similarly simple system which we need to describe using a (two-...
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3answers
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Are pseudospinors valid or useful?

We all know that in addition to scalars and vectors, there are pseudoscalars and pseudovectors, which have an additional sign flip under parity. These are useful and necessary when constructing ...
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A Naive Question about SUSY Variation

I am following BUSSTEPP Lectures on Supersymmetry to learn supersymmetry. My simple question is the following. My Lagrangian for the Wess-Zumino model in $4D$ is $$\mathcal{L}=-\frac{1}{2}(\...
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1answer
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A few questions about spinors and gamma matrices

I am following BUSSTEPP Lectures on Supersymmetry and trying to show that the Wess-Zumino action is invariant under SUSY transformations. I encountered the following questions about spinors and gamma ...
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1answer
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Transpose of fermion bilinears

TL;DR When we take the transpose of two Grassmann-valued spinors (fermions), should we add a minus sign because we end up anticommutating the two spinors? More details. I'm studying the behavior of ...
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What does it mean to define a spin-structure on a manifold? [closed]

I'm trying to think about what information I need to add to a manifold that it describes a spin structure? I know you can have spin-structure on a 2d plane, a 2-sphere. I also know you can define a ...
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What is difference between fermions and spins?

A spin model i.e. $H_s = \sum_i^{L-1} S_i^x\cdot S_{i+1}^x$ can be written in matrix form as following $$H_s = \big(S_1^x \otimes S_2^x \otimes I_3^2 \otimes I_4^2\otimes \cdots\otimes I_{L-1}^2\big)...
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Intuitively understanding complex projective space or twistor space

I'm studying momentum twistor variables, which I understand can be seen to be defined projectively from dual complexified Minkowski space to this complex projective space $\mathbb{C}\mathbb{P}^3$. ...
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Are there supersymmetry algebras with higher spinor representations?

The super-Poincare algebra contains supersymmetry generators $Q^I$ which satisfy fermionic anticommutation relations. By the higher-dimensional analogue of the spin-statistics theorem, they must ...
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Question about Spinors and Probability Densities

So I was toying around attempting to simulate some relativistic wave equations for a recreational project. Now I have never studied spinors in dept and the knowledge I have is from reading online (...
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1answer
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Dirac matrices in 1+1 dimensions

Given $\gamma^\mu$ in 1+3 dimensions with signature $(+,-,-,-)$, how can I obtain Dirac matrices in 1+1 dimensions expressed in terms of the $\gamma^\mu$?
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Do I need Gamma matrices in Majorana representation in the Lagrangian of a Majorana fermion?

I understand that the Majorana representation of the Gamma matrices are the real representations of the associated Clifford algebra. A Majorana fermion is defined as a fermion that equals to its ...
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How does canonical quantization work with Grassmann variables?

Every quantum field theory textbook I've encountered seems to have the same logical oversight, because of the particular order they cover topics. First, the books introduce the Dirac Lagrangian, $$\...
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1answer
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Peskin and Schroeder: derivation of Dirac fields commutator

I'm perplexed by the following non numbered equation at page 54 of Peskin & Schroeder, right between $(3.92)$ and $(3.93)$ $$ [\psi_a(x),\overline{\psi}_b(x)]=\int\frac{d^3p}{(2\pi)^3}\frac{1}{...
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What precisely is a *classical* spin-1/2 particle?

I was recently having a Twitter conversation with a UC Riverside Prof. John Carlos Baez about Geometric Quantization, and he said (about his work) that "Right. For example, you can get the ...
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How to build an antisymmetric selfdual tensor out of two 4-vectors?

In problem C of section 1.4 of Ramon's Field Theory: A Modern Primer, we are asked to build a field bilinear in $\chi_L$ and $\psi_L$, two left-handed weyl spinors, which transforms as the (1,0) ...
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Do we need really need tetrads? OR: Does the No-Go theorem for spinors in curved space only apply to a linear connection?

When I first learned General Relativity, the tetrad formalism was introduced with near simultaneity. I was immediately taught that, to utilize spinors in any way, I had to formulate a local ...
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1answer
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Does the external leg contraction of gluon in QCD carry group generator index?

While I am trying to compute the amplitude for the following Feynman diagram I realized that the external leg contraction of $g$ should carry group generator index $A$ or $B$, is that right? If so, ...
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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, ...
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Would Left and Right Weyl spinor components mix to become massive in an expanding space?

Sorry, this might be a dumb question. I was just reading a very old paper by Schrodinger where he's talking about different frequency modes mixing in an expanding universe. Basically he says if the ...
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What is the Newman-Penrose (NP) form of Rarita-Schwinger (RS) equation?

Does anyone knows how to write down the Rarita-Schwinger equation in Newman-Penrose formalism?
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Equivalence between Dirac and Majorana action in CFT

In Mussardo's Statistical field theory Chapter 12, section 12.3 about the conformal field theory of a free fermion field he talks about the complex fermion field (Dirac field) $$ \Psi(z,\bar{z}) = \...
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Majorana fermions

If you write the Majorana spinors as $$\chi = \begin{pmatrix}\psi_L\\ i\sigma_2\psi_L^* \end{pmatrix} \tag1$$ It satisfies the Dirac equation that leads you to the Majorana equation $$i\bar{\sigma}^\...
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Commute covariant derivatives of spinors

Consider a spinor field $\psi$ on a general smooth Lorentzian manifold. Let $\Sigma_{ab}$ be a matrix representation of the Lorentz group, and let Greek/Latin letters represent world/Lorentz indices. ...
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Could there be a pseudovector kinetic term for fermions?

Could there be a kinetic term of the form $\bar{\Psi} \gamma_5 \gamma^\mu \partial_\mu \Psi $ in addition to the usual one? Or is this forbidden by a symmetry?