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Questions tagged [spinors]

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4
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2answers
388 views

Relation between spinors and anticommutation relation of fermions

I read that the state of a pair of particles is the tensor product of the single states of both, and you will get a wavefunction with the parameters of both, if you swap the parameters you will get a ...
0
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0answers
37 views

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 ...
0
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1answer
42 views

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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0answers
45 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
25 views

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 ...
1
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0answers
33 views

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 = ...
4
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2answers
211 views

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 ...
0
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2answers
87 views

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\...
0
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2answers
156 views

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 ...
0
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1answer
122 views

Connection between two Petrov classification schemes

For the Weyl scalars of all spacetimes, at any point, possess one special structure, the so called principal null directions. Consider a general null tetrad $\{ l_a,n_a,m_a,\overline{m}_a \}$, we ...
7
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1answer
315 views

What are Killing spinors?

What are Killing spinors? How can they be motivated? Are they directly related to Killing vectors and Killing tensors and is there an overarching motivation for all three objects? Any answer is ...
11
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4answers
3k views

Covariant derivative for spinor fields

scalars (spin-0) derivatives is expressed as: $$\nabla_{i} \phi = \frac{\partial \phi}{ \partial x_{i}}.$$ vector (spin-1) derivatives are expressed as: $$\nabla_{i} V^{k} = \frac{\partial V^{k}}{ \...
0
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1answer
55 views

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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0answers
25 views

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 ...
0
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2answers
55 views

Algebraic operation on a bilinear expression of Weyl-Spinors

In the book of Srednicki in equation (35.29) a rather peculiar algebraic operation is carried out on spinors that I am not able to understand. It's $$ [\psi_\dot{a}^{\dagger} \overline{\sigma}^{\mu\...
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0answers
40 views

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 ...
0
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1answer
157 views

Lorentz invariance from Dirac spinor

I have a really naive question that I didn't manage to explain to myself. If I consider SUSY theory without R-parity conservation there exist an operator that mediates proton decay. This operator is $...
0
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1answer
43 views

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 ...
1
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1answer
55 views

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?
0
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1answer
49 views

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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0answers
100 views

Parke-Taylor formula in the $n=4$ simple case

I am trying to do ex. 2.23 of http://arxiv.org/abs/1308.1697. I have chosen as reference spinors $q_1,q_2 = p_3$ and $q_3,q_4 = p_1$. Therefore if I compute $A^4[1^- 2^- 3^+ 4^+]$ the computation ...
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1answer
31 views

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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0answers
32 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_{\...
0
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1answer
36 views

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 ...
5
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1answer
136 views

Coupling a spinor field to a preexisting scalar field?

So I'm not a physicist, but I'm thinking about a mathematical problem where I think physical insight might be useful. We're working on a Riemannian manifold $(M,g)$ (positive definite metric) with a ...
1
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1answer
139 views

Massless Weyl spinor components

Let's say there is a massless Weyl spinor moving in an arbitrary direction $n$. What are the analytical solutions for the two components of this spinor?
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0answers
20 views

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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0answers
33 views

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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0answers
23 views

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}\...
1
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1answer
42 views

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{\...
2
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0answers
28 views

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{\...
0
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1answer
153 views

Finding the direction of a Spin-State

Given a non normalized state, find the direction of the state $\chi$?: $$ \chi = (1+i)\chi_{+}^{z}-(1+i\sqrt{3})\chi_{-}^{z}$$ Where $\chi_{+}^{z}, \chi_{-}^{z}$ are eigenstates of $S_{z}$. I know ...
2
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2answers
682 views

Orthogonality relations for spinors of plane wave solution

I was looking for an explanation that doesn't depend on the representation of the gamma matrices that shows that the orthogonality relations are fulfilled. Let me situate my problem: So the Dirac ...
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0answers
42 views

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\...
2
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3answers
110 views

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 ...
1
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1answer
182 views

What's the reasoning behind propagators definitions (specifically fermionic propagators)

I'm studying QFT by David Tong's lecture notes. When he discusses causility with real scalar fields, he defines the propagator as $$D(x-y)=\left\langle0\right|\phi(x)\phi(y)\left|0\right\rangle=\int\...
2
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0answers
33 views

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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0answers
19 views

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 ...
4
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1answer
161 views

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-...
0
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0answers
39 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)-\...
0
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1answer
81 views

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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0answers
29 views

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}(\...
1
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1answer
163 views

Different definitions of the parity transformation for the Dirac spinors

There are two definitions of the parity transformation acting on the Dirac spinors: $\Psi_P = \eta \gamma^0 \Psi$ with $\eta = i$ ($P^2=-1$ as in Srednicki) and $\eta=1$ ($P^2=+1$ as in Peskin & ...
0
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1answer
188 views

Do the eigenstates of the Pauli operators correspond to the six directions of the 3D world?

I understand that the six eigenstates of the three Pauli operators $X, Y, Z$ correspond to the six poles of the Bloch sphere. By fixing an orthonormal basis of our physical word, does "measuring Pauli ...
0
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1answer
56 views

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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0answers
53 views

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 ...
2
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1answer
78 views

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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1answer
193 views

expanding a Dirac spinor in Weyl basis

For a massless electron Dirac spinor in Weyl basis (where $\chi$ is the left-handed spinor and $\eta$ is the right-handed spinor): \begin{equation} \begin{pmatrix} \chi \\ \eta \end{pmatrix} \end{...
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2answers
283 views

Diffeomorphisms and the Dirac action

I have a question concerning fermions in curved space-time. Please read it to the end before suggesting the spin-connection and vierbein-based approach. I heard that there is a special way of ...
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0answers
20 views

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 $\mathbb{C}\mathbb{P}^3$. ...