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Questions tagged [spinors]

The tag has no usage guidance.

4
votes
4answers
229 views

The symmetry group and representation of spin-$N$ particle

I am confused with the symmetry group and the representation of spin-$N$ particles, and will appreciate any help or suggestions of reference. There are $2N+1$ internal states for a (massive) spin-$N$ ...
1
vote
1answer
66 views

Weak isospin current

I cannot understand the product of a Dirac gamma matrix and a Pauli matrix in this formula of the weak isospin current: $$J_α^i(x)=\frac12\bar \psi_L(x)\gamma_\alpha\tau^i\psi_L(x),$$ where $γ_α$ is ...
0
votes
0answers
60 views

Fermions redefinition in switching from Einstein-frame to string-frame

Recently I have been working on 10D supergravity. My question regards fields redefinition in passing from Einstein-frame to string-frame. I was wondering if the fermionic fields (gravitinos and ...
2
votes
0answers
260 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
votes
1answer
49 views

How is chirality defined for row vectors?

When working with Dirac spinors, the chirality of a spinor field is determined by its $\gamma_5$ eigenvalue, so if $\psi_L$ is left-handed then $$\gamma_5 \psi_L = - \psi_L.$$ Some sources define a ...
4
votes
2answers
335 views

What spinor field corresponds to a forwards moving positron?

When we search for spinor solutions to the Dirac equation, we consider the 'positive' and 'negative' frequency ansatzes $$ u(p)\, e^{-ip\cdot x} \quad \text{and} \quad v(p)\, e^{ip\cdot x} \,,$$ ...
1
vote
1answer
267 views

Direct product of spin representations

Consider a system of two 1/2-spins. Under some conditions the Hilbert space can be decomposed into the direct sum of spin-0 and spin-1 representations: $\frac12\otimes\frac12=0\oplus1$. When I write ...
0
votes
3answers
319 views

Meaning of the vector expectation value $\langle \mathbf{S} \rangle $

The Cartesian components of the spin operators $S_x, S_y$ and $S_z$ don't commute $[S_i,S_j] \neq 0 \ (i \neq j)$. Hence we can't simultaneously determine all Cartesian components of the spin ...
1
vote
0answers
123 views

Pauli matrices in curved space-time

I am studying the chapter on spin-1/2 particles in Kerr geometry from "The Mathematical Theory of Black Holes" by S. Chandrasekhar. I have trouble in understanding how he arrives at the generalized ...
0
votes
1answer
197 views

What is the dimension/unit of a spinor?

I am interested in getting the physical units of a spinor for the usual $(1,3)$ Minkowski spacetime. I am getting 2 different answers, using 2 different approaches! On one hand, using the Lagrangian (...
6
votes
3answers
288 views

How are the spinor indices different from the spacetime or Lorentz indices?

A spinor $\zeta$ transforms under $SU(2)$ transformation as $$\zeta^\prime_a=U_{ab}\zeta_b.$$ Why are the spinor indices kept different from the spacetime indices $\mu,\nu$? After all the $SU(2)$ we ...
1
vote
0answers
175 views

Weyl and Majorana-Weyl spinors why need commutation?

Let $\psi$ denote a Dirac spinor then Weyl spinors are defined by: $$\psi_{L,R}=\frac{1}{2} (I\pm \gamma)\psi$$ on even dimensions $\gamma$ commutes with $\sigma_{\mu \nu}$ (generators used to define ...
1
vote
2answers
150 views

Wave function in tensor product of Hilbert spaces

If I had the wave function $$\Psi\equiv\psi(r,\theta,\phi)\otimes\chi \in \mathscr{L}^2(\mathbb{R}^3)\otimes\mathbb{C}^{2S+1},$$ where $S$ is the spin of the state, is it correct to normalize the ...
0
votes
1answer
46 views

Is the Heighest weight vector in the Spinor rep of $SO(1,d-1)$ zero?

Consider the highest weight vector of the Spinor rep of $SO(1,d-1)$ where $d=2m+1$. It can be shown that: $$\gamma_i \gamma_{m+i}v=v\tag{*}$$ I cannot see why this relation does not imply that $v=0$? ...
0
votes
1answer
75 views

Is a (Dirac) Particle Where $\vec{p} = (p^1,0,0)$ in an Eigenstate of Helicity? [closed]

Is a particle where $\vec{p} = (p^1,0,0)$ an eigenstate of the helicity operator? First, can I determine this without doing the math? Second, I also wanna prove it mathematically but doing the math ...
1
vote
1answer
125 views

Weyl Spinors and Lorentz Invariance

Let $\phi_a$ and $\chi_{\dot{a}}$ be two component commuting spinors, where $\chi$ is an anti-spinor. In terms of some spinor basis, these can both be written in some arbitrary frame as $$ \phi_a(P) =...
1
vote
1answer
471 views

Weyl transformation of Dirac equation

The Dirac Equation is given by $$\left(i\gamma^\mu\partial_\mu- \frac{mc}{\hbar}\right)\Psi_D = 0,$$ where $\gamma^\mu$ are the Dirac $\gamma$-matrices and $\Psi_D$ is a Dirac spinor. I would like to ...
2
votes
1answer
80 views

Proof of inequivalence right and left spinor representation

I'm asked to prove the inequivalence of $\Lambda_L$ and $\Lambda_R$ for transformations close to the identity. So I start with the definition \begin{equation} \Lambda_R=S\Lambda_LS^{-1} \end{equation} ...
2
votes
1answer
212 views

Majorana mass term

I see somewhere that we can add a Majorana mass term, $m\psi^Ti\sigma^2\psi+h.c.$, to a Weyl fermion Lagrangian, where $\psi$ is a two-component spinor. However, it seems that this term simply ...
3
votes
1answer
146 views

Building $\mathfrak{so}(1,3)$ reps using $\mathfrak{so}(1,3)\cong \mathfrak{su}(2)\oplus \mathfrak{su}(2)$

I'm going through the representation theory of $\mathfrak{so}(1,3)$, building Dirac/Weyl spinors and vectors, and I'm a bit confused on the mathematical definitions involved. We have $\mathfrak{so}(1,...
0
votes
3answers
170 views

Spin Up with Indefinite Helicity

Imagine we are studying the spin quantization along the same axis as the momentum. What if I have a Dirac spinor with a spin up but no definite helicity ($\psi_L,\psi_R\neq0$): $$ u(p)= \left(\begin{...
2
votes
0answers
123 views

Transformation of Weyl spinors

I usually see Weyl spinor and Weyl equations as derived from Dirac equation, like in Peskin. Now, I'm following a course where the professor actually builds Weyl spinor lagrangians BEFORE talking ...
1
vote
2answers
607 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 ...
1
vote
1answer
42 views

The space of physical states of the spin-$s$ system

In the case of a spin-1 particle, the space of its possible (spin) states is a $2$-sphere. Mathematically, it is obtained as follows. We act with the whole $SO(3)$ on a given vector and end up with ...
2
votes
0answers
62 views

Spinors, punctured plane and principle frame bundle

I am reading Applied Conformal Field Theory by Ginsparg. On page 72, while describing different boundary conditions on fermion he states the following. We shall choose to consider periodic $(P)$ ...
1
vote
1answer
216 views

Spin expectation values in Dirac theory

The wave function of a free Dirac particle moving with momentum $\vec{p}=(p,0,0)^T$ is given in the rest frame and the laboratory frame as $$\Psi_r=N_r\left(\begin{array}{c}1\\0\\0\\0\end{array}\...
4
votes
2answers
148 views

Tetrad formalism and fields with half integer spin

The main aim of the tetrad formalism is to apply action principle in general theory of relativity. But why to incorporate general relativity with field theory of particle with half integer spin?
-1
votes
1answer
177 views

Representation $(1/2,1/2)$ of Lorentz group

I want to show that the Lorentz representation $(1/2,1/2)$ corresponds to the normal vectorial representation $A^\mu$. For this I need to show that the double spinors $A_{ij}=(A_\mu\sigma^\mu\sigma^2)...
1
vote
2answers
92 views

Rotations of eigenstates of $S_z$

I have a question regarding the rotation of spinors in a spin-1/2 system. We have a Spin generator $\hat{S}$ for rotations of spinors. A rotation around the axis $\vec{n}$ with the angle $\phi$ is ...
2
votes
0answers
233 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 ...
4
votes
1answer
316 views

Covariant derivative of a Dirac spinor and Kosmann lift

In [1] I have found a definition of the covariant derivative of a Dirac field with a general connection $\omega_{\mu a}{}^{b}$ (with torsion and non-metricity) [see eq. (29)]: $$\nabla_{\mu}\psi=\...
4
votes
2answers
374 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
votes
1answer
106 views

Separation of term in Møller scattering cross section

In trying to calculate the Møller scattering cross section, I arrived at the following term$^1$: $$\frac{e^4}{(p_3-p_1)^4}\bar u(p_3)\gamma^\mu u(p_1)\bar u(p_4)\gamma_\mu u(p_2)\bar u(p_2)\gamma_\nu ...
5
votes
3answers
522 views

When an electron orbits in a magnetic field, how exactly does its spin precess?

In the case of a cyclotron, with a constant magnetic field $B$ in the vertical direction, a moving electron circles in a horizontal orbit. The cyclotron frequency is $\omega = eB/m$. At the same ...
0
votes
1answer
85 views

Determine spin-state with $B=B_x$ in the $S_z$ base

We're a group trying to determine the spin-state at time t with respect to the $S_z$ axis. Initially we have a particle with spin $s=1/2$ in the z-direction. The problem for us is that the magnetic ...
0
votes
2answers
281 views

Property of Charge Conjugation Operator

In class, we have defined the Charge Conjugation Operator ($C$) such that: \begin{equation} C \left(\gamma^\mu\right)^T C^{-1} = - \gamma ^\mu , \end{equation} \begin{equation} \psi^C \equiv C\,\...
8
votes
2answers
309 views

Are there projective representations of the Lorentz Group NOT coming from a Clifford algebra?

Let $\mathrm{SO}(1,d-1)_{+}$ be the restricted Lorentz Group in $d$ dimensions. Are there projective irreducible representations of this group that do not descend from a representation of $\mathrm{C}\...
1
vote
1answer
56 views

Super Field Strength Identity

I work on a introduction into Super-symmetry. In the course we define \begin{equation} D_{\alpha} = \frac{\partial}{\partial \theta^{\alpha}} - i \sigma^{\mu}_{\alpha \dot{\alpha}} \bar{\theta}^{\dot{\...
2
votes
0answers
259 views

Matching Dirac/Majorana/Weyl Spinor Degrees of Freedom in Minkowski signature

Question: How do we match the real degrees of freedom (DOF) of Dirac/Majorana/Weyl Spinor in terms of their quantum numbers (spin, momentum, etc) in any dimensions [1+1 to 9+1] in Minkowski signature?...
0
votes
0answers
87 views

How are the covariant Pauli matrices defined?

When doing calculations with Weyl spinors, terms like $\theta\sigma^\mu\theta^\dagger$ appear. I know that for 3+1 spacetime dimensions, $\sigma^\mu = (\textbf{1}, \sigma^i)$ with $i=1,2,3$ the usual ...
0
votes
1answer
136 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 ...
0
votes
1answer
152 views

Spinor rotation around the $z$-axis

In Schwartz's QFT textbook (eq. 10.118), he gives the equation for the Lorentz transformation matrix of a rotation around the $z$-axis for as spinor as $$ \Lambda_s(\theta_z)=\left( \begin{array}{...
10
votes
2answers
717 views

Lorentz transformations for spinors

The lorentz transform for spinors is not unitary, that is $S(\Lambda)^{\dagger}\neq S(\Lambda)^{-1}$. I understand that this is because it is impossible to choose a representation of the Clifford ...
10
votes
2answers
541 views

Why do lattice models of fermions need a spin structure?

It is well-known that in order to define a relavistic quantum-field theory containing fermions on a general manifold $M$, the manifold $M$ needs to be equipped with a spin structure. The spin ...
3
votes
1answer
190 views

Massless limit for Dirac field

I'm a little bit confused about how to take the massless limit of the Dirac field: \begin{align} \psi(x)=\int\frac{d^3p}{(2\pi)^2}\frac{1}{\sqrt{E_p}}\sum_{s}\left(a_p^su^s(p)e^{-i p x}+b_p^{s\dagger}...
1
vote
1answer
83 views

Extract Weyl curvature spinor

Eq. (27) in http://arxiv.org/abs/1110.2662 says I can construct the Weyl spinor according to $$\Psi_{ABCD} = \frac 14 C{}_{\mu\nu\lambda\rho} \left( \sigma^\mu \right){}_A{}^{\dot A} \left( \sigma^\...
4
votes
5answers
224 views

Do spins have spatial directions?

When we consider a spin-1/2 particle and try to write down it's wave function, we have $$|\psi\rangle = a|+\rangle + b|-\rangle,$$ where in a reference about two-level system, the author wrote ...
1
vote
0answers
60 views

Matrix representation of Clifford algebra - steps [closed]

In (Vaz and da Rocha, 2016;pg108) the following two step process is given for finding the matrix representation of a Clifford algebra: (verbatim; except for notation) (1) Choose a set of $N$ ...
1
vote
0answers
88 views

Parity transformation of spin-3/2 field

In conventional quantum field theory textbook, we can find the expression of parity transformation of spin-0, 1/2 or 1 fields. For example, for spin-1/2 fields, we have $$U^{-1}(\mathcal{P})\Psi(x) U(...
-2
votes
1answer
117 views

Linear momentum of a Weyl spinor

Let's say we have a left-handed Weyl spinor as follows: \begin{equation} \chi = \begin{pmatrix} \alpha \\ \beta \end{pmatrix} \end{equation} where $\alpha$ and $\beta$ are complex components. What ...