# Questions tagged [spinors]

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518 questions
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### 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-$N$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $$u(p)\, e^{-ip\cdot x} \quad \text{and} \quad v(p)\, e^{ip\cdot x} \,,$$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 (...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $v=0$? ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 $(P)$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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}...
Eq. (27) in http://arxiv.org/abs/1110.2662 says I can construct the Weyl spinor according to $$\Psi_{ABCD} = \frac 14 C{}_{\mu\nu\lambda\rho} \left( \sigma^\mu \right){}_A{}^{\dot A} \left( \sigma^\... 5answers 224 views ### Do spins have spatial directions? When we consider a spin-1/2 particle and try to write down it's wave function, we have$$|\psi\rangle = a|+\rangle + b|-\rangle,$$where in a reference about two-level system, the author wrote ... 0answers 60 views ### Matrix representation of Clifford algebra - steps [closed] In (Vaz and da Rocha, 2016;pg108) the following two step process is given for finding the matrix representation of a Clifford algebra: (verbatim; except for notation) (1) Choose a set of N ... 0answers 88 views ### Parity transformation of spin-3/2 field In conventional quantum field theory textbook, we can find the expression of parity transformation of spin-0, 1/2 or 1 fields. For example, for spin-1/2 fields, we have$$U^{-1}(\mathcal{P})\Psi(x) U(...
Let's say we have a left-handed Weyl spinor as follows: $$\chi = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}$$ where $\alpha$ and $\beta$ are complex components. What ...