Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [spinors]

The tag has no usage guidance.

0
votes
0answers
60 views

Anti-commutative coordinates or Grassmann coordinates

In Supersymmetry, we have spinor generators. We have usual spacetime coordinates like $(x^\mu:t,x,y,z)$ and additional grassmann coordinates like $\theta^{\alpha},\bar{\theta}^\dot{\alpha}$.We know ...
3
votes
0answers
125 views

Formal definition of gauge field and spinors in QFT

I am trying to pin down what spaces a spinor and gluon gauge field exactly occupy. I know that the spinor is a quantity $\psi_{i\alpha f}(\vec x, t)$ where $i$ is a color index in the fundamental ...
0
votes
2answers
185 views

Does d/d(spinor) anticommute with a spinor?

Weyl spinors anticommute (see, e.g. Why isn't the anticommutativity of spinors sufficient as "spin-statistics-theorem"?). If we consider the derivative, with respect to a Weyl spinor $\...
-1
votes
1answer
178 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{...
-1
votes
1answer
138 views

What do you lose if you rewrite the Dirac equation in terms of $\mid\Psi\mid^{2}=\Phi$?

Taking a look at the Dirac equation (taking $\hbar$ to be unity): $$\bar{\Psi}(i\gamma^{a}e_{a}^{\mu}\partial_{\mu}-m)\Psi=0$$ The operator is Hermitian and and hence we may rewrite it as: $$\Psi(i\...
10
votes
4answers
1k views

Why does spin arise in non-relativistic quantum mechanics?

In my study of quantum mechanics thus far, I have not yet encountered the Dirac equation, but to the best of my knowledge, the Dirac equation is the first place where you can show mathematically that ...
0
votes
1answer
125 views

An identity of the product of three spinors

In Aitchison's "Supersymmetry in particle physics" I found the identity for the product of three spinors: $$ \lambda(\zeta.\rho) + \zeta (\rho.\lambda) + \rho (\lambda.\zeta)=0 $$ Does anyone have an ...
1
vote
1answer
215 views

Hamiltonian of two level spin system

Why is it the case that for a two level system, say a particle which is a spin $1/2$ system (hence can either be spin up or spin down), in the absence of any external perturbation by a magnetic field ...
3
votes
2answers
216 views

Expansion of a Majorana field operator in creation and annihilation operators

A Dirac spinor $\Psi=\left(\begin{array}{c}\chi_\alpha\\\psi^\dot{\alpha}\end{array} \right)$ can be expanded in the following way:$\Psi=\int \frac{d^3p}{(2\pi)^3}\sqrt{\frac{1}{2E_p}} \sum_s\left( ...
1
vote
1answer
168 views

Spin part of the Dirac spinors

I am reading Peskin's book on QFT and he defines the spin component associated with the particle initially by $\xi^1=\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $\xi^2=\begin{...
1
vote
0answers
52 views

A confusing 2-component spinor notation

In Michael Dine's Supersymmetry and String Theory, the Lagrangian for QCD of one single quark is written as (in equation (5.1)) $$ \mathcal{L} = -\frac{1}{4g^2} F^2 + i\bar{q} D^\mu \sigma_\mu \bar{q}...
15
votes
2answers
494 views

Why does spin appear in quantum systems but not classical systems?

It is often claimed that spin is a purely quantum property with no classical analogue. However (as was very recently pointed out to me), there is a classical analogue to spin whose action is given (in ...
4
votes
1answer
795 views

Does the creation/annihilation operator commute with the spinors?

I am self-studying QFT and I came to the point of quantizing the Dirac field. The Dirac field expanded in terms of creation/annihilation operators is: $$\psi(\vec{x})=\sum_{s=1}^{2}\int{\frac{d^3p}{(...
5
votes
5answers
1k views

Total spin of two spin-$1/2$ particles

On my book I read: $S_{z-tot}\chi_+(1)\chi_+(2)=[S_{1z}+S_{2z}]\chi_+(1)\chi_+(2)=[S_{1z}\chi_+(1)]\chi_+(2)+[S_2\chi_+(2)]\chi_+(1)=...$ Now, I have two questions: What's $\chi_+(1)\chi_+(2)$ ? I ...
2
votes
1answer
203 views

Link between the Grassmann algebra and spinors

What is the exact link between spinors and the Grassmann algebra? I'm pretty sure there's one, based on the following: The Berezin integral in path integrals is done over the Grassmann algebra of $\...
2
votes
2answers
227 views

Is there a bi-4-vector representation of the Dirac gamma matrices and the spinor?

I learned recently that if you have the Dirac spinor represented in the Weyl (chiral) basis $\Psi = \begin{pmatrix} \psi_L \\ \psi_R \end{pmatrix}$, then given a Lorentz Transformation $\Lambda = exp[\...
0
votes
1answer
69 views

What is the $4\times 2\times 2$ matrix $\sigma_{A \dot B}^{\mu}$ explicitly?

In Tales of 1001 Gluons by Stefan Weinzierl, in the end of page 36, (163), $\sigma_{A \dot B}^{\mu} = (1, -\sigma_x, -\sigma_y, -\sigma_z)$. It seems that $\sigma_{A \dot B}^{\mu} = (1, -\sigma_x, -\...
2
votes
1answer
168 views

Massive spin-$s$ representations of the Poincare group on the space of spin tensor fields

Context The following is from the book "Ideas and methods in supersymmetry and supergravity" by I.L. Buchbinder and S.M Kuzenko, pg 56-60. It is about realizing the irreducible massive ...
1
vote
0answers
72 views

Dirac operator with torsion

On a spin manifold $M$, (in local coordinates) the Dirac operator $D_M$ is of the form $$-i\gamma^{\mu}(\partial_{\mu}-\frac{1}{4}\tilde{\Gamma}^b_{\mu a}\gamma^a\gamma_b),$$ where the (torsion-free) ...
1
vote
0answers
76 views

Converting field equation from position to momentum space

Question I would like to convert the following equation on position space to an algebraic equation on momentum space. $$\partial^{\alpha\dot{\alpha}}\Phi_{\alpha\alpha_1\cdots\alpha_{A-1}\dot{\alpha}...
1
vote
1answer
69 views

The notation $p_{a \dot{b}}$

I am a mathematician and trying to learn scattering amplitude by reading Henriette Elvang and Yu-tin Huang's review Scattering Amplitudes (arXiv:1308.1697). I have a question about the notation $p_{...
3
votes
0answers
110 views

How this spinor identity is shown?

In one QFT problem it is asked to prove the following identity: $$\overline{u}_\sigma(p)\gamma^\mu u_{\sigma'}(p)=2\delta_{\sigma\sigma'}p^\mu.$$ Considering $u_\sigma$ the basis solutions to the ...
3
votes
2answers
508 views

Coupling fermions with gravity

Einstein's gravity does not incorporate the "spinor" nature of fermions. The tetrad formulation or Cartan's theory is suggested as the way to go around this problem - by allowing the spin connection ...
5
votes
1answer
482 views

Physical intuition of spin connection and spinor bundles?

I've been trying to learn about how to express the Dirac equation in curved spacetime, and in looking through various resources, I've found that the concepts of the spinor bundle and spin connection ...
1
vote
0answers
75 views

Weinberg soft factor in spinor helicity

The leading order Weinberg soft factor is found to be $$\displaystyle\sum_{a=1}^{n}\frac{\epsilon^{\mu\nu}p_{\mu}p_{\nu}}{q.k_a},$$ where $a$ labels the external particles and $p$ denotes the ...
1
vote
0answers
114 views

Six dimensional Weyl spinors and dimensional reduction

I am trying to understand spinors in 5+1 dimensions but I have troubles when going through dimensional reduction to 4+1 dimensions. The dimension of six dimensional Weyl representation is 4 and the ...
2
votes
1answer
204 views

What, in natural language terms, does “spinor representation” mean (for students)?

Please look away if you’re an expert in Representation Theory… unless you’re happy to offer a helping hand to some tyros. To those of us – especially physical chemists and experimental physicists (...
2
votes
1answer
743 views

What is the difference between a Pauli spinor, a Weyl spinor, and a Cartan spinor?

I know that a spinor is a complex two components "vector", which is acted on by the $SU(2)$ group under a rotation. In the physics litterature, I often read "Weyl spinors", "Pauli spinors", "Cartan ...
0
votes
1answer
85 views

Question about the Dirac Field: determining the scalars inside the integral on the spinors

You can read only this first paragraph to understand my problem. In the second one I explain how I tried to understand but it is not successful. In my QFT course we wrote : $$ \psi(x)= 2m \int d\...
5
votes
1answer
226 views

Antiparticles vs. conjugate particles

In lectures on the standard model I recently saw, in writing down the SM Lagrangian the professor was careful to refer to fields like $e^c$ as a conjugate electron rather than an antielectron. He ...
0
votes
1answer
198 views

Why spin manifolds?

In physics people usually work with spin manifolds: this is quite deep and elaborate mathematical notion therefore I would like to ask: what is the reason to consider spin manifolds in physics?
0
votes
1answer
84 views

Is there a little mistake in this expression in Griffiths' book about elementary particles?

I was looking through my old, well-known book of David Griffiths, Intrododuction to elementary particles, and couldn't tell if I made a mistake in interpreting an expression, or Griffiths made a ...
5
votes
1answer
143 views

How do experiments prove that fermion wavefunctions really pick up a minus sign when rotated by $2\pi$?

Theoretically, after a rotation of $2\pi$, a fermion wavefunction picks up a minus sign, and it is after a rotation through $4\pi$ that it returns to its initial quantum state. Now, the wave-functions ...
0
votes
1answer
198 views

Axial symmetry, “left handed and right handed” and “particle and antiparticle”?

In Srednicki's text book the axial symmetry is described as invariance under, $$\Psi (x) \rightarrow e^{-i\alpha (x) \gamma_5}\Psi(x)$$ and $$\bar \Psi (x) \rightarrow \bar \Psi e^{-i\alpha (x) \...
2
votes
1answer
218 views

Why do we say that in non-relativistic limit we need only two component spinor?

Why do we say that in non-relativistic limit we need only two component spinor? (As in Schrödinger equation, we do not even talk of spinors,... they are one component object) I have read this ...
1
vote
1answer
77 views

Spinor quantization: contradiction between covariant anticommutator and canonical rules?

Starting from the free lagrangian $$\mathscr L = \bar\Psi(i\displaystyle{\not}\partial - m)\Psi$$ I compute the canonical momenta $$\Pi =\frac{\partial \mathscr L}{\partial\dot{\Psi}}=i\Psi^\dagger ...
1
vote
0answers
38 views

Extrinsic curvature spinor

I know that we can write the tensor quantities in the form of spinor indices. I can found $R_{AA'BB'}$ from Penrose & Rindler 's book. But I'm curious to know that how about the extrinsic ...
1
vote
1answer
132 views

Spinor decomposition

I just read in the book "Covariant loop quantum gravity" of Rovelli about spinors (section 1.7.1) I'm confused about the decomposition of an spinor of two indices that it is done: $$z^{AB}= z_0 \...
3
votes
0answers
204 views

Spin Connection derivation in dirac equation

I'm learning about the spin connection, more specifically how it is derived in the dirac equation through this notes: http://web.phys.ntnu.no/~mika/CPP/ch15.pdf In the third page of the document it ...
2
votes
0answers
438 views

Variation of the Spin Connection

I am trying to get an explicit expression for the variation $\delta \omega_{\mu}^{\ ab} / \delta e_\mu^a$, but when doing the actual variation I end up with a series of 16 terms that I cannot simplify ...
0
votes
1answer
103 views

polarized trace

Lets say I want to calculate the following Trace $\mathrm{Tr}[u^{s_1}(p)~\bar{u}^{s_2}(p)~\gamma^{\mu}\not p~ \not q~ \not p~ \gamma^\nu]$ Now if I consider unpolaized case then $s_1=s_2=s$ and I ...
6
votes
2answers
351 views

Does the form of fermion multiplet change with gauge symmetry representation?

The covariant derivative for a fermion with a symmetry group $SU(N)$ is given by $$ D_\nu \psi = \partial_\nu \psi -i g A^A_\nu t_A \, \psi, \tag{1} $$ where $A^A_\nu$ is a gauge field, $g$ is a ...
2
votes
0answers
216 views

Geometric meaning of killing spinor?

I know about killing vectors- They forms the basis for the isometries of a metric space. I am not clear about the definition of killing spinors, in the sense the one there is for killing vectors, ...
2
votes
0answers
123 views

How to transform Dirac fermion coupled with $SU(2)/Z_2$ gauge field to the real fermion coupled with $SO(3)$?

For instance, I am considering a free (3+1)-D Dirac fermion coupled with $SU(2)$ (fundamental representation) gauge field. Then I do a "gauging" the center $Z_2$ to the associated principal bundle by ...
0
votes
1answer
67 views

How is a particular form of the Dirac Spinor derived by a boost from the system of rest?

In Peskin-Schroeder Chapter 3.3: Free Particle Solutions of the Dirac Equation the form of the general Dirac Spinor $u(p)$ along 3-direction is derived from $u(p_0)$ in the system of rest by applying ...
0
votes
1answer
67 views

Spin states in a finite potential well

i have a question concerning an electron in an attractive potential well. Let's suppose the potential function is defined as $$V = \left\{ \begin{array}{cl}0, & \mbox{for } z < 0\\ ...
32
votes
2answers
1k views

Is there an elegant proof of the existence of Majorana spinors?

Almost all standard sources on the existence of Majorana spinors (e.g. Appendix B.1 to Polchinski's "String Theory", Vol. 2) do so in a way I consider inherently ugly: A priori, we are dealing with ...
1
vote
1answer
144 views

Does spinor formalism work just in four dimensions?

Does spinor formalism work just in four dimensions? I am reading the book "Spinors and space-time" written by Penrose. I didn't understand a statement. Penrose says that besides tensor formalism and ...
1
vote
1answer
509 views

Pauli matrices and Lorentz transformations

Consider the Weyl equations: \begin{align} i\sigma^{\mu} \partial_{\mu} \psi_{L} & = 0 \\ i\overline{\sigma}^{\mu} \partial_{\mu} \psi_{R} & = 0, \end{align} where $\sigma^{\mu} = \left ( \...
0
votes
1answer
42 views

What's the generator of spinor field shifts?

The shift of a scalar field $\Phi$: $$ \Phi \rightarrow \Phi'=\Phi - i \epsilon $$ is generated by $$ G = -i \frac{d}{d\Phi},$$ because $$ \mathrm{e}^{-i \epsilon \frac{d}{d\Phi} } \Phi = (1-i\...