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Questions tagged [spinors]

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4
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1answer
299 views

Spinor indices manipulation in the Noether's current for free fermions

I can't solve this apparent paradox; I have the free lagrangian for massive fermions $\mathscr L = i\bar\Psi\gamma^\mu\partial_\mu\Psi - m\bar\Psi\Psi$ which is invariant under the global phase ...
3
votes
0answers
307 views

4-vector from a spinor

Currently reading Aitchison's book on SUSY, and on page 35 (section 2.2) he asks the reader to prove that $\bar{\Psi}\gamma^\mu\Psi=\psi^\dagger\sigma^\mu\psi+\chi^\dagger\bar{\sigma}^\mu\chi$ ...
2
votes
1answer
739 views

Partial completeness relation for Dirac spinors

in studying trace techniques to obtain matrix elements, I came across a problem when we treat scattering of neutrinos on protons. Indeed, since those neutrinos are supposedly created in a weak decay, ...
5
votes
1answer
217 views

How is the invariant speed of light encoded in $SL(2, \mathbb C)$?

In quantum field theory, we use the universal cover of the Lorentz group: $SL(2, \mathbb C)$, instead of $SO(3,1)$. The reason for this is, of course, that $SO(3,1)$ representations aren't able to ...
4
votes
0answers
109 views

Interpretation of inequivalent spin structures (on the circle)

I was wondering about the physical interpretation of inequivalent spin structures on a given configuration space. For simplicity, I'd be satisified by only discussing the case of the circle. There ...
3
votes
1answer
661 views

How to prove that Weyl spinors equations are Lorentz invariant? [duplicate]

The Dirac equation is given by: $[iγ^μ ∂_μ − m] ψ(x) = 0$ . We can prove that it's Lorentz invariant when: $ψ(x) \to S^{-1} \psi'(x')$ and $\partial_\mu \to \Lambda^\nu_\mu \partial'_\nu$, where ...
0
votes
1answer
205 views

Why do we say that photons have spin 1 comparing with the fermoins?

I am learning QFT, and I find it very difficult to understand the spin of photon. Firstly I have some facts listed here: We can get the spin of an electron from Dirac equation. The reasons for ...
0
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0answers
84 views

Which fields/particles correspond to higher dimensional representations of the Lorentz group?

To establish the Dirac equation we use Clifford algebra to find the spinor representation of the Lorentz group. A four dimensional representation tells us that the Dirac spinor describes a quantum ...
0
votes
1answer
225 views

Solution of the Dirac equation by Pauli four-vector

Reading through David Tong lecture notes on QFT. On page 100, he solves the Dirac equation by Pauli four-vector. See below link: QFT notes by Tong, Chapter 4 In (4.107) he gives the solution in ...
4
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0answers
60 views

How similar are the spin states and the matter/antimatter states?

Within the Dirac formalism, we have bispinors that represent both if a particle is spin up or spin down, and if a particle is an electron or a positron. And these representations are very similar. (...
-1
votes
1answer
232 views

Transformation of the spinor indices of Hermitian $2\times 2$ matrices under the Lorentz group

The left-handed Weyl operator is defined by the $2\times 2$ matrix $$p_{\mu}\bar{\sigma}_{\dot{\beta}\alpha}^{\mu} = \begin{pmatrix} p^0 +p^3 & p^1 - i p^2\\ p^1 + ip^2 & p^0 - p^3 \end{...
0
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0answers
83 views

Understanding the Dirac spinor representation $(1/2,0) \bigoplus (0,1/2)$ of the Lorentz group?

I read that the generators of the Lie algebra in this representation are $$ J_k= \begin{pmatrix} \frac{1}{2}\sigma_k & 0 \\ 0 & \frac{1}{2}\sigma_k \end{pmatrix} $$ (Rotations) and $...
3
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0answers
271 views

Why parity exchanges right handed and left handed spinors

Reading through David Tong lecture notes on QFT. On pages 94, he shows the action of parity on spinors. See below link: QFT notes by Tong In (4.75) he confirms that parity exchanges right handed ...
4
votes
1answer
460 views

Normalization of Dirac bispinors

Let $u_\lambda(\vec{k})$ and $v_\lambda (\vec{k})$ be solutions of the following equations $$(\not k-m)u_\lambda(\vec{k})=0$$ $$(\not k+m)v_\lambda(\vec{k})=0$$ Suppose that $u_\lambda(\vec{k})^\...
1
vote
1answer
624 views

Rotation of a spin-1/2 system

The Hamiltonian of a spin 1/2 system in a magnetic field $\mathbf{B} = B \hat{\mathbf{n}}$ is \begin{equation}\hat{H} = - \frac{e}{mc} \hat{\boldsymbol{\sigma}} \cdot \mathbf{B} \end{equation} where ...
1
vote
1answer
152 views

General definition of an $n$-rank spinor

I have been looking around for a formal (and easy to comprehend) definition of a general $n$-rank spinor. I have had no luck trying to find such a definition, or any definition for that matter. So ...
0
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0answers
45 views

Simplifying Z-Component of Angular Momentum in Dirac Field

Peskin and Schroeder gives the definition of the $z$ component of the angular momentum to be $$J_z = \int \mathrm d^3x ~\psi^{\dagger}\frac{1}{2}\Sigma_3\psi.$$ Using the field expansions, $$\psi(x)...
12
votes
2answers
976 views

If the mass of neutrino is not zero, why we cannot find right-handed neutrinos and left-handed anti-neutrinos?

I am learning P&S's Introduction of quantum field theory. My teacher said that if the mass of neutrino is exactly 0, then we should not observe any right-handed neutrinos and left-handed anti-...
3
votes
1answer
218 views

Derivation of conformal generators in spinor helicity formalism

I've been trying for some time to find the expressions for conformal generators of Witten's paper in perturbative Yang-Mills. Given $P_{\alpha \dot{\alpha}} = \lambda_{\alpha} \overline{\lambda}_{\...
14
votes
6answers
959 views

Does spin have anything to do with a rate of change?

The orbital angular momentum of a particle can be related to the revolution of that particle about some external axis. But in quantum mechanics, the spin angular momentum of a particle can't really ...
0
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0answers
81 views

Writing spinor bilinear explicitly in terms of polar and axial four vectors

I need help writing the fermion bilinear $$\bar{u}_s(\mathbf{p},m) \gamma^\mu\gamma_5 u_{s'}(\mathbf{p},m) = ?$$ which transforms like an axial vector, in terms of polar and axial four-vectors such as ...
1
vote
2answers
606 views

Dirac spinors in 2+1 dimensions

In 3+1 dimensions, Dirac spinors have four complex components. In 2+1 dimensions, the representation of the Clifford algebra by $\sigma^3$ and $-i\sigma^3\sigma^i$, with $i\in\{1,2\}$ is 2-dimensional,...
11
votes
1answer
344 views

Is it possible to define off-shell spinors?

For the sake of concreteness, let us consider the Dirac spinor $u_s(\boldsymbol p)$. Is it possible to covariantly extend this to a function $u_s(p)$, such that it matches $u_s(\boldsymbol p)$ on the ...
2
votes
1answer
187 views

Different Lorentz transformations for classical and quantized Dirac fields?

Under the Lorentz transformation, classical Dirac fields transform like: $$ \psi(x)\rightarrow\psi^\prime(x^\prime)=\Lambda_{1/2}\psi(\Lambda^{-1}x) $$ where $\Lambda_{1/2}=e^{-i\omega_{\mu\nu}S^{\mu\...
2
votes
2answers
265 views

Can Dirac equation be reformulated in an equivalent tensor form?

1) Can Dirac equation (including bispinors) be represented by a tensor formalism? 2) If yes, what kind of tensors could be the components of the wave function in Dirac equation in such formulation? ...
-1
votes
1answer
266 views

Dimensionality of Gamma Matrices

If I express the Dirac equation in the form of $$i\hbar \frac{\partial}{\partial t} \psi_a(x) = \left(-i\hbar c(\alpha^j)_{ab}\partial _j + mc^2(\beta)_{ab}\right)\psi_b(x),$$ with the constraints $...
1
vote
1answer
126 views

Confusion about the solution and inner product properties of Dirac equation

I'v just read the solutions of Dirac equation and haven't yet known anything about the quantization of the equation, so my confusion is about this level.The problem is trivial. The solutions of Dirac ...
0
votes
1answer
68 views

Measurement of Spin

Suppose we have a spin state for a spin 1 particle in $S_z$ basis defined by a column matrix $|Ψ\rangle=(a,b,c)$.What is the probability of getting $\hbar$ if we measure $S_z$ for a state $S_x|Ψ\...
3
votes
1answer
428 views

Spinor representation and Lorentz transformation in Peskin &Schroeder

I am a newbie in group theory. In Peskin & Schroeder's QFT P.42 (3.29) it says that, since we have $$\Lambda_{\frac{1}{2}}^{-1}\gamma ^{\mu}\Lambda_{\frac{1}{2}}~=~\Lambda^{\mu}_{~~\nu }\gamma^{\...
0
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0answers
105 views

Spinor manipulation for Feynman Diagram

I am stuck on evaluating a feynman diagram. Essential I have a scalar which decays to an outgoing right handed particle ($\overline{u_{R}}$) and an outgoing left handed anti-particle ($v_{L}$). The ...
0
votes
0answers
242 views

Spinors in dimensions greater than $4$

The Dirac equation describes the behaviour of non-interacting spin-$1/2$ fermions in a quantum-field-theoretic framework and is given by $$i\gamma^{\mu}\partial_{\mu}\psi=-m\psi,$$ where $\gamma^{\...
1
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0answers
91 views

A Hamiltonian for a left-handed spinor field

In finding a lagrangian for a left-handed spinor field , a textbook claims that a kinetic term such as $ \partial_{\mu} \psi^a \partial^{\mu} \psi_a =\epsilon^{ab}\partial_{\mu}\psi_a \partial^{\mu}\...
3
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1answer
661 views

A more general completeness relation for Dirac spinors

Assume that we have two 1/2-spin particles with four-momenta $p$ and $p'$. Particle Dirac spinors satisfy the completeness relation $$ \sum_{s=1}^2u_s(p)\overline{u}_s(p)=\not p+m $$ My goal now is to ...
1
vote
1answer
133 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?
1
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0answers
87 views

Question about complex conjugation, and dotted/undotted indices in the 4d $\mathcal{N} = 1$ superconformal algebra

The 4-dimensional $\mathcal{N} = 1$ superconformal algebra as presented in equations (2.2, 2.3, and 2.4) of the paper, "Counting chiral primaries in N = 1 d = 4 superconformal algebras (arXiv:hep-th/...
1
vote
1answer
103 views

Is there a geometric object analagous to a spinor that encodes projections onto bivectors?

The most sensible geometric interpretation of spinors that I've come across is that they encode projections in the Clifford algebra. So if $\mathbf A$ is a vector with components $A_i$ and $\psi$ is ...
1
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0answers
135 views

The representation of Lorentz boost for two component spinor

It is known that the two components spinor $\chi$ is transformed under the $(\frac{1}{2},0)$ representation of lorentz group. This transformation can be written as $$\chi\rightarrow \exp[-\frac{i}{2}\...
1
vote
1answer
238 views

Why are they called bispinors?

In multiple sources I found the notion of bispinor as a label of one-electron wave functions in Dirac theory. In Mathematical Aspects of Quantum Field Theory page 107-108 it is stated that for every $...
3
votes
2answers
89 views

Can the tensor product of two function spaces be regarded as a function space?

Let $K,T$ be fields and $V:=\{g:K\to T\}$ be a vector space over T. Then take $W:=V\otimes V$, is this $W$ isomorphic to some function space? Little background: In quantum mechanics the the state of ...
4
votes
1answer
111 views

How does the expression $P=\frac12\left(1+\vec\sigma_1\cdot\vec\sigma_2\right)$ for the exchange of two spins work?

According to wikipedia, the spin exchange operator can be expressed as $$P = \frac{1}{2}\big(1+\vec{\sigma}_1\cdot \vec{\sigma}_2 \big) $$ I am not sure I understand what this means. I think spin ...
1
vote
1answer
511 views

How to treat charge conjugation and time reversal operators for Dirac Field in representation invariant way?

Since manipulations with charge conjugation and time reversal operators involve taking complex conjugate of bispinors, most formulas are not invariant under change of representation of $\gamma$ ...
3
votes
2answers
467 views

What are the actual transformation properties of Dirac spinors $u_\sigma(p)$?

Let $u_\sigma(p)$ be a Dirac spinor. As far as I know, it transforms under changes of reference frame according to $$ u_\sigma(p)=S(\Lambda)u_\sigma(\Lambda p)\tag{1} $$ where the $\sigma$ label doesn'...
1
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0answers
52 views

Left-handed and right-handed helicity, can you explained well the phenomenon of chirality simply? [duplicate]

Chirality, helicity as a projection of its spin vector .. that becomes a kind of 'virtual spin'? I often confuse them: but the particles 'spin' on the right or left, or is the projection of .. what? ...
3
votes
1answer
191 views

Non-minimal coupling of the gauge fields to the matter

Does any one know the physical meaning of the following gauge invariant gauge coupling to the spinors? $$\bar \psi F_{\mu \nu} [\gamma^\mu, \gamma^\nu] \psi$$ This coupling is not minimal, as $$\bar \...
3
votes
1answer
371 views

What are possible ways to construct J-matrices (higher order Pauli matrices)?

I'm looking for possible ways to construct $J$-matrices. $J$-matrices are the higher-order version of Pauli matrices. Pauli matrices are suited for spin-1/2 systems, while J-matrices can be for any ...
0
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0answers
164 views

Basic calculus of the adjoint spinor being transformed under parity

In Modern Particle Physics (p.287) Thompson says that under the parity transformation of the adjoint spinor we have $$\bar u=u^\dagger\gamma^0\rightarrow^p (\hat Pu)^\dagger\gamma^0= u^\dagger\gamma^{...
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0answers
111 views

Acting for a covariant derivative on charged spinor [closed]

For field, theory what i know $i.e$,complex scalar QED \begin{align} D_\mu \phi = \partial_{\mu} \phi - i Q A_{\mu} \phi \end{align} and \begin{align} D_\mu \phi^{\dagger} = \partial_{\mu} \phi^{\...
3
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1answer
629 views

Spinors in 2+1 dimensions

I am trying to understand representations of the Poincare/Lorentz group, and in particular spinors, in 2+1 dimensions. I know some of the math, but I'm not sure about the physical interpretation of it ...
1
vote
3answers
1k views

Measuring different components of spin simultaneously

I'm reading Griffiths Introduction to QM and I'm having trouble understanding why you can't simultaneously measure the x,y and z components of spin. I know that the uncertainty principle prevents this ...
-1
votes
1answer
156 views

Majorana fields and attributed supercharges [closed]

Numbr of components for a Majorana field in D-dim is equal to $2^([D/2]-1)$. Now, what is the number of attributed supercharges to the Majorana field?