[Paper Review] An introduction to spinors

Andrew Steane|arXiv (Cornell University)|Dec 13, 2013

Algebraic and Geometric Analysis8 citations

TL;DR

This paper provides a pedagogical introduction to spinors for students in relativity, particle physics, or quantum field theory, emphasizing their role in describing rotational and Lorentz invariance. It explains the SU(2)–SO(3) homomorphism, constructs 4-vectors from 2-component spinors, derives the Dirac equation via spinor bilinears, and shows how spinors naturally yield the g=2 gyromagnetic ratio for electrons, confirming quantum predictions.

ABSTRACT

We introduce spinors, at a level appropriate for an undergraduate or first year graduate course on relativity, astrophysics or particle physics. The treatment assumes very little mathematical knowledge (mainly just vector analysis and some idea of what a group is). The SU(2)--SO(3) homomorphism is presented in detail. Lorentz transformation, chirality, and the spinor Minkowski metric are introduced. Applications to electromagnetism, parity violation, and to Dirac spinors are presented. A classical form of the Dirac equation is obtained, and the (quantum) prediction that $g=2$ for Dirac particles is presented.

Motivation & Objective

• To provide a self-contained, mathematically accessible introduction to spinors for advanced undergraduates or beginning graduate students in physics.
• To clarify the geometric and algebraic structure of spinors, particularly their relationship to rotations and Lorentz transformations via the SU(2)–SO(3) homomorphism.
• To demonstrate how spinors can be used to derive the Dirac equation and explain the g=2 gyromagnetic factor in a classical-quantum hybrid framework.
• To connect spinor formalism to physical observables such as 4-velocity, 4-spin, and parity violation in electromagnetic interactions.

Proposed method

• Uses 2-component complex spinors as the fundamental object, representing them as flagpoles with rigid flags to visualize spatial orientation and phase.
• Constructs 4-vectors from spinors via the map $ V^ u = \langle u|\sigma^\nu|u\rangle $, linking spinors to null and non-null 4-vectors.
• Applies the SU(2) group structure to describe rotations and boosts, with unitary matrices for rotations and Hermitian matrices for boosts.
• Derives the Weyl equations by requiring Lorentz invariance of $ (W^\alpha \sigma_\alpha)w = 0 $, leading to helicity constraints.
• Constructs the Dirac spinor as a pair of Weyl spinors (right- and left-handed), and derives the Dirac equation from a 4×4 matrix eigenvalue problem.
• Uses the bilinear forms $ \Psi^\dagger \gamma^0 \gamma^\mu \Psi $ and $ \Psi^\dagger \gamma^0 \gamma^\mu \gamma^5 \Psi $ to extract 4-velocity and 4-spin, confirming orthogonality and normalization.

Experimental results

Research questions

• RQ1 How can spinors be used to represent Lorentz transformations and rotations in 3+1 spacetime without relying on tensors?
• RQ2 What is the precise mathematical relationship between SU(2) and SO(3), and how does it underlie spinor behavior?
• RQ3 How do spinors lead to the derivation of the Dirac equation and the prediction of g=2 for the electron's magnetic moment?
• RQ4 Why is the Dirac equation parity-invariant, and how do the left- and right-handed components transform under parity?
• RQ5 How can the 4-velocity and 4-spin of a particle be extracted from a Dirac spinor using bilinear forms?

Key findings

• The SU(2) group provides a double cover of SO(3), explaining why a 4π rotation is required to return a spinor to its original state.
• A 2-component spinor corresponds to a null 4-vector via $ V^\mu = \langle u|\sigma^\mu|u\rangle $, with the flagpole picture capturing spatial orientation and phase.
• The Dirac equation is derived as a 4×4 eigenvalue problem: $ \begin{pmatrix} -m & E - \mathbf{\sigma} \cdot \mathbf{p} \\ E + \mathbf{\sigma} \cdot \mathbf{p} & -m \end{pmatrix} \begin{pmatrix} \phi_R \\ \chi_L \end{pmatrix} = 0 $, which ensures Lorentz covariance.
• The bilinear form $ \Psi^\dagger \gamma^0 \gamma^\mu \Psi $ yields a 4-vector representing the 4-velocity, and $ \Psi^\dagger \gamma^0 \gamma^\mu \gamma^5 \Psi $ gives the 4-spin, both orthogonal and normalized.
• The paper confirms that the Dirac equation predicts $ g=2 $ for the electron’s magnetic moment, consistent with quantum electrodynamics, by analyzing the spinor structure and current conservation.
• Under parity, the right- and left-handed components of a Dirac spinor swap, and $ \mathbf{\sigma} \to -\mathbf{\sigma} $, preserving the form of the Dirac equation and ensuring parity invariance.