Spinors, Lorentz Symmetry, and Particle States

Spinors are representations of spacetime symmetry, not four-vectors with unfamiliar entries. This chapter first constructs their finite-component transformations, then explains how spin labels transform in a unitary one-particle Hilbert space. The connection resolves common confusions about nonunitary component boosts, chirality, Majorana reality, and momentum-dependent Wigner rotations.

Enter this chapter

Use the Lorentz transformations and Poincaré group for the spacetime conventions. Familiarity with Pauli matrices, angular momentum, complex conjugation, and unitary symmetry transformations is assumed. The Dirac chapter supplies the wave equation and normalized free modes.

The chapter has two connected paths. Start with the component path if the meaning of a spinor is the question. Start with the particle-state path if the immediate task is transforming momentum and spin labels in a calculation.

Component representations

Page	Question it answers
SL(2, ℂ) and the Lorentz Group	How does a two-by-two matrix produce a Lorentz transformation, and why are there two lifts?
Weyl Spinors	Why do two chiralities have the same rotations and opposite boost signs?
Dirac Spinors	How does a direct sum of Weyl components support a mass and a conserved current?
Majorana Spinors	What is an antilinear reality condition, and what does it require of charge and frequency sectors?
Spinor Bilinears	Which chiral components do vector currents, masses, and tensor couplings connect?
Helicity and Chirality	How do chiral fractions depend on mass and momentum, and what can a boost change?

The gamma-matrix ledger fixes the explicit Dirac and chiral bases. The classification of all Dirac bilinears remains on Bilinear Covariants; this chapter’s bilinear page applies that classification to chiral couplings.

Particle-state representations

Page	Calculation to carry forward
Massive and Massless Representations	Find the standard-momentum stabilizer and count spin or helicity states
Wigner Classification Preview	Transform normalized momentum-spin kets and packets with the induced little-group matrix
Wigner Rotations	Extract the rotation from two noncollinear boosts and apply it to a spin state

The central distinction is between a finite set of field components at one event and an entire state with amplitudes over a continuous momentum shell. The first admits nonunitary boost matrices in its component norm. The second transforms unitarily through its momentum argument, invariant measure, and little-group action together.

A consistent set of signs

The explicit boost convention sends coordinates to a frame moving with velocity $c\tanh\xi\,\mathbf n$. For the Hermitian matrix $X=x^0I+\mathbf x\cdot\boldsymbol\sigma$, its lift is $A=e^{-\xi\mathbf n\cdot\boldsymbol\sigma/2}$. The left and right chiral blocks transform with $(A^\dagger)^{-1}$ and $A$, respectively.

A standard boost used to construct a particle state instead satisfies $L(p)(m,\mathbf0)=p$. It sends a rest momentum toward $+\mathbf p$, so it is the inverse of the passive frame boost with velocity $\mathbf p/E$. Keeping these two definitions distinct avoids a sign error in the Wigner rotation.

Questions to answer before continuing

1. Why does a $2\pi$ rotation change a spinor’s sign without changing its probability density?
2. Why is a Dirac spinor a direct sum of two Weyl representations, whereas a four-vector has a different representation?
3. Can a massive positive-energy state have both chiralities without containing a negative-energy component?
4. Why can a Majorana field identity about anticommuting bilinears fail for a commuting classical amplitude?
5. Which normalization removes the square-root energy factor from the momentum-ket transformation?
6. What choice of standard boosts is implicit in a quoted Wigner rotation?

The worked solutions in the linked articles answer these questions through explicit examples. They prepare the spinor input to external-field dynamics and the field-theory continuation. Representation theory constrains that continuation, but does not supply its interactions or quantum statistics by itself.

References

• H. K. Dreiner, H. E. Haber, and S. P. Martin, “Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry,” Physics Reports 494, 1–196, 2010, doi:10.1016/j.physrep.2010.05.002; corrected arXiv:0812.1594v6, 2022.
• W. N. Polyzou, W. Glöckle, and H. Witała, “Spin in Relativistic Quantum Theory,” Few-Body Systems 54, 1667–1704, 2013, doi:10.1007/s00601-012-0526-8.
• N. Straumann, “Unitary Representations of the inhomogeneous Lorentz Group and their Significance in Quantum Physics,” 2008, arXiv:0809.4942.