Representations of the Lorentz Group and Weyl Spinors

Core Idea

Relativistic fields are classified by how they transform under Lorentz transformations, that is, by representations of the Lorentz group. The six generators, three rotations J and three boosts K, can be recombined into two commuting sets J± = (J ± iK)/2, each obeying the angular momentum algebra. Finite-dimensional representations are therefore labelled by a pair of spins (j₁, j₂), with dimension (2j₁ + 1)(2j₂ + 1). The scalar is (0, 0), the two Weyl spinors are (½, 0) and (0, ½), the four-vector is (½, ½), and the Dirac spinor is the sum (½, 0) ⊕ (0, ½).

How It's Best Learned

Write the Lorentz algebra from the generators of rotations and boosts, check the commutators in a 4 × 4 matrix representation, and then form J±. Use the known spin-j representations of the angular momentum algebra to build the (j₁, j₂) list. Construct the Weyl spinor transformations explicitly with Pauli matrices and see how a rotation by 2π gives a factor of −1. Finally, show that parity exchanges the two Weyl representations, which is why parity-conserving theories of massive spin-½ particles use Dirac spinors.

Common Misconceptions

• A spinor is just a four-vector written in a different basis. (A Weyl spinor has two complex components and transforms under (½, 0) or (0, ½); a four-vector is (½, ½), a different representation.)
• The finite-dimensional representations of the Lorentz group are unitary. (The Lorentz group is non-compact, so its finite-dimensional representations, other than the trivial one, are not unitary; the unitary representations that describe particle states are infinite-dimensional.)
• The Dirac spinor is an irreducible representation of the proper Lorentz group. (It splits into two Weyl spinors; it becomes irreducible only when parity is included.)

Explainer

A Lorentz transformation acts on spacetime coordinates through a 4 × 4 matrix Λ. A field must transform in a way consistent with this, φ(x) → D(Λ)φ(Λ⁻¹x), where the matrices D(Λ) form a representation of the Lorentz group. Finding all such representations amounts to listing every kind of relativistic field.

The group has six generators: three rotation generators Jᵢ and three boost generators Kᵢ. In a common convention their commutators are

[Jᵢ, Jⱼ] = iεᵢⱼₖJₖ, [Jᵢ, Kⱼ] = iεᵢⱼₖKₖ, [Kᵢ, Kⱼ] = −iεᵢⱼₖJₖ.

The last relation says that two boosts in different directions combine into a rotation, the origin of Thomas precession. Define J₊ = (J + iK)/2 and J₋ = (J − iK)/2. A short calculation shows that J₊ and J₋ commute with each other, and that each obeys the angular momentum algebra [Aᵢ, Aⱼ] = iεᵢⱼₖAₖ.

The representations of the angular momentum algebra are known from quantum mechanics: one for each spin j = 0, ½, 1, and so on, of dimension 2j + 1. A representation of the Lorentz algebra is a choice of spin for each copy, written (j₁, j₂), with dimension (2j₁ + 1)(2j₂ + 1). The ordinary angular momentum J = J₊ + J₋ then contains spins from |j₁ − j₂| to j₁ + j₂. The simplest cases are:

• (0, 0): a scalar field, one component.
• (½, 0) and (0, ½): the two Weyl spinors, each with two complex components, usually called left-handed and right-handed.
• (½, ½): a four-vector, such as the electromagnetic potential A^μ.
• (1, 0) ⊕ (0, 1): an antisymmetric tensor such as F^μν, with six components.

On a Weyl spinor, rotations act as exp(−iθ·σ/2) and boosts as exp(∓η·σ/2), with opposite signs for the two kinds. A rotation by 2π multiplies the spinor by −1, which reflects the fact that spinors are representations of SL(2, C), the double cover of the proper Lorentz group.

Parity reverses K but leaves J unchanged, so it swaps J₊ and J₋. It therefore turns a (½, 0) spinor into a (0, ½) spinor. A theory that conserves parity and describes a massive spin-½ particle needs both, combined into the four-component Dirac spinor (½, 0) ⊕ (0, ½). In the weak interaction, which violates parity, only the left-handed Weyl component of each fermion couples to the W boson.

These finite-dimensional representations are not unitary, because the Lorentz group is non-compact. Particle states instead form unitary, infinite-dimensional representations of the full Poincaré group. Eugene Wigner classified them in 1939 by mass and by spin or helicity.