Questions: Representations of the Lorentz Group and Weyl Spinors

Question 1

What is the dimension of the (½, ½) representation, and what familiar object transforms this way?

• A. 2, a Weyl spinor
• B. 4, a four-vector
• C. 4, a Dirac spinor
• D. 6, the electromagnetic field tensor

Answer: (B)

The dimension is (2·½ + 1)(2·½ + 1) = 4. Four-vectors such as x^μ and the four-momentum transform as (½, ½). The Dirac spinor is also four-dimensional but is (½, 0) ⊕ (0, ½).

Question 2

Every finite-dimensional representation of the Lorentz group in which the boosts act nontrivially is unitary.

• T. True
• F. False

Answer: False

The group is non-compact, so its nontrivial finite-dimensional representations cannot be unitary. Boost generators are represented by anti-Hermitian matrices in, for example, the spinor representations.

Question 3

A Weyl spinor is rotated by 2π about any axis. What happens to it?

• A. It returns unchanged
• B. It is multiplied by −1
• C. It is multiplied by i
• D. It becomes a right-handed spinor

Answer: (B)

Rotations act on Weyl spinors as exp(−iθ·σ/2). For θ = 2π this gives −1. A rotation by 4π is needed to return the spinor to itself.

Question 4

Why does quantum electrodynamics use four-component Dirac spinors rather than two-component Weyl spinors for the electron?

Model answer: Parity exchanges the (½, 0) and (0, ½) representations because it reverses boosts but not rotations. Electromagnetism conserves parity and the electron has mass, so the field must contain both Weyl components, which together form a Dirac spinor.

A mass term couples the left- and right-handed Weyl spinors. In the weak interaction, which violates parity, the two components couple differently, and the Weyl description is natural.