Spinors

Spinors first appear in quantum mechanics as states transforming under the double cover of the rotation group. In relativistic field theory, spinors become representations of the Lorentz group and fields whose quanta are spin-$1/2$ particles.

Quantum Mechanics Starting Point

A spin-$1/2$ state is a two-component vector acted on by rotations

$$U(R) = \exp\!\left( -\frac{i}{\hbar}\boldsymbol\theta\cdot\mathbf S \right), \qquad \mathbf S=\frac{\hbar}{2}\boldsymbol\sigma.$$

A rotation by $2\pi$ changes the sign of a spinor, while the physical ray is unchanged.

Relativistic Continuation

Relativistic spinors transform under representations of the Lorentz group or its double cover. The Dirac equation can be written

$$(i\gamma^\mu\partial_\mu-m)\psi=0,$$

using units with $\hbar=c=1$ on this bridge page.

The slash convention should be written as

$$p\!\!\!/ \equiv \gamma^\mu p_\mu,$$

not with unsupported slash macros.

Dictionary

Quantum Mechanics	Relativistic/QFT Continuation
spin-$1/2$ Hilbert space	spinor representation space
Pauli matrices	gamma matrices and Lorentz generators
SU(2) rotations	Lorentz spinor transformations
spin observable	spin and helicity labels
two-component spinor	Weyl spinor in relativistic notation
finite-dimensional state vector	spinor field component or external spinor, depending on context

Cautions

A spinor wavefunction and a spinor field are not the same object. In one-particle relativistic quantum mechanics, $\psi$ may be treated as a wavefunction. In QFT, a Dirac field is operator-valued, and particle states are created by field-mode operators acting on the vacuum.

Also distinguish:

• spin from helicity,
• chirality from helicity,
• Pauli matrices from gamma matrices,
• rotations from Lorentz boosts,
• state spinors from field operators.

Common Mistakes

• Treating a sign change under $2\pi$ rotation as a directly measurable phase by itself.
• Importing Pauli-matrix identities into gamma-matrix calculations without checking dimensions and metric convention.
• Confusing a two-component nonrelativistic spin state with a relativistic Weyl spinor.
• Forgetting that field-theory spinor normalization differs from ordinary Hilbert-space normalization.

Canonical Links

• What Spin Is
• From Spin to Relativistic Representations
• From SU(2) Spinors to Lorentz Spinors
• From Angular Momentum to Helicity
• Spin-Half Hilbert Space
• Spin Rotations
• Pauli Matrices
• Projective Representations
• Dirac Equation
• Gamma Matrix Identities

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• S. Weinberg, The Quantum Theory of Fields, Volume I, Cambridge University Press, 1995.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Westview Press, 1995.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014.

Exercises

1. Why is a two-component spin-$1/2$ state not automatically a relativistic spinor field?

Solution

The two-component state describes a finite-dimensional internal degree of freedom in ordinary quantum mechanics. A relativistic spinor field has Lorentz-transformation properties, spacetime dependence, and in QFT is operator-valued. The representation spaces are related, but the physical and mathematical objects are different.