From SU(2) Spinors to Lorentz Spinors

A nonrelativistic spin-$1/2$ state is a two-component spinor for spatial rotations. A Lorentz spinor is a finite-dimensional representation of the double cover of the proper Lorentz group. The former teaches the double-cover idea; the latter adds boosts, chirality, and relativistic field transformation laws.

The conceptual bridge is:

$$\begin{array}{rcl} SU(2)\text{ spinor} &:& \text{two-component object for rotations}, \\ SL(2,\mathbb C)\text{ spinor} &:& \text{two-component object for rotations and boosts}. \end{array}$$

This page prepares the notation used by Weyl and Dirac spinors. It does not replace a full construction of the Lorentz group, the Dirac equation, or quantum field theory.

The SU(2) Starting Point

For a spin-$1/2$ Hilbert space, a rotation by angle $\theta$ about $\hat{\mathbf n}$ acts as

$$U(\hat{\mathbf n},\theta) = \exp\left( -\frac{i}{2} \theta\, \hat{\mathbf n}\cdot\boldsymbol\sigma \right).$$

The Pauli matrices generate $SU(2)$, and $SU(2)$ is the double cover of $SO(3)$. Thus a $2\pi$ rotation gives $-I$ on spinor components while leaving the physical ray unchanged.

This nonrelativistic spinor is a state vector in an internal two-dimensional Hilbert space. It is not yet a relativistic field, and it has no boost transformation law by itself.

Why Lorentz Spinors Need More Structure

Relativity requires transformations between inertial frames. Spatial rotations are only part of the Lorentz group. Boosts mix space and time, so a relativistic spinor must say how its components transform under both rotations and boosts.

For the proper orthochronous Lorentz group $SO^+(1,3)$, the relevant double cover is

$$SL(2,\mathbb C).$$

The rotation subgroup appears inside $SL(2,\mathbb C)$ as $SU(2)$. The new feature is that generic $SL(2,\mathbb C)$ matrices are not unitary. This is not a contradiction: finite-dimensional Lorentz spinor representations used for fields are not the same as the unitary Hilbert-space representation carried by physical states.

Four-Vectors as Hermitian Matrices

A compact way to see why $SL(2,\mathbb C)$ appears is to package a spacetime vector into a Hermitian two-by-two matrix. With

$$\sigma^\mu = (I,\sigma_x,\sigma_y,\sigma_z),$$

define schematically

$$X = x_\mu\sigma^\mu.$$

Up to metric and index-placement convention, the determinant of $X$ is the Minkowski length:

$$\det X = (x^0)^2-\mathbf x^2.$$

If $A\in SL(2,\mathbb C)$, then

$$X \longmapsto X' = A X A^\dagger$$

preserves the determinant because $\det A=1$. This transformation therefore induces a Lorentz transformation on $x^\mu$. The matrices $A$ and $-A$ induce the same Lorentz transformation, giving another double-cover relation.

The construction also explains why Pauli matrices keep appearing. They are not merely spin observables; they also provide the two-by-two matrices that connect spinor indices to spacetime vector indices.

Rotations Versus Boosts

For rotations, the spinor matrices are unitary:

$$A_R(\boldsymbol\theta) = \exp\left( -\frac{i}{2} \boldsymbol\theta\cdot\boldsymbol\sigma \right).$$

For a boost of rapidity $\eta$ along the $z$ direction, one common convention gives a nonunitary matrix of the form

$$A_B(\eta) = \exp\left( \frac{\eta}{2}\sigma_z \right) = \begin{pmatrix} e^{\eta/2}&0\\ 0&e^{-\eta/2} \end{pmatrix}.$$

The contrast is useful:

$$\begin{array}{rcl} \text{rotation} &:& \text{unitary finite-dimensional spinor matrix}, \\ \text{boost} &:& \text{nonunitary finite-dimensional spinor matrix}. \end{array}$$

The nonunitarity here is about the finite-dimensional Lorentz-index transformation of a field component. It does not mean time evolution is nonunitary, and it does not replace the unitary action of spacetime symmetries on the physical Hilbert space.

Two Inequivalent Weyl Spinors

The complexified Lorentz algebra separates into two $SU(2)$-like factors. Finite-dimensional irreducible Lorentz representations are often labeled

$$(j_L,j_R).$$

The two basic spinor representations are

$$\left(\frac12,0\right), \qquad \left(0,\frac12\right).$$

These are the left-handed and right-handed Weyl spinor representations, up to convention. They behave the same way under ordinary rotations but differently under boosts. In a generator convention with the same rotation generators

$$J_i = \frac12\sigma_i,$$

the two Weyl representations have opposite boost generators:

$$K_i = \frac{i}{2}\sigma_i, \qquad K_i = -\frac{i}{2}\sigma_i.$$

This is the mathematical seed of chirality. Chirality is not the same as spin projection, and for massive particles it is not the same as helicity.

Sigma and Sigma-Bar Notation

Relativistic two-component spinor notation uses two related packages of Pauli matrices:

$$\sigma^\mu = (I,\boldsymbol\sigma), \qquad \bar\sigma^\mu = (I,-\boldsymbol\sigma).$$

The relative sign is convention-dependent in detail because it interacts with metric signature and whether indices are raised or lowered. The invariant lesson is that one set naturally contracts one Weyl type, while the barred set naturally contracts the conjugate Weyl type.

This is why formulas such as

$$p_\mu\sigma^\mu, \qquad p_\mu\bar\sigma^\mu$$

appear before gamma matrices are introduced. Gamma matrices assemble these two-component structures into a four-component Dirac notation; the compact reference is Gamma-Matrix Identities.

Dirac Spinors as a Direct Sum

A Dirac spinor combines the two Weyl types. In a chiral basis one writes schematically

$$\Psi = \begin{pmatrix} \psi_L\\ \psi_R \end{pmatrix}.$$

The Lorentz transformation acts separately on the two chiral pieces, while a mass term couples them. This is why the massless theory can separate into left- and right-handed Weyl equations, whereas the massive Dirac equation relates both components.

The free Dirac equation is summarized in Dirac Equation. Its nonrelativistic limit is not obtained by simply deleting two components; one expands positive- and negative-energy degrees of freedom in a controlled low-velocity regime.

State Spinors and Spinor Fields

There are three related but distinct objects that are easy to blur:

• a two-component Pauli spinor state in nonrelativistic quantum mechanics,
• a Weyl or Dirac spinor wavefunction in relativistic one-particle theory,
• a spinor field operator in quantum field theory.

The same algebraic symbols may appear in all three settings, but the transformation law and physical interpretation differ. A state vector belongs to a Hilbert space. A classical spinor field is a spacetime-dependent field with spinor components. In QFT, a spinor field is operator-valued and creates or annihilates particle states.

The page From Spin to Relativistic Representations explains the larger distinction between particle-state representations and field representations.

Nonrelativistic Limit

At energies small compared with $mc^2$, a massive Dirac particle admits a Pauli-spinor description. The leading spin dynamics are described by a two-component wavefunction acted on by Pauli matrices. Electromagnetic coupling gives the Pauli magnetic term, summarized in Pauli Equation.

This limiting relation is important pedagogically:

$$\text{Dirac spinor} \quad \longrightarrow \quad \text{Pauli spinor plus relativistic corrections}.$$

It should not be read backward too aggressively. A Pauli spinor does not by itself determine the full Lorentz transformation law, antiparticle degrees of freedom, or field quantization.

Chirality and Helicity

Chirality labels which Weyl representation a spinor component belongs to. Helicity is the projection of angular momentum along momentum:

$$h = \frac{\mathbf J\cdot\mathbf p}{|\mathbf p|}.$$

For massless particles, chirality and helicity are closely related in standard relativistic theories. For massive particles, they are distinct: a boost can change the sign of helicity by overtaking the particle, while chirality remains a representation label.

This distinction is a common source of wrong statements about neutrinos, Dirac spinors, and relativistic spin. From Angular Momentum to Helicity treats helicity and polarization as their own topic.

Worked Example: A z Boost

Start with the two-component object

$$\chi = \begin{pmatrix} a\\ b \end{pmatrix}.$$

Under the $z$-boost convention above,

$$\chi \longmapsto \chi' = \begin{pmatrix} e^{\eta/2}&0\\ 0&e^{-\eta/2} \end{pmatrix} \begin{pmatrix} a\\ b \end{pmatrix} = \begin{pmatrix} e^{\eta/2}a\\ e^{-\eta/2}b \end{pmatrix}.$$

This is unlike a spin rotation about $z$, which multiplies the two components by phases. A boost changes relative magnitudes in this finite-dimensional spinor representation. That is the simplest visible difference between $SU(2)$ spinors and Lorentz spinors.

Common Mistakes

• Treating every two-component spinor as the same kind of object.
• Assuming a Pauli spinor automatically has a Lorentz boost law.
• Confusing $SU(2)$, $SO(3)$, $SL(2,\mathbb C)$, and $SO^+(1,3)$.
• Expecting finite-dimensional Lorentz spinor matrices to be unitary.
• Using chirality and helicity interchangeably for massive particles.
• Importing gamma-matrix identities into two-component notation without checking conventions.
• Forgetting that a Dirac spinor is built from two Weyl pieces, not from two unrelated Pauli spinors.

Related Pages

• Spin-1/2 Hilbert Space
• Spin Rotations
• Pauli Matrices
• Projective Representations
• From Spin to Relativistic Representations
• From Angular Momentum to Helicity
• From Projective Representations to Anomalies Preview
• Spinors
• Dirac Equation
• Gamma-Matrix Identities
• Pauli Equation
• Weyl and Dirac Semimetals explains the useful Weyl/Dirac analogy for crystal quasiparticles and the limits imposed by lattice pseudospin and broken Lorentz invariance.

References

• P. A. M. Dirac, “The quantum theory of the electron”, Proceedings of the Royal Society A 117, 610-624, 1928.
• H. Weyl, “Elektron und Gravitation. I”, Zeitschrift fur Physik 56, 330-352, 1929.
• J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, 1964.
• S. Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
• M. Srednicki, Quantum Field Theory, Cambridge University Press, 2007.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.

Exercises

1. Show that the determinant of $X$ is preserved under $X\mapsto AXA^\dagger$ when $A\in SL(2,\mathbb C)$.

Solution

Use multiplicativity of the determinant:

$$\det(AXA^\dagger) = \det A\, \det X\, \det A^\dagger.$$

For $A\in SL(2,\mathbb C)$, $\det A=1$. Therefore $\det A^\dagger=(\det A)^*=1$, so

$$\det(AXA^\dagger) = \det X.$$

Since $\det X$ encodes the Minkowski length in this construction, the induced transformation is Lorentzian.

2. Compare a spin rotation and a Lorentz boost along $z$ for a two-component spinor.

Solution

A rotation about $z$ has the form

$$A_R(\theta) = \begin{pmatrix} e^{-i\theta/2}&0\\ 0&e^{i\theta/2} \end{pmatrix},$$

so it changes component phases. A boost convention can use

$$A_B(\eta) = \begin{pmatrix} e^{\eta/2}&0\\ 0&e^{-\eta/2} \end{pmatrix},$$

so it rescales the two components. The boost matrix is not unitary, reflecting that this is a finite-dimensional Lorentz spinor transformation, not a Hilbert-space time-evolution operator.

3. Why are chirality and helicity not the same notion for a massive particle?

Solution

Chirality labels the Lorentz representation type: roughly, whether a component transforms as a left- or right-handed Weyl spinor. Helicity is the spin projection along momentum,

$$h = \frac{\mathbf J\cdot\mathbf p}{|\mathbf p|}.$$

For a massive particle, an observer can boost to a frame that reverses the momentum direction relative to the spin, changing helicity. Chirality remains a representation label, and a mass term couples the two chiral components.

4. Why is a Pauli spinor not already a Weyl field?

Solution

A Pauli spinor is a two-component state object used for nonrelativistic spin. A Weyl field is a spacetime-dependent relativistic spinor with a definite Lorentz transformation law under boosts and rotations. The two share $SU(2)$ rotation structure, but the Weyl field carries additional spacetime and Lorentz-representation data.