From Spin to Relativistic Representations

Spin begins in nonrelativistic quantum mechanics as intrinsic angular momentum: states carry representations of the rotation group. In relativistic quantum theory, the same idea expands. Particle states and fields are organized by representations of spacetime symmetry.

The bridge is:

$$\begin{array}{rcl} \text{nonrelativistic QM} &:& \text{spin labels representations of rotations}, \\ \text{relativistic quantum theory} &:& \text{particle states and fields transform under spacetime symmetry}. \end{array}$$

This page is a conceptual bridge. It does not build the full Lorentz group, derive the Dirac equation, or prove Wigner’s classification theorem. It explains why those topics are the natural continuation of the spin formalism.

Spin Is Already Representation Theory

In ordinary spin quantum mechanics, a rotation acts by a unitary operator

$$U(R) = \exp\left( -\frac{i}{\hbar}\boldsymbol\theta\cdot\mathbf J \right).$$

For spin alone,

$$\mathbf J=\mathbf S,$$

and a spin-$s$ Hilbert space carries a $(2s+1)$-dimensional irreducible representation of $SU(2)$. For spin-$1/2$,

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

Thus spin is not an extra classical vector bolted onto the theory. It is how quantum states transform under rotations. The pages What Spin Is and Is Not and Spin Rotations develop this nonrelativistic side.

Relativity Changes the Symmetry Group

Nonrelativistic spin focuses on spatial rotations. Relativistic quantum theory must also treat boosts consistently. The spacetime symmetry group is no longer just rotations and translations in space; for flat relativistic spacetime it is the Poincaré group:

$$\text{Poincaré} = \text{translations} \rtimes \text{Lorentz transformations}.$$

The Lorentz part contains rotations and boosts. Rotations form a familiar subgroup, but boosts do not commute with rotations in the same simple way. This means spin cannot be treated as merely a fixed three-vector independent of momentum and frame.

The conceptual shift is:

$$\text{spin as }SU(2)\text{ rotation representation} \quad \longrightarrow \quad \text{state representations of the Poincaré group}.$$

Particle States: Mass and Spin

For a stable relativistic particle in flat spacetime, one classifies one-particle states by irreducible unitary representations of the Poincaré group. The invariant labels include mass and spin for massive particles.

A schematic state label is

$$\lvert p,\sigma\rangle,$$

where $p^\mu$ is four-momentum and $\sigma$ labels spin projection or polarization data. The invariant mass is determined by

$$p^\mu p_\mu = m^2$$

in the mostly-minus convention, or by the corresponding sign-adjusted formula in the opposite metric convention.

For a massive particle, one can go to the rest frame

$$p^\mu = (m,\mathbf 0)$$

in units with $c=1$. The transformations that leave this rest momentum fixed form the little group, which is rotational. This is why massive relativistic particles still have familiar spin labels

$$s=0,\frac12,1,\frac32,\ldots.$$

The nonrelativistic spin multiplet is therefore not discarded. It appears as the rest-frame little-group representation inside the relativistic representation.

Wigner Rotations

The relativistic transformation law of spin states is momentum dependent. Schematically,

$$U(\Lambda) \lvert p,\sigma\rangle = \sum_{\sigma'} D_{\sigma'\sigma} \!\left(W(\Lambda,p)\right) \lvert \Lambda p,\sigma'\rangle,$$

where $\Lambda$ is a Lorentz transformation and $W(\Lambda,p)$ is a Wigner rotation. Normalization factors depend on convention and are suppressed here.

The important lesson is not the formula’s details. It is that boosts and rotations combine in a way that can rotate the spin labels depending on the particle momentum. Relativistic spin is still representation theory, but the representation acts on momentum and internal labels together.

Massless Particles and Helicity

Massless particles have no rest frame. That changes the little-group analysis. For physically observed finite-helicity particles, the useful spin-like label is helicity:

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

Helicity is the angular momentum projection along the direction of motion. For massless particles, it is invariant under proper Lorentz transformations in the usual finite-helicity representations. This is why photon polarization and graviton helicity are not just ordinary rest-frame spin projections.

The dedicated bridge is From Angular Momentum to Helicity. The point here is only that the massive spin story and massless helicity story are both representation-theoretic.

Fields Versus Particle States

Relativistic quantum theory uses two related but distinct representation ideas.

Particle states carry unitary representations of the Poincaré group on Hilbert space. These are the representations used to classify particles by mass, spin, and helicity.

Fields transform in finite-dimensional representations of the Lorentz group or its double cover. Examples include:

$$\phi(x) \quad \text{scalar field},$$ $$A^\mu(x) \quad \text{vector field},$$

and spinor fields such as Weyl or Dirac fields. A field creates or annihilates particle states, but a field component is not the same object as a one-particle spin state.

This distinction prevents a common confusion. A two-component Pauli spinor in nonrelativistic quantum mechanics is not automatically a relativistic Weyl field. The page Spinors gives the compact reference bridge, while From SU(2) Spinors to Lorentz Spinors focuses on the route from rotation spinors to Lorentz spinors.

Lorentz Spinors in One Line

The double cover of the proper Lorentz group is closely related to $SL(2,\mathbb C)$. Its finite-dimensional spinor representations include the two inequivalent Weyl spinor types, often denoted

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

A Dirac spinor combines both chiral pieces. The Dirac equation reference card is Dirac Equation, and gamma-matrix conventions are collected in Gamma Matrix Identities.

The bridge lesson is modest: the Pauli spinor teaches the double-cover idea, but Lorentz spinors add boosts, chirality, metric conventions, and relativistic field transformation laws.

Nonrelativistic Limit

Relativistic spinor theory reduces to Pauli spin in appropriate low-energy regimes. Roughly, positive-energy components of a Dirac spinor become the two spin components of a Pauli wavefunction, while relativistic corrections generate effects such as spin–orbit coupling and magnetic-moment terms.

This is why nonrelativistic spin is both self-contained and provisional. It is a complete effective framework for many systems, but it also points toward the relativistic representation theory behind spinor fields.

Common Mistakes

• Treating spin as a literal rotating object rather than a representation label.
• Assuming boosts leave spin labels untouched in the same way as ordinary rotations.
• Confusing a Pauli spinor with a Weyl spinor or Dirac field.
• Treating field representations and particle-state representations as the same object.
• Using helicity and spin projection interchangeably for massive particles.
• Forgetting that metric, gamma-matrix, and normalization conventions matter in relativistic formulas.
• Saying “spin comes from relativity” in a way that erases the nonrelativistic representation theory of spin.

Related Pages

• Why Symmetry Becomes Central in QFT
• From Quantum Generators to Noether Currents
• What Spin Is and Is Not
• Spin Rotations
• Projective Representations
• From SU(2) Spinors to Lorentz Spinors
• From Angular Momentum to Helicity
• From Discrete Symmetries to CPT
• Spinors
• Spin-Statistics Preview
• Dirac Equation

References

• E. P. Wigner, “On unitary representations of the inhomogeneous Lorentz group,” Annals of Mathematics 40, 149-204, 1939.
• 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.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.

Exercises

1. Why does a massive relativistic particle still have ordinary spin labels?

Solution

A massive particle has a rest frame. The subgroup of Lorentz transformations that leaves the rest four-momentum fixed is rotational, so the internal labels are representations of the rotation little group. This gives the familiar spin values $s=0,1/2,1,\ldots$ and $2s+1$ spin states.

2. Why is helicity especially natural for massless particles?

Solution

Massless particles have no rest frame, so one cannot define spin by going to rest and using ordinary rotations there. The projection of angular momentum along the direction of motion,

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

is the natural finite-helicity label for massless particles.

3. In the schematic transformation

$$U(\Lambda)|p,\sigma\rangle = \sum_{\sigma'} D_{\sigma'\sigma}(W(\Lambda,p)) |\Lambda p,\sigma'\rangle,$$

what is the conceptual role of $W(\Lambda,p)$?

Solution

$W(\Lambda,p)$ is the Wigner rotation induced by combining the Lorentz transformation with the standard way of assigning spin states to momenta. It shows that the spin labels can rotate in a momentum-dependent way under Lorentz transformations. This is the relativistic refinement of treating spin as a fixed internal label.

4. Why is a field representation not the same thing as a one-particle state representation?

Solution

Particle states transform in unitary representations on Hilbert space and are classified by mass, spin, or helicity. Fields transform as local operator-valued objects in finite-dimensional Lorentz representations. A field can create or annihilate particle states, but its components and transformation law are not identical to the Hilbert-space particle-state representation.