Follow spin when the momentum frame changes

In units with $\hbar=1$, helicity is the eigenvalue of $\mathbf J\cdot\widehat{\mathbf p}$ for a nonzero momentum. A spin-one-half particle has eigenvalues plus or minus one half in this convention; some references rescale the label to plus or minus one. A fixed z-spin label instead refers to a chosen quantization axis and need not equal helicity when the momentum points elsewhere. The zero-momentum state has no preferred momentum direction, so its spin remains definable while helicity as a directional projection is not.

For a massive particle moving along z, an observer can overtake it using a boost parallel to that axis. The momentum reverses in the observer's frame while a canonical spin state along z remains aligned with the same quantization axis under the collinear boost. Helicity can therefore reverse. A massless particle cannot be overtaken by a subluminal observer, and its physical helicity is invariant under proper orthochronous Lorentz transformations. Parity is a different transformation and reverses helicity, so the invariance statement must name its transformation class.

For a massive particle, choose rotation-free standard boosts $L(p)$ that send the rest momentum $k=(m,\mathbf0)$ to p. After applying a Lorentz transformation $\Lambda$, compare the transformed state with the standard state at momentum $\Lambda p$. The transformation $L^{-1}(\Lambda p)\Lambda L(p)$ leaves k unchanged and belongs to the massive little group. It is a spatial rotation, represented on spin states by the appropriate SU(2) matrix. Another convention for standard boosts changes the basis representation consistently, not the physical predictions.

Two boosts along different axes generally do not compose to a rotation-free boost. For active boosts first along z and then along y, the magnitude of the resulting canonical Wigner angle has a simple rapidity formula. Its axis is perpendicular to the boost plane; its orientation depends on ordering and sign conventions. Reversing the order changes the transformation. It is therefore incorrect to treat a Lorentz boost as merely redirecting a classical spin arrow while leaving every quantum amplitude unchanged, or to identify the Wigner angle with the laboratory momentum's deflection angle.

A spin-one-half rotation through an angle has half-angle amplitudes. If the rotation axis is transverse to the initial z-spin direction, the probability to retain the positive z label is the square of cosine of half the angle. This fixed-axis probability becomes a helicity probability only after relating the final momentum axis to the chosen spin basis. In decay chains, those axis transformations matter because intermediate polarization correlations can survive into measured daughter angles. Summing spin probabilities too early can discard interference between the necessary amplitudes.

Chirality is another concept: it labels transformation properties of spinor components and enters the structure of interactions. For massive fermions it is not a synonym for a helicity measurement. Avoid translating a frame-dependent massive helicity reversal into a claim that the underlying interaction changed its chiral coupling. A calculation can use helicity states, canonical spin states, or another consistent basis, but the momenta, boosts, spin density matrices, and analyzer axes must all follow that choice. Observable probabilities agree only after the same physical measurement is represented.