Lorentz Boost of a Spin-Singlet: A Worked Example with Wigner Rotations




Lorentz Boost of a Spin-Singlet: A Worked Example with Wigner Rotations

We show explicitly how a Lorentz boost acts on a two–spin-1/2 singlet state by inducing Wigner rotations that depend on the particle momenta.
The key takeaway: the full bipartite state remains entangled in every inertial frame, but the distribution of entanglement between spin and momentum,
and therefore the spin-only correlations accessible to a given observer, can change across frames.

1) Setup in the Center-of-Momentum (CM) Frame

Consider two spin-1/2 particles with equal and opposite momenta along the \(z\)-axis in the CM frame:
\(\mathbf{p}_A = +p\,\hat{\mathbf{z}}\), \(\mathbf{p}_B = -p\,\hat{\mathbf{z}}\).
Let the spin state be the singlet:

\[
\lvert \Psi^- \rangle
= \frac{1}{\sqrt{2}}\big(\lvert \uparrow \rangle_A \lvert \downarrow \rangle_B
– \lvert \downarrow \rangle_A \lvert \uparrow \rangle_B\big).
\]

Assume (for clarity) sharply peaked momenta (delta-like wavepackets) so that we can track definite momenta through the boost.

2) Apply a Pure Lorentz Boost Along x

Boost the whole system by rapidity \(\eta\) along the \(x\)-axis, i.e., a boost \(\Lambda(\eta\,\hat{\mathbf{x}})\).
For a particle with 4-momentum \(p^\mu\), a pure boost followed by the standard momentum alignment induces a Wigner rotation
on its spin. Because the two particles carry opposite momenta in the CM frame, they experience opposite Wigner rotation angles.

Let \(\xi\) be the rapidity corresponding to the particle’s CM-frame momentum magnitude \(p\) and mass \(m\):

\[
\cosh \xi = \frac{E}{m}, \qquad \sinh \xi = \frac{p}{m}, \qquad \tanh \xi = \frac{p}{E} = v.
\]

For this geometry (boost along \(x\), momenta along \(\pm z\)), the induced Wi