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Thomas Precession


Thomas precession normally occurs when a spinning particle such as an electron orbits around an atomic nucleus (figure 2); this contributes to its effective spin-orbit coupling, which in turn affects spectral lines (see e.g. [41, section 11.8]).


In this section things should be explained by analogy and with pictures and, if necessary, some formulas.


Abstractly, the Thomas precession is due to an additional rotation the spin of the particle pics up as a result of of the orbit movement. In this sense, the phenomenon is due to a geometric phase/Berry phase.

From a group theoretic perspective, Thomas precession occurs because Lorentz boost do not commute. For example, we have $$[K_y,\,K_x] = i\,J_z \, .$$ The rotation that we get by combining boosts is known as a Wigner rotation.


[T]he revolution of the particle around the nucleus can be interpreted as a sequence of Lorentz boosts.[…] Abstractly, the effect is due to Wigner rotations [3] that appear in representations of the Poincaré group, so it is purely kinematical—it follows solely from space-time symmetries.

To get the Lorentz transformation mapping from frame 1 to 3 directly, we compose boosts:

$$\exp\left(\frac{a_y}{c} \,\delta t\, K_y + \frac{a_x}{c} \,\delta t\, K_x\right) \exp\left(\frac{v_x}{c}\,K_x\right) = \exp\left(\frac{v_x}{c}\,K_x + \frac{a_x}{c}\,\delta t\,K_x + \frac{a_y}{c}\,\delta t\, K_y + \frac{a_y\,v_x}{2\,c^2}\,[K_y,\,K_x]\,\delta t + \cdots\right)\tag{1}$$

by the Dynkin formula version of the Baker-Campbell-Hausdorff theorem. From the commutation relationships for the Lorentz group we get $[K_y,\,K_x] = i\,J_z$, so that $\frac{a_y\,v_x}{2\,c^2}\,[K_y,\,K_x]\,\delta t$ corresponds to a rotation about the $z$ axis of $a_y\,v_x\,\delta t / (2\,c^2)$ radians (here, somewhat obviously $J_z,\,J_y,\,J_z$ are "generators of rotations", *i.e.* tangents to rotations about the $x,\,y,\,z$ axes at the identity, respectively, whilst $J_z,\,J_y,\,J_z$ are "generators" of boosts).

Why is it interesting?

[T]he phenomenon of Thomas precession is one of the best established kinematical consequences of Poincaré symmetry

Contributing authors:

Jakob Schwichtenberg
advanced_notions/thomas_precession.txt · Last modified: 2019/06/10 08:20 by jakobadmin