✎ Thomas Precession [The Physics Travel Guide]

You've loaded an old revision of the document! If you save it, you will create a new version with this data.

====== Thomas Precession ======

<blockquote>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]).<cite>https://arxiv.org/abs/1711.05753</cite></blockquote>
  
<tabbox Concrete>

<note tip>
In this section things should be explained by analogy and with pictures and, if necessary, some formulas.
</note>
 
<tabbox Abstract> 
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 [[advanced_tools:geometric_phase|geometric phase]]/Berry phase.

From a group theoretic perspective, Thomas precession occurs because Lorentz boost do not commute. For example, we have 
$$[K_y,\,K_x] = i\,J_z \, .$$

---

<blockquote>{{ :advanced_notions:thomas_precession.jpg?nolink&200|}}[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. <cite>https://arxiv.org/abs/1711.05753</cite></blockquote>

<blockquote>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).<cite>https://physics.stackexchange.com/a/99459/37286</cite></blockquote>
<tabbox Why is it interesting?>

<blockquote>[T]he phenomenon of Thomas precession is one of the best established kinematical consequences of [[advanced_tools:group_theory:poincare_group|Poincaré symmetry]]<cite>https://cqgplus.com/2018/02/15/can-you-see-asymptotic-symmetries/</cite></blockquote>

/*<tabbox FAQ>*/

/*<tabbox History>*/

</tabbox>

Please solve the following equation to prove you're human. 133 +13 = Please keep this field empty:

Edit summary

Note: By editing this page you agree to license your content under the following license: CC Attribution-Share Alike 4.0 International

User Tools

Site Tools

Page Tools