My professor is talking about how Lorentz boosts do not commute and how it relates to Thomas Precession, but I am struggling to wrap my head around the implications of that and how it works. Also confused about how commutation in general is applied to areas other than quantum mechanics.
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Boosts are proper Lorentz transformations, so are rotations (defined herein as determinant-$1$ metric-preserving linear transformations) in Minkowski spacetime. Just like rotations in Euclidean space, the order matters.
Consider a vector pointing along $x$. If you rotate about $x$, it does nothing. So if you rotate about $x$ and then about $y$ you get a different outcome than if you rotate about $y$ and then $x$.
This is the physical interpretation. A boost is legitimately a rotation in spacetime.
