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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Qmechanic
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hollowhills
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Commutation is all about testing if the order of things matter. They don't just appear in quantum mechanics. Take the braid group for example on some number of strings. Non-commutativity there is telling you something about which ways to braid can be done in any order to achieve the same result. – JamalS Feb 13 '22 at 23:28
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2Related : (1) Combining two Lorentz boosts. (2) General matrix Lorentz transformation. (3) Show that any proper homogeneous Lorentz transformation may be expressed as the product of a boost times a rotation. – Frobenius Feb 13 '22 at 23:42
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Due diligence. Linked. – Cosmas Zachos Feb 14 '22 at 01:10
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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.
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1Dale, this is an excellent explanation I had never grasped before. You have done me, at age 70, a valuable service! – niels nielsen Feb 14 '22 at 05:00
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I've edited in the requested precision about rotation terminology, then flagged as no longer needed several existing comments. – J.G. Feb 14 '22 at 07:57
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@WillO the Minkowski metric is not a metric though, which is why boosts indeed don't quite fit the definition of a rotation. – leftaroundabout Feb 14 '22 at 09:37