Is there any physical significance of the Lorentz transformations not having a group structure?

Asked Jan 05 '21 at 14:35

Active Jan 05 '21 at 15:21

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I was studying transformations in QFT and I saw that Lorentz transformations do not have a closed group structure. If $K_i's$ are the generators of the Lorentz transformations and $J_i's$ are the generators of rotations, we have

$$[K_i,K_j]=-\epsilon_{ijk}J_k$$

However, the generators of both the Lorentz transformations and rotations together give us the Lorentz group (which is a closed group).

This seems fairly straightforward in terms of the derivations involved, but does this have some more non-trivial and deeper significance which I am unable to decipher?

edited Jan 05 '21 at 15:21

Prof. Legolasov

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asked Jan 05 '21 at 14:35

VBtheHun

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This fact has a name ‘Thomas precession’. You can read up the wiki page for it to learn more. – Prahar Jan 05 '21 at 14:40
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One possible "deeper significance" is the observation that it is impossible to have a theory invariant under boosts that is not also rotation invariant. – Prof. Legolasov Jan 05 '21 at 14:43
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Related: https://physics.stackexchange.com/a/68586/276737 Does this answer your question? – Cream Jan 05 '21 at 14:46
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What you’re calling Lorentz transformations are known as Lorentz boosts. Both boosts and rotations are considered Lorentz transformations, as in Wikipedia. – G. Smith Jan 05 '21 at 18:07
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In addition to reading about Thomas precession, you should read about Wigner rotation, which is more fundamental. – G. Smith Jan 05 '21 at 18:12

Is there any physical significance of the Lorentz transformations not having a group structure?

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