Is There a Universal Group Contraction for Lorentz and Galilean Groups?

[Kevin_spencer2]

Is Lorentz group correct?, my question is let's be a group A so the Lorentz Groups is a subgroups of it so $$A>L$$ (L=Lorentz group , G= Galilean group) of course if we had an element tending to 0 so:

$$A(\hbar)\rightarrow L$$ (Group contraction)

so for small h the groups A and L are the same and the laws of physics are invariant under L or A transform, but a pure quantum level when Planck's h is different from 0 the A and L group would be completely different.

this 'Group contraction' would be an analogue of:

$$L(\beta)\rightarrow G$$ where 'beta' is v/c for small velocities we find that law of physics are invariant under Galilean or Lorentz transform.

the question is what would be the 'A' group?, could we find a group A so contracted in elements b=v/c or e=h gives us the Lorentz or Galiean groups and that the transformations (Lorentz, galilean,..) are linear?.

 

[mma]

It's interesting.

I think you mean something like this:http://www.physics.umd.edu/robot/einstein/eicontr.html", isn't it?

I am afraid that I hear the phrase "group contraction" now at the first time, so I know nothing about this, but it seems to me very interesting. I hope that you will find an expert of this topic here; I will read you with pleasure.

 

[Chris Hillman]

I am familiar with contraction, but Kevin's post appears too incoherent for me to make out his question.

A possible reference for the notion of contraction would be Sharpe, Differential Geometry, a textbook on Cartan geometry. This is a notion which arises naturally in Lie theory and which can be used to relate various Lie groups by examining their Lie algebras.

 

[mma]

Could you give a web reference for a brief summary of group contraction (perhaps with special regard to Lorentz-group)? I mean something like a Wikipedia article.

 

[Chris Hillman]

[screetch]Wikipedia?![/screetch]

Could you give a web reference for a brief summary of group contraction (perhaps with special regard to Lorentz-group)? I mean something like a Wikipedia article.

Two seconds with Google yielded this:

http://eom.springer.de/C/c025850.htm

 

[robphy]

And another two seconds yields
http://www.pnas.org/cgi/reprint/39/6/510
E. Inönü, E.P. Wigner, "On the contraction of groups and their representations" Proc. Nat. Acad. Sci. USA , 39 (1953) pp. 510–524

Instead of contraction, a similar term [in the "opposite direction"] used in the literature is "deformation",
e.g., http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1998__S88__73_0
Faddeev, Ludvig, "A mathematician's view of the development of physics." Publications Mathématiques de l'IHÉS, S88 (1998), p. 73-79

 

[mma]

Thank you, robphy and Chris!
Now reading...