- #1

- 679

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(1)In page 64, he says

[itex]L^{-1}({\Lambda}p){\Lambda}L(p)[/itex]...(2.5.10) ((L(p) is defined in (2.5.4))

belongs to the subgroup(of lorentz group) that leaves the standard momentum k invariant, i.e. [itex]W^{\mu}_{\;\nu}k^{\nu}=k^{\mu}[/itex]...(2.57), and this subgroup is called the little group.

My question is about the converse of the statement: can all little group elements be written as (2.5.10)?In other words, can we use (2.5.10) as an alternative definition of little group?If no, what are the exceptions?If yes, how do I prove it?

(2)Suppose little group can be defined by (2.5.10), does the matrix of [itex]W(\Lambda,p)[/itex]depend on the particular choice of the "standard boost“ of L(p)?

Take an example from page 68, the standard boost given in (2.5.24) (which is a pure boost without rotation), can be written as [itex]L(p)=R(\hat {\mathbf{p}})B(|\mathbf{p}|)R^{-1}(\hat {\mathbf{p}})[/itex], but I find we may just as well define L(p) to be [itex]L(p)=R(\hat {\mathbf{p}})B(|\mathbf{p}|)[/itex] and still we'll get the same p from the standard momentum k, but would W still be the same if I choose L(p) to be the latter way(and this L(p) is later used for mass 0 particles, see (2.5.44))? I checked for pure rotation these two L(p) indeed give the same W, but I don't know how to prove it for a general [itex]\Lambda[/itex]