turin
Apr18-04, 08:03 PM
I'm putting this question here because I can't get any help from the HW forum (It's actually not a HW, but it looks a lot like a HW, so I won't be surprised if it gets moved there).
Source: Anderson, Principles of Relativity Physics
p. 13, prob. 1.4
"Reparametrize the rotation group by taking, as new infinitesimal parameters, ε1 = ε23, ε2 = ε31, and ε3 = ε12 and calculate the structure constants for these parameters."
My assumptions:
(1)
The εij mentioned in the problem are the infinitesimal Cartesian parameters of the 3-D rotation group such that εij = -εji, and yi = xi + Σjεijxj, where x is the original point and y is the transformed point.
(2)
To generalize this to non-Cartesian coordinates and still maintain the Lie group-ness, the transformation takes the general form:
yi = xi + Σkεkfki(x)
where the fki(x) satisfy the following condition.
(3)
The request for structure constants is a request for constants ckmn such that:
yi = xi + ΣkΣmΣn(εBmεAn - εAmεBn)ckmnfki(x)
(4)
The parameters εk are the non-Cartesian parameters, and so, they should multiply some functions fki(x), and these functions determine the structure constants.
My problem with understanding:
I don't know how to find the fki(x). I have:
Σjεijxj = Σkεkfki(x)
but I don't see how this tells me fki(x). Am I supposed to assume some kind of orthogonality or something?
Source: Anderson, Principles of Relativity Physics
p. 13, prob. 1.4
"Reparametrize the rotation group by taking, as new infinitesimal parameters, ε1 = ε23, ε2 = ε31, and ε3 = ε12 and calculate the structure constants for these parameters."
My assumptions:
(1)
The εij mentioned in the problem are the infinitesimal Cartesian parameters of the 3-D rotation group such that εij = -εji, and yi = xi + Σjεijxj, where x is the original point and y is the transformed point.
(2)
To generalize this to non-Cartesian coordinates and still maintain the Lie group-ness, the transformation takes the general form:
yi = xi + Σkεkfki(x)
where the fki(x) satisfy the following condition.
(3)
The request for structure constants is a request for constants ckmn such that:
yi = xi + ΣkΣmΣn(εBmεAn - εAmεBn)ckmnfki(x)
(4)
The parameters εk are the non-Cartesian parameters, and so, they should multiply some functions fki(x), and these functions determine the structure constants.
My problem with understanding:
I don't know how to find the fki(x). I have:
Σjεijxj = Σkεkfki(x)
but I don't see how this tells me fki(x). Am I supposed to assume some kind of orthogonality or something?