- #1
AdrianMay
- 121
- 4
Why do people try against all odds to make SU(2) isometric with SO(3) when it's clear from the definition that it's actually isometric with SO(4). Either way you've got 4 variables and the same constraint between them.
It's interesting to see all the dodgy tricks that go into this deception. First, the definition of Lie groups has got that i sneaked inside the exponential for no apparent reason. For SO(2) or SO(3), the generators are defined as what you'd expect them to be, only divided by i to get rid of it again, and the system is just the same as if the i had never been there.
Almost the same. The difference is that you can say that the commutator [Jx,Jy] = iJz. This is also a deception. You can twiddle a globe around in your fingers all day long (or use google Earth if you don't trust your fingers) and you'll never find a meaningful sense in which [Jx, Jy] is related to Jz. It's just a trick of the i, but we never needed to enter the complex domain in order to describe rotations.
Nevertheless, this fictitious commutation relation is then used as the definition of ... of what? Of whatever we're trying to shoehorn SU(2) into. Even this is not enough though. We also have to rob SU(2) of one of it's generators. It's the one that due to the i obfuscation trick looks like the identity. Without that trick, it's got i on it's diagonal, which is not the identity at all. Some authors dismiss it as just an equal phase shift of all the wavefunctions which is supposedly not significant.
Like hell it's not significant. If you phase shift a square well solution you break the boundary conditions. The scalar and vector potentials act only on this phase, so declaring that missing generator insignificant is tantamount to denying the existence of electromagnetism.
It doesn't stop there either. SU(2), like SO(4) actually has 6 generators (because you can rotate about the planes wx, wy, wz, xy, xz and yz, which is a a lot simpler than the Pauli matrices) so we've robbed it of an entire 50% of its generators.
Even after dragging this poor group into a space it doesn't need and stealing half it's generators, rather than fitting the square hole we prepared for it, it expresses it's derision by refusing to return home after 360 degrees. Instead of recognising this as the reductio ad absurdum we deserved, we invent a name for it (double cover) and still insist that our fictitious commutation relation is closer to the essence of rotation than the fact that there are 360 degrees in a circle.
The only guy who found a use for all this spaghetti is Dirac, but even then it wasn't enough. He had to pop out into four dimensions, which brings me full circle.
Adrian.
It's interesting to see all the dodgy tricks that go into this deception. First, the definition of Lie groups has got that i sneaked inside the exponential for no apparent reason. For SO(2) or SO(3), the generators are defined as what you'd expect them to be, only divided by i to get rid of it again, and the system is just the same as if the i had never been there.
Almost the same. The difference is that you can say that the commutator [Jx,Jy] = iJz. This is also a deception. You can twiddle a globe around in your fingers all day long (or use google Earth if you don't trust your fingers) and you'll never find a meaningful sense in which [Jx, Jy] is related to Jz. It's just a trick of the i, but we never needed to enter the complex domain in order to describe rotations.
Nevertheless, this fictitious commutation relation is then used as the definition of ... of what? Of whatever we're trying to shoehorn SU(2) into. Even this is not enough though. We also have to rob SU(2) of one of it's generators. It's the one that due to the i obfuscation trick looks like the identity. Without that trick, it's got i on it's diagonal, which is not the identity at all. Some authors dismiss it as just an equal phase shift of all the wavefunctions which is supposedly not significant.
Like hell it's not significant. If you phase shift a square well solution you break the boundary conditions. The scalar and vector potentials act only on this phase, so declaring that missing generator insignificant is tantamount to denying the existence of electromagnetism.
It doesn't stop there either. SU(2), like SO(4) actually has 6 generators (because you can rotate about the planes wx, wy, wz, xy, xz and yz, which is a a lot simpler than the Pauli matrices) so we've robbed it of an entire 50% of its generators.
Even after dragging this poor group into a space it doesn't need and stealing half it's generators, rather than fitting the square hole we prepared for it, it expresses it's derision by refusing to return home after 360 degrees. Instead of recognising this as the reductio ad absurdum we deserved, we invent a name for it (double cover) and still insist that our fictitious commutation relation is closer to the essence of rotation than the fact that there are 360 degrees in a circle.
The only guy who found a use for all this spaghetti is Dirac, but even then it wasn't enough. He had to pop out into four dimensions, which brings me full circle.
Adrian.