On Fri, Apr 22, 2011 at 12:35:52PM -0700, Gregory Woodhouse wrote about clifford algebras and such.
This discussion is completely off topic on the Racket main=ling list, but please don't stop. I find it fascinating. I would ask everyone in this thread to prefix the subject with OT: so that those who are uninterested can skip it easily. -- hendrik > Clifford algebras (which are non-associative) are higher dimensional > generalizations of H (the quaternions), so in the case of SU(2), you don't > really need the machinery of Clifford algebras. That being said, I'm not > entirely sure what you are looking for. Ordinary rotations (the circle group) > can be embedded in SU(2) in various ways. Beyond that, you need a set of > generators, such as the Pauli spin matrices (the chemistry connection). These > slides > > http://people.virginia.edu/~mah7cd/Math552/SU2SO3SU3.pdf > > may be helpful there. Otherwise, the main thing to look at is whether > representations are real or not (i.e., map group elements to real matrices). > A complex rotation may look like dilation or a shear in real coordinates. > Topologically, SU(2) is a double cover of SO(3) and the covering map p: SU(2) > -> SO(3) associates two distinct complex rotations to each (3-dimensional) > real rotation. That's where the split between fermions and bosons comes from. > > On Apr 22, 2011, at 12:08 PM, Paul Ellis wrote: > > > Many thanks for the continuing interest. > > > > As you say, the 2nd is closer to my level and more accessible to me. > > (The first is likely to take me rather longer to be able to tackle.) > > > > Something I came across in the late Pertti Lounesto's "Clifford > > Algebras and Spinors" (2e, 2001; ISBN 0-521-00551-5), which resembles > > your 2nd ref, makes statements such as: > > > > "In general, a rotation in R^4 has two invariant planes which are > > completely orthogonal, in particular they have only one point in > > common." ( p. 83) > > > > and > > > > "There are three different kinds of rotations in four dimensions..." (p. > > 89). > > > > Do you think there are any references to spaces of two complex > > dimensions that make statements of this sort? Or maybe I should be > > working them out for myself. > > > > _________________________________________________ > For list-related administrative tasks: > http://lists.racket-lang.org/listinfo/users _________________________________________________ For list-related administrative tasks: http://lists.racket-lang.org/listinfo/users
