summary:

The group U(1)xSU(2) can be represented using all four parts of a quaternion, three in the exponential, the fourth as a normalized quaternion. The group covers the entire unit sphere, but has a bias for the past.

description:

Quaternions do not commute in general, but they will commute if two quaternions point in the same direction. The common way to represent the group U(1) is with a normalized complex number. The same thing can be done with a quaternion. This will commute with a unitary quaternion if they both use the same quaternion pointing in the same direction. Electroweak symmetry uses all the degrees freedom available in a quaternion.

command:

q_graph -dir vp -out group_u1xsu2 -loop 0 -box 1.1 -command 'q_group -group U1xSU2 -n 20000'

youtube link:

The Group U(1)xSU(2)
iPod downloads:

equation:

sage notebook:

The Group U(1)xSU(2)
numbers:

Physics Tag:

standard model

electroweak force

gauge symmetry

electromagnetism

weak force

Math Tag:

groups

U(1)

SU(2)

U(1)xSU(2)

Programming Tag:

command line quaternions

q_group

q_random_n_11