electromagnetism | visualphysics.org

Visualizing a Student's Guide to the Maxwell Equations

23 May 2009

Posted by doug

Hello:

My focus has always been on the best field equations in all of physics, the Maxwell equations. My proposal for gravity is a variation on Maxwell: change the algebra for forming the action from Hamilton's quaternions to hypercomplex numbers (what I call the even representation of quaternions). Now that I have launched this site, my hope is to create visualizations of any applications of the Maxwell equations. It would be quite a site to understand visually the difference between the E and the B field!

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Group U(1)xSU(2) for Electroweak Symmetry

Posted by doug

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)

youtube embed:

iPod downloads:

group_u1xsu2.povray.100.1000.animation.scan_.m4v

math

equation:

$\frac{A}{|A|} exp(A - A^*) \in U(1) \times SU(2)$

sage notebook:

The Group U(1)xSU(2)

numbers:

group_u1xsu2.povray.100.1000.numbers

The Group U(1), Electromagnetism's Gauge Symmetry

Posted by doug

summary:

The group for the unit circle in a complex plane, U(1), can be at an arbitrary angle in 3D space. The transverse waves of EM have this symmetry.

description:

If one picks a quaternion at random, normalize it, then take n powers of that number, one ends up with this animation. It looks like tilted circles in the complex planes. The motion is fasted at the creation and annihilation. The velocity of the dots is slowest when the two have their largest separation. This group is symmetric in both time and space reflection. Recall that time reflection requires recall, memories of the path taken, while space reflection involves mirrors.

<p>On pleasing aspect of this animation is that it starts to make sense of a transverse wave. Mapping that wave to electric or magnetic fields will require considerably more work.