The Standard Model Symmetries

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

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.

command: 
q_graph -dir vp -out group_u1 -loop 0 -box 1.1 -command "g_u1 1 2 3 4 1000"
youtube embed: 
Outside video: 
math
equation: 

\frac{A}{|A|} \in U(1)

tags
Physics Tag: 
standard model
electromagnetism
gauge symmetry
U(1)
Math Tag: 
groups
U(1)
Programming Tag: 
command line quaternions
g_u1

The Group SU(2), Weak Force Symmetry

summary: 

The group of unitary quaternions is the symmetry underlying the weak force of radioactive decay. Who would have thought the symmetry looks like this?

description: 

If one takes the vector part of a quaternion and takes the exponential, the norm is always equal to 1. The animation starts out at 8 points, those for exp(0, +/-1, +/-1, +/-1). These points grow into each other until they form a sphere. That sphere then shrinks to the point (1, 0, 0, 0), the furthest into the future the exponential can reach.

command: 
q_graph -dir vp -out group_SU2 -loop 0 -box 1.1 -command 'q_random_n_11 50000 | q_vector | q_exp'
youtube embed: 
Outside video: 
math
equation: 

exp(A - A^*)\in SU(2)

tags
Physics Tag: 
standard model
gauge symmetry
weak force
Math Tag: 
groups
SU(2)
Programming Tag: 
command line quaternions
g_su2

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

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 embed: 
math
equation: 

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

tags
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

The Group SU(3), Symmetry of the Strong Force

summary: 

The group SU(3) is created by taking the Euclidean product of two electroweak symmetries. Nature may need less tools than the standard model suggests.

description: 

This groups is the completely uniform unit quaternion sphere, starting from t=-1, expanding to its maximal size at t=0, then contracting to t=+1. For an observer is now at the center of their private Universe - (0, 0, 0, 0) - when they see an event, no matter what the cause, the event can be scaled to fit on this sphere. The norm of any event in the unit sphere is exactly 1, even if with rulers and atomic clocks a big or small sized measurement could be made.

<p>Because the symmetries U(1), SU(2) and U(1)xSU(2) are formally subgroups of SU(3), there is no need for a larger group to unify these groups. A rather large effort is still required to connect to all we know of the standard model.

command: 
q_graph -dir vp -out group_su3 -loop 0 -box 1.1 -command q_group -group U1xSU2xSU3 -n 50000
youtube embed: 
Outside video: 
math
equation: 

(\frac{A}{|A|} exp(A - A^*))^* \frac{B}{|B|} exp(B - B^*) \in SU(3)

tags
Physics Tag: 
standard model
gauge symmetry
Math Tag: 
groups
Programming Tag: 
command line quaternions
q_group
g_su3

The Lorentz Group: Circles and hyperbolas

summary: 

The Lorentz group has one subgroup of 3D rotations, and another for inertial reference frame boosts where time effectively rotates into a spatial dimension.

description: 

The spacial rotations are in yellow, while the boosts are in red. The hyperbolas can cover all of spactime while the rotations are limited to a circle about the origin. Out at infinity, there will be two points that approach the yellow circle. They split so that one set of red points can be there are the creation and annihilation of the yellow points, and another set can greet the yellow points when they are furthest apart, about to change directions.

command: 
q_graph -box 3 -dir trig -out circle -command 'g_u1_n 4 1 2 3 1000' -color yello -command 'g_u1h_n 4 1 2 3 4000' -color red [note: the function for generating the hyperbola has not been released, and its name will probably change]
math
equation: 

for the circle: \frac{q}{\sqrt{q q*}} = \frac{(t, \vec{R})}{t^2 + R^2}
for the hyperbola: \frac{q}{\sqrt{\pm scalar(q q)}} = \frac{(t, \vec{R})}{t^2 - R^2}

tags
Physics Tag: 
Lorentz transformations
special relativity
Math Tag: 
Lorentz group
Programming Tag: 
command line quaternions
g_u1