Visualizing the Symmetries in the Standard Model

Intro S0 S1 S2 S3 SU(2) U(1)xSU(2) U(1)xSU(2)xSU(3) Diff(M)xU(1)xSU(2)xSU(3)

An Introduction to the Standard Model

The Standard Model of physics was developed in the 1970's to explain the ~270 different types of particles seen in colliders (a general introduction is available on wikipedia, and a one page cheat sheet). The part we need to understand is the groups that describe the symmetry of the standard model.

What is a group? A group has an identity, an inverse, and a binary operation (multiplication). One member of the group times another member of the group generates yet another member of the same group. This is a case where the math name is accurate: once in a group, you are always in a group.

The standard model has three continuous groups that characterize three of the four known fundamental forces of nature. The simplest group is known as U(1) and governs electromagnetism via the photon. The reason there is one photon is that the Lie algebra u(1) - note that was a small u! - has one degree of freedom. This group is called the unary group, complex numbers with a norm of 1. The members of this group commute, so it does not matter the order things are written in. Quaternions have this property only when all point in the same or opposite directions, which is the case for when using one quaternion times itself.

The continuous group SU(2) rules the weak force, the stuff driving radioactive decay. Mathematically this is call unitary quaternions, quaternions with a norm of 1. The Lie algebra used to generate this group has three degrees of freedom. That is why the weak force is mediated by three particles, the W+, W-, and the Z.

The group SU(3) is for the strong force whose residual interactions keep nuclei together. Its Lie algebra has eight members, and there are eight gluons.

Animations of Groups

Start with a simple picture, layer pictures together, and we will be able to see what the standard model of particle physics looks like.

S0 - So simple!

This is the symmetry of +/-R, one number.

What was up/down	What is	What can be


What was near/far	What was left/right	What can be that is

This is (-1, 0, 0, 0) and (+1, 0, 0, 0). It sits in the center, as quaternions of the form (n, 0, 0, 0) like to do.

S1 - The Circle

Now we let the sum of squares of 2 numbers equal 1. This creates a circle.

What was up/down	What is	What can be


What was near/far	What was left/right	What can be that is

The circle could have any orientation in 3D space. The program choose one at random.

S2 - Slice of an Expanding/Contracting Sphere

The sum of squares of 3 numbers equal 1. A quaternion has four numbers. One approach to representing S2 is to set t=0. You get the standard sphere, but only at the instant of t=0. Blink!

What was up/down	What is	What can be


What was near/far	What was left/right	What can be that is

That looks like a typical sphere, except it doesn't last long. Three straight lines appear in the "what was" graph because time is fixed.

Another way to represent S2 is to set x=0. Then you have an edge view of an expanding circle.

What was up/down	What is	What can be


What was near/far	What was left/right	What can be that is

Only the "what was left/right" graph has a fixed, straight line graph, because x=0.

If z=0, at least you can see the "circleness"

What was up/down	What is	What can be


What was near/far	What was left/right	What can be that is

In the "what was" graphs, it is easy to spot which dimension is set to zero: it is the straight line.

S3 - A Quaternion Sphere

Now use all 4 terms, and fill in the sphere in both time and space!

What was up/down	What is	What can be


What was near/far	What was left/right	What can be that is

SU(2) - Like S2, the Partial Sphere

The graphs of S2 were all paper thin or fleeting. They do not "fill up" spacetime. The next graph, generated by putting random quaternions into the expression exp(q-q*), fills up spacetime.

What was up/down	What is	What can be


What was near/far	What was left/right	What can be that is

Although more of spacetime appears filled, only places were time is greater than zero have a chance to have an event.

U(1)xSU(2) - Like the Complete Quaternion Sphere S3

SU(2) has only three of the four degrees of freedom available to a quaternion. There is no way to fill up all of spacetime with just SU(2). Now fill spacetime in by multiplying by itself, or q/|q| exp(q-q*)

What was up/down	What is	What can be


What was near/far	What was left/right	What can be that is

Most of the points cluster on the negative side of the time line.

U(1)xSU(2)xSU(3) - The Standard Model

The question is how to generate SU(3)? It has a Lie algebra su(3) that has eight elements. Based on work done on quaternion quantum mechanics, it is clear I need to work with the conjugate of one quaternion times another, what I call the Euclidean product, because q* q generates the norm of a quaternion q, (t² + x² + y² + z², 0, 0, 0). If we have 2 different quaternions, q and q', we can write them as q* q' as U(1)xSU(2): (q/|q| exp(q-q*))* (q'/|q'| exp(q'-q'*)). Here is its animation:

What was up/down	What is	What can be


What was near/far	What was left/right	What can be that is

Notice how all of spacetime is filled evenly with events. A product of two quaternions that uses a conjugate different from a standard product because multiplication is no longer associative ((a b)*c does not equal a* (b c)). The norms are preserved, so the norm will remain 1. Eight independent number are used to make something with a norm of one. The identity is 1, and all elements have an inverse under what I call "Euclidean multiplication", q* q'. Based on the animation, the group is compact and simply connected. All of this traits contribute to the conclusion that the symmetry of the standard model can be represented by quaternions in this way.

It would be great to include gravity, which is all about how measurements change as one moves around a differentiable 4D manifold. Include the metric as part of the calculation of a quaternion product.

The group Diff(M) is all diffeomorphisms of a compact smooth manifold. It is at the heart of general relativity. One can imagine this spacetime filling sphere on any compact smooth manifold.

What happens if q=q'? That is shown below:

What was up/down	What is	What can be


What was near/far	What was left/right	What can be that is

The standard model is about the group symmetry of the quaternion multiplication identity in spacetime.

OK, but what does that mean? Here is my take. Observers sit at here-now in spacetime, or numerically at (0, 0, 0, 0). An observer sees something out there, and tries to characterize the "thingie". The basic bit of information it can classify is an event. Whatever set of events is collected, they are all tied up in describing this one thingie out there. Every event contributes to the description of the thingie, and so makes a group. The multiplicative identity of a quaternion, (1, 0, 0, 0) is a way to represent the thingie. Almost none of the events map to (1, 0, 0, 0). The events are scattered all around spacetime. U(1)xSU(2)xSU(3) is the way to cow-rope all the events and bring them home, while remaining part of the same group, the one thing being observed.

Quaternion s Question and Answer website

Next: Time Reversal