Thursday, June 23, 2011

Symmetry Everywhere

I'm taking a break from engineering and getting back physics today...

This post is intended to expand upon an earlier discussion on the symmetry of the laws of physics. In previous posts, I pointed out that nature appears to follow the Rosen-Curie principle: that the symmetry of the effect (future) is greater than the symmetry of the cause (past). And this helps to explain the 2nd Law of Thermodynamics, which states that the number of equivalent (symmetric) microstates of the future is greater then the number of equivalent (symmetric) microstates of the past. In this post, I'd like to expand upon my discussion of  the underlying symmetries in the laws of nature.

Since recently reading "The Force of Symmetry" by Vincent Icke and "Why Beauty is Truth" by Ian Stewart, I've expanding a previous post on symmetry and the forces of nature. In that post, I summarized the relationship between the known forces of nature and the symmetry groups that generate those forces. I highly recommend reading these books if you have at least taken freshman or sophomore level physics in college. I'll summarize some of the major points of both books and my previous post.

Note: an explanation of the U(1), SU(2), SU(3), SU(4) terminology that follows that be found in this link on Sophius Lie's groups.

1) Gravity is a force generated by local Lorentz transformations. The curvature of space-time itself is what we call the gravitational force. All objects with energy, and hence mass, feel the curvature of space-time, and hence feel the gravitational force. There is still a fair amount of debate regarding the symmetry group of gravity because all attempts to unite gravity, quantum mechanics, and the other forces of nature have failed. So, in speculation, I suggest that there might not be a particle for the transmission of the gravitational force. But if there is, then that particle can't have mass (or else it couldn't get out of a black hole in order to communicate with mass outside of the black hole that the black hole exists.)

2) Electromagnetism is a force generated by local U(1) transformations of space-time. U(1) is the symmetry of a circle (i.e. there is one angle to describe the angular position on the circle.) The photon is the carrier of the force. The photon is like a wrinkle in a table cloth, when you locally twist the table cloth while holding the ends fixed. The wrinkles point towards the local of the twist. (i.e. the charge on an object.) Electric charge is the ability to locally twist space-time in a U(1) direction (circular). The photon communicates the amount of twisting of space-time at the location of the charge. Magnetism is related to this wrinkle due to the Lorentz transformation of relativity.

3) The Weak Nuclear Force is a wrinkle (i.e. a force) generated by local SU(2) transformations of space-time. SU(2) symmetry is similar to the symmetry of the surface of a sphere, SO(3). There are two angles to describe the position on a sphere and one angle to describe the angle of the sphere with respect to an observer. This is why there are three particles that exchange the weak nuclear force (W+, W-, & Z) The weak nuclear force describes how quarks change their flavor (but not color, i.e. down into up, but not green into red), and it is difficult to understand because it a Non-Abelian force, meaning that order of operations matters. The same is true for rotations of a sphere. You have to remember the order of operations if you rotation an object in 3-D about different axes.

4) The Strong Nuclear Force (also called the color force) is a wrinkle (i.e. a force) generated by local SU(3) transformations of space-time. There are eight particles, called gluons, that transmit this force. The eight particles correspond to the eight generators of the symmetry group SU(3).

5) Is there another force of nature that we have yet to describe?  The answer is most likely yes. Here's what's missing from our current understanding of physics:
a) Why do the particles (such as electron, muons, tauns, and quarks) have the masses that they do? Why do the charges of the elementary particles have values of: +/- 1 (electrons family), +/- 2/3 (up quark family), +/- 1/3 (down quark family), and +/- 0  (neutrino family) ? There appears to be some underlying structure here that we do not understand. It's almost as if the 'elemental' particles we understand today are really composed of smaller particles, and depending on how you arrange the smaller particles, you get either electrons, quarks, or neutrinos. The heavier members of each family (such as muons and taons) appear to be excited states of the ground state member (such as electrons.)

Given the trend of U(0), U(1), SU(2), SU(3), I suggested earlier that SU(4) might be a force that relates electrons, neutrinos, up quarks, and down quarks. The fact that there are four groups of objects and the earlier trend high symmetry groups led me to think that the symmetry group would be SU(4). But I'm not sure how one explains electric charges of the electron, neutrino, up quark, and down quark using the SU(4) symmetry group. Clearly, the grouping of four objects don't get you the +/-1, +/-2/3, +/-1/3, and +/-0 charge of the electron, quarks, and neutrinos.

If you know how to explain the masses of 'elementary' particles or their values of electric charge, please let me know. I try to keep up to date on what's going on the field of particle physics as best I can, but it's not as much as when I was in grad school.

So, I'd like to end by saying, if you find the connection between symmetry and the forces of nature interesting, pick up a copy of "The Force of Symmetry" by Vincent Icke and "Why Beauty is Truth" by Ian Stewart. And keep in mind that there are probably still more symmetries to be found, and hence more forces of nature to discover, and it might just be a force generated by local SU(4) transformations (twisting) of space-time.

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