Wednesday, August 4, 2010

Global and local symmetries

An important distinction in physics is global and local.

A global property is something which is inherent to a system as a whole. A local property is something attached to a particular point in space and time. Assume for the moment that the earth would be a perfect sphere, which it is to a rather good approximation. Then the rate at which the earth's surface bends under one's feet is a global property, because it is the same on the whole planet. On the other hand, whether there is water and land under the feet is a local property, and depends on where on the earth one stands.

So far, this is a static situation, which permits to divide between global and local properties. Even more important in physics is the difference between local and global changes. A local change modifies something at a given place. E. g., the property whether there is land or water below one's feet is changed locally by the tides. A local change is not limited to a certain point, but it can affect many (or all) points at the same time, but something different may go on at every point. The tides all over the world are an example of a local change, which let the water rise at some point and removes it at another point. A global change is then a special case of a local change in that it makes the same change at each and every point. For example covering the earth's surface everywhere by a meter of sand would be a global change.

This leads back to symmetries. It is now possible to divide between a global and a local symmetry. A global symmetry is something inherent to the system as a whole. A global symmetry transformation would then be a symmetry transformation applied to every point which leaves the system unchanged.

A local symmetry transformation is much more complicated to visualize. Take a rectangular grid of the billiard balls from the last post, say ten times ten. Each ball is spherical symmetric, and thus invariant under a rotation. The system now has a global and a local symmetry. A global symmetry transformation would rotate each ball by the same amount in the same direction, leaving the system unchanged. A local symmetry transformation would rotate each ball about a different amount and around a different axis, still leaving the system to the eye unchanged. The system has also an additional global symmetry. Moving the whole grid to the left or to the right leaves the grid unchanged. However, no such local symmetry exists: Moving only one ball will destroy the grid's structure.

Such global and local symmetries play an important role in physics. The global symmetries are found to be associated with properties of particles, e. g., whether they are matter or antimatter, whether they carry electric charge, and so on. Local symmetries are found to be associated with forces. In fact, all the fundamental forces of nature are associated with very special local symmetries. For example, the weak force is actually associated in a very intricate way with local rotations of a four-dimensional sphere. The reason is that, invisible to the eye, everything charged under the weak force can be characterized by a arrow pointing from the center to the surface of such a four-dimensional sphere. This arrow can be rotated in a certain way and at every individual point, without changing anything which can be measured. It is thus a local symmetry. This will become more clearer over time, as at the moment of first encounter this appears to be very strange indeed.