Thursday, April 22, 2010

Are vector bundles fundamental in Physics?

Vector bundles and gauge theory

The idea in Gauge Theory is that the fields of the known forces can be expressed starting with some principal bundles and their associated vector bundles. To be more precise, let's consider Maxwell's electromagnetic field $F_{ab}$. It can be represented with the help of a principal bundle of group $U(1)$, and a connection on this bundle. The connection corresponds to the electromagnetic potential, and the curvature to the electromagnetic field. It is known that we can modify the potential to $A_a(t,x)\mapsto A_a(t,x) + \partial_a \theta(t,x)$, and obtain the same $F_{ab}$. In terms of bundles, this transformation corresponds to a gauge transformation of $\mathbb C$ by the action of $e^{i\theta(t,x)}$. The connection will appear to depend on the gauge, but the curvature is gauge invariant.

A bundle is just another manifold

Both principal bundles and vector bundles are differential manifolds (that is, topological spaces which looks locally, from topological viewpoint, like a vector space with with a fixed number of dimensions, and on which we can define partial derivatives). A fiber bundle over spacetime looks locally like the cartesian product between the spacetime and a fixed manifold named fiber. For the vector bundles the fiber is a vector space, for the principal bundle it is a Lie group. The $U(1)$ bundle looks locally like a cartesian product between the spacetime and a circle. This space is 5-dimensional, and it was used by Kaluza and Klein in their attempt to unify electromagnetism with gravity by using a 5-dimensional version of general relativity.

After the electromagnetic force was understood as a gauge field, Yang and Mills provided a generalization which allowed us to see as gauge fields also the strong and electroweak forces. It seemed as easy as replacing the $U(1)$ group with a non-abelian group like $U(2)$ for the electroweak force, and $SU(3)$ for the color force. New bundles resulted, and they can be viewed as well as spacetimes with more dimensions, from which some are compactified.

The obvious problem with these extra dimensions is that we cannot "see" them. What explanation is that we cannot test? To avoid this questions, these dimensions are referred as corresponding to "internal spaces", and the Kaluza-Klein interpretation is in general avoided, being preferred that in terms of bundles.

What is more fundamental, the field or the connection?

It was believed that the potential is only a mathematical trick to simplify Maxwell's equations, and that it has no correspondent in reality. There are some reasons to change this view.

One is, as I explained here, in chapter III., the following. Maxwell's equations contain constraints imposed on the field for equal time, that is, between the values of the field at spacelike separated points (Gauss' law). This may seem a little bit acausal, because requires the initial conditions at two spacelike separated points to be related. Of course, the separation between the two points is infinitesimal, but it still exists, and has non-local consequences. In terms of the potential, these constraints are no longer needed. If we consider the connection as fundamental, then the curvature will be a derived field. It will still obey Gauss' law, but this time just as a consequence of being associated to the connection, which is the true fundamental field. And the connection is not constrained.

Taking a charged field, such as the Dirac electron field, under a gauge transformation it is multiplied by $e^{i\theta(t,x)}$. The Dirac-Maxwell equations maintain their form, if we apply the corresponding gauge transformation to the potential. This allows us to perform an experiment to see whether the potential is a real field, or just a mathematical trick. This experiment was imagined by Werner Ehrenberg and Raymond E. Siday, and Aharonov and Bohm, a decade later. It was verified experimentally by S. Olariu and I. Iovitzu Popescu, and confirmed two years later by Osakabe et. al..

Basically, this effect shows that the electromagnetic potential has a fundamental nature. But how can a potential be the fundamental quantity? Which potential, considering that there can be an infinity such choices, related by $A_a(t,x)\mapsto A_a(t,x) + \partial_a \theta(t,x)$? The only way known for this is if it represents a connection on a $U(1)$-bundle. This way, the potential is just the expression of the connection, in a particular frame on the bundle. Gauge transformations are just changes of that frame.

The Aharonov-Bohm effect is interpreted topologically as an effect of the holonomy of a connection on this bundle (which is the electromagnetic potential). These properties are captured by Wilson’s loops.

Are those "internal spaces" real?

It is easy to check the number of dimensions of our space: it is the number of coordinates required to indicate the position of a point, that is, 3. The number of numbers needed to express a rotation, 3(3-1)/2=3, indicates also that we live in a 3-dimensional space. How can we check the extra, "internal" dimensions? We just count the numbers needed to represent them. Since the electromagnetic potential can be changed in a way indicating a rotation of a circle, we conclude that the internal space has one dimension. It is the same as in the case of the 3-dimensional space. The only difference is that we can actually move in this space, and this is why we consider it real. We cannot move in the internal dimensions. But can we, at least, send particles to move in those dimensions?

In fact we can. The Aharonov-Bohm effect shows that we can rotate the wavefunction of an electron. We can compare the rotation of a part of the wavefunction of an electron with that of another part. To do this, we just make them interfere, and see the relative rotation between them. Isn't this remind us of comparing the speed of light in two arms of the Michaelson-Morley interferometer? Only that the Aharonov-Bohm effect succeded, and showed that there is an "internal rotation".

Now, it is time to remember the notion of existence as it is used by mathematicians. Something exists from a mathematical viewpoint if it is logically consistent. The 5-dimensional spacetime (3+1+1) of the electromagnetism exists, in this respect. Did the rotation verified by the Aharonov-Bohm effect confirm its physical existence? In fact, we can take for the internal space, instead of a circle, the complex space $\mathbb C$. The group $U(1)$ acts as well on this space, and we can think that the physical spacetime is in fact 6-dimensional (3+1+2). What is the true number of dimensions? I would say that this number is given by the number of dimensions of the $U(1)$-bundle, that is, 3+1+1. And the internal space happens to be a circle because the $U(1)$ group itself is, topologically, a circle. It has one dimension too. And both the circle bundle and the $\mathbb C$ bundle are associated to this principal bundle, that is, they are obtained from representations of the $U(1)$ group.

OK, so the space dimensions are more real for us, because we can move almost freely in these dimensions. Time is the fourth, at least mathematically, and some people can accept that it is the fourth physically too. They think that this is true, because of the great beauty and symmetry of the Lorentz group. But the internal dimensions, have they more than a mathematical existence? We can ask as well whether the three space dimensions are true or not. What if the real number of space dimensions is two, as the holographic principle suggests?

Do we have a criterion to distinguish between real dimensions and simple mathematical constructions in physics? Can this criterion be the experiment?