discussion

The complex structure on the phase space and unitary gauge groups

The gauge groups of the SM are unitary groups. That means, they preserve a metric together with a complex structure. A complex structure in a vector space (which is the only case interesting here) is simply a rule which defines a special linear operator - multiplication with the complex unit i.

In the case of a phase space, there seems to be a simple way to define such a complex structure. We introduce complex variables \[ z = q + i p.\]

This already defines how to multiply them with i. Unfortunately, this complex structure is not natural, it depends on our particular choice of coordinates q and p. For example, we could have used, instead of q, another coordinate \(q' = a q\). If we want to have the same symplectic structure, we would have to use another momentum variable too, and the rule for the replacement of the momentum variable would be \(p' = p / a\). But this gives already another complex structure \[ z' = q' + i p' = a q + i p / a.\]

Thus, the question is how to obtain a "natural" complex structure, which is preserved by the gauge groups. The problem may be reformulated in the following way:

Complex structure is symplectic structure + metric

If we have a complex structure (more accurate, a Kähler metric), we can use it to define two other structures. First, a metric, and second, a symplectic structure. And the reverse is also true: If we have a metric as well as a symplectic structure, we can reconstruct the complex structure.

On the phase space, we already have a natural symplectic structure. But we have no natural metric. How to construct an appropriate metric on the phase space?

Constructing a metric preserved by a compact group

In gauge theory, usually only compact gauge groups are considered. Especially all gauge groups of the SM are compact. (That's not a necessity. One approach to the theory of gravity is to interprete it as a gauge theory for the Lorentz group, which is noncompact. But it simplifies a lot of things if we restrict ourself to compact groups.)

Once we have a compact group, we can already construct an invariant metric \(\langle.,.\rangle\). The idea is to start with an arbitrary, non-invariant metric \( \langle .,. \rangle_0\) and to average over the action of the group: \[ \langle x,y \rangle = \int_{g\in G} \langle gx,gy \rangle_0 dg\]

The point of this formula is that for compact groups there exists such a measure dg with finite volume on the group which is right-invariant, that means, dgg' = dg. Because the volume is finite, and the group compact, the integral exists always. And the resulting averaged scalar product is already invariant, preserved by the action of the group.

Thus, using this construction, we obtain, for every compact group which preserves the symplectic structure of the phase space, a metric as well as a complex structure which is preserved by this group.

Application

Unitarity is important if we consider the possible actions of the Wilson gauge group. We have found that it preserves the generation and has to leave the component a0 unchanged. Without additional restrictions, the largest possible group would be O(7) acting on the three \(a_j\) for the color \(j > 0\) and the four \(\pi_\mu\). But these rotations would not preserve the symplectic structure on the phase space. With this additional restriction, \(a_0\) has to remain untouched, together with \(\pi_0\), because they form the complex number \(z_0 = a_0+i\pi_0\). Moreover, the group O(6) has to be reduced to U(3). This is already much more close to (in other words, almost identical with) the observed SU(3) of strong interactions.

Summary

Thus, if we postulate that

we can derive that the gauge groups have to be unitary groups.