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[...] one does not yet have a mathematically complete example of a quantum gauge theory in four-dimensional space-time, nor even a precise definition of quantum gauge theory in four dimensions. Will this change in the 21st century? We hope so! ”

—From the Clay Institute's official problem description by Arthur Jaffe and Edward Witten.

Can you interprete this mathematically to me? I mean why can't we represent or define precisely " quantum gauge theory in four-dimensional space-time"

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up vote 7 down vote accepted

This is an extremely complicated question and a lot of work has been done on the subject over the last century. Although a quick precise answer is impossible, I believe I can provide some intuition.

Let's start with a classical field theory. In particular, I'm going to look at electromagnetism. Let $X$ be a manifold of pseudo Riemannian dimension $d=3+1$. That is, the metric has one negative and three positive eigenvalues. Suppose we have an electromagnetic field present (induced by some current). We can view this electromagnetic field as a closed curvature 2-form $F$ on a $U(1)$ principal bundle $P\to X$. The group $U(1)$ should by thought of as LOCAL symmetries of the Lagrangian. Now Let $\mathcal{A}$ denote the space of connections on this bundle $P$ and $Aut(P)$ be the automorphism group of $P$. This group of automorphisms is called the gauge group and should be thought of as GLOBAL symmetries. $Aut(P)$ induces an action on $\mathcal{A}$ and in particular an action on the subspace $\mathcal{M}\subset\mathcal{A}$ which minimize the Lagrangian for electrodynamics (this condition is given by one of Maxwell's equations).

Now this space $\mathcal{M}$ can be seen as the space of solutions to Maxwell's equations on $X$. However, we only really care about solutions up to global gauge symmetry. The reason being, two solutions which differ only by a gauge symmetry are dynamically indistinguishable and really represent the same physical situation. So we take $\mathcal{M}$ and mod out by the action of $Aut(P)$ on $\mathcal{M}$. The resulting space $$\mathcal{P}=\frac{\mathcal{M}}{Aut(P)}$$ are the solutions we really care about.

Ok, so already, one can see why the situation is complicated. This space $\mathcal{P}$ depends both on the underlying space and the electromagnetic field present. So to understand all of electrodynamics, one needs to understand all pseudo Riemannian manifolds of dimension $3+1$ (which we don't) and all principal $U(1)$ bundles over $X$.

Now add quantization to the picture. To quantize, one wants to exponentiate the Lagrangian and integrate over the space of connections....yeah... This space is a huge infinite dimensional manifold and the topology is way too fine to have any "reasonable" measure on it. Feynman was able to get a partial answer to the quantization problem of electrodynamics by approximating the integral by a series and examining terms of the series (which correspond to Feynman diagrams).

Needless to say, the story is far from over. This is only classical electromagnetism were trying to quantize! Now one could be interested in nonabelian gauge theories such as Yang Mills theory or more scary things like a 10 dimensional $SU(32)$ gauge theory.

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