There are at least two areas in physics where gauge-type invariances appear. The first area is field theory, where the gauge-invariant quantity is the action, and the gauge coordinates are the vector potentials. One practical reason that gauge invariance is useful in field theory is that (depending on the problem) we can choose a gauge that makes calculations shorter and easier. A deeper reason the gauge-invariance of field equations generates conservation laws that constrain the solutions; for example, the gauge invariance of quantum electrodynamics guarantees charge conservation. Deepest of all (and not well understood by me, or perhaps anyone) is that Nature seems to prefer gauge-invariant field theories (perhaps because they are renormalizable?).

Physicists have been struggling to understand these dynamical gauge invariances for at least the past 130 years (ever since Helmholtz proved celebrated Helmholtz Theorem), and there is still plenty of work to do.

More recently, gauge-type invariances have begun to appear ubiquitously in quantum information science. Here the gauge-invariant quantity is the time-dependent density matrix, and the gauge-coordinates are the probability densities that appear in Fokker-Planck equations (also called forward Kolmogorov equations). Here too at least three levels of mathematical understanding can be distinguished. From a practical point of view, we can exploit these informatic gauge invariances (in a more-or-less *ad hoc* way at present) to more efficiently solve practical problems in quantum simulation. At a deeper level, we appreciate that informatic gauge invariances are responsible for enforcing informatic causality (the natural law that quantum correlations cannot transmit information outside the light code). And at the very deepest level, we recognize that Nature is ubiquitously fond of quantum informatic gauge invariances, just as she is ubiquitously fond of quantum dynamical gauge invariances … and the deep reason for this ubiquity (if there is a deep reason) is simply not understood at present.

The mathematical and physical depth of these gauge-theoretic questions is so great, and they are so far from being answered at present, that I won’t even hazard a guess at what the coming century of research might unveil, but instead will refer you to Feynman’s 1982 article *Simulating Physics With Computers*, whose concluding sentence is “If you want to make a simulation of nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy!”

If gauge transformations are connections over some mathematical object -fiber bundles?-and connections-differential forms- are curvature,what do gauge transformations have to do with curvature.

The reason I ask this question is because gauge transformations have something to do with particle physics which is Quantum Merchanics which is defined on hilbert spaces which are flat-noncurved infinite dimensional spaces. So are spaces where gauge transformations can take place automatically curved?

Do gauge transformations “glue” together fibers which are infintesmially close.

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]]>Dusa McDuff makes a point that is new (as far as I know) when she asserts: “As is increasingly understood, the real question is how any young person can build a satisfying personal life while still managing to be a creative mathematician. Once people start working on this in a serious way, we will have truly come a long way.”

Two key phrases in McDuff’s passage are “any young person” and “work on this in a serious way”. On a planet with (about) two billion young people, any serious endeavor to create technical jobs for (say) five percent of these young people has to contemplate creating on the order of one hundred million jobs. If we follow Dusa McDuff’s advice and work on this in a “serious way”, then these are the numbers we have to seriously contemplate.

Dusa McDuff’s essay deserves credit for starting us on this path.

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