Wednesday, September 26, 2012

Hoping for a mathematical understanding

The Standard Model of particle physics is a very successful mathematical model for all of physics. But it has not truly reduced physics to math. Here is what a couple of our leading mathematical physicists, Arthur Jaffe and Edward Witten, say:
Quantum Yang-Mills theory is now the foundation of most of elementary particle theory, and its predictions have been tested at many experimental laboratories, but its mathematical foundation is still unclear. ...

2 Quest For Mathematical Understanding ... On the other hand, one does not yet have a mathematical understanding of the quantum behavior of four-dimensional gauge theory, or even a precise definition of quantum gauge theory in four dimensions. Will this change in the twenty-first century? We hope so!
Okay, they hope so. But we may never have a purely mathematical understanding of quantum reality.

5 comments:

  1. Who cares? As long as it works to make predictions, it's a good theory.

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  2. I agree that it is a good theory. I am just pointing out that it is not a theory that has been completely reduced to mathematics, as there are mathematical questions about the foundations.

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  3. The numerical approximations to the equations work, but it is unclear whether the actual equations are good. So my point is why care if the actual equations are good?

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  4. "Okay, they hope so. But we may never have a purely mathematical understanding of quantum reality."

    I don't think it's unreasonable to hope for a mathematically rigorous formulation of four-dimensional gauge theory. For one thing, people like Arthur Jaffe have already made huge strides towards a rigorous understanding of quantum field theory. We also have a good mathematical understanding of two-dimensional conformal field theory and three-dimensional topological field theory, and we can formulate classical Yang-Mills theory in a completely rigorous way. Finally, we have many theoretical tools at our disposal: various field theoretic dualities and the AdS/CFT correspondence.

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  5. I am not criticizing this foundational work. It is good and important work. It would be great is someone found a mathematical justification for the mass gap. I am just pointing out limits to what has been accomplished.

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