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.


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

  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.

  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?

  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.

  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.