Mass Gap Revisited
There appears to be a lot of work involved in completely solving the mass gap Millennium Problem. The official problem statement, by Jaffe and Witten, asks:
"Prove that for any compact simple gauge group $G$, a non-trivial quantum Yang-Mills theory exists on $\mathbb{R}^4$ and has a mass gap $\Delta > 0$. Existence includes establishing axiomatic properties at least as strong as those cited in [Osterwalder + Schrader, Streater + Wightman]."
Unlike the purely mathematical problems, not all terms in this statement are well defined. What is a Yang-Mills theory? The article suggests that a rigorous formulation of QCD would be adequate. The remaining difficulty with the statement is the one thing that people seem so comfortable with: the use of $\mathbb{R}^4$.
In M theory, as in Algebraic Geometry, we don't like fixing number fields unnecessarily. In fact, since the numbers depend on the class of experimental question, we had better go a long way with the rationals before we even contemplate any kind of continuum limit. Fortunately, Grothendieck understood this a long time ago. Now when we discuss twistor geometry and the division algebras, we understand that these things are simply convenient models for the underlying operadic axioms.
It is still necessary, however, to display in detail how this geometry arises from the axioms. This is why we need Motivic Cohomology, because without it we will never get close to anything looking like a path integral.
"Prove that for any compact simple gauge group $G$, a non-trivial quantum Yang-Mills theory exists on $\mathbb{R}^4$ and has a mass gap $\Delta > 0$. Existence includes establishing axiomatic properties at least as strong as those cited in [Osterwalder + Schrader, Streater + Wightman]."
Unlike the purely mathematical problems, not all terms in this statement are well defined. What is a Yang-Mills theory? The article suggests that a rigorous formulation of QCD would be adequate. The remaining difficulty with the statement is the one thing that people seem so comfortable with: the use of $\mathbb{R}^4$.
In M theory, as in Algebraic Geometry, we don't like fixing number fields unnecessarily. In fact, since the numbers depend on the class of experimental question, we had better go a long way with the rationals before we even contemplate any kind of continuum limit. Fortunately, Grothendieck understood this a long time ago. Now when we discuss twistor geometry and the division algebras, we understand that these things are simply convenient models for the underlying operadic axioms.
It is still necessary, however, to display in detail how this geometry arises from the axioms. This is why we need Motivic Cohomology, because without it we will never get close to anything looking like a path integral.
posted by Kea | 8:16 AM
4 Comments:
I would not have included mass gap problem to Millenium problems. Quantum field theories are too poorly defined as mathematical structures. Giving mass gap problem the status of Riemann Hypothesis can badly mis-direct efforts. My feeling is that one must first try to understand physics and try to transcend the conceptual limitations of existing QFT framework. In physics this is time for visionaries, not for solvers of precisely defined technical problems.
Number theoretical universality of physics is a really fascinating idea. Especially so, when one asks seriously what would be the interpretation of say p-adic physics with a particular value of prime p. Second fascination is that number theoretic constraints such as algebraic number character of S-matrix elements provide immensely powerful constraints on theory.
Best,
Matti
Kea
What do you think of the AMS monograph on Motivic Cohomology based on Voevodsky's lecture notes?
In fact, since the numbers depend on the class of experimental question, we had better go a long way with the rationals before we even contemplate any kind of continuum limit.
Since the M-theory U-duality groups are defined over the integers, one can go a long way even before invoking the full set of rationals.
U-duality groups such as E8(8)(Z) and E7(7)(Z) are discrete groups inducing integer shifts on the charge lattice. Schroeder gives a nice overview of discrete U-duality groups in hep-th/9909157.
From the Jordan algebra perspective, E7(7)(Z) arises from the Freudenthal triple system defined over the Jordan algebra of 3x3 Hermitian matrices over the split-octonions (with integer coefficients).
Thanks for the link, kneemo. I had a copy of V's notes at one point, but unfortunately with so much moving around I don't have anything printed out at present. Sometime soon hopefully...
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