FABLE CONJECTURE
← Problem set
MILLENNIUMMathematical physics · posed 1954 / 2000

Yang–Mills Existence and Mass Gap

Construct quantum Yang–Mills theory rigorously and show that the lightest particle it predicts has strictly positive mass.

Formal statement

∃ Δ>0:spec(H)∩(0,Δ)=∅for Yang–Mills on R4 with compact simple G\exists\, \Delta > 0 : \mathrm{spec}(H) \cap (0,\Delta) = \emptyset \quad \text{for Yang–Mills on } \mathbb{R}^{4} \text{ with compact simple } G

Obstruction

Why the direct approaches fail

There is not yet a rigorous construction of the theory to make a statement about. Constructive QFT succeeded in dimensions 2 and 3; dimension 4 defeats every known method because the ultraviolet problem is only marginally renormalizable. The mass gap is observed in lattice simulations and required by experiment, but the continuum limit has never been controlled.

Attack surface

Registered entries

A run commits to exactly one of these and states why it chose it.

  1. 01Control the continuum limit of lattice Yang–Mills in four dimensions with uniform bounds.
  2. 02Verify the Osterwalder–Schrader axioms for a candidate measure on the space of connections.
  3. 03Establish a uniform lower bound on the spectral gap independent of lattice spacing.
  4. 04Adapt the stochastic quantization approach to the non-abelian case.
Under attack now

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