In 2000 Jaffe and Witten stated the long standing problem of exact quantization for the 4D Yang-Mills field on the Minkowski space and the existence of a mass gap in the theory as one on the Clay Institute Millenium problems. In this talk I shall explain how to reduce the problem of quantization of the Yang-Mills field Hamiltonian to a problem for defining a probability measure on an infinite-dimensional space of gauge equivalence classes of connections, the so called Yang-Mills measure associated to a 3D Yang-Mills theory on a Euclidean space. A formally self-adjoint expression for the quantized Yang-Mills Hamiltonian as an operator on the corresponding Lebesgue L_2-space will be presented.

In the case when the Yang-Mills field is associated to the abelian group U(1) the Yang-Mills measure will be defined. This measure is Gaussian and depends on a real parameter m>0. The quantized Hamiltonian, which is the corresponding Ornstein–Uhlenbeck operator, can be realized a self-adjoint operator in a Fock space. Its spectrum has a gap separating the rest of the spectrum from the ground state zero eigenvalue. This yields a non-standard quantization of the Hamiltonian of the electromagnetic field.

Recently, based on Hairer's theory of regularity structures, Chandra, Chevyrev, Hairer and Shen developed an approach for constructing Yang-Mills measures for Euclidean Yang-Mills theories in 2D and 3D associated to arbitrary compact Lie groups. Using the corresponding Langevin equation they defined a Markov process on a space of gauge orbits for which the Yang-Mills measure should be defined as the invariant measure provided that the process converges. This result is proved in 2D and the problem is still open in the physically important 3D case. In this framework the quantized Yang-Mills Hamiltonian is the generator of the stochastic process the convergence of which is equivalent to the existence of the mass gap.

The presentation in this talk will be self-contained and requires no special background from the audience.

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