Towards non-perturbative quantization and the mass gap problem for the Yang–Mills field
Abstract
In this paper, we 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 on ℝ3. We suggest a formally self-adjoint expression for the quantized Yang–Mills Hamiltonian as an operator on the corresponding Lebesgue L2-space. In the case when the Yang–Mills field is associated to the abelian group U(1), we define the probability measure which depends on two real parameters m>0 and c≠0. This yields a non-standard quantization of the Hamiltonian of the electromagnetic field, and the associated probability measure is Gaussian. The corresponding quantized Hamiltonian is a self-adjoint operator in a Fock space the spectrum of which is {0}∪[12m,∞), i.e. it has a gap.
