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We reduce the problem of quantization of the Yang-Mills field Hamiltonian toa problem for defining a probability measure on an infinite-dimensional spaceof gauge equivalence classes of connections on $\mathbb{R}^3$. We suggest aformally self-adjoint expression for the quantized Yang-Mills Hamiltonian as anoperator on the corresponding Lebesgue $L^2$-space. In the case when theYang-Mills field is associated to the Abelian group $U(1)$ we define theprobability measure which depends on two real parameters $m>0$ and $c\neq 0$.This yields a non-standard quantization of the Hamiltonian of theelectromagnetic field, and the associated probability measure is Gaussian. Thecorresponding quantized Hamiltonian is a self-adjoint operator in a Fock spacethe spectrum of which is $\{0\}\cup[\frac12m, \infty)$, i.e. it has a gap.

Towards non-perturbative quantization and the mass gap problem for the Yang-Mills Field