We generalize Koopman-von Neumann classical mechanics to relativistic field theory. The manifestly covariant Koopman-von Neumann mechanics formulated over polysympletic fields leads to De Donder-Weyl mechanics. Comparing this polysymplectic formulation with Dirac's quantization leads to a new Hamiltonian density that is canonical and covariant with symplectic structure. We provide the commutation relations for these classical and quantum fields with a new type of canonical momentum that has the covariant structure of the De Donder-Weyl poly-momentum, yet has the symplectic geometry of Dirac's conjugate momentum. For the first time, we clarify how 1st and 2nd quantization relates to an algebraic deformation of commutation relations of 1st and 2nd Fourier-phase spaces over coordinates and fields, respectively.