  In this paper the deformation quantization is constructed in the case of
scalar fields on Minkowski space-time. We construct the star products at three
level concerning fields, Hamiltonian functionals and their underlying structure
called Hamiltonian functions in a special sense. Which mean the star products
of fields, functionals, Hamiltonian functions, and ones between the fields and
functionals. As bases of star products the Poisson brackets at different level
are generalized, constructed and discussed in a systematical way, where the
Poisson brackets like canonical and time-equal ones. For both of the star
products and Poisson brackets the construction at level of Hamiltonian
functions plays the essential role. Actually, the discussion for the case of
Hamiltonian functions includes the key information about Poisson brackets and
the star products in our approach. All of other Poisson brackets and star
products in this paper are based on ones of Hamiltonian functions, and the
Poisson brackets and the star products at three level are compatible. To
discuss the Poisson brackets and the star products in the case of scalar
fields, we introduce the notion of algebra of Euler-Lagrange operators which
related to variation calculus closely.
