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

The deformation quantization of the scalar fields