physical system has still great challenges (see the third reference from [@reviews]).

The canonical realization of deformation quantization is done with the use of the Weyl-Wigner-Moyal (WWM) formalism introduced originally in Refs. [@wwmoriginal], which establishes a one-one correspondence between the operator algebra (on certain Hilbert space) and the algebra of symbols of operators via the so called WWM correspondence. The product of operators is mapped into an associative and noncommutative product of the respective symbols called the $\star$-product. The simplest example is here the famous Moyal $\star$-product. Quantum theory in terms of deformation quantization has been intensively studied for physical systems with a finite number of degrees of freedom.

Recently, much more interesting and difficult case of classical fields (with an infinite number of degrees of freedom) has been also considered by some authors. For example, deformation quantization of scalar field is done in Refs. [@dito; @ditothree; @ditodos; @cfz; @zachosone]. In some of these references it has been shown that the $\star$-product can be modified to absorb the divergences of the field theory with the $\lambda \phi^4$-interaction. This modification is accomplished by a $\lambda$ dependent redefinition of the $\star$-product consistent with a cocycle condition. Free electromagnetic field in deformation quantization 