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A gauge invariant quantization in a closed integral form is developed over a linear phase space endowed with an inhomogeneous Faraday electromagnetic tensor. An analog of the Groenewold product formula (corresponding to Weyl ordering) is obtained via a membrane magnetic area, and extended to the product of N symbols. The problem of ordering in quantization is related to different configurations of membranes: a choice of configuration determines a phase factor that fixes the ordering and controls a symplectic groupoid structure on the secondary phase space. A gauge invariant solution of the quantum evolution problem for a charged particle in an electromagnetic field is represented in an exact continual form and in the semiclassical approximation via the area of dynamical membranes.

The symplectic geometry of the phase space associated with a charged particle is determined by the addition of the Faraday 2-form to the standard dp ∧ dq structure on R 2n . In this paper we describe the corresponding algebra of Weyl-symmetrized functions in operatorsq,p satisfying nonlinear commutation relations. The multiplication in this algebra generates an associative * product of functions on the phase space. This * product is given by an integral kernel whose phase is the symplectic area of a groupoid-consistent membrane. A symplectic phase space connection with non-trivial curvature is extracted from the magnetic reflections associated with the Stratonovich quantizer. Zero and constant curvature cases are considered as examples. The quantization with both static and time dependent electromagnetic fields is obtained. The expansion of the * product by the deformation parameter , written in the covariant form, is compared with the known deformation quantization formulas.

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