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Fedosov Quantization and Perturbative Quantum Field Theory

Fedosov has described a geometro-algebraic method to construct in a canonical way a deformation of the Poisson algebra associated with a finite-dimensional symplectic manifold ("phase space"). His algorithm gives a non-commutative, but associative, product (a so-called "star-product") between smooth phase space functions parameterized by Planck's constant $\hbar$, which is treated as a deformation parameter. In the limit as $\hbar$ goes to zero, the star product commutator goes to $\hbar$ times the Poisson bracket, so in this sense his method provides a quantization of the algebra of classical observables. In this work, we develop a generalization of Fedosov's method which applies to the infinite-dimensional symplectic "manifolds" that occur in Lagrangian field theories. We show that the procedure remains mathematically well-defined, and we explain the relationship of this method to more standard perturbative quantization schemes in quantum field theory.