Mathematics of Quantization and Quantum Fields

belong to a relatively small group Spj(Y), resp. Oj(Y). Other interesting objects in the case of an infinite number of degrees of freedom are the analogs of the metaplectic and Pin representation. CCR and CAR representations provide a convenient setting to describe various forms of quantization. By a quantization we usually mean a map that transforms a function on a classical phase space into an operator and has some good properties. Of course, this is not a precise definition – actually, there seems to be no generally accepted definition of the term “quantization”. Clearly, some quantizations are better and more useful than others. We describe a number of the most important and useful quantizations. In the case of CCR, they include the Weyl, Wick, anti-Wick, x, D- and D, xquantizations. In the case of CAR, we discuss the anti-symmetric, Wick and anti-Wick quantizations. Among these quantizations, the Weyl, resp. the antisymmetric quantization play a distinguished role, since they preserve the underlying symmetry of the CCR, resp. CAR – the symplectic, resp. orthogonal group. However, they are not very useful for an infinite number of degrees of freedom, in which case the Wick quantization is much better behaved. The x, D-quantization is a favorite tool in the microlocal analysis of partial differential equations. The non-uniqueness of CCR or CAR representations for an infinite number of degrees of freedom is a motivation for adopting a purely algebraic point of view, without considering a particular representation. This leads to the use of operator algebras in the description of the CCR and CAR. This is easily done in the case of the CAR, where there exists an obvious candidate for the CAR C∗-algebra corresponding to a given Euclidean space. This algebra belongs to the well-known class of uniformly hyper-finite algebras, the so-called UHF(2∞) algebra. We also have a natural CAR W∗-algebra. It has the structure of the well-known injective type II1 factor. In the case of the CCR, the choice of the corresponding C∗-algebra