Renormalization and critical phenomena in the Parisi and Wu stochastic quantization scheme

In this thesis, I will focus my attention on the problem of the renormalization in the stochastic quantization scheme. in Part II. I give a mathematical introduction to the stochastic processes (II.1) and to the diffusion equations (II.2). The main purpouse of this introduction is to make clear and natural the idea underlying the stochastic quantization. The stochastic quantization is described in detail in the Sect.111.3 for both the perturbative and the non perturbative aspects. In Sect.111.4 a formal discussion of the emergence of the Ward identities in the stochastic quantization scheme is reported. In Part Ill. the problem of the renormalization is analyzed. In particular I discuss the consistence of the stochastic regularization (III.1) and the possibility of renormalizing the stochastic theory also in presence of some peculiar features of the stochastic regularization (III.2). To this end we make a non trivial use of a hidden BRS symmerty. In Sect.III.3 a first example of renormalization procedure is given. Moreover the background field method for the stochastic quantization scheme is introduced. In Sect. III.4 a detailed analysis of the renormalization group equations and the interplay between stochastic quantization and the theory of ( dynamical ) critical phenomena is reported. In Part IV. I show the possibility of using the new methods produced by the stochastic quantization to get new numerical values for the critical exponents of λΦ^4 in d=3. In Sect.IV.1 I remind the main features of the ε-expansion and set the problem. In Sect.IV.2 I explain the idea on which the computations are based, while in Sect.IV.3 the computations and the results are reported