Stochastic Quantization

1. Front Matter

   Pages I-X

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2. Background Ideas

   Pages 1-4

3. Elements of the Theory of Stochastic Processes

   Pages 5-30

4. General Prescription of Stochastic Quantization

   Pages 31-61

5. Perturbative Approach to Scalar Field Theory

   Pages 62-77

6. Perturbative Approach to Gauge Fields

   Pages 78-94

7. Stochastic Quantization of Constrained Systems

   Pages 95-107

8. Superfield Formulation

   Pages 108-116

9. Renormalization Scheme in Stochastic Quantization

   Pages 117-153

10. New Regularizations in Stochastic Quantization

    Pages 154-163

11. Generalized Langevin Equation and Anomaly

    Pages 164-175

12. Application to Numerical Simulations

    Pages 176-182

13. Minkowski Stochastic Quantization and Complex Langevin Equation

    Pages 183-203

14. Back Matter

    Pages 204-217

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This is a textbook on stochastic quantization which was originally proposed by G. Parisi and Y. S. Wu in 1981 and then developed by many workers. I assume that the reader has finished a standard course in quantum field theory. The Parisi-Wu stochastic quantization method gives quantum mechanics as the thermal-equilibrium limit of a hypothetical stochastic process with respect to some fictitious time other than ordinary time. We can consider this to be a third method of quantization; remarkably different from the conventional theories, i. e, the canonical and path-integral ones. Over the past ten years, we have seen the technical merits of this method in quantizing gauge fields and in performing large numerical simulations, which have never been obtained by the other methods. I believe that the stochastic quantization method has the potential to extend the territory of quantum mechanics and of quantum field theory. However, I should remark that stochastic quantization is still under development through many mathematical improvements and physical applications, and also that the fictitious time of the theory is only a mathematical tool, for which we do not yet know its origin in the physical background. For these reasons, in this book, I attempt to describe its theoretical formulation in detail as well as practical achievements.

• Minkowski spacetime
• quantization
• renormalization

• Department of Physics, Waseda University, Tokyo, Japan

  Mikio Namiki

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