Shastic mechanics is a theory that holds great promise in resolving the mathematical and interpretational issues encountered in the canonical and path integral formulations of quantum theories. It provides an equivalent formulation of quantum theories, but substantiates it with a mathematically rigorous shastic interpretation by means of a shastic quantization prescription.
The book builds on recent developments in this theory, and shows that quantum mechanics can be unified with the theory of Brownian motion in a single mathematical framework. Moreover, it discusses the extension of the theory to curved spacetime using second order geometry, and the induced Itô deformations of the spacetime symmetries.
The book is self-contained and provides an extensive review of shastic mechanics of the single spinless particle. The book builds up the theory on a step by step basis. It starts, in chapter 2, with a review of the classical particle subjected to scalar and vector potentials. In chapter 3, the theory is extended to the study of a Brownian motion in any potential, by the introduction of a Gaussian noise. In chapter 4, the Gaussian noise is complexified. The result is a complex diffusion theory that contains both Brownian motion and quantum mechanics as a special limit. In chapters 5, the theory is extended to relativistic diffusion theories. In chapter 6, the theory is further generalized to the context of pseudo-Riemannian geometry. Finally, in chapter 7, some interpretational aspects of the shastic theory are discussed in more detail. The appendices concisely review relevant notions from probability theory, shastic processes, shastic calculus, shastic differential geometry and shastic variational calculus.
The book is aimed at graduate students and researchers in theoretical physics and applied mathematics with an interest in the foundations of quantum theory andBrownian motion. The book can be used as reference material for courses on and further research in shastic mechanics, shastic quantization, diffusion theories on curved spacetimes and quantum gravity.