Quantum physics, relativity, and complex spacetime. Towards a new synthesis.

*(English)*Zbl 0709.53061
North-Holland Mathematics Studies, 163. Amsterdam etc.: North-Holland. xvi, 359 p. Dfl. 150.00 (1990).

This ambitious book is devoted to the idea of complex spacetime as a possible geometric basis for the synthesis of relativity and quantum theory. The first chapter deals with some simple examples of coherent states and time-frequency localization, including the original coherent canonical states, windowed Fourier transforms and wavelet transforms. The second chapter treats a new algebraic approach to orthonormal bases of wavelets, which are closely related with the concept of a complex structure. The third chapter is devoted to reviewing the concept of generalized coherent states based on Lie groups and their homogeneous spaces. In the fourth chapter the author investigates the interactions between phase space, quantum mechanics and relativity. In the following chapter the author extends his formalism to quantized fields by using the analytic-signal transform. In the last chapter the reader can find a brief account of such ongoing work as holomorphic gauge theory in which general relativity and gauge theory become much closer than in the real case.

Reviewer: H.Nishimura

##### MSC:

53C80 | Applications of global differential geometry to the sciences |

83-02 | Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

32L81 | Applications of holomorphic fiber spaces to the sciences |