Abstract
We present a brief overview of some key concepts in the theory of generalized complex manifolds. This new geometry interpolates, so to speak, between symplectic geometry and complex geometry. As such it provides an ideal framework to analyze thermodynamical fluctuation theory in the presence of gravitational fields. To illustrate the usefulness of generalized complex geometry, we examine a simplified version of the Unruh effect: the thermalising effect of gravitational fields on the Schroedinger wavefunction.
Highlights
The theory of thermodynamical fluctuations provides a solid link between macroscopic and microscopic physics.Classical fluctuation theory [1] often sheds light on counterintuitive quantum-mechanical phenomena, helping to bridge the gap between the classical world and the quantum world
The gravitational field considered here must be weak in order to rule out effects such as, e.g., relativistic speeds, or the likely breakdown of standard quantum mechanics in the presence of very strong gravitational fields [20]
The relation just derived between the Unruh effect and the B-transformation of the Generalized complex structures (GCS) on phase space was based on the assumption that the gravitational field was static and spatially constant
Summary
The theory of thermodynamical fluctuations provides a solid link between macroscopic and microscopic physics. The theory of thermodynamical fluctuations can be recast using the geometric language of differential manifolds [3,4,5,6,7,8,9]. This reexpression of a physical discipline in more abstract mathematical language goes a long way beyond a mere rewriting of the concepts involved. We would like to report on another recent development in geometry with implications on the thermodynamics of fluctuations: the theory of generalized complex manifolds [14,15]. This raises the fundamental question: How is one to treat thermal and quantum fluctuations on the same footing? Is it altogether possible? We will see here that generalized complex manifolds provide one viable answer to this question, one that appears not to have been explored yet in the geometrical approach to thermodynamics
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