Abstract
It is argued that quantum mechanics is fundamentally a geometric theory. This is illustrated by means of the connection and symplectic structures associated with the projective Hilbert space, using which the geometric phase can be understood. A prescription is given for obtaining the geometric phase from the motion of a time dependent invariant along a closed curve in a parameter space, which may be finite dimensional even for nonadiabatic cyclic evolutions in an infinite dimensional Hilbert space. Using the natural metric on the projective space, we reformulate Schrödinger's equation for an isolated system. This metric is generalized to the space of all density matrices, and a physical meaning is proposed.
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References and notes
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Based on the talk delivered by the author at the Conference on Fundamental Aspects of Quantum Theory to celebrate 30 years of the Aharonov-Bohm effect, Columbia, South Carolina, December 14–16, 1989, published inQuantum Coherence, J. S. Anandan, ed. (World Scientific, 1990).
Alexander von Humboldt fellow.
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Anandan, J. A geometric approach to quantum mechanics. Found Phys 21, 1265–1284 (1991). https://doi.org/10.1007/BF00732829
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DOI: https://doi.org/10.1007/BF00732829