nLab geometrical formulation of quantum mechanics

Idea

What is called the “geometrization of quantum mechanics” by Kibble (1979) who introduced the idea, or “geometrical formulation of quantum mechanics” [Ashtekar & Schilling (1999)] or just “geometric quantum mechanics” [Brody & Hughston (2001)], but which ought to be called the “symplectic formulation”, is the observation that, in standard quantum mechanics:

1. the complex projective space $P \mathscr{H}$ of the Hilbert space of pure quantum states $\mathscr{H}$ is canonically a Kähler manifold (via the Fubini-Study metric) and so in particular a symplectic manifold,

2. the curves in $P \mathscr{H}$ which correspond to solutions of the Schrödinger equation are Hamiltonian flows with respect to this symplectic structure.

This is fairly immediate to see from mathematical inspection, but the perspective is somewhat surprising from the point of view of standard accounts of classical/quantum physics, which tend to frame symplectic geometry and its Hamiltonian flows as the hallmark of classical phase spaces and classical physics, and to highlight that quantization is a deformation of this symplectic structure (whence: deformation quantization).

Now, the standard perspective is certainly not wrong, but various authors have inevitably suggested that the “geometrical formulation” (more accurate would be: “symplectic formulation”!) of quantum mechanics may point to some deeper truth, and if only to show some kind of conceptual unity where one is used to amplifying the dichotomy.

References

The observation is due to:

• Tom W. B. Kibble, Geometrization of quantum mechanics, Communications in Mathematical Physics 65 (1979) 189–201 [doi:10.1007/BF01225149]

The idea was picked up in:

• Lane P. Hughston, Geometry of stochastic state vector reduction, Proceedings of the Royal Society A 452 1947 (1996) [doi:10.1098/rspa.1996.0048, jstor:52944]

  (with speculations on the implication for quantum state collapse)

• Abhay Ashtekar, Troy A. Schilling, Geometrical Formulation of Quantum Mechanics, in: On Einstein’s Path Springer (1999) [arXiv:gr-qc/9706069, doi:10.1007/978-1-4612-1422-9_3]

Further discussion in:

• Dorje C. Brody, Lane P. Hughston, Geometric Quantum Mechanics, J. Geom. Phys. 38 (2001) 19-53 [arXiv:quant-ph/9906086, doi:10.1016/S0393-0440(00)00052-8]

• A Benvegnù, N Sansonetto, M Spera, Remarks on geometric quantum mechanics, Journal of Geometry and Physics 51 2 (2004) 229-243 [doi:10.1016/j.geomphys.2003.10.008]

• J. F. Cariñena, J. Clemente-Gallardo & G. Marmo, Geometrization of quantum mechanics, Theoretical and Mathematical Physics 152 (2007) 894–903 [doi:10.1007/s11232-007-0075-3]

• Secret Blogging Seminar, Quantum mechanics and geometry (2009)

• Hoshang Heydari, Geometric formulation of quantum mechanics [arXiv:1503.00238]