Mikhail Leonidovich,

I wonder if you would be interested in

http://www.God-does-not-play-dice.net/Einstein.html#addendum

Regards,

Dimi Chakalov
--
http://www.God-does-not-play-dice.net
http://www.God-does-not-play-dice.net/download.html
 
 

Note: It is very unlikely that Mikhail Gromov (currently at Institut des Hautes Études Scientifiques; web site here) will be interested in my efforts. The textbook lore says that "nothing moves in GR"; more from Bob Geroch here. We have the Cauchy problem (more from Bob Geroch here) and the initial value problem in GR, which can be traced back to the famous Hole Argument, and yet people stick to textbooks, and that's it.

But since Misha Gromov is one of the leading experts in symplectic geometry, perhaps he can elucidate a peculiar fact from the geometric formulation of QM [Ref. 1]: the correct space of quantum states, the projective Hilbert space  P , is a symplectic manifold. If so, it should be orientable, correct?

Now, where do we find this extra degree of freedom in the standard QM? The way I see it, the only chance to unravel this hidden "orientability" of Hilbert space -- recall that we don't have Hilbert space-time -- is in the Berry phase [Ref. 2]. It is a genuine geometrical phase, not a dynamical phase derived from Schrödinger's equation. However, we kill both the Berry phase and the dynamical phase from Schrödinger's equation with the so-called collapse. Hence my question:

What if the projective Hilbert space  P  is "orientable" in the global mode of spacetime only? Just a wild guess; I'm not at all good in math, which is why I decided to contact Misha Gromov.

Regrettably, I cannot post here his reply (if I ever get one), but if Misha Gromov proves that I've said something really stupid, I will immediately correct my conjecture.
 

D. Chakalov
March 29, 2005
 

References

[Ref. 1] Abhay Ashtekar, Troy A. Schilling, Geometrical Formulation of Quantum Mechanics, Mon, 23 Jun 1997 10:55:02 -0500,
http://xxx.lanl.gov/abs/gr-qc/9706069

"The geometric formulation shows that the linear structure which is at the forefront in text-book treatments of quantum mechanics is, primarily, only a technical convenience and the essential ingredients -- the manifold of states, the symplectic structure and the Riemannian metric -- do not share this linearity.
...

"Also, even in the finite-dimensional case, we do not know if there exist any Kahler manifolds other than projective Hilbert spaces for which a satisfactory measurement theory can be developed. Even isolated examples of such manifolds would be very illuminating.
...

"(D)eeper reflection shows that quantum mechanics is in fact not as linear as it is advertised to be. For, the space of physical states is not the Hilbert space H but the space of rays in it, i.e., the projective Hilbert space P. And P is a genuine, non-linear manifold. Furthermore, it turns out that the Hermitian inner-product of H naturally endows P with the structure of a Kahler manifold. Thus, in particular, like the classical state space Gamma, the correct space of quantum states, P, is a symplectic manifold!"
 

[Ref. 2] Jeeva Anandan, Joy Christian, Kazimir Wanelik, Resource Letter GPP-1: Geometric Phases in Physics, Mon, 24 Mar 1997 20:12:32 +0000,
http://xxx.lanl.gov/abs/quant-ph/9702011