The uncertainty principle and the energy identity for holomorphic maps in geometric quantum mechanics

TL;DR

This paper links the uncertainty principle in quantum mechanics to the energy identity of holomorphic maps, revealing geometric and topological insights into quantum state spaces using symplectic topology and harmonic map theory.

Contribution

It extends geometric quantum mechanics by connecting the Robertson-Schrödinger uncertainty relation to the energy identity of J-holomorphic curves, introducing a geometric framework for quantum uncertainties.

Findings

Uncertainty relation expressed as an energy equality for holomorphic maps.

Saturation occurs when the map is conformal and off-diagonal covariances vanish.

Holomorphic maps from Riemann surfaces are harmonic and their images are minimal surfaces.

Abstract

The theory of geometric quantum mechanics describes a quantum system as a Hamiltonian dynamical system with a complex projective Hilbert space as its phase space, thus equipped with a Riemannian metric in addition to a symplectic structure. This paper extends the geometric quantum theory to include aspects of the symplectic topology of the state space by identifying the Robertson-Schr\"{o}dinger uncertainty relation for pure quantum states as the differential version of the energy identity in the theory of $J$-holomorphic curves. We consider a family of maps from a Riemann surface into a finite-dimensional quantum state space by using the vector fields generated by two quantum observables,and show that the Fubini-Study metric tensor pulls back by such a map to the covariance tensor for the two observables. By calculating the map energy density in the pull-back metric, the uncertainty relation is represented as an equality that compares the map energy differential to the sum of the pull-back of the symplectic form and a positive definite term that vanishes when the map is holomorphic. Saturation of the Robertson-Schr\"{o}dinger inequality occurs when the map is conformal and the off-diagonal covariance terms vanish. For compact Riemann surfaces where such a map can be globally defined, if the map is holomorphic, it is harmonic and its image is a minimal surface. In this case, the uncertainty product integral is a topological invariant that depends only on the homology class of the curve modulo its boundary.