Abstract
A description of differential geometrical structure of the quantum pure-states space, the projective Hilbert space, is provided. It is shown that this space is endowed with a natural symplectic structure which allows us to reformulate quantum mechanics in classical mechanical language as a classical field theory.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Another, more intuitive and more detailed approach to the structure of quantum state space can be found in [16]. For geometry and dynamics (also nonlinear) of general—not only pure—states see also [37, Sect. 2.1].
- 2.
For an alternative proof valid also for unitary orbits of density matrices see [37, Theorem 2.1.19].
- 3.
A brief review of the theory of unbounded operators is present in [37, C], or in [37, Textbook] in detail.
- 4.
A certain, more detailed, account of the geometry and interpretation questions of the set of density matrices is given in [37, 2.1-e].
- 5.
A core \(D\subset {\mathcal H}\) of a closable operator C is such a subset \(D\subset D(C)\subset {\mathcal H}\), that the closure of the restriction \(\overline{C\upharpoonright D}=\overline{C}\), cf. also [37, C1].
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Bóna, P. (2020). Geometry of the State Space of Quantum Mechanics. In: Classical Systems in Quantum Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-030-45070-0_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-45070-0_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-45069-4
Online ISBN: 978-3-030-45070-0
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)