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A remark on natural quantum Lagrangians and natural generalized Schrödinger operators in Galilei quantum mechanics. (English) Zbl 0983.81027
Slovák, Jan (ed.) et al., The proceedings of the 20th winter school “Geometry and physics”, Srní, Czech Republic, January 15-22, 2000. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 66, 117-128 (2001).
Cariñena: The classification of first and second order natural quantum Lagrangians appearing in the covariant quantum mechanics for curved Galilei spacetime is given by usig methods of natural operators [see I. Kolár, P. W. Michor and J. Slovák, Natural operations in differential geometry, Springer, Berlin, 1993; Zbl 0782.53013)]. It is proved that all natural generalized Schrödinger operators are linear and can be induced from natural quantum Lagrangians.
Summary: The natural quantum Lagrangians which appear in Galilei general relativistic quantum mechanics are classified by using methods of natural operators. It is proved that all first order natural quantum Lagrangians are linear combinations (with real coefficients) of the canonical quantum Lagrangian and the product of the scalar curvature of the spacetime vertical connection and the Hermitian product. The classification of all natural generalized Schrödinger operators is given and it is proved that all natural generalized Schrödinger operators can be induced from natural quantum Lagrangians.
For the entire collection see [Zbl 0961.00020].
MSC:
81Q70 Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory
53D50 Geometric quantization
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53C80 Applications of global differential geometry to the sciences
58A20 Jets in global analysis
81S10 Geometry and quantization, symplectic methods
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
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