TY - GEN
T1 - From Quantum Hydrodynamics to Koopman Wavefunctions I
AU - Gay-Balmaz, François
AU - Tronci, Cesare
N1 - Publisher Copyright:
© 2021, Springer Nature Switzerland AG.
PY - 2021/1/1
Y1 - 2021/1/1
N2 - Based on Koopman’s theory of classical wavefunctions in phase space, we present the Koopman-van Hove (KvH) formulation of classical mechanics as well as some of its properties. In particular, we show how the associated classical Liouville density arises as a momentum map associated to the unitary action of strict contact transformations on classical wavefunctions. Upon applying the Madelung transform from quantum hydrodynamics in the new context, we show how the Koopman wavefunction picture is insufficient to reproduce arbitrary classical distributions. However, this problem is entirely overcome by resorting to von Neumann operators. Indeed, we show that the latter also allow for singular δ -like profiles of the Liouville density, thereby reproducing point particles in phase space.
AB - Based on Koopman’s theory of classical wavefunctions in phase space, we present the Koopman-van Hove (KvH) formulation of classical mechanics as well as some of its properties. In particular, we show how the associated classical Liouville density arises as a momentum map associated to the unitary action of strict contact transformations on classical wavefunctions. Upon applying the Madelung transform from quantum hydrodynamics in the new context, we show how the Koopman wavefunction picture is insufficient to reproduce arbitrary classical distributions. However, this problem is entirely overcome by resorting to von Neumann operators. Indeed, we show that the latter also allow for singular δ -like profiles of the Liouville density, thereby reproducing point particles in phase space.
KW - Hamiltonian dynamics
KW - Koopman wavefunctions
KW - Momentum map
KW - Prequantization
U2 - 10.1007/978-3-030-80209-7_34
DO - 10.1007/978-3-030-80209-7_34
M3 - Conference contribution
AN - SCOPUS:85112524703
SN - 9783030802080
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 302
EP - 310
BT - Geometric Science of Information - 5th International Conference, GSI 2021, Proceedings
A2 - Nielsen, Frank
A2 - Barbaresco, Frédéric
PB - Springer Science and Business Media Deutschland GmbH
T2 - 5th International Conference on Geometric Science of Information, GSI 2021
Y2 - 21 July 2021 through 23 July 2021
ER -