Download Minimal separable quantizations of Stäckel systems

Minimal separable quantizations of Stäckel
systems
Krzysztof Marciniak
Department of Science and Technology, Linköping University, Sweden
In this talk I will addresses the issue of separable and integrable quantizations
of commuting sets of quadratic in momenta Hamiltonian of the form
H(x; p) =
1 ij
A (x)pi pj + V (x)
2
b =
The Hamilton operator (quantum Hamiltonian) H
2
ing on the Hilbert space L
1=2
1=2
Q; jdet gj
(1)
h2
ij
2 ri A rj
+ V (x) act-
dx of square integrable (in the meas-
ure ! g = jdet gj dx) complex functions on Q is called a minimal quantization
of the Hamiltonian (1) in the metric g (that also de…nes the operators ri of the
asociated Levi-Civita connection).
In the standard approach to the quantization of (1) one assumes that g =
A 1 (as it has been done in the classical works [1] and [2] devoted to the problem
of separability of classical Hamilton-Jacobi equation associated with (1)) This
assumption it leads to severe limitations on the process of quantization of (1).
I this talk I …rst explain the notion of minimal quantization and its relation
to the more general quantization theory developed recently in [3, 4, 5]. Then
I demonstrate that many Hamiltonian systems of the form (1) - that can not
be separably quantized in the classical approach of Robertson and Eisenhardt
- can be separably quantized if we extend the class of admissible quantizations
through a suitable choice of Riemann space adapted to the Poisson geometry
of the system I also explain the origin of so called quantum correction terms,
observed - but not explained - in [6] and [7]
This is a joint work with prof. Maciej Blaszak and dr Ziemowit Domanski,
Faculty of Physics, Poznan University, Poland. The results presented in this
talk can be to some extent found in [8].
References
[1] Robertson, H. P. Bemerkung über separierbare Systeme in der Wellenmechanik. Math. Ann. 98 (1928), no. 1, 749–752.
[2] Eisenhart, L. P. Separable systems of Stäckel. Ann. of Math. 35 (1934), no.
2, 284–305.
[3] B÷
aszak, M.; Domański, Z. Phase space quantum mechanics. Ann. Phys. 327
(2012), no. 2, 167–211.
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[4] B÷
aszak, M.; Domański, Z. Canonical quantization of classical mechanics in
curvilinear coordinates. Invariant quantization procedure. Ann. Phys. 339
(2013), 89–108.
[5] B÷
aszak, M.; Domański, Z. Natural star-products on symplectic manifolds
and related quantum mechanical operators. Ann. Phys. 344 (2014), 29–42.
[6] Hietarinta, J. Classical versus quantum integrability. J. Math. Phys. 25
(1984), no. 6, 1833–1840.
[7] Hietarinta, J.; Grammaticos, B. On the h2 correction terms in quantum
integrability. J. Phys. A 22 (1989), no. 9, 1315–1322.
[8] M. Blaszak, K. Marciniak, Z. Domanski. Quantizations preserving separability of Stäckel systems. arXiv:1501.00576
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