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Transcript
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. 1 [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 2
