Sharp correspondence principle and quantum measurements

L. Charles ; L. Polterovich

Algebra i analiz, Tome 29 (2017) no. 1, pp. 237-278.

Résumé

Sharp remainder bounds are proved for the Berezin–Toeplitz quantization and applications to semiclassical quantum measurements are presented.

Détail

Keywords: Berezin–Toeplitz quantization, symplectic manifold, quantum measurement.

Bibliographie

[1] Barron T., Ma X., Marinescu G., Pinsonnault M., “Semi-classical properties of Berezin–Toeplitz operators with $C^k$-symbol”, J. Math. Phys., 55:4 (2014), 042108 | DOI | MR | Zbl

[2] Berezin F., “General concept of quantization”, Comm. Math. Phys., 40 (1975), 153–174 | DOI | MR | Zbl

[3] Berezin F. A., Shubin M. A., Uravnenie Shredingera, Izd-vo MGU, M., 1983 | MR

[4] Berman R., Berndtsson B., Sjöstrand J., “A direct approach to Bergman kernel asymptotics for positive line bundles”, Ark. Mat., 46:2 (2008), 197–217 | DOI | MR | Zbl

[5] Bordemann M., Meinrenken E., Schlichenmaier M., “Toeplitz quantization of Kähler manifolds and $\mathrm{gl}(N)$, $N\to\infty$ limits”, Comm. Math. Phys., 165:2 (1994), 281–296 | DOI | MR | Zbl

[6] Borthwick D., Uribe A., “Almost complex structure and geometric quantization”, Math. Res. Lett., 3:6 (1996), 845–861 | DOI | MR | Zbl

[7] Boutet de Monvel L., Guillemin V., The spectral theory of Toeplitz operators, Ann. of Math. Stud., 99, Princeton Univ. Press, Princeton, NJ, 1981 | MR | Zbl

[8] Boutet de Monvel L., Sjöstrand J., “Sur la singularité des noyaux de Bergman et de Szegő”, Journées: Équations aux Dérivées Partielles de Rennes (1975), Astérisque, 34–35, 1976, 123–164 | MR | Zbl

[9] Busch P., Lahti P., Werner R. F., “Quantum root-mean-square error and measurement uncertainty relations”, Rev. Modern Phys., 86 (2014), 1261–1281 | DOI | MR

[10] Charles L., “Berezin–Toeplitz operators, a semi-classical approach”, Comm. Math. Phys., 239:1–2 (2003), 1–28 | DOI | MR | Zbl

[11] Charles L., “Symbolic calculus for Toeplitz operators with half-form”, J. Symplectic Geom., 4:2 (2006), 171–198 | DOI | MR | Zbl

[12] Charles L., “Semi-classical properties of geometric quantization with metaplectic correction”, Comm. Math. Phys., 270:2 (2007), 445–480 | DOI | MR | Zbl

[13] Charles L., “Quantization of compact symplectic manifold”, J. Geom. Anal., 26:4 (2016), 2664–2710 | DOI | MR | Zbl

[14] Chen Q., Zhou X., “Generalization of interpolation inequalities”, J. Math. Anal. Appl., 226:1 (1998), 130–142 | DOI | MR | Zbl

[15] Chihara H., “Bounded Berezin–Toeplitz operators on the Segal–Bargmann space”, Integral Equations Operator Theory, 63:3 (2009), 321–335 | DOI | MR | Zbl

[16] Coburn L. A., “Deformation estimates for the Berezin–Toeplitz quantization”, Comm. Math. Phys., 149:2 (1992), 415–424 | DOI | MR | Zbl

[17] Ginzburg V. L., Montgomery R., “Geometric quantization and no-go theorems”, Poisson Geometry (Warsaw, 1998), Banach Center Publ., 51, Polish Acad. Sci. Inst. Math., Warsaw, 2000, 69–77 | MR | Zbl

[18] Folland G. B., Harmonic analysis in phase space, Ann. of Math. Stud., 122, Princeton Univ. Press, Princeton, NJ, 1989 | MR | Zbl

[19] Halmos P. R., Sunder V. S., Bounded integral operators on $L^2$ spaces, Ergeb. Math. Grenzgeb. (3), 96, Springer-Verlag, Berlin–New York, 1978 | MR

[20] Guillemin V., “Star products on compact pre-quantizable symplectic manifolds”, Lett. Math. Phys., 35:1 (1995), 85–89 | DOI | MR | Zbl

[21] Kostant B., “Quantization and unitary representations. I. Prequantization”, Lectures in Modern Analysis and Applications, v. III, Lecture Notes in Math., 170, Springer, Berlin, 1970, 87–208 | DOI | MR

[22] Landsman N. P., Mathematical topics between classical and quantum mechanics, Springer Monogr. Math., Springer-Verlag, New York, 1998 | DOI | MR

[23] Lerner N., Metrics on the phase space and non-selfadjoint pseudo-differential operators, Pseudo-Differential Operators. Theory and Applications, 3, Birkhäuser Verlag, Basel, 2010 | MR | Zbl

[24] Le Floch Y., Berezin–Toeplitz operators on compact Kähler manifolds: an introduction, Preprint, , 2016 https://sites.google.com/site/yohannleflochhomepage/english-version/teaching

[25] Le Floch Y., Polterovich L., Shabtai O., In preparation

[26] Ma X., Marinescu G., Holomorphic Morse inequalities and Bergman kernels, Progr. Math., 254, Birkhäuser Verlag, Basel, 2007 | MR | Zbl

[27] Ma X., Marinescu G., “Toeplitz operators on symplectic manifolds”, J. Geom. Anal., 18:2 (2008), 565–611 | DOI | MR | Zbl

[28] Polterovich L., “Symplectic geometry of quantum noise”, Comm. Math. Phys., 327:4 (2014), 481–519 | DOI | MR | Zbl

[29] Polterovich L., Rosen D., Function theory on symplectic manifolds, CRM Monogr. Ser., 34, Amer. Math. Soc., Providence, RI, 2014 | DOI | MR | Zbl

[30] Shiffman B., Zelditch S., “Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds”, J. Reine Angew. Math., 544 (2002), 181–222 | MR | Zbl

[31] Shubin M. A., Psevdodifferentsialnye operatory i spektralnaya teoriya, Nauka, M., 1978 | MR

[32] Sjöstrand J., “Pseudodifferential operators and weighted normed symbol spaces”, Serdica Math. J., 34:1 (2008), 1–38 | MR | Zbl

[33] Souriau J.-M., Structure des systèmes dynamiques, Maîtrises de mathématiques, Dunod, Paris, 1970 | MR | Zbl

[34] Tuynman G. M., “Quantization: towards a comparison between methods”, J. Math. Phys., 28:12 (1987), 2829–2840 | DOI | MR | Zbl

[35] Zworski M., Semiclassical analysis, Grad. Stud. Math., 138, Amer. Math. Soc., Providence, RI, 2012 | DOI | MR | Zbl