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Journal articles on the topic 'Geometric quantization'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

Camosso, Simone. "Prequantization, Geometric Quantization, Corrected Geometric Quantization." Journal of Applied Mathematics and Physics 09, no. 09 (2021): 2290–320. http://dx.doi.org/10.4236/jamp.2021.99146.

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2

Gracia‐Bondía, José M., and Joseph C. Várilly. "From geometric quantization to Moyal quantization." Journal of Mathematical Physics 36, no. 6 (1995): 2691–701. http://dx.doi.org/10.1063/1.531059.

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3

Paradan, Paul-Émile. "Formal geometric quantization." Annales de l’institut Fourier 59, no. 1 (2009): 199–238. http://dx.doi.org/10.5802/aif.2429.

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4

ALI, S. TWAREQUE, and MIROSLAV ENGLIŠ. "QUANTIZATION METHODS: A GUIDE FOR PHYSICISTS AND ANALYSTS." Reviews in Mathematical Physics 17, no. 04 (2005): 391–490. http://dx.doi.org/10.1142/s0129055x05002376.

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This survey is an overview of some of the better known quantization techniques (for systems with finite numbers of degrees-of-freedom) including in particular canonical quantization and the related Dirac scheme, introduced in the early days of quantum mechanics, Segal and Borel quantizations, geometric quantization, various ramifications of deformation quantization, Berezin and Berezin–Toeplitz quantizations, prime quantization and coherent state quantization. We have attempted to give an account sufficiently in depth to convey the general picture, as well as to indicate the mutual relationshi

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5

ABRIKOSOV, A. A., E. GOZZI, and D. MAURO. "TIME AND GEOMETRIC QUANTIZATION." Modern Physics Letters A 18, no. 33n35 (2003): 2347–54. http://dx.doi.org/10.1142/s0217732303012568.

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In this paper we briefly review the functional version of the Koopman-von Neumann operatorial approach to classical mechanics. We then show that its quantization can be achieved by freezing to zero two Grassmannian partners of time. This method of quantization presents many similarities with the one known as Geometric Quantization.

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6

Śniatycki, Jędrzej. "Lectures on Geometric Quantization." Geometry, Integrability and Quantization 17 (2016): 95–129. http://dx.doi.org/10.7546/giq-17-2016-95-129.

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7

Rota, Gian-Carlo. "Geometric quantization in action." Advances in Mathematics 58, no. 3 (1985): 322. http://dx.doi.org/10.1016/0001-8708(85)90128-8.

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8

Mykytyuk, I. V., and A. K. Prykarpats'kyy. "Reduction and geometric quantization." Ukrainian Mathematical Journal 44, no. 9 (1992): 1116–22. http://dx.doi.org/10.1007/bf01058372.

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9

Paradan, Paul-Émile. "Formal geometric quantization II." Pacific Journal of Mathematics 253, no. 1 (2011): 169–211. http://dx.doi.org/10.2140/pjm.2011.253.169.

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10

Bordemann, Martin, Jens Hoppe, Peter Schaller, and Martin Schlichenmaier. "gl(∞) and geometric quantization." Communications in Mathematical Physics 138, no. 2 (1991): 209–44. http://dx.doi.org/10.1007/bf02099490.

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11

Marcolli, Matilde, and Roger Penrose. "Gluing Non-commutative Twistor Spaces." Quarterly Journal of Mathematics 72, no. 1-2 (2021): 417–54. http://dx.doi.org/10.1093/qmath/haab024.

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Abstract We describe a general procedure, based on Gerstenhaber–Schack complexes, for extending to quantized twistor spaces the Donaldson–Friedman gluing of twistor spaces via deformation theory of singular spaces. We consider in particular various possible quantizations of twistor spaces that leave the underlying spacetime manifold classical, including the geometric quantization of twistor spaces originally constructed by the second author, as well as some variants based on non-commutative geometry. We discuss specific aspects of the gluing construction for these different quantization proced

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12

Cushman, Richard, and Jędrzej Śniatycki. "Shifting Operators in Geometric Quantization." Axioms 9, no. 4 (2020): 125. http://dx.doi.org/10.3390/axioms9040125.

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The original Bohr-Sommerfeld theory of quantization did not give operators of transitions between quantum quantum states. This paper derives these operators, using the first principles of geometric quantization.

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13

Nunes, João P. "Degenerating Kähler structures and geometric quantization." Reviews in Mathematical Physics 26, no. 09 (2014): 1430009. http://dx.doi.org/10.1142/s0129055x1430009x.

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We review some recent results on the problem of the choice of polarization in geometric quantization. Specifically, we describe the general philosophy, developed by the author together with his collaborators, of treating real polarizations as limits of degenerating families of holomorphic polarizations. We first review briefly the general framework of geometric quantization, with a particular focus on the problem of the dependence of quantization on the choice of polarization. The problem of quantization in real polarizations is emphasized. We then describe the relation between quantization in

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14

KLAUDER, J. R., and E. ONOFRI. "LANDAU LEVELS AND GEOMETRIC QUANTIZATION." International Journal of Modern Physics A 04, no. 15 (1989): 3939–49. http://dx.doi.org/10.1142/s0217751x89001606.

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The geometrical approach to phase-space quantization introduced by Klauder [KQ] is interpreted in terms of a universal magnetic field acting on a free particle moving in a higher dimensional configuration space; quantization corresponds to freezing the particle to its first Landau level. The Geometric Quantization [GQ] scheme appears as the natural technique to define the interaction with the magnetic field for a particle on a general Riemannian manifold. The freedom of redefining the operators' ordering makes it possible to select that particular definition of the Hamiltonian which is adapted

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15

Bergeron, Hervé, and Jean-Pierre Gazeau. "Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane." Entropy 20, no. 10 (2018): 787. http://dx.doi.org/10.3390/e20100787.

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Any quantization maps linearly function on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with the symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral means that we use all resources of integral calculus, in order to implement the method when we apply it to singular functions, or distributions, for which the integral calculus is an essential ingredient. We first review this quantization scheme

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16

HEDREA, CIPRIAN, ROMEO NEGREA, IOAN ZAHARIE, and MIRCEA PUTA. "ON A PROBLEM OF GEOMETRIC QUANTIZATION." International Journal of Geometric Methods in Modern Physics 08, no. 06 (2011): 1259–68. http://dx.doi.org/10.1142/s0219887811005683.

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The problem of geometric Kostant quantization of this paper is the role played by "½-correction forms" in order to arrive at the result given by the classical Schrödinger quantization, in the one-dimensional harmonic oscillator study.

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17

BOS, ROGIER. "GEOMETRIC QUANTIZATION OF HAMILTONIAN ACTIONS OF LIE ALGEBROIDS AND LIE GROUPOIDS." International Journal of Geometric Methods in Modern Physics 04, no. 03 (2007): 389–436. http://dx.doi.org/10.1142/s0219887807002077.

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We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose, we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line bundle. Next, we discuss a version of Kähler quantization suitable for this setting. We proceed by defining a Marsden–Weinstein quotient for our setting and prove a "quantization commutes with reduction" theorem. We explain how our geometric quantization procedure relates to a p

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18

Qiang, Zhao. "Quantum kinematics and geometric quantization." Journal of Geometry and Physics 21, no. 1 (1996): 34–42. http://dx.doi.org/10.1016/s0393-0440(96)00008-3.

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19

Popov, A. D. "Generalized twistors and geometric quantization." Theoretical and Mathematical Physics 87, no. 1 (1991): 331–44. http://dx.doi.org/10.1007/bf01016571.

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20

Kirwin, William D. "Coherent states in geometric quantization." Journal of Geometry and Physics 57, no. 2 (2007): 531–48. http://dx.doi.org/10.1016/j.geomphys.2006.04.007.

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21

Fradkin, E. S., and V. Ya Linetsky. "BFV approach to geometric quantization." Nuclear Physics B 431, no. 3 (1994): 569–621. http://dx.doi.org/10.1016/0550-3213(94)90216-x.

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22

Mathai, Varghese, and Weiping Zhang. "Geometric quantization for proper actions." Advances in Mathematics 225, no. 3 (2010): 1224–47. http://dx.doi.org/10.1016/j.aim.2010.03.023.

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23

Ali, S. Twareque, Amine M. El Gradechi, and Gérard G. Emch. "Modular algebras in geometric quantization." Journal of Mathematical Physics 35, no. 12 (1994): 6237–43. http://dx.doi.org/10.1063/1.530672.

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24

Chan, Kwokwai, and Yat-Hin Suen. "Geometric quantization via SYZ transforms." Advances in Theoretical and Mathematical Physics 24, no. 1 (2020): 25–66. http://dx.doi.org/10.4310/atmp.2020.v24.n1.a2.

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25

Camosso, Simone. "Quantum Logic and Geometric Quantization." Journal of Quantum Information Science 07, no. 01 (2017): 35–42. http://dx.doi.org/10.4236/jqis.2017.71003.

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26

Ashtekar, Abhay, and Matthew Stillerman. "Geometric quantization and constrained systems." Journal of Mathematical Physics 27, no. 5 (1986): 1319–30. http://dx.doi.org/10.1063/1.527138.

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27

Hirota, Yuji. "Geometric quantization of Dirac manifolds." Journal of Mathematical Physics 57, no. 12 (2016): 123507. http://dx.doi.org/10.1063/1.4972779.

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28

Yue, Yu, and Zhu Zhongyuan. "Geometric Quantization of Anyon Systems." Communications in Theoretical Physics 15, no. 3 (1991): 339–46. http://dx.doi.org/10.1088/0253-6102/15/3/339.

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29

Hitchin, N. J. "Flat connections and geometric quantization." Communications in Mathematical Physics 131, no. 2 (1990): 347–80. http://dx.doi.org/10.1007/bf02161419.

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30

Odzijewicz, Anatol. "Coherent states and geometric quantization." Communications in Mathematical Physics 150, no. 2 (1992): 385–413. http://dx.doi.org/10.1007/bf02096666.

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31

Figueroa-O'Farrill, José M., and Takashi Kimura. "Geometric BRST quantization, I: Prequantization." Communications in Mathematical Physics 136, no. 2 (1991): 209–29. http://dx.doi.org/10.1007/bf02100022.

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32

Darvas, Tamás, Chinh H. Lu, and Yanir A. Rubinstein. "Quantization in Geometric Pluripotential Theory." Communications on Pure and Applied Mathematics 73, no. 5 (2019): 1100–1138. http://dx.doi.org/10.1002/cpa.21857.

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33

Lian, Dingkun, Liangdong Hu, and Quanhui Liu. "Geometric Potential and Dirac Quantization." Annalen der Physik 530, no. 5 (2018): 1700415. http://dx.doi.org/10.1002/andp.201700415.

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34

Miranda, Antonio Díaz. "Wave functions in geometric quantization." International Journal of Theoretical Physics 35, no. 10 (1996): 2139–68. http://dx.doi.org/10.1007/bf02302234.

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35

Karzhemanov, I. V. "Projective-Geometric Aspects of Quantization." Lobachevskii Journal of Mathematics 43, no. 7 (2022): 1651–54. http://dx.doi.org/10.1134/s1995080222100171.

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36

Fulp, Ronald. "BRST Extension of Geometric Quantization." Foundations of Physics 37, no. 1 (2007): 103–24. http://dx.doi.org/10.1007/s10701-006-9090-8.

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37

Andersen, J�rgen Ellegaard. "Deformation Quantization and Geometric Quantization of Abelian Moduli Spaces." Communications in Mathematical Physics 255, no. 3 (2005): 727–45. http://dx.doi.org/10.1007/s00220-004-1244-y.

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38

HAJRA, K., and P. BANDYOPADHYAY. "EQUIVALENCE OF STOCHASTIC, KLAUDER AND GEOMETRIC QUANTIZATION." International Journal of Modern Physics A 07, no. 06 (1992): 1267–85. http://dx.doi.org/10.1142/s0217751x92000545.

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The relativistic generalization of stochastic quantization helps us to introduce a stochastic-phase-space formulation when a relativistic quantum particle appears as a stochastically extended one. The nonrelativistic quantum mechanics is obtained in the sharp point limit. This also helps us to introduce a gauge-theoretical extension of a relativistic quantum particle when for a fermion the group structure of the gauge field is SU(2). The sharp point limit is obtained when we have a minimal contribution of the residual gauge field retained in the limiting procedure. This is shown to be equivale

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39

Robson, M. A. "Geometric quantization on homogeneous spaces and the meaning of “inequivalent” quantizations." Physics Letters B 335, no. 3-4 (1994): 383–87. http://dx.doi.org/10.1016/0370-2693(94)90368-9.

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40

Galasso, Andrea, and Mauro Spera. "Remarks on the geometric quantization of Landau levels." International Journal of Geometric Methods in Modern Physics 13, no. 10 (2016): 1650122. http://dx.doi.org/10.1142/s021988781650122x.

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In this note, we resume the geometric quantization approach to the motion of a charged particle on a plane, subject to a constant magnetic field perpendicular to the latter, by showing directly that it gives rise to a completely integrable system to which we may apply holomorphic geometric quantization. In addition, we present a variant employing a suitable vertical polarization and we also make contact with Bott’s quantization, enforcing the property “quantization commutes with reduction”, which is known to hold under quite general conditions. We also provide an interpretation of translationa

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41

SARDANASHVILY, G. "GEOMETRIC FORMULATION OF NON-AUTONOMOUS MECHANICS." International Journal of Geometric Methods in Modern Physics 10, no. 10 (2013): 1350061. http://dx.doi.org/10.1142/s0219887813500618.

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We address classical and quantum mechanics in a general setting of arbitrary time-dependent transformations. Classical non-relativistic mechanics is formulated as a particular field theory on smooth fiber bundles over a time axis ℝ. Connections on these bundles describe reference frames. Quantum non-autonomous mechanics is phrased in geometric terms of Banach and Hilbert bundles and connections on these bundles. A quantization scheme speaking this language is geometric quantization.

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42

Hurtado, P., A. Leones, and J. B. Moreno. "Strict Deformation Quantization via Geometric Quantization in the Bieliavsky Plane." Abstract and Applied Analysis 2020 (August 25, 2020): 1–7. http://dx.doi.org/10.1155/2020/6794709.

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Using standard techniques from geometric quantization, we rederive the integral product of functions on ℝ2 (non-Euclidian) which was introduced by Pierre Bieliavsky as a contribution to the area of strict quantization. More specifically, by pairing the nontransverse real polarization on the pair groupoid ℝ2×ℝ¯2, we obtain the well-defined integral transform. Together with a convolution of functions, which is a natural deformation of the usual convolution of functions on the pair groupoid, this readily defines the Bieliavsky product on a subset of L2ℝ2.

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43

Yu, Yue, and Han-Ying Guo. "On the geometric quantization and BRST quantization for bosonic strings." Physics Letters B 216, no. 1-2 (1989): 68–74. http://dx.doi.org/10.1016/0370-2693(89)91370-1.

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44

Rawnsley, J., M. Cahen, and S. Gutt. "Quantization of Kähler manifolds I: geometric interpretation of Berezin's quantization." Journal of Geometry and Physics 7, no. 1 (1990): 45–62. http://dx.doi.org/10.1016/0393-0440(90)90019-y.

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45

Dey, Rukmini. "Geometric quantization of the Hitchin system." International Journal of Geometric Methods in Modern Physics 14, no. 04 (2017): 1750064. http://dx.doi.org/10.1142/s0219887817500645.

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This paper is about geometric quantization of the Hitchin system. We quantize a Kahler form on the Hitchin moduli space (which is half the first Kahler form defined by Hitchin) by considering the Quillen bundle as the prequantum line bundle and modifying the Quillen metric using the Higgs field so that the curvature is proportional to the Kahler form. We show that this Kahler form is integral and the Quillen bundle descends as a prequantum line bundle on the moduli space. It is holomorphic and hence one can take holomorphic square integrable sections as the Hilbert space of quantization of the

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46

Braverman, Maxim, Yiannis Loizides, and Yanli Song. "Geometric quantization of $b$-symplectic manifolds." Journal of Symplectic Geometry 19, no. 1 (2021): 1–36. http://dx.doi.org/10.4310/jsg.2021.v19.n1.a1.

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47

Wang, Jian, and Yong Wang. "GEOMETRIC QUANTIZATION OF ODD DIMENSIONAL SPINcMANIFOLDS." Bulletin of the Korean Mathematical Society 49, no. 2 (2012): 223–34. http://dx.doi.org/10.4134/bkms.2012.49.2.223.

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48

García-Compeán, H., J. F. Plebanski, M. Przanowski, and F. J. Turrubiates. "Deformation quantization of geometric quantum mechanics." Journal of Physics A: Mathematical and General 35, no. 19 (2002): 4301–19. http://dx.doi.org/10.1088/0305-4470/35/19/311.

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49

Fischer, T. R. "Geometric source coding and vector quantization." IEEE Transactions on Information Theory 35, no. 1 (1989): 137–45. http://dx.doi.org/10.1109/18.42184.

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50

Brylinski, R., and B. Kostant. "Minimal representations, geometric quantization, and unitarity." Proceedings of the National Academy of Sciences 91, no. 13 (1994): 6026–29. http://dx.doi.org/10.1073/pnas.91.13.6026.

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