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Journal articles on the topic 'Quantization'

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Author: Grafiati

Published: 4 June 2021

Last updated: 7 July 2024

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1

Borisov, Leonid A., and Yuriy N. Orlov. "On the generalization of moyal equation for an arbitrary linear quantization." Infinite Dimensional Analysis, Quantum Probability and Related Topics 24, no. 01 (March 2021): 2150003. http://dx.doi.org/10.1142/s021902572150003x.

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For an arbitrary linear combination of quantizations, the kernel of the inverse operator is constructed. An equation for the evolution of the Wigner function for an arbitrary linear quantization is derived and it is shown that only for Weyl quantization this equation does not contain a source of quasi-probability. Stationary solutions for the Wigner function of a harmonic oscillator are constructed, depending on the characteristic function of the quantization rule. In the general case of Hermitian linear quantization these solutions are real but not positive. We found the representation of Weyl quantization in the form of the limit of a sequence of linear Hermitian quantizations, such that for each element of this sequence the stationary solution of the Moyal equation is positive.

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2

Klauder, John R. "A New Proposal to Create a Valid Quantization of Einstein’s Gravity." Entropy 24, no. 10 (September 27, 2022): 1374. http://dx.doi.org/10.3390/e24101374.

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Canonical quantization has created many valid quantizations that require infinite-line coordinate variables. However, the half-harmonic oscillator, which is limited to the positive coordinate half, cannot receive a valid canonical quantization because of the reduced coordinate space. Instead, affine quantization, which is a new quantization procedure, has been deliberately designed to handle the quantization of problems with reduced coordinate spaces. Following examples of what affine quantization is, and what it can offer, a remarkably straightforward quantization of Einstein’s gravity is attained, in which a proper treatment of the positive definite metric field of gravity has been secured.

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3

ALI, S. TWAREQUE, and MIROSLAV ENGLIŠ. "QUANTIZATION METHODS: A GUIDE FOR PHYSICISTS AND ANALYSTS." Reviews in Mathematical Physics 17, no. 04 (May 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 relationships between various methods, their relative successes and shortcomings, mentioning also open problems in the area. Finally, even for approaches for which lack of space or expertise prevented us from treating them to the extent they would deserve, we have tried to provide ample references to the existing literature on the subject. In all cases, we have made an effort to keep the discussion accessible both to physicists and to mathematicians, including non-specialists in the field.

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4

Pavšič, Matej. "A novel view on successive quantizations, leading to increasingly more “miraculous” states." Modern Physics Letters A 34, no. 23 (July 30, 2019): 1950186. http://dx.doi.org/10.1142/s0217732319501861.

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A series of successive quantizations is considered, starting with the quantization of a non-relativistic or relativistic point particle: (1) quantization of a particle’s position, (2) quantization of wave function, (3) quantization of wave functional. The latter step implies that the wave packet profiles forming the states of quantum field theory are themselves quantized, which gives new physical states that are configurations of configurations. In the procedure of quantization, instead of the Schrödinger first-order equation in time derivative for complex wave function (or functional), the equivalent second-order equation for its real part was used. In such a way, at each level of quantization, the equation a quantum state satisfies is just like that of a harmonic oscillator, and wave function(al) is composed in terms of the pair of its canonically conjugated variables.

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5

Trusov, Anton, Elena Limonova, Dmitry Nikolaev, and Vladimir V. Arlazarov. "4.6-Bit Quantization for Fast and Accurate Neural Network Inference on CPUs." Mathematics 12, no. 5 (February 23, 2024): 651. http://dx.doi.org/10.3390/math12050651.

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Quantization is a widespread method for reducing the inference time of neural networks on mobile Central Processing Units (CPUs). Eight-bit quantized networks demonstrate similarly high quality as full precision models and perfectly fit the hardware architecture with one-byte coefficients and thirty-two-bit dot product accumulators. Lower precision quantizations usually suffer from noticeable quality loss and require specific computational algorithms to outperform eight-bit quantization. In this paper, we propose a novel 4.6-bit quantization scheme that allows for more efficient use of CPU resources. This scheme has more quantization bins than four-bit quantization and is more accurate while preserving the computational efficiency of the later (it runs only 4% slower). Our multiplication uses a combination of 16- and 32-bit accumulators and avoids multiplication depth limitation, which the previous 4-bit multiplication algorithm had. The experiments with different convolutional neural networks on CIFAR-10 and ImageNet datasets show that 4.6-bit quantized networks are 1.5–1.6 times faster than eight-bit networks on the ARMv8 CPU. Regarding the quality, the results of the 4.6-bit quantized network are close to the mean of four-bit and eight-bit networks of the same architecture. Therefore, 4.6-bit quantization may serve as an intermediate solution between fast and inaccurate low-bit network quantizations and accurate but relatively slow eight-bit ones.

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6

Klauder, John R. "A Valid Quantization of a Half-Harmonic Oscillator Field Theory." Axioms 11, no. 8 (July 24, 2022): 360. http://dx.doi.org/10.3390/axioms11080360.

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The usual full- and half-harmonic oscillators are turned into field theories, and that behavior is examined using canonical and affine quantization. The result leads to a valid affine quantization of the half harmonic oscillator field theory, which points toward further valid quantizations of more realistic field theory models.

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7

BATES, L., R. CUSHMAN, M. HAMILTON, and J. ŚNIATYCKI. "QUANTIZATION OF SINGULAR REDUCTION." Reviews in Mathematical Physics 21, no. 03 (April 2009): 315–71. http://dx.doi.org/10.1142/s0129055x09003633.

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This paper creates a theory of quantization of singularly reduced systems. We compare our results with those obtained by quantizing algebraically reduced systems. In the case of a Kähler polarization, we show that quantization of a singularly reduced system commutes with reduction, thus generalizing results of Sternberg and Guillemin. We illustrate our theory by treating an example of Arms, Gotay and Jennings where algebraic and singular reduction at the zero level of the momentum mapping differ. In spite of this, their quantizations agree.

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8

AURICH, R., and J. BOLTE. "QUANTIZATION RULES FOR STRONGLY CHAOTIC SYSTEMS." Modern Physics Letters B 06, no. 27 (November 20, 1992): 1691–719. http://dx.doi.org/10.1142/s0217984992001393.

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We discuss the quantization of strongly chaotic systems and apply several quantization rules to a model system given by the unconstrained motion of a particle on a compact surface of constant negative Gaussian curvature. We study the periodic-orbit theory for distinct symmetry classes corresponding to a parity operation which is always present when such a surface has genus two. Recently, several quantization rules based on periodic orbit theory have been introduced. We compare quantizations using the dynamical zeta function Z(s) with the quantization condition [Formula: see text] where a periodic-orbit expression for the spectral staircase N(E) is used. A general discussion of the efficiency of periodic-orbit quantization then allows us to compare the different methods. The system dependence of the efficiency, which is determined by the topological entropy τ and the mean level density [Formula: see text], is emphasized.

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9

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 (October 13, 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 before revisiting the cases where symmetry covariance is described by the Weyl-Heisenberg group and the affine group respectively, and we emphasize the fundamental role played by Fourier transform in both cases. As an original outcome of our generalisations of the Wigner-Weyl transform, we show that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations.

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10

Losev, Ivan. "Isomorphisms of quantizations via quantization of resolutions." Advances in Mathematics 231, no. 3-4 (October 2012): 1216–70. http://dx.doi.org/10.1016/j.aim.2012.06.017.

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11

Marcolli, Matilde, and Roger Penrose. "Gluing Non-commutative Twistor Spaces." Quarterly Journal of Mathematics 72, no. 1-2 (April 26, 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 procedures.

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12

Klauder, J. R. "Quantization Without Quantization." Annals of Physics 237, no. 1 (January 1995): 147–60. http://dx.doi.org/10.1006/aphy.1995.1007.

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13

Klauder, John R. "A Valid Quantization of the Particle in a Box Field Theory, and Well Beyond." Axioms 11, no. 10 (October 19, 2022): 567. http://dx.doi.org/10.3390/axioms11100567.

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The usual particle in a box is turned into a field theory, and its behavior is examined using canonical and affine quantizations. The result leads to a valid affine quantization of the particle in a box field theory, which points toward further valid quantizations of more realistic field theory models.

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14

Sher, Artem, Anton Trusov, Elena Limonova, Dmitry Nikolaev, and Vladimir V. Arlazarov. "Neuron-by-Neuron Quantization for Efficient Low-Bit QNN Training." Mathematics 11, no. 9 (April 29, 2023): 2112. http://dx.doi.org/10.3390/math11092112.

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Quantized neural networks (QNNs) are widely used to achieve computationally efficient solutions to recognition problems. Overall, eight-bit QNNs have almost the same accuracy as full-precision networks, but working several times faster. However, the networks with lower quantization levels demonstrate inferior accuracy in comparison to their classical analogs. To solve this issue, a number of quantization-aware training (QAT) approaches were proposed. In this paper, we study QAT approaches for two- to eight-bit linear quantization schemes and propose a new combined QAT approach: neuron-by-neuron quantization with straight-through estimator (STE) gradient forwarding. It is suitable for quantizations with two- to eight-bit widths and eliminates significant accuracy drops during training, which results in better accuracy of the final QNN. We experimentally evaluate our approach on CIFAR-10 and ImageNet classification and show that it is comparable to other approaches for four to eight bits and outperforms some of them for two to three bits while being easier to implement. For example, the proposed approach to three-bit quantization of the CIFAR-10 dataset results in 73.2% accuracy, while baseline direct and layer-by-layer result in 71.4% and 67.2% accuracy, respectively. The results for two-bit quantization for ResNet18 on the ImageNet dataset are 63.69% for our approach and 61.55% for the direct baseline.

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15

Calaque, Damien, Giovanni Felder, Andrea Ferrario, and Carlo A. Rossi. "Bimodules and branes in deformation quantization." Compositio Mathematica 147, no. 1 (August 11, 2010): 105–60. http://dx.doi.org/10.1112/s0010437x10004847.

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AbstractWe prove a version of Kontsevich’s formality theorem for two subspaces (branes) of a vector space X. The result implies, in particular, that the Kontsevich deformation quantizations of S(X*) and ∧(X) associated with a quadratic Poisson structure are Koszul dual. This answers an open question in Shoikhet’s recent paper on Koszul duality in deformation quantization.

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16

HORIE, KENICHI. "INEQUIVALENT QUANTIZATIONS OF GAUGE THEORIES." International Journal of Modern Physics A 14, no. 13 (May 20, 1999): 2023–36. http://dx.doi.org/10.1142/s0217751x99001020.

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It is known that the quantization of a system defined on a topologically nontrivial configuration space is ambiguous in that many inequivalent quantum systems are possible. This is the case for multiply connected spaces as well as for coset spaces. Recently, a new framework for these inequivalent quantizations approach has been proposed by McMullan and Tsutsui, which is based on a generalized Dirac approach. We employ this framework to the quantization of the Yang–Mills theory in the simplest fashion. The resulting inequivalent quantum systems are labelled by quantized nondynamical topological charges.

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17

DOEBNER, H. D., P. ŠŤOVÍČEK, and J. TOLAR. "QUANTIZATION OF KINEMATICS ON CONFIGURATION MANIFOLDS." Reviews in Mathematical Physics 13, no. 07 (July 2001): 799–845. http://dx.doi.org/10.1142/s0129055x0100079x.

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This review paper is devoted to topological global aspects of quantal description. The treatment concentrates on quantizations of kinematical observables — generalized positions and momenta. A broad class of quantum kinematics is rigorously constructed for systems, the configuration space of which is either a homogeneous space of a Lie group or a connected smooth finite-dimensional manifold without boundary. The class also includes systems in an external gauge field for an Abelian or a compact gauge group. Conditions for equivalence and irreducibility of generalized quantum kinematics are investigated with the aim of classification of possible quantizations. Complete classification theorems are given in two special cases. It is attempted to motivate the global approach based on a generalization of imprimitivity systems called quantum Borel kinematics. These are classified by means of global invariants — quantum numbers of topological origin. Selected examples are presented which demonstrate the richness of applications of Borel quantization. The review aims to provide an introductory survey of the subject and to be sufficiently selfcontained as well, so that it can serve as a standard reference concerning Borel quantization for systems admitting localization on differentiable manifolds.

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18

HAJRA, K. "EQUIVALENCE OF STOCHASTIC AND HYDRODYNAMICAL QUANTIZATION." International Journal of Modern Physics A 04, no. 13 (August 10, 1989): 3163–78. http://dx.doi.org/10.1142/s0217751x8900128x.

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On the basis of earlier work on relativistic generalization of Nelson’s stochastic quantization procedure introducing an anisotropy in the internal space it is shown here that in the non-relativistic limit the equivalence of stochastic and hydrodynamical quantizations formulated respectively by Nelson and Takabayashi can be achieved. Some difficulties regarding interpretation in both the formalisms may possibly be removed from the geometry of internal space-time.

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19

IKEMORI, HITOSHI, SHINSAKU KITAKADO, HAJIME NAKATANI, HIDEHARU OTSU, and TOSHIRO SATO. "INEQUIVALENT QUANTIZATION IN THE SKYRME MODEL." International Journal of Modern Physics A 14, no. 19 (July 30, 1999): 2977–91. http://dx.doi.org/10.1142/s0217751x99001445.

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Quantum mechanics on manifolds is not unique, and in general an infinite number of inequivalent quantizations can be considered. These are specified by the induced spin and the induced gauge structures on the manifold. The configuration space of the collective mode in the Skyrme model can be identified with S3 and thus the quantization is not unique. This leads to the different predictions for the physical observables.

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20

BASHKIROV, D. "BV QUANTIZATION OF COVARIANT (POLYSYMPLECTIC) HAMILTONIAN FIELD THEORY." International Journal of Geometric Methods in Modern Physics 01, no. 03 (June 2004): 233–52. http://dx.doi.org/10.1142/s0219887804000149.

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Covariant (polysymplectic) Hamiltonian field theory is the Hamiltonian counterpart of classical Lagrangian field theory. They are quasi-equivalent in the case of almost-regular Lagrangians. This work addresses Batalin–Vilkoviski (BV) quantization of polysymplectic Hamiltonian field theory. We compare BV quantizations of associated Lagrangian and polysymplectic Hamiltonian field systems in the case of almost-regular quadratic Lagrangians.

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21

Bieliavsky, Pierre, Victor Gayral, Sergey Neshveyev, and Lars Tuset. "On deformations of C∗-algebras by actions of Kählerian Lie groups." International Journal of Mathematics 27, no. 03 (March 2016): 1650023. http://dx.doi.org/10.1142/s0129167x16500233.

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We show that two approaches to equivariant strict deformation quantization of C[Formula: see text]-algebras by actions of negatively curved Kählerian Lie groups, one based on oscillatory integrals and the other on quantizations maps defined by dual 2-cocycles, are equivalent.

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22

Marc Kesseböhmer and Sanguo Zhu. "Quantization Dimension via Quantization Numbers." Real Analysis Exchange 29, no. 2 (2004): 857. http://dx.doi.org/10.14321/realanalexch.29.2.0857.

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23

Hulle, Marc M. Van, and Dominique Martinez. "On an Unsupervised Learning Rule for Scalar Quantization following the Maximum Entropy Principle." Neural Computation 5, no. 6 (November 1993): 939–53. http://dx.doi.org/10.1162/neco.1993.5.6.939.

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A novel unsupervised learning rule, called Boundary Adaptation Rule (BAR), is introduced for scalar quantization. It is shown that the rule maximizes information-theoretic entropy and thus yields equiprobable quantizations of univariate probability density functions. It is shown by simulations that BAR outperforms other unsupervised competitive learning rules in generating equiprobable quantizations. It is also shown that our rule can do better or worse than the Lloyd I algorithm in minimizing average mean square error, depending on the input distribution. Finally, an application to adaptive non-uniform analog to digital (A/D) conversion is considered.

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24

Gray, R. M., and D. L. Neuhoff. "Quantization." IEEE Transactions on Information Theory 44, no. 6 (1998): 2325–83. http://dx.doi.org/10.1109/18.720541.

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25

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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26

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

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27

LANDSMAN, N. P. "QUANTIZATION AND SUPERSELECTION SECTORS I: TRANSFORMATION GROUP C*-ALGEBRAS." Reviews in Mathematical Physics 02, no. 01 (January 1990): 45–72. http://dx.doi.org/10.1142/s0129055x9000003x.

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Quantization is defined as the act of assigning an appropriate C*-algebra [Formula: see text] to a given configuration space Q, along with a prescription mapping self-adjoint elements of [Formula: see text] into physically interpretable observables. This procedure is adopted to solve the problem of quantizing a particle moving on a homogeneous locally compact configuration space Q=G/H. Here [Formula: see text] is chosen to be the transformation group C*-algebra corresponding to the canonical action of G on Q. The structure of these algebras and their representations are examined in some detail. Inequivalent quantizations are identified with inequivalent irreducible representations of the C*-algebra corresponding to the system, hence with its superselection sectors. Introducing the concept of a pre-Hamiltonian, we construct a large class of G-invariant time-evolutions on these algebras, and find the Hamiltonians implementing these time-evolutions in each irreducible representation of [Formula: see text]. “Topological” terms in the Hamiltonian (or the corresponding action) turn out to be representation-dependent, and are automatically induced by the quantization procedure. Known “topological” charge quantization or periodicity conditions are then identically satisfied as a consequence of the representation theory of [Formula: see text].

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28

Kakushadze, Z. "Quantization Rules for Dynamical Systems." Ukrainian Journal of Physics 61, no. 2 (February 2016): 95–97. http://dx.doi.org/10.15407/ujpe61.02.0095.

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29

Chudakov, D., A. Goncharenko, S. Alyamkin, and A. Densidov. "Quantization noise in low bit quantization and iterative adaptation to quantization noise in quantizable neural networks." Journal of Physics: Conference Series 2134, no. 1 (December 1, 2021): 012004. http://dx.doi.org/10.1088/1742-6596/2134/1/012004.

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Abstract Quantization is one of the most popular and widely used methods of speeding up a neural network. At the moment, the standard is 8-bit uniform quantization. Nevertheless, the use of uniform low-bit quantization (4- and 6-bit quantization) has significant advantages in speed and resource requirements for inference. We present our quantization algorithm that offers advantages when using uniform low-bit quantization. It is faster than quantization-aware training from scratch and more accurate than methods aimed only at selecting thresholds and reducing noise from quantization. We also investigated quantization noise in neural networks for low-bit quantization and concluded that quantization noise is not always a good metric for quantization quality.

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30

Jianping Pan and T. R. Fischer. "Two-stage vector quantization-lattice vector quantization." IEEE Transactions on Information Theory 41, no. 1 (1995): 155–63. http://dx.doi.org/10.1109/18.370111.

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31

Floyd, Edward R. "Action Quantization, Energy Quantization, and Time Parametrization." Foundations of Physics 47, no. 3 (February 10, 2017): 392–429. http://dx.doi.org/10.1007/s10701-017-0067-6.

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32

Xu, Xinbiao, Liyan Ma, Tieyong Zeng, and Qinghua Huang. "Quantized Graph Neural Networks for Image Classification." Mathematics 11, no. 24 (December 11, 2023): 4927. http://dx.doi.org/10.3390/math11244927.

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Researchers have resorted to model quantization to compress and accelerate graph neural networks (GNNs). Nevertheless, several challenges remain: (1) quantization functions overlook outliers in the distribution, leading to increased quantization errors; (2) the reliance on full-precision teacher models results in higher computational and memory overhead. To address these issues, this study introduces a novel framework called quantized graph neural networks for image classification (QGNN-IC), which incorporates a novel quantization function, Pauta quantization (PQ), and two innovative self-distillation methods, attention quantization distillation (AQD) and stochastic quantization distillation (SQD). Specifically, PQ utilizes the statistical characteristics of distribution to effectively eliminate outliers, thereby promoting fine-grained quantization and reducing quantization errors. AQD enhances the semantic information extraction capability by learning from beneficial channels via attention. SQD enhances the quantization robustness through stochastic quantization. AQD and SQD significantly improve the performance of the quantized model with minimal overhead. Extensive experiments show that QGNN-IC not only surpasses existing state-of-the-art quantization methods but also demonstrates robust generalizability.

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33

DOSI, ANAR. "NONCOMMUTATIVE MACKEY THEOREM." International Journal of Mathematics 22, no. 04 (April 2011): 535–44. http://dx.doi.org/10.1142/s0129167x11006891.

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In this note we investigate quantizations of the weak topology associated with a pair of dual linear spaces. We prove that the weak topology admits only one quantization called the weak quantum topology, and that weakly matrix bounded sets are precisely the min-bounded sets with respect to any polynormed topology compatible with the given duality. The technique of this paper allows us to obtain an operator space proof of the noncommutative bipolar theorem.

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Dias, Nuno Costa. "Half quantization." Journal of Physics A: Mathematical and General 34, no. 4 (January 18, 2001): 771–91. http://dx.doi.org/10.1088/0305-4470/34/4/306.

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35

Wang, Jingdong, and Ting Zhang. "Composite Quantization." IEEE Transactions on Pattern Analysis and Machine Intelligence 41, no. 6 (June 1, 2019): 1308–22. http://dx.doi.org/10.1109/tpami.2018.2835468.

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36

Guillemin, V., S. Sternberg, and J. Weitsman. "Signature quantization." Proceedings of the National Academy of Sciences 100, no. 22 (October 20, 2003): 12559–60. http://dx.doi.org/10.1073/pnas.1735527100.

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37

Faizal, Mir. "Fourth quantization." Physics Letters B 727, no. 4-5 (December 2013): 536–40. http://dx.doi.org/10.1016/j.physletb.2013.10.069.

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38

Kämpke, Thomas. "Constrained quantization." Signal Processing 83, no. 9 (September 2003): 1839–58. http://dx.doi.org/10.1016/s0165-1684(03)00104-x.

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39

Batalin, I. A., K. Bering, and P. H. Damgaard. "Superfield quantization." Nuclear Physics B 515, no. 1-2 (March 1998): 455–87. http://dx.doi.org/10.1016/s0550-3213(97)00806-7.

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40

Finkelstein, David Ritz. "General Quantization." International Journal of Theoretical Physics 45, no. 8 (September 26, 2006): 1397–427. http://dx.doi.org/10.1007/s10773-006-9132-1.

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41

Damgaard, Poul H., and Helmuth Hüffel. "Stochastic quantization." Physics Reports 152, no. 5-6 (August 1987): 227–398. http://dx.doi.org/10.1016/0370-1573(87)90144-x.

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42

Jain, Brijnesh J., and Klaus Obermayer. "Graph quantization." Computer Vision and Image Understanding 115, no. 7 (July 2011): 946–61. http://dx.doi.org/10.1016/j.cviu.2011.03.004.

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43

Yen, Yu-Chuan, Cing-Yu Chu, Chien-Nan Chen, Su-Ling Yeh, Hao-Hua Chu, and Polly Huang. "Exponential quantization." ACM SIGCOMM Computer Communication Review 43, no. 4 (September 19, 2013): 551–52. http://dx.doi.org/10.1145/2534169.2491708.

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44

Peter J. Olver. "Dispersive Quantization." American Mathematical Monthly 117, no. 7 (2010): 599. http://dx.doi.org/10.4169/000298910x496723.

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45

Becchi, Carlo. "Second quantization." Scholarpedia 5, no. 6 (2010): 7902. http://dx.doi.org/10.4249/scholarpedia.7902.

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46

Guillemin, Victor, Shlomo Sternberg, and Jonathan Weitsman. "Signature Quantization." Journal of Differential Geometry 66, no. 1 (January 2004): 139–68. http://dx.doi.org/10.4310/jdg/1090415031.

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47

Khrennikov, Andrei. "Hyperbolic Quantization." Advances in Applied Clifford Algebras 18, no. 3-4 (June 16, 2008): 843–52. http://dx.doi.org/10.1007/s00006-008-0105-8.

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48

Berger, R. "Algebraic quantization." Letters in Mathematical Physics 17, no. 4 (May 1989): 275–83. http://dx.doi.org/10.1007/bf00399750.

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49

Klauder, John R. "Understanding quantization." Foundations of Physics 27, no. 11 (November 1997): 1467–83. http://dx.doi.org/10.1007/bf02551494.

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50

Koch, Benjamin, and Enrique Muñoz. "Earthquake Quantization." Quantum 8 (January 2, 2024): 1216. http://dx.doi.org/10.22331/q-2024-01-02-1216.

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Abstract:

In this homage to Einstein's 144th birthday we propose a novel quantization prescription, where the paths of a path-integral are not random, but rather solutions of a geodesic equation in a random background. We show that this change of perspective can be made mathematically equivalent to the usual formulations of non-relativistic quantum mechanics. To conclude, we comment on conceptual issues, such as quantum gravity coupled to matter and the quantum equivalence principle.

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