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Recipe for constructing a quantum analog of a classical physical theory
In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts
Geometric_quantization
Topics referred to by the same term
processing) Color quantization Vector quantization Quantization (music) Quantization (physics) Canonical quantization Geometric quantization Discrete spectrum
Quantization
Systematic procedure of turning a classical theory into a quantum one
generalization involving infinite degrees of freedom is field quantization, as in the "quantization of the electromagnetic field", referring to photons as field
Quantization_(physics)
Process in quantum mechanical theories
context, it is also called the second quantization of fields, in contrast to the semi-classical first quantization of single particles. When it was first
Canonical_quantization
geometric quantization, quantization commutes with reduction states that the space of global sections of a line bundle L satisfying the quantization condition
Quantization commutes with reduction
Quantization_commutes_with_reduction
in topological and geometric terms. It plays an important role in the theory of Fourier integral operators, geometric quantization, Hamiltonian systems
Maslov_index
Converting classical mechanics to quantum mechanics
single particle either. Canonical quantization Geometric quantization Quantization Second quantization This statement is not unique since it can be argued
First_quantization
Formulation of the quantum many-body problem
Canonical quantization First quantization Geometric quantization Quantization (physics) Schrödinger functional Scalar field theory Second quantization on Wikiversity
Second_quantization
Formulation to quantize gauge field theories in physics
relatively rigorous mathematical approach to quantizing a field theory with a gauge symmetry. Quantization rules in earlier quantum field theory (QFT)
BRST_quantization
Any sheaf whose value is based on an eigenfunction
conjecture" (PDF). Max Planck Institute for Mathematics. July 19, 2024. "QUANTIZATION OF HITCHIN'S INTEGRABLE SYSTEM AND HECKE". University of Chicago. 1991
Hecke_eigensheaf
Mathematical theory
In mathematics, the geometric Langlands correspondence relates algebraic geometry and representation theory. It is a reformulation of the Langlands correspondence
Geometric Langlands correspondence
Geometric_Langlands_correspondence
Partial differential equations whose solutions are instantons
and geometric quantization. Communications in mathematical physics, 131(2), 347–380. Axelrod, S., Della Pietra, S., & Witten, E. (1991). Geometric quantization
Yang–Mills_equations
Study of vector bundles, principal bundles, and fibre bundles
Witten, E., 1991. Geometric quantization of Chern–Simons gauge theory. representations, 34, p. 39. Witten, E., 1991. Quantization of Chern-Simons gauge theory
Gauge_theory_(mathematics)
Space of all possible states that a system can take
abstractions include deformation quantization and geometric quantization.) Expectation values in phase-space quantization are obtained isomorphically to
Phase_space
Phase of a cycle
insight into the Landau level quantization. This alternative way is based on the semiclassical Bohr–Sommerfeld quantization condition ℏ ∮ d r ⋅ k − e ∮
Geometric_phase
Search algorithm
In computer science, geometric hashing is a method for efficiently finding two-dimensional objects represented by discrete points that have undergone
Geometric_hashing
Physical theory with fields invariant under the action of local "gauge" Lie groups
variety of means. Methods for quantization are covered in the article on quantization. The main point to quantization is to be able to compute quantum
Gauge_theory
American Jewish mathematician
E8. He has been one of the principal developers of the theory of geometric quantization. His introduction of the theory of prequantization has led to the
Bertram_Kostant
well-defined) without some transversality conditions. Notes He means geometric quantization. Sources Weinstein, Alan (2009). "Symplectic Categories". arXiv:0911
Symplectic_category
completely integrable system Darboux chart deformation quantization deformation quantization. dilating derived symplectic geometry Derived algebraic
Glossary of symplectic geometry
Glossary_of_symplectic_geometry
American mathematician (1936–2024)
1002/cpa.3160310405. Guillemin, V.; Sternberg, S. (October 1, 1982). "Geometric quantization and multiplicities of group representations". Inventiones Mathematicae
Shlomo_Sternberg
Construction in representation theory
view has been significantly advanced by Kostant in his theory of geometric quantization of coadjoint orbits. For a Lie group G {\displaystyle G} , the Kirillov
Orbit_method
British mathematician (born 1949)
2020 New Year Honours for services to mathematics. Lectures on Geometric Quantization, with D J Simms, Lecture Notes in Physics, Springer, 1976 Introduction
Nick_Woodhouse
Online vector quantization algorithm
TurboQuant is an online vector quantization algorithm for compressing high-dimensional Euclidean vectors while preserving their geometric structure. It was proposed
TurboQuant
Topological variant of the loop
Quasigroup Brylinski, Jean-Luc: Loop spaces, characteristic classes and geometric quantization. Reprint of the 1993 edition. Modern Birkhäuser Classics. Birkhäuser
Free_loop
Mathematical structures that allow quantum mechanics to be explained
renormalization of the norm). This is related to the quantization of constrained systems and quantization of gauge theories. It is also possible to formulate
Mathematical formulation of quantum mechanics
Mathematical_formulation_of_quantum_mechanics
Assignment of a tensor continuously varying across a region of space
in areas such as defining integral operators on manifolds, and geometric quantization. When M is a Euclidean space and all the fields are taken to be
Tensor_field
French-American mathematician
his wife Ranee Brylinski. Loop Spaces, Characteristic Classes and Geometric Quantization (1992) Brylinski, Jean-Luc; Kashiwara, Masaki (October 1981), "Kazhdan-Lusztig
Jean-Luc_Brylinski
Foliation of symplectic manifolds
Lagrangian submanifolds. It is one of the steps involved in the geometric quantization of a square-integrable functions on a symplectic manifold. Kenji
Lagrangian_foliation
Formulation of classical mechanics in terms of Hilbert spaces
Jauslin, D. Sugny, Dynamics of mixed classical-quantum systems, geometric quantization and coherent states, Lecture Note Series, IMS, NUS, Review Vol.
Koopman–von Neumann classical mechanics
Koopman–von_Neumann_classical_mechanics
Ratio of the desired signal to the background noise
possible noise level is the error caused by the quantization of the signal, sometimes called quantization noise. This noise level is non-linear and signal-dependent;
Signal-to-noise_ratio
Canadian mathematician
ISSN 0040-9383. S2CID 18573269. Guillemin, V.; Sternberg, S. (1982-10-01). "Geometric quantization and multiplicities of group representations". Inventiones Mathematicae
Eckhard_Meinrenken
Branch of geometry
broad applications in physics, e.g. geometrical optics, classical mechanics, thermodynamics, geometric quantization, integrable systems and to control
Contact_geometry
Generalization of Hamiltonian mechanics involving multiple Hamiltonians
helicity. From the view point of Zariski quantization, Takhtajan et al. propose quantization of Nambu dynamics. Quantizing Nambu dynamics leads to intriguing
Nambu_mechanics
Algebra over a field where binary multiplication is not necessarily associative
algebras and many more. The Poisson algebras are considered in geometric quantization. They carry two multiplications, turning them into commutative algebras
Non-associative_algebra
Topological space which is a generalization of certain compact spaces
Brylinski, Jean-Luc (2007), Loop Spaces, Characteristic Classes and Geometric Quantization, Progress in Mathematics, vol. 107, Springer, p. 32, ISBN 9780817647308
Paracompact_space
Broad concept generalizing scalars in mathematics and physics
Vector meson, a meson with total spin 1 and odd parity Vector quantization, a quantization technique used in signal processing Vector soliton, a solitary
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Description of gravity using discrete values
Carlo Rovelli and Lee Smolin to derive naturally from a non-perturbative quantization of general relativity. Spin networks do not represent quantum states
Quantum_gravity
General relativity in 2+1 dimensions
large diffeomorphisms, and then obtaining the Hilbert space by geometric quantization. The Hilbert space turns out to be generated from Virasoro conformal
(2+1)-dimensional topological gravity
(2+1)-dimensional_topological_gravity
Lie group whose manifold is complex and whose group operation is holomorphic
[clarification needed] Guillemin, Victor; Sternberg, Shlomo (1982). "Geometric quantization and multiplicities of group representations". Inventiones Mathematicae
Complex_Lie_group
Jean-Luc (2008) [1993], Loop spaces, characteristic classes and geometric quantization, Modern Birkhäuser Classics, Boston, MA: Birkhäuser Boston, doi:10
Deligne_cohomology
Jedrzej Śniatycki, the author of Geometric quantization and quantum mechanics, developed an invariant geometrical formulation of jet bundles, building
De_Donder–Weyl_theory
storm Geomechanics Geomelting Geometric algebra Geometric phase Geometric quantization Geometrical frustration Geometrically frustrated magnet Geometrized
Index_of_physics_articles_(G)
Branch of mathematics
sheaves. Deformation quantization and quantum groups are related areas when their noncommutative algebras are interpreted geometrically, although they are
Noncommutative_geometry
Complex structures in matter physics
In condensed matter physics, geometrical frustration (or in short, frustration) is a phenomenon where the combination of conflicting inter-atomic forces
Geometrical_frustration
Vector quantization algorithm minimizing the sum of squared deviations
k-means clustering is a method of vector quantization, originally from signal processing, that aims to partition n observations into k clusters in which
K-means_clustering
Theory in theoretical physics
to a structure that one finds geometrically quantizing the space of complex structures. Once this space has been quantized, only half of the dimensions
Topological_string_theory
Theory of quantum gravity merging quantum mechanics and general relativity
develop a quantum theory of gravity based directly on Albert Einstein's geometric formulation, general relativity. As a theory, LQG postulates that the
Loop_quantum_gravity
Mathematical group of loops in a Lie group
index-theoretic constructions link loop-group representation theory with geometric quantization, central extensions, and the topology of the group G itself. Loop
Loop_group
Concept in algebraic geometry
with < replaced by ≤. GIT quotient Geometric complexity theory Geometric quotient Categorical quotient Quantization commutes with reduction K-stability
Geometric_invariant_theory
Construct in mathematics
Brylinski, Jean-Luc (1993), Loop space, characteristic classes and geometric quantization, Birkhäuser Verlag, ISBN 0-8176-3644-7. Constructions with Bundle
Gerbe
State of matter at low temperatures
_{n}} is the viscosity of the normal component, Z {\textstyle Z} some geometrical factor, and V ˙ n {\textstyle {\dot {V}}_{n}} the volume flow. The normal
Superfluid_helium-4
Chinese mathematician
Ray–Singer torsion, Eta forms, elliptic genera), Bergman kernels and geometric quantization. He is editor of Science in China A (Mathematics) and of International
Xiaonan_Ma
American mathematician (1931–2015)
at UCLA working on harmonic analysis, representation theory, and geometric quantization, who introduced Blattner's conjecture. Born in Milwaukee, Blattner
Robert_James_Blattner
and geometric quantisation. Lie Groups and Quantum Mechanics, Springer Lecture Notes in Mathematics Number 52, 1968 Lectures on Geometric Quantization, (with
David_J._Simms
Aspect of theoretical physics
Quantum geometry in condensed matter physics refers to gauge-invariant geometric properties of quantum states as functions of external parameters—most
Quantum geometry (condensed matter)
Quantum_geometry_(condensed_matter)
Formulation of general relativity
canonical quantization techniques for constrained Hamiltonian systems invented by Paul Dirac.[3] Dirac's approach allows the quantization of systems
Canonical_quantum_gravity
Polish physicist and mathematician (1931–2022)
described as particles moving backward in time. His work prefigured geometric quantization, later developed by Jean-Marie Souriau and Bertram Kostant. Tulczyjew
Włodzimierz_Marek_Tulczyjew
Polish mathematician and physicist
S2CID 123138009. Odzijewicz, Anatol (1 November 1992). "Coherent states and geometric quantization". Communications in Mathematical Physics. 150 (2): 385–413. Bibcode:1992CMaPh
Anatol_Odzijewicz
Concept in differential geometry
handle the infinite-dimensional groups arising from his work in geometric quantization. Thus the notion of diffeological group preceded the more general
Diffeology
Phase in insulators
noncontractible loop in the Brillouin zone. In other words, it's the geometric phase acquired by an electron in a crystal as its wave vector, k {\displaystyle
Zak_phase
French mathematician
which led to the first geometric interpretation of spin at a classical level. He also suggested a program of geometric quantization and developed a more
Jean-Marie_Souriau
Forger, H. Hess (1979). "Universal metaplectic structures and geometric quantization" (PDF). Commun. Math. Phys. 64: 269–278. doi:10.1007/bf01221734
Metaplectic_structure
Application of K-theory in string theory
One needs to choose a half of the fluxes to quantize, or a polarization in the geometric quantization-inspired language of Diaconescu, Moore, and Witten
K-theory_(physics)
Superconductivity theory
all its parameters.[not in body] The theory can also be given a general geometric setting, placing it in the context of Riemannian geometry, where in many
Ginzburg–Landau_theory
Infinite series summing alternating 1 and -1 terms
Grandi's series as a divergent geometric series and using the same algebraic methods that evaluate convergent geometric series to obtain a third value:
Grandi's_series
Hypothetical elementary particle that mediates gravity
detecting single gravitons possible. Even quantum events may not indicate quantization of gravitational radiation. LIGO and Virgo collaborations' observations
Graviton
Set of mathematical concepts in quantum gravity
quantum geometry to describe exotic phenomena such as T-duality and other geometric dualities, mirror symmetry, topology-changing transitions, minimal possible
Quantum_geometry
French mathematician
Mathematicians in Beijing, where he gave a talk on bubbling, quantization and regularity issues in geometric non-linear analysis. "Everywhere discontinuous Harmonic
Tristan_Rivière
Algorithm used for points in euclidean space
method was originally used for scalar quantization, but it is clear that the method extends for vector quantization as well. As such, it is extensively
Lloyd's_algorithm
Concept in physics
associated with the Berry phase or geometric phase. The concept was first introduced by S. Pancharatnam as geometric phase and later elaborately explained
Berry connection and curvature
Berry_connection_and_curvature
Closed degenerate differential 2-form of constant rank
systems with constraints, and control theory. Vaisman, Izu (1983). "Geometric quantization on presymplectic manifolds". Monatshefte für Mathematik. 96 (4):
Presymplectic_form
photographic images Vector quantization: technique often used in lossy data compression TurboQuant: online vector quantization algorithm for lossy compression
List_of_algorithms
American mathematician (born 1943)
Riemannian geometry, symplectic geometry, Lie groupoids, geometric mechanics and deformation quantization. Among his most important contributions, in 1971 he
Alan_Weinstein
Russian and French mathematician (born 1964)
member. His work concentrates on geometric aspects of mathematical physics, most notably on knot theory, quantization, and mirror symmetry. One of his
Maxim_Kontsevich
Moment of inertia of diff geometric shapes
(1982). "Voronoi regions of lattices, second moments of polytopes, and quantization" (PDF). IEEE Transactions on Information Theory. 28 (2): 211–226. Bibcode:1982ITIT
List_of_moments_of_inertia
Elementary particle or quantum of light
electrodynamics, i.e., the quantization of the electromagnetic field itself. Dirac's approach is also called second quantization or quantum field theory;
Photon
German mathematician
German-Luxembourgish mathematician whose research deals with algebraic, geometric and analytic mathematical methods which partly have relations to theoretical
Martin_Schlichenmaier
Curve simplification algorithm
made non-parametric by using the error bound due to digitization and quantization as a termination condition. Assuming the input is a one-based array:
Ramer–Douglas–Peucker algorithm
Ramer–Douglas–Peucker_algorithm
Electromagnetic quantum-mechanical effect in regions of zero magnetic and electric field
same phase effect is responsible for the quantized-flux requirement in superconducting loops. This quantization occurs because the superconducting wave
Aharonov–Bohm_effect
Quantum analog of probabilistic automata
understood as the quantization of subshifts of finite type, or as a quantization of Markov chains. QFAs are, in turn, special cases of geometric finite automata
Quantum_finite_automaton
Lossless data compression method
codes invented by Solomon W. Golomb in the 1960s. Alphabets following a geometric distribution will have a Golomb code as an optimal prefix code, making
Golomb_coding
Italian mathematician
Poisson and coisotropic structures with applications to deformation quantization. Lately Toën and Vezzosi (partly in collaboration with Anthony Blanc
Gabriele_Vezzosi
Open API by Pixar Animation Studios
was that it allowed using high-level geometric primitives, like quadrics or bicubic patches, to specify geometric primitives implicitly, rather than relying
RenderMan Interface Specification
RenderMan_Interface_Specification
Identity in abelian theories due to gauge invariance
operator and plays a central role in providing a geometric description of the consistent quantization of gauge theories. The Ward–Takahashi identity applies
Ward–Takahashi_identity
Means of studying the interaction of light and matter
silicon (Si) or sapphire (Al2O3). The line impedance is given by the geometric properties, which are chosen to match the 50 Ω {\displaystyle \Omega }
Circuit quantum electrodynamics
Circuit_quantum_electrodynamics
Romanian mathematician (1950–2007)
in 1993. Puta, Mircea (1993). Hamiltonian Mechanical Systems and Geometric Quantization. Mathematics and its Applications. Vol. 260. Dordrecht: Kluwer Academic
Mircea_Puta
Quantum field theory
contribute something to the discussions, especially with regard to the quantization procedures, and to a small degree in working out the formalism; however
Yang–Mills_theory
Symmetry of physical laws under a charge-conjugation transformation
on a U(1) fiber bundle, the so-called circle bundle. This provides a geometric interpretation of electromagnetism: the electromagnetic potential A μ
C-symmetry
Extended physical object in string theory
Inami, C. N. Pope, E. Sezgin [de], and K. S. Stelle, "Semiclassical quantization of the supermembrane", Nucl. Phys. B297 (1988), 515. Moore 2005, p. 214
Brane
Area of mathematics
one by one via their common faces in Rn. DDGNS focuses primarily on "quantization" rather than "discretization" of classical differential geometry. Just
Discrete differential geometry
Discrete_differential_geometry
Field theory of a point particle confined to move on a fixed manifold
The model may or may not be quantized. An example of the non-quantized version is the Skyrme model; it cannot be quantized due to non-linearities of power
Sigma_model
Trait of artistic and architectural styles
physical labor. In music, this aesthetic manifests through hyper-precision, quantization, and looping. Rhythmic rigidity: This style mimics the "grid" of early
Machine_aesthetic
Approximate nearest neighbor search algorithm
Other approximate methods include locality-sensitive hashing and product quantization. HNSW builds on research into small-world networks and navigable graphs
Hierarchical navigable small world
Hierarchical_navigable_small_world
Laser interferometer at Fermilab
tested and conclusively excluded a general class of models of quantum geometrical shear noise correlations". An upgraded version of the holometer was used
Fermilab_Holometer
German mathematician (1885–1955)
group, were also streamlined in that specific context, in his 1927 Weyl quantization, the best extant bridge between classical and quantum physics to date
Hermann_Weyl
Interpretation of quantum mechanics
action in combination with a quantization prescription. This quantization prescription can be compared to canonical quantization and the path integral formulation
Stochastic_quantum_mechanics
Indian theoretical physicist (1926–2005)
supersymmetric classical mechanics, supersymmetric quantum mechanics, stochastic quantization, quark stars, continued fraction theory, role of parastatistics in statistical
Samarendra_Nath_Biswas
Finite, symmetry-reduced model of loop quantum gravity
(2017-10-05). "Immirzi parameter without Immirzi ambiguity: Conformal loop quantization of scalar-tensor gravity". Physical Review D. 96 (8) 084011. arXiv:1705
Loop_quantum_cosmology
French scientist
1142/p752. ISBN 978-1-84816-650-9. Nottale, L. (1996). "Scale relativity and quantization of extra-solar planetary systems". Astronomy and Astrophysics. 315. Bibcode:1996A&A
Laurent_Nottale
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