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In mathematics and physics, deformation quantization roughly amounts to finding a (quantum) algebra whose classical limit is a given (classical) algebra
Deformation_quantization
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
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)
the deformation quantization of the corresponding Poisson algebra. It is due to Maxim Kontsevich. Given a Poisson algebra (A, {⋅, ⋅}), a deformation quantization
Kontsevich quantization formula
Kontsevich_quantization_formula
Space of all possible states that a system can take
modern abstractions include deformation quantization and geometric quantization.) Expectation values in phase-space quantization are obtained isomorphically
Phase_space
Branch of mathematics
noncommutative rings and graded algebras; and constructions related to deformation quantization, groupoid C*-algebras, cyclic homology, and K-theory. A standard
Noncommutative_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
Italian mathematician and physicist (born 1967)
invited speaker, with the talk From topological field theory to deformation quantization and reduction, at the International Congress of Mathematicians
Alberto_Cattaneo
Recipe for constructing a quantum analog of a classical physical theory
quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization
Geometric_quantization
Typically linear operator defined in terms of differentiation of functions
appears, for instance, in an associative algebra structure on a deformation quantization of a Poisson algebra. A microdifferential operator is a type of
Differential_operator
Russian and French mathematician (born 1964)
most notably on knot theory, quantization, and mirror symmetry. One of his results is a formal deformation quantization that holds for any Poisson manifold
Maxim_Kontsevich
Mathematical structure in differential geometry
(1993–1994). "Deformation quantization". Séminaire Bourbaki. 36: 389–409. ISSN 0303-1179. Kontsevich, Maxim (2003-12-01). "Deformation Quantization of Poisson
Poisson_manifold
Indian mathematician
in the areas of algebraic geometry, differential geometry, and deformation quantization. In 2006, the Government of India awarded him the Shanti Swarup
Indranil_Biswas
coisotropic completely integrable system Darboux chart deformation quantization deformation quantization. dilating derived symplectic geometry Derived algebraic
Glossary of symplectic geometry
Glossary_of_symplectic_geometry
Pyramid vector quantization (PVQ) is a method used in audio and video codecs to quantize and transmit unit vectors, i.e. vectors whose magnitudes are
Pyramid_vector_quantization
Relation satisfied by conjugate variables in quantum mechanics
equivalent mathematical representation of quantum mechanics known as deformation quantization. According to the correspondence principle, in certain limits the
Canonical commutation relation
Canonical_commutation_relation
Formulation of quantum mechanics
into mathematical offshoots such as Kontsevich's deformation-quantization (see Kontsevich quantization formula) and noncommutative geometry.[citation needed]
Phase-space_formulation
Correspondence in functional analysis
Stefan Waldmann: On the representation theory of deformation quantization, In: Deformation Quantization: Proceedings of the Meeting of Theoretical Physicists
Gelfand–Naimark–Segal construction
Gelfand–Naimark–Segal_construction
Mapping between functions in the quantum phase space
Weyl quantization. It is now understood that Weyl quantization does not satisfy all the properties one would require for consistent quantization and therefore
Wigner–Weyl_transform
Russian mathematician
A. Rossi, Charles Torossian, Thomas Willwacher: Logarithms and Deformation Quantization, Inventiones Mathematicae, vol. 206, 2016, pp. 1–26, Arxiv with
Anton Alekseev (mathematician)
Anton_Alekseev_(mathematician)
Israeli mathematician
mathematician, working in noncommutative algebra, algebraic geometry and deformation quantization. He is a professor of mathematics at the Ben-Gurion University
Amnon_Yekutieli
Semiring arising in tropical analysis
{\displaystyle b\to 0} (min-plus semiring), and thus can be viewed as a deformation ("quantization") of the tropical semiring. Notably, the addition operation, logadd
Log_semiring
algebra but instead an associative algebra that can be regarded as a deformation of the universal enveloping algebra of s l 2 {\displaystyle {\mathfrak
Quantized_enveloping_algebra
Ring that is also a vector space or a module
{\mathfrak {a}}[\![u]\!]} is called a deformation quantization of a {\displaystyle {\mathfrak {a}}} . A quantized enveloping algebra. The dual of such
Associative_algebra
Approximation or recovery of classical mechanics in certain theories
reduced Planck constant ħ, so the "deformation parameter" ħ/S can be effectively taken to be zero (cf. Weyl quantization.) Thus typically, quantum commutators
Classical_limit
American theoretical physicist
Liouville theory, geometrostatic sigma models, quantum algebras, and deformation quantization. Curtright is a Fellow of the American Physical Society (1998)
Thomas_Curtright
Algebraic construct of interest in theoretical physics
"deformed", although the deformation will no longer remain a group algebra or enveloping algebra. More precisely, deformation can be accomplished within
Quantum_group
Generalization of associativity properties
operads. Operads have since found many applications, such as in deformation quantization of Poisson manifolds, the Deligne conjecture, or graph homology
Operad
Japanese mathematician
beyond typical retirement age, focusing particularly on problems of deformation quantization beginning in 1999. His retirement from Tokyo University of Science
Hideki_Omori
British mathematician and theoretical physicist
solutions of gauge theories, higher-dimensional gauge theories, and deformation quantization. He has co-authored several volumes, notably on quantum mechanics
David_Fairlie
algebra BV formalism Simplicial Lie algebra Hochschild homology Deformation quantization Lie n-algebra Lurie, Jacob. "Derived Algebraic Geometry X: Formal
Homotopy_Lie_algebra
The famous result of Boris Vasilievich Fedosov gives a canonical deformation quantization of a Fedosov manifold. For example, R 2 n {\displaystyle \mathbb
Fedosov_manifold
Italian mathematician
version of Poisson and coisotropic structures with applications to deformation quantization. Lately Toën and Vezzosi (partly in collaboration with Anthony
Gabriele_Vezzosi
Mathematics timeline
Francois Bayen–Moshe Flato–Chris Fronsdal–André Lichnerowicz–Daniel Sternheimer Deformation quantization, later to be a part of categorical quantization
Timeline_of_manifolds
History of maths
categorical noncommutative geometry, etc. Quantization related to category theory, in particular categorical quantization; Categorical physics relevant for mathematics
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
American mathematician (born 1943)
geometry, symplectic geometry, Lie groupoids, geometric mechanics and deformation quantization. Among his most important contributions, in 1971 he proved a tubular
Alan_Weinstein
Science prizes established by Run Run Shaw
in algebra, geometry and mathematical physics and in particular deformation quantization, motivic integration and mirror symmetry. 2013 David L. Donoho
Shaw_Prize
generalizations are index theorems based on spectral triples and deformation quantization of Poisson structures. An elliptic operator D on a compact smooth
Cyclic_homology
Mathematical concept
classical semisimple Lie algebra was correspondingly replaced by the deformation quantization of the affine Poisson variety. Kamnitzer, Joel (2022-02-08). "Symplectic
Symplectic_resolution
Swiss physicist and mathematician
in 2000 he gave a path integral interpretation of Kontsevich's deformation quantization of Poisson manifolds as well as a description of the symplectic
Giovanni_Felder
Complex Geometry" with a talk "Derived Algebraic Geometry and Deformation Quantization". He was awarded an ERC Advanced Grant in 2016. In 2019 he received
Bertrand_Toën
Example of a phase-space star product in mathematics
to have emerged only in the 1970s, in homage to his flat phase-space quantization picture. The product for smooth functions f and g on R 2 n {\displaystyle
Moyal_product
Belgian mathematician (born 1970)
Science, Letters and Fine Arts of Belgium (2015) with Victor Gayral, Deformation Quantization for Actions of Kählerian Lie Groups, Volume 236, Number 1115, Memoirs
Pierre_Bieliavsky
Theory of quantum gravity merging quantum mechanics and general relativity
{E}}_{i}^{3}{\tilde {E}}^{3i}}}.} According to the rules of canonical quantization the triads E ~ i 3 {\displaystyle {\tilde {E}}_{i}^{3}} should be promoted
Loop_quantum_gravity
Austrian mathematician and mathematical physicist
(2011), no. 1, 115–139 (with N. Dias F. Luef, J. Prata, João) A deformation quantization theory for noncommutative quantum mechanics. J. Math. Phys. 51
Maurice_A._de_Gosson
Quasiparticle of mechanical vibrations
mechanical quantization of the modes of vibrations for elastic structures of interacting particles. Phonons can be thought of as quantized sound waves
Phonon
Formalism in string theory
action found by second-quantizing the free string and adding interaction terms. As is usually the case in second quantization, a classical field configuration
String_field_theory
Op(L2(Rn)). This property is fully transferred to the phase space upon deformation quantization and, in the limit of ħ → 0, to the classical mechanics. Table compares
Method of quantum characteristics
Method_of_quantum_characteristics
American annual mathematics conference
Tribute to Louis Nirenberg Akito Futaki (Yau Center, Tsinghua) - Deformation Quantization, and Obstructions to the Existence of Closed Star Products Jean-Pierre
Geometry_Festival
Mathematical discipline
affine algebra (or affine quantum group) is a Hopf algebra that is a q-deformation of the universal enveloping algebra of an affine Lie algebra. They were
Quantum_affine_algebra
German mathematician
Schlichenmaier, Martin (2001), "Identification of Berezin-Toeplitz deformation quantization" (PDF), J. reine angew. Math., 2001 (540): 49–76, doi:10.1515/crll
Martin_Schlichenmaier
Suitably normalized antisymmetrization of the phase-space star product
equations. Mathematically, it is a deformation of the phase-space Poisson bracket (essentially an extension of it), the deformation parameter being the reduced
Moyal_bracket
Isomorphism of symplectic manifolds
the group of symplectomorphisms (after ħ-deformations, in general) on Hilbert spaces are called quantizations. When the Lie group is the one defined by
Symplectomorphism
Operation in Hamiltonian mechanics
giving the desired result. Poisson brackets deform to Moyal brackets upon quantization, that is, they generalize to a different Lie algebra, the Moyal algebra
Poisson_bracket
Alteration of the original shape of a signal
(hum, interference) is not considered distortion, though the effects of quantization distortion are sometimes included in noise. Quality measures that reflect
Distortion
French mathematical physicist (1915–1998)
Lichnerowicz, A.; Sternheimer, D. (1978-03-01). "Deformation theory and quantization. I. Deformations of symplectic structures". Annals of Physics. 111
André_Lichnerowicz
Strong-weak duality in supersymmetric theories of theoretical physics
of the usual electrodynamics and it leads to the quantization of electricity. [...] The quantization of electricity is one of the most fundamental and
Montonen–Olive_duality
Concept in theoretical mathematical physics
34.2045M, doi:10.1063/1.530154, S2CID 3138714. 't Hooft, G. (1996), "Quantization of point particles in (2 + 1)-dimensional gravity and spacetime discreteness"
Quantum_spacetime
Type of user interface
deformable) user interfaces: When flexible displays are deployed, shape deformation, e.g., through bends, is a key form of input for OUI. Flexible display
Organic_user_interface
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
Breakdown of parity at the quantum level
Chern–Simons level is even. In the case n=1, this statement is the half-integer quantization condition in N = 1 {\displaystyle {\mathcal {N}}=1} supersymmetric Chern–Simons
Parity_anomaly
Intensive quantity, heat capacity per amount of substance
are called the rigid degrees of freedom, since they do not involve any deformation of the molecule. Because of those two extra degrees of freedom, the molar
Molar_heat_capacity
Topological quantum field theory
Adam; Seiberg, Nathan (30 October 1989). "Remarks on the canonical quantization of the Chern-Simons-Witten theory". Nuclear Physics B. 326 (1): 108–134
Chern–Simons_theory
Construct in mathematics
convenient, if highly abstract, language for dealing with many types of deformation questions especially in modern algebraic geometry. In addition, special
Gerbe
Mathematical condition
Yan (2000). "Deformations of algebras over operads and Deligne's conjecture". Conférence Moshé Flato 1999: Quantization, Deformations, and Symmetries
Poincaré_lemma
Partial differential equations whose solutions are instantons
geometric quantization. Communications in mathematical physics, 131(2), 347–380. Axelrod, S., Della Pietra, S., & Witten, E. (1991). Geometric quantization of
Yang–Mills_equations
Type of topological order in condensed matter physics
smoothly deformed into each other without a phase transition, if the deformation preserves the symmetry. (b) however, they all can be smoothly deformed
Symmetry-protected topological order
Symmetry-protected_topological_order
Property of physical systems that stays somewhat constant through slow changes
processes in thermodynamics. In mechanics, an adiabatic change is a slow deformation of the Hamiltonian, where the fractional rate of change of the energy
Adiabatic_invariant
Wigner distribution function in physics as opposed to in signal processing
operators, in Weyl quantization. Thus, the Wigner function is the cornerstone of quantum mechanics in phase space. In second quantization the phase-space
Wigner quasiprobability distribution
Wigner_quasiprobability_distribution
Smallest unit of a chemical element
of energy corresponding to absorption or radiation of a photon. This quantization was used to explain why the electrons' orbits are stable and why elements
Atom
Quantum bit
must be limited to a suitably low rate. For electron-on-helium qubits, deformations of the helium surface due to surface or bulk excitations (ripplons or
Electron-on-helium_qubit
Smooth approximation of one-hot arg max
values. In the language of tropical analysis, the softmax is a deformation or "quantization" of arg max and arg min, corresponding to using the log semiring
Softmax_function
destination pixels. Bone Coordinate systems used to control surface deformation (via Weight maps) during skeletal animation. Typically stored in a hierarchy
Glossary_of_computer_graphics
for ranking data, Inverse scattering on the line, Deformation theory of algebras and quantization with applications to physics, Strategies for sequential
Robbins'_problem
Physics generalization
"Quantum gravitational decoherence from fluctuating minimal length and deformation parameter at the Planck scale" (PDF). Nature Communications. 12 (1):
Generalized uncertainty principle
Generalized_uncertainty_principle
skein modules in knot theory. The skein module is roughly a deformation (or quantization) of the character variety. It is closely related to topological
Character_variety
American scientist
investigators of the CeNTREX collaboration with David DeMille to search for the deformation in the shape of atomic nuclei known as a Schiff moment using the thallium
Tanya_Zelevinsky
American mathematician (1936–2024)
mathematical treatment of what is known in the physics literature as the BRST quantization procedure. Together with David Kazhdan and Bertram Kostant, he showed
Shlomo_Sternberg
Graphics programming language
Displacement shaders manipulate surface geometry independent of color. Deformation shaders transform the entire space. Only one RenderMan implementation
Shading_language
Method of fabricating nanometer scale patterns using a special stamp
high throughput and high resolution. It creates patterns by mechanical deformation of imprint resist and subsequent processes. The imprint resist is typically
Nanoimprint_lithography
Berman–Boucksom–Jonsson and the so-called quantized delta invariants of Fujita–Odaka, Zhang produced a short quantization-based proof of the YTD conjecture for
K-stability_of_Fano_varieties
Mathematics glossary
of symplectic geometry for the topics in symplectic topology such as quantization. Contents: !$@ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z *
Glossary of algebraic topology
Glossary_of_algebraic_topology
Problem in applied mathematics
model of electrons, nor for QCD i.e. the theory of quarks. Stochastic quantization: The sum over configurations is obtained as the equilibrium distribution
Numerical_sign_problem
British physicist and mathematician (1886–1975)
work on fluid mechanics and solid mechanics, including research on the deformation of crystalline materials which followed from his war work at Farnborough
G._I._Taylor
Integrable classical system
2022. Reshetikhin, N. (1992). "The Knizhnik-Zamolodchikov system as a deformation of the isomonodromy problem". Lett. Math. Phys. 26: 166–177. doi:10.1007/BF00420750
Garnier_integrable_system
Chinese mathematician (born 1958)
sphere into N, called "bubbles." Ding and Tian proved a certain "energy quantization," meaning that the defect between the Dirichlet energy of u(T) and the
Tian_Gang
Equations relating to massless particles in AdS space
properly implemented and the equations give a solution of certain formal deformation procedure, which is difficult to map to field theory language. The higher-spin
Vasiliev_equations
Branch of mathematics
that affine space admits non-commutative deformations to the space determined by the Weyl algebra. This deformation is related to the symbol of a differential
Noncommutative algebraic geometry
Noncommutative_algebraic_geometry
Kazakh mathematician and physicist (born 1956)
Dzhumadildaev A.S. Identities and derivations for Jacobian algebras//"Quantization, Poisson brackets and beyond", Contemp. Math. v.315, 245–278, 2002. Preprint
Askar_Dzhumadildayev
Theory of subatomic structure
Friedan, Emil Martinec and Stephen Shenker further developed the covariant quantization of the superstring using conformal field theory techniques. David Gross
String_theory
Canadian engineer (1941–2014)
lossy, still image transmission. He contributed to the study of vector quantization for lossy image compression and developed a number of hierarchical coding
Anastasios_Venetsanopoulos
Equation describing the evolution of the vorticity of a fluid particle as it flows
57262/die/1356039440. S2CID 50701138. Barbu, V.; Sritharan, S. S. (2000). "M-Accretive Quantization of the Vorticity Equation" (PDF). In Balakrishnan, A. V. (ed.). Semi-Groups
Vorticity_equation
Canadian-American physicist and academic
While investigating the deformation of the Ising model and its ultraviolet completion, his study established that such deformations are generally incomplete
André_LeClair
Physical quantities taking values at each point in space and time
point, is an example of a vector field. Strain tensor, representing the deformation of matter caused by stress, is an example of a tensor field. Field theories
Field_(physics)
Mathematical concept
algebra: one can think of the central extension as corresponding to quantization or deformation. Formally, the symmetric algebra of a vector space V over a field
Symplectic_vector_space
Field theory involving topological effects in physics
functions are metric-independent, so they remain unchanged under any deformation of spacetime and are therefore topological invariants. Topological field
Topological quantum field theory
Topological_quantum_field_theory
Study of categorified structures
Note on Quantum Groupoids". C*-algebras, deformation theory, groupoids, noncommutative geometry, quantization. Theoretical Atlas. Brown, R.; Higgins, P
Higher-dimensional_algebra
Classification of Artificial Neural Networks (ANNs)
the output space, and SOM attempts to preserve these. Learning vector quantization (LVQ) can be interpreted as a neural network architecture. Prototypical
Types of artificial neural networks
Types_of_artificial_neural_networks
Numerical method in quantum field theory
{\text{vol}}(M)\sim R^{d-1}} up to some c-number coefficient. If the deformation V is the integral of a local operator of dimension Δ {\displaystyle \Delta
Hamiltonian_truncation
Hypothetical types of stars
these have quantized angular momentum, and their energy density profiles are torus-shaped, which can be understood as a result of deformation due to centrifugal
Exotic_star
DEFORMATION QUANTIZATION
DEFORMATION QUANTIZATION
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Good Information
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Decoration. Beauty.
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Information; News
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Address; Information
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DEFORMATION QUANTIZATION
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n.
Defedation.
n.
The act of deflouring; as, the defloration of a virgin.
n.
The act of giving shape or form.
n.
Dissuasion; advice against something.
n.
That which is chosen as the flower or choicest part; careful culling or selection.
n.
Transformation; change of shape.
v. t.
The act of informing, or communicating knowledge or intelligence.
n.
Decoration.
n.
The act of reforming, or the state of being reformed; change from worse to better; correction or amendment of life, manners, or of anything vicious or corrupt; as, the reformation of manners; reformation of the age; reformation of abuses.
v. t.
A proceeding in the nature of a prosecution for some offens against the government, instituted and prosecuted, really or nominally, by some authorized public officer on behalt of the government. It differs from an indictment in criminal cases chiefly in not being based on the finding of a grand juri. See Indictment.
n.
Same as Deforcement, n.
n.
An old theory of the preexistence of germs. Cf. Embo/tement.
n.
A group of beds of the same age or period; as, the Eocene formation.
n.
Specifically (Eccl. Hist.), the important religious movement commenced by Luther early in the sixteenth century, which resulted in the formation of the various Protestant churches.
n.
The manner in which a thing is formed; structure; construction; conformation; form; as, the peculiar formation of the heart.
n.
The act of deforming, or state of anything deformed.
v. t.
News, advice, or knowledge, communicated by others or obtained by personal study and investigation; intelligence; knowledge derived from reading, observation, or instruction.
n.
The act of forming anew; a second forming in order; as, the reformation of a column of troops into a hollow square.
n.
The separation of ripened leaves from a branch or stem; the falling or shedding of the leaves.
n.
Mineral deposits and rock masses designated with reference to their origin; as, the siliceous formation about geysers; alluvial formations; marine formations.