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Converting classical mechanics to quantum mechanics
First quantization is a procedure for converting equations of classical particle equations into quantum wave equations. The companion concept of second
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
Process in quantum mechanical theories
In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries
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)
Signal encoder
the quantized signal. The incorporation of the decoder inside the encoder allows quantization of the differences, including nonlinear quantization, in
Differential pulse-code modulation
Differential_pulse-code_modulation
Type of machine learning model
preserving most of its performance. Quantization can be further classified as static quantization if the quantization parameters are determined beforehand
Large_language_model
Image processing technique
In computer graphics, color quantization or color image quantization is quantization applied to color spaces; it is a process that reduces the number of
Color_quantization
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
Topics referred to by the same term
Look up quantization in Wiktionary, the free dictionary. Quantization is the process of constraining an input from a continuous or otherwise large set
Quantization
Predecessor to modern quantum mechanics (1900–1925)
orientations, so that the process of quantization does not pick out a preferred axis. For this reason, the name "space quantization" fell out of favor, and the
Old_quantum_theory
Continuous progression from past to future
down in the figure depending on the acceleration of the observer. Time quantization is a hypothetical concept that time has a smallest possible unit. In
Time
Quantization giving rise to photons
function is derived from a "first" quantized Hamiltonian, the quantization of the Schrödinger field is the second time quantization is performed, hence the
Quantization of the electromagnetic field
Quantization_of_the_electromagnetic_field
Atomic model introduced by Niels Bohr in 1913
the quantized angular momentum condition of the Bohr model with an additional radial quantization condition, the Wilson–Sommerfeld quantization condition
Bohr_model
Physical model of propagating energy
sometimes known as the Planck–Einstein equation. In quantum theory (see first quantization) the energy of the photons is thus directly proportional to the frequency
Electromagnetic_radiation
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
Video encoding algorithm
Adaptive Quantization Not Suck". 2008-02-22. Archived from the original on 2020-08-07. Implementation of variance-based adaptive quantization in x264 The
Variance Adaptive Quantization
Variance_Adaptive_Quantization
Method for converting signals between digital and analog
feedback loop during quantization to the lower bit depth that continuously corrects quantization errors and moves quantization noise to higher frequencies
Delta-sigma_modulation
Charge carried by one electron
multiples of the charge on the down quark, an observation known as charge quantization. Thus the charge on the down quark, 1/3 e can be considered the fundamental
Elementary_charge
Hypothetical particle with one magnetic pole
charge quantization has appeared. (The concept of local gauge invariance—see gauge theory—provides a natural explanation of charge quantization, without
Magnetic_monopole
Online vector quantization algorithm
searched efficiently. Earlier approaches to vector quantization include product quantization, scalar quantization, and data-dependent k-means codebook construction
TurboQuant
Lossy compression technique
Specific applications include DCT data quantization in JPEG and DWT data quantization in JPEG 2000. Color quantization reduces the number of colors used in
Quantization (image processing)
Quantization_(image_processing)
Minimum amount of a physical entity involved in an interaction
levels within an atom. Quantization is one of the foundations of the much broader physics of quantum mechanics. Quantization of energy and its influence
Quantum
Quantized unit of magnetic flux
on the values of the magnetic flux is called magnetic flux quantization. The quantization of the magnetic flux originates from the requirement that the
Magnetic_flux_quantum
Binary file format for storing machine-learning models
2-bit to 8-bit quantized integer types, common floating-point data formats such as float32, float16 and bfloat16, and 1.58-bit quantization. Several "K-quant"
GGUF
also be "water". Hence, the predicate "apple" is quantized, while "water" is not. Formally, a quantization predicate QUA can be defined as follows, where
Quantization_(linguistics)
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
Digital video coder/decoder
the quantization stage. There, a best-effort approximation of inversion is performed. This part of the process is often called inverse quantization or
Video_codec
Noise that reduces quantization error
produce. Dither is added before any quantization or re-quantization process, in order to decorrelate the quantization noise from the input signal and to
Dither
System that converts an analog signal into a digital signal
signal. The conversion involves quantization of the input, so it necessarily introduces a small amount of quantization error. Furthermore, instead of continuously
Analog-to-digital_converter
Quantum mechanical operator related to rotational symmetry
{\displaystyle \hbar } is reduced Planck constant: A common way to derive the quantization rules above is the method of ladder operators. The ladder operators for
Angular_momentum_operator
Number of bits of information recorded for each digital audio sample
determined by quantization error. The bit depth has no impact on the frequency response, which is constrained by the sample rate. Quantization error introduced
Audio_bit_depth
Extension of the Bohr model
the quantized angular momentum condition of the Bohr model with a radial quantization (condition by William Wilson, the Wilson–Sommerfeld quantization condition):
Bohr–Sommerfeld_model
German-born theoretical physicist (1879–1955)
received skeptically at first. In another major paper from this era, Einstein observed that de Broglie waves could explain the quantization rules of Bohr and
Albert_Einstein
Image dithering algorithm
meaning it pushes (adds) the residual quantization error of a pixel onto its neighboring pixels, to be quantized after. It spreads the debt out according
Floyd–Steinberg_dithering
Tabular arrangement of the chemical elements
number. The Danish physicist Niels Bohr applied Max Planck's idea of quantization to the atom. He concluded that the energy levels of electrons were quantised:
Periodic_table
State of matter at low temperatures
the rotating state consists of quantized vortices. That is, when the container is rotated at speeds below the first critical angular velocity, the liquid
Superfluid_helium-4
Physical hypothesis
Redshift quantization, also referred to as redshift periodicity, redshift discretization, preferred redshifts and redshift-magnitude bands, is the hypothesis
Redshift_quantization
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
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
Chinese artificial intelligence company
release of GLM-4.6 in late September 2025 marked the first integration of FP8 and Int4 quantization on Cambricon chips. It also supports native FP8 on Moore
Z.ai
contains all the information about the state of that system. This is the first quantization approach and historically Bose–Einstein and Fermi–Dirac correlations
Bose–Einstein_correlations
Theoretical framework in physics
systems. The process of quantizing an arbitrary number of particles instead of a single particle is often also called second quantization. The foregoing procedure
Quantum_field_theory
Quantized flux circulation of some physical quantity
Fritz London published a very closely related work on magnetic flux quantization in superconductors. London's fluxoid can also be viewed as a quantum
Quantum_vortex
First-order fluid First-order transition First Step to Nobel Prize in Physics First class constraint First law of thermodynamics First quantization First
Index_of_physics_articles_(F)
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
The quantized state systems (QSS) methods are a family of numerical integration solvers based on the idea of state quantization, dual to the traditional
Quantized state systems method
Quantized_state_systems_method
Mathematics award
with the first-ever IMU silver plaque in recognition of his proof of Fermat's Last Theorem. Don Zagier referred to the plaque as a "quantized Fields Medal"
Fields_Medal
Secure speech system
"firsts"; this list is taken from (Bennett, 1983): The first realization of enciphered telephony The first quantized speech transmission The first transmission
SIGSALY
Description of physical properties at the atomic and subatomic scale
of quantum field theory, which applies quantization to a field (rather than a fixed set of particles). The first complete quantum field theory, quantum
Quantum_mechanics
Special collection of states in closed string theory
configuration of sectors in orbifold conformal field theories. In the first quantized formalism of string theory (or in two-dimensional conformal field theory)
Twisted_sector
deformation quantization of the corresponding Poisson algebra. It is due to Maxim Kontsevich. Given a Poisson algebra (A, {⋅, ⋅}), a deformation quantization is
Kontsevich quantization formula
Kontsevich_quantization_formula
Digital representation of sampled analog signals
type of PCM in which the quantization levels are linearly uniform. This is in contrast to PCM encodings in which quantization levels vary as a function
Pulse-code_modulation
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
brackets generated. First- and second-class constraints were introduced by Dirac (1950, p. 136, 1964, p. 17) as a way of quantizing mechanical systems
First-class_constraint
Mathematical description of quantum state
annihilation of particles and not external potentials as in ordinary "first quantized" quantum theory. In string theory, the situation remains analogous
Wave_function
Elementary particle or quantum of light
energy quantization resulted from some unknown constraint on the matter that absorbs or emits the radiation. In 1905, Einstein was the first to propose
Photon
Algebraic construct of interest in theoretical physics
though they are in some sense 'close' to a group. The term "quantum group" first appeared in the theory of quantum integrable systems, which was then formalized
Quantum_group
classical mechanics and result derived from Schrödinger equation. first quantization x → x ^ , p → i ℏ ∂ ∂ x {\displaystyle x\to {\hat {x}},\,p\to i\hbar
Glossary of elementary quantum mechanics
Glossary_of_elementary_quantum_mechanics
Solitons in Euclidean spacetime
(Euclidean, i. e., with imaginary time) (1 + 1)-dimensional field theory – first quantized quantum mechanical description – allows to be interpreted as a tunneling
Instanton
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
Theory of interwoven space and time by Albert Einstein
interactions.[citation needed] Instead, a theory of particles interpreted as quantized fields, called quantum field theory, becomes necessary; in which particles
Special_relativity
Commuting Lie algebra operator
typically have many operators of this kind. In string theory, in the first quantized formalism, these operators also have the interpretation of winding
Central_charge
Open source amateur radio mode
US-based company Connect Systems, Inc. released the CS7000-M17, being the first commercial off-the-shelf handheld transceiver with native M17 support. Links
M17_(amateur_radio)
American theoretical physicist (1918–1988)
the prescient comment that the theory would be extremely difficult to quantize, and Einstein said that one might try to apply this method to gravity in
Richard_Feynman
Physical law for entropy and heat
blackbody energy formula, Planck postulated that the energy of a photon was quantized (partly to simplify the mathematics), thereby starting quantum theory
Second_law_of_thermodynamics
Amateur radio communications software
compound callsign and/or 6 digit locator use a two-transmission sequence. The first transmission carries compound callsign and power level, or standard callsign
WSPR_(amateur_radio_software)
Software
propagation and testing moon bounce echo. FSK441, introduced in 2001 as the first communications mode included with WSJT, is designed to support meteor scatter
WSJT_(amateur_radio_software)
Chinese artificial intelligence company
execute 200-300 sequential tool calls autonomously and uses native INT4 quantization for efficiency. Moonshot reported scores above GPT-5 and Claude Sonnet
Moonshot_AI
Intrinsic quantum property of particles
momentum. The relativistic spin–statistics theorem connects electron spin quantization to the Pauli exclusion principle: observations of exclusion imply half-integer
Spin_(physics)
Data structure in computer science
results of octree color quantization. The first image is the original (532818 distinct colors), while the second is the quantized image (184 distinct colors)
Octree
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
Large language model developed by Google
Gemini Pro was said to have outperformed GPT-3.5. Gemini Ultra was also the first language model to outperform human experts on the 57-subject Massive Multitask
Gemini_(language_model)
experiment: quantization should not have been observable. At least five years would elapse before this mystery was resolved: quantization was observed
History_of_quantum_mechanics
Technique in computational quantum field theory
The light-front quantization of quantum field theories provides a useful alternative to ordinary equal-time quantization. In particular, it can lead to
Light-front computational methods
Light-front_computational_methods
Lossy compression method for reducing the size of digital images
domain, the process of reducing information is called quantization. In simpler terms, quantization is a method for optimally reducing a large number scale
JPEG
GPU microarchitecture designed by Nvidia
architecture introduced the Transformer Engine, software to facilitate quantization of higher-precision models (e.g., FP32) to lower precision, for which
Blackwell_(microarchitecture)
Semi-classical method for computing quantum eigenvalues
eigenvalues in quantum-mechanical systems. EBK quantization is an improvement from Bohr–Sommerfeld quantization which did not consider the caustic phase jumps
Einstein–Brillouin–Keller method
Einstein–Brillouin–Keller_method
Replacing a number with a simpler value
as "the table-maker's dilemma". Rounding has many similarities to the quantization that occurs when physical quantities must be encoded by numbers or digital
Rounding
Something that has mass and volume
ISBN 978-0-521-55002-4. Masujima, M. (2008). Path Integral Quantization and Stochastic Quantization. Springer. p. 103. ISBN 978-3-540-87850-6. Hale, Bob (19
Matter
Hungarian and American mathematician and physicist (1903–1957)
Doran, Robert S.; Kadison, Richard V., eds. (2004). Operator Algebras, Quantization, and Noncommutative Geometry: A Centennial Celebration Honoring John
John_von_Neumann
Digital audio format
when a transient is detected. Doing so limits the temporal spread of quantization noise accompanying the transient (see psychoacoustics). Frequency resolution
MP3
Conjecture on zeros of the zeta function
Michael Berry and Jonathan Keating conjectured that there is some unknown quantization H ^ {\displaystyle {\hat {H}}} of the classical Hamiltonian H = xp so
Riemann_hypothesis
Chemical element with atomic number 1 (H)
According to quantum theory, this behavior arises from the spacing of the (quantized) rotational energy levels, which are particularly wide-spaced in H2 because
Hydrogen
1922 physical experiment demonstrating that atomic spin is quantized
to their quantized spin. Historically, this experiment was decisive in convincing physicists of the reality of angular-momentum quantization in all atomic-scale
Stern–Gerlach_experiment
Force resulting from the quantisation of a field
field in question. The second quantization of quantum field theory requires that each such ball-spring combination be quantized, that is, that the strength
Casimir_effect
Gauge fixing procedure
In quantum field theory, the Gupta–Bleuler formalism is a way of quantizing the electromagnetic field. The formulation is due to theoretical physicists
Gupta–Bleuler_formalism
Family of large language models by Google
of Gemma 3 1B, which has a context length of 32K. Quantized versions fine-tuned using quantization-aware training (QAT) are also available, offering sizable
Gemma_(language_model)
numbers and the arithmetic operations done with them. This is a form of quantization error. When using approximation equations or algorithms, especially when
Glossary_of_computer_science
Average uncertainty in variable's states
distribution is to describe than a distribution that is uniform over its quantization scheme. Another useful measure of entropy that works equally well in
Entropy_(information_theory)
Mathematical transform that expresses a function of time as a function of frequency
treated as a separate harmonic oscillator and then quantized, a procedure known as "second quantization". Fourier methods have been adapted to also deal
Fourier_transform
Italian-American physicist (1901–1954)
his exclusion principle in 1925, Fermi responded with a paper "On the quantization of the perfect monoatomic gas" (Sulla quantizzazione del gas perfetto
Enrico_Fermi
Video encoding standard
absence of any quantization and rounding errors). In reality, there are some (sometimes large) rounding errors introduced both by quantization in the encoder
MPEG-1
Type of continuous-phase frequency-shift keying
standard minimum-shift keying (MSK); however, the digital data stream is first shaped with a Gaussian filter before being applied to a frequency modulator
Minimum-shift_keying
Hiding messages in other messages
work on steganography is the first-ever effort to improve the codebook partition by using Graph theory along with Quantization Index Modulation in low bit-rate
Steganography
Video or display resolutions with a width of around 4,000 pixels
characteristics as defined in the standard, including color primaries, quantization parameters, and the electro-optical transfer function. These are the
4K_resolution
Asymmetry of classical and quantum action
conserved as a classical symmetry of electrodynamics, but is broken by the quantized theory. The relationship of this anomaly to the Atiyah–Singer index theorem
Anomaly_(physics)
Digital audiovisual data interface
CEA-861-E document defines "video formats and waveforms; colorimetry and quantization; transport of compressed and uncompressed LPCM audio; carriage of auxiliary
HDMI
Type of analog-to-digital converter
continuous analog waveform using a binary search through all possible quantization levels. A successive-approximation analog-to-digital converter contains
Successive-approximation_ADC
Telecommunications and signal processing technique
limited-range uniform quantizer, and a (continuous-domain) signal dynamic range expander that inverts the compressor function. This type of quantization is frequently
Companding
Theory of forces and subatomic particles
ISSN 0218-2718. Carney, Daniel (2024). "Graviton detection and the quantization of gravity". Physical Review D. 109 (4) 044009. arXiv:2308.12988. Bibcode:2024PhRvD
Standard_Model
travel, tourism, insurance
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