In

Physical phenomena
Theoretical physics
Quantum field theory
Mathematical quantization
Mathematical physics

physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which re ...

, quantization (in British English quantisation) is the systematic transition procedure from a classical understanding of physical phenomena to a newer understanding known as quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

. It is a procedure for constructing quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

from classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical m ...

. A generalization involving infinite degrees of freedom is field quantization, as in the "quantization of the electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classica ...

", referring to photons
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...

as field " quanta" (for instance as light quanta). This procedure is basic to theories of atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned wit ...

, chemistry, particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...

, nuclear physics
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter.
Nuclear physics should not be confused with atomic physics, which studies t ...

, condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the ...

, and quantum optics
Quantum optics is a branch of atomic, molecular, and optical physics dealing with how individual quanta of light, known as photons, interact with atoms and molecules. It includes the study of the particle-like properties of photons. Photons have b ...

.
Historical overview

In 1901, whenMax Planck
Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918.
Planck made many substantial contributions to theoretical ...

was developing the distribution function of statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...

to solve ultraviolet catastrophe
The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium would emit an unbounded quantity of energ ...

problem, he realized that the properties of blackbody radiation can be explained by the assumption that the amount of energy must be in countable fundamental units, i.e. amount of energy is not continuous but discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a g ...

. That is, a minimum unit of energy exists and the following relationship holds $E\; =\; h\; \backslash nu$
for the frequency $\backslash nu$. Here, $h$ is called Planck's constant and is a unique constant representing the amount of the quantum mechanical effect. It means a fundamental change of mathematical model of physical quantities.
In 1905, Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...

published a paper "On a heuristic viewpoint concerning the emission and transformation of light" explaining photoelectric effect
The photoelectric effect is the emission of electrons when electromagnetic radiation, such as light, hits a material. Electrons emitted in this manner are called photoelectrons. The phenomenon is studied in condensed matter physics, and solid sta ...

based on quantized electromagnetic waves
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) ligh ...

. The ''energy quantum'' referred to in this paper was later called "photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...

". In July 1913, Niels Bohr
Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922. ...

used quantization to describe the spectrum of a hydrogen atom in his paper "'On the Constitution of Atoms and Molecules''.
The preceding theories have been successful, but they are very phenomenological theories. However, the French mathematician Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...

first gave a systematic and rigorous definition of what quantization is in his 1912 paper ''"Sur la théorie des quanta"''.
The term "quantum physics" was first used in Johnston's ''Planck's Universe in Light of Modern Physics''. (1931).
Canonical quantization

Canonical quantization developsquantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

from classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical m ...

. One introduces a commutation relation
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...

among canonical coordinate
In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cla ...

s. Technically, one converts coordinates to operators, through combinations of creation and annihilation operators. The operators act on quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...

s of the theory. The lowest energy state is called the vacuum state
In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used ...

.
Quantization schemes

Even within the setting of canonical quantization, there is difficulty associated to quantizing arbitrary observables on the classical phase space. This is the ordering ambiguity: Classically, the position and momentum variables ''x'' and ''p'' commute, but their quantum mechanical operator counterparts do not. Various ''quantization schemes'' have been proposed to resolve this ambiguity, of which the most popular is the Weyl quantization scheme. Nevertheless, the ''Groenewold–van Hove theorem'' dictates that no perfect quantization scheme exists. Specifically, if the quantizations of ''x'' and ''p'' are taken to be the usual position and momentum operators, then no quantization scheme can perfectly reproduce the Poisson bracket relations among the classical observables. See Groenewold's theorem for one version of this result.Covariant canonical quantization

There is a way to perform a canonical quantization without having to resort to the non covariant approach of foliating spacetime and choosing a Hamiltonian. This method is based upon a classical action, but is different from the functional integral approach. The method does not apply to all possible actions (for instance, actions with a noncausal structure or actions with gauge "flows"). It starts with the classical algebra of all (smooth) functionals over the configuration space. This algebra is quotiented over by the ideal generated by theEuler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...

s. Then, this quotient algebra is converted into a Poisson algebra by introducing a Poisson bracket derivable from the action, called the Peierls bracket
In theoretical physics, the Peierls bracket is an equivalent description of the Poisson bracket. It can be defined directly from the action and does not require the canonical coordinates and their canonical momenta to be defined in advance.
The ...

. This Poisson algebra is then ℏ -deformed in the same way as in canonical quantization.
In quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles a ...

, there is also a way to quantize actions with gauge "flows". It involves the Batalin–Vilkovisky formalism, an extension of the BRST formalism BRST may refer to:
* BRST Films, a Serbian video production company
* BRST algorithm, an optimization algorithm suitable for finding the global optimum of black box functions
* BRST quantization in Yang-Mills theories, a way to quantize a gauge-symm ...

.
Deformation quantization

One of the earliest attempts at a natural quantization was Weyl quantization, proposed by Hermann Weyl in 1927. Here, an attempt is made to associate a quantum-mechanical observable (a self-adjoint operator on a Hilbert space) with a real-valued function on classical phase space. The position and momentum in this phase space are mapped to the generators of the Heisenberg group, and the Hilbert space appears as a group representation of the Heisenberg group. In 1946, H. J. Groenewold considered the product of a pair of such observables and asked what the corresponding function would be on the classical phase space. This led him to discover the phase-space star-product of a pair of functions. More generally, this technique leads to deformation quantization, where the ★-product is taken to be a deformation of the algebra of functions on a symplectic manifold or Poisson manifold. However, as a natural quantization scheme (afunctor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...

), Weyl's map is not satisfactory.
For example, the Weyl map of the classical angular-momentum-squared is not just the quantum angular momentum squared operator, but it further contains a constant term . (This extra term offset is pedagogically significant, since it accounts for the nonvanishing angular momentum of the ground-state Bohr orbit in the hydrogen atom, even though the standard QM ground state of the atom has vanishing .)
As a mere ''representation change'', however, Weyl's map is useful and important, as it underlies the alternate ''equivalent'' phase space formulation
The phase-space formulation of quantum mechanics places the position ''and'' momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position ''or'' momentum representations (see also position and mome ...

of conventional quantum mechanics.
Geometric quantization

In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in. A more geometric approach to quantization, in which the classical phase space can be a general symplectic manifold, was developed in the 1970s by Bertram Kostant and Jean-Marie Souriau. The method proceeds in two stages. Chapters 22 and 23 First, once constructs a "prequantum Hilbert space" consisting of square-integrable functions (or, more properly, sections of a line bundle) over the phase space. Here one can construct operators satisfying commutation relations corresponding exactly to the classical Poisson-bracket relations. On the other hand, this prequantum Hilbert space is too big to be physically meaningful. One then restricts to functions (or sections) depending on half the variables on the phase space, yielding the quantum Hilbert space.Loop quantization

SeeLoop quantum gravity
Loop quantum gravity (LQG) is a theory of quantum gravity, which aims to merge quantum mechanics and general relativity, incorporating matter of the Standard Model into the framework established for the pure quantum gravity case. It is an attem ...

.
Path integral quantization

A classical mechanical theory is given by anaction
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...

with the permissible configurations being the ones which are extremal with respect to functional variations of the action. A quantum-mechanical description of the classical system can also be constructed from the action of the system by means of the path integral formulation
The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional ...

.
Quantum statistical mechanics approach

SeeUncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...

.
Schwinger's variational approach

SeeSchwinger's quantum action principle
The Schwinger's quantum action principle is a variational approach to quantum mechanics and quantum field theory. This theory was introduced by Julian Schwinger in a series of articles starting 1950.
Approach
In Schwingers approach, the a ...

.
See also

*First quantization
A first quantization of a physical system is a possibly semiclassical treatment of quantum mechanics, in which particles or physical objects are treated using quantum wave functions but the surrounding environment (for example a potential well ...

* Feynman path integral
* Light front quantization
* Photon polarization
* Quantum Hall effect
* Quantum number
In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be ...

* Stochastic quantization
References

* Abraham, R. & Marsden (1985): ''Foundations of Mechanics'', ed. Addison–Wesley, *Ali, S. T., & Engliš, M. (2005). "Quantization methods: a guide for physicists and analysts". ''Reviews in Mathematical Physics'' 17 (04), 391-490. * * G. Giachetta, L. Mangiarotti, G. Sardanashvily, ''Geometric and Algebraic Topological Methods in Quantum Mechanics'' (World Scientific, 2005) * * M. Peskin, D. Schroeder, ''An Introduction to Quantum Field Theory'' (Westview Press, 1995) {{ISBN, 0-201-50397-2 * Todorov, Ivan (2012). "Quantization is a mystery." arXiv preprint arXiv:1206.3116 (2012) * Weinberg, Steven, ''The Quantum Theory of Fields'' (3 volumes)Notes