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In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, quantization is a procedure for constructing a quantum field theory
Quantum field theory

Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
 starting from a classical field theory
Field (physics)

In physics, a field is a physical quantity associated to each point of spacetime. A field can be classified as a scalar field, a vector field, or a tensor field, according to whether the value of the field at each point is a scalar , a vector , or, more generally, a tensor, respectively....
. This is a generalization of the procedure for building quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
 from classical mechanics
Classical mechanics

Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies....
. One also speaks of field quantization, as in the "quantization of the electromagnetic field
Electromagnetic field

The electromagnetic field is a physical field produced by electric charge. It affects the behavior of charged objects in the vicinity of the field....
", where one refers to photons as field "quanta" (for instance as light quanta). This procedure is basic to theories of particle physics
Particle physics

Particle physics is a branch of physics that studies the elementary particle constituents of matter and radiation, and the interactions between them....
, nuclear physics
Nuclear physics

Nuclear physics is the field of physics that studies the building blocks and interactions of atomic nuclei.The most commonly known applications of nuclear physics are nuclear power and nuclear weapons, but the research field is also the basis for a far wider range of applications, including in the medical sector , in materials engineering...
, condensed matter physics
Condensed matter physics

Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter. In particular, it is concerned with the "condensed" phase that appear whenever the number of constituents in a system is extremely large and the interactions between the constituents are strong....
, and quantum optics
Quantum optics

Quantum optics is a field of research in physics, dealing with the application of quantum mechanics to phenomena involving light and its interactions with matter....
.

Some quantization methods
Quantization converts classical field
Field (physics)

In physics, a field is a physical quantity associated to each point of spacetime. A field can be classified as a scalar field, a vector field, or a tensor field, according to whether the value of the field at each point is a scalar , a vector , or, more generally, a tensor, respectively....
s into operators acting on quantum states of the field theory
Field theory

Field theory may refer to:*Field theory , the theory of the algebraic concept of field*Field theory , a physical theory which employs fields in the physical sense...
.






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In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, quantization is a procedure for constructing a quantum field theory
Quantum field theory

Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
 starting from a classical field theory
Field (physics)

In physics, a field is a physical quantity associated to each point of spacetime. A field can be classified as a scalar field, a vector field, or a tensor field, according to whether the value of the field at each point is a scalar , a vector , or, more generally, a tensor, respectively....
. This is a generalization of the procedure for building quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
 from classical mechanics
Classical mechanics

Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies....
. One also speaks of field quantization, as in the "quantization of the electromagnetic field
Electromagnetic field

The electromagnetic field is a physical field produced by electric charge. It affects the behavior of charged objects in the vicinity of the field....
", where one refers to photons as field "quanta" (for instance as light quanta). This procedure is basic to theories of particle physics
Particle physics

Particle physics is a branch of physics that studies the elementary particle constituents of matter and radiation, and the interactions between them....
, nuclear physics
Nuclear physics

Nuclear physics is the field of physics that studies the building blocks and interactions of atomic nuclei.The most commonly known applications of nuclear physics are nuclear power and nuclear weapons, but the research field is also the basis for a far wider range of applications, including in the medical sector , in materials engineering...
, condensed matter physics
Condensed matter physics

Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter. In particular, it is concerned with the "condensed" phase that appear whenever the number of constituents in a system is extremely large and the interactions between the constituents are strong....
, and quantum optics
Quantum optics

Quantum optics is a field of research in physics, dealing with the application of quantum mechanics to phenomena involving light and its interactions with matter....
.

Some quantization methods


Quantization converts classical field
Field (physics)

In physics, a field is a physical quantity associated to each point of spacetime. A field can be classified as a scalar field, a vector field, or a tensor field, according to whether the value of the field at each point is a scalar , a vector , or, more generally, a tensor, respectively....
s into operators acting on quantum states of the field theory
Field theory

Field theory may refer to:*Field theory , the theory of the algebraic concept of field*Field theory , a physical theory which employs fields in the physical sense...
. The lowest energy state is called the vacuum state
Vacuum state

In quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The term "zero-point field" is sometimes used as a synonym for the vacuum state of an individual quantized field....
 and may be very complicated. The reason for quantizing a theory is to deduce properties of materials, objects or particles through the computation of quantum amplitudes. Such computations have to deal with certain subtleties called renormalization
Renormalization

In quantum field theory, the statistical mechanics of fields, and the theory of self-similarity geometric structures, renormalization refers to a collection of techniques used to take a continuum limit....
, which, if neglected, can often lead to nonsense results, such as the appearance of infinities in various amplitudes. The full specification of a quantization procedure requires methods of performing renormalization.

The first method to be developed for quantization of field theories was canonical quantization
Canonical quantization

In physics, canonical quantization is one of many procedures for quantization a classical theory. Historically, this was the earliest method to be used to build quantum mechanics....
. While this is extremely easy to implement on sufficiently simple theories, there are many situations where other methods of quantization yield more efficient procedures for computing quantum amplitudes. However, the use of canonical quantization
Canonical quantization

In physics, canonical quantization is one of many procedures for quantization a classical theory. Historically, this was the earliest method to be used to build quantum mechanics....
 has left its mark on the language and interpretation of quantum field theory
Quantum field theory

Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
.

Canonical quantization

Main article canonical quantization
Canonical quantization

In physics, canonical quantization is one of many procedures for quantization a classical theory. Historically, this was the earliest method to be used to build quantum mechanics....
.


Canonical quantization of a field theory is analogous to the construction of quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
 from classical mechanics
Classical mechanics

Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies....
. The classical field
Field (physics)

In physics, a field is a physical quantity associated to each point of spacetime. A field can be classified as a scalar field, a vector field, or a tensor field, according to whether the value of the field at each point is a scalar , a vector , or, more generally, a tensor, respectively....
 is treated as a dynamical variable called the canonical coordinate, and its time-derivative is the canonical momentum. One introduces a commutation relation between these which is exactly the same as the commutation relation between a particle's position and momentum in quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
. Technically, one converts the field to an operator, through combinations of creation and annihilation operators
Creation and annihilation operators

In physics, an annihilation operator is an operator that lowers the number of particles in a given state by one.A creation operator is an operator that increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator....
. The field operator acts on quantum state
Quantum state

In quantum physics, a quantum State is a mathematical object that fully describes a Quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus....
s of the theory. The lowest energy state is called the vacuum state
Vacuum state

In quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The term "zero-point field" is sometimes used as a synonym for the vacuum state of an individual quantized field....
. The procedure is also called second quantization
Quantum field theory

Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
.

This procedure can be applied to the quantization of any field
Field (physics)

In physics, a field is a physical quantity associated to each point of spacetime. A field can be classified as a scalar field, a vector field, or a tensor field, according to whether the value of the field at each point is a scalar , a vector , or, more generally, a tensor, respectively....
 theory: whether of fermion
Fermion

In particle physics, fermions are subatomic particle which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In contrast to bosons, which have Bose-Einstein statistics, only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle....
s or boson
Boson

In particle physics, bosons are subatomic particle which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein....
s, and with any internal symmetry. However, it leads to a fairly simple picture of the vacuum state
Vacuum state

In quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The term "zero-point field" is sometimes used as a synonym for the vacuum state of an individual quantized field....
 and is not easily amenable to use in some quantum field theories
Quantum field theory

Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
, such as quantum chromodynamics
Quantum chromodynamics

Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons ....
 which is known to have a complicated vacuum
QCD vacuum

The QCD vacuum is the vacuum state of quantum chromodynamics . It is an example of a non-perturbative vacuum state, characterized by many non-vanishing condensate s such as the gluon condensate or the quark condensate....
 characterized by many different condensates
Vacuum expectation value

In quantum field theory the vacuum expectation value of an Operator is its average, expected value in the Vacuum#The quantum-mechanical vacuum....
.

Covariant canonical quantization


It turns out there is a way to perform a canonical quantization without having to resort to the noncovariant 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 (like for instance actions with a noncausal structure or actions with gauge "flows"
Analysis of flows

In theoretical physics, an analysis of flows is the study of "gauge" or "gaugelike" "symmetries" . It is generally agreed that flows indicate nothing more than a redundancy in the description of the dynamics of a system, but often, it is simpler computationally to work with a redundant description....
). It starts with the classical algebra of all (smooth) functionals over the configuration space. This algebra is quotiented over by the ideal generated by the Euler–Lagrange equations. Then, this quotient algebra is converted into a Poisson algebra by introducing a Poisson bracket derivable from the action, called the Peierls bracket
Peierls bracket

In theoretical physics, the Peierls bracket is an equivalent description of the Poisson bracket. It directly follows from the action and does not require the canonical coordinates and their canonical momenta to be defined in advance....
. This Poisson algebra is then -deformed in the same way as in canonical quantization.

Actually, there is a way to quantize actions with gauge "flows"
Analysis of flows

In theoretical physics, an analysis of flows is the study of "gauge" or "gaugelike" "symmetries" . It is generally agreed that flows indicate nothing more than a redundancy in the description of the dynamics of a system, but often, it is simpler computationally to work with a redundant description....
. It involves the Batalin-Vilkovisky formalism
Batalin-Vilkovisky formalism

In theoretical physics, Batalin-Vilkovisky formalism was developed as a method for determining the Faddeev-Popov ghost structure for theories, such as gravity and supergravity, whose Hamiltonian formalism has constraints not related to a Lie algebra action....
, an extension of the BRST formalism
BRST formalism

In theoretical physics, the BRST formalism is a method of implementing first class constraints. The letters BRST stand for Becchi, Rouet, Stora, and Tyutin who discovered this formalism....
.

Path integral quantization

A classical mechanical theory is given by an action
Action (physics)

In modern physics, action is an attribute of the development of a physical system over a period of time, namely amount by which the Phase of the wave function has changed....
 with the permissible configurations being the ones which are extremal with respect to functional
Functional

Generally, functional refers to something able to fulfill its purpose or function.* Functional form and functionalism apply to architectural design....
 variation
Variation

Variation means a change within a population, or between sub-populations.* Biodiversity* Genetic diversity, differences within a speciesPhysics:...
s of the action. A quantum-mechanical desription of the classical system can also be constructed from the action of the system by means of the path integral formulation
Path integral formulation

The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action of classical mechanics. It replaces the classical notion of a single, unique trajectory for a system with a sum, or functional integral, over an infinity of possible trajectories to compute a probability amplitude....
.

Geometric quantization


See geometric quantization
Geometric quantization

In mathematical physics, geometric quantization is a mathematical approach to defining a Quantum mechanics corresponding to a given classical theory....


Schwinger's variational approach


See quantum action

Deformation Quantization


See
  • Weyl quantization
    Weyl quantization

    In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for systematically associating a "quantum mechanical" Hermitian operator with a "classical" distribution in phase space invertibly....
  • Moyal bracket
    Moyal bracket

    In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space Moyal product.The Moyal Bracket was introduced in 1946 by Hip Groenewold and reprised in 1949 by Jos? Enrique Moyal ....
  • star product
    Moyal product

    In mathematics, the Moyal product, named after Jos? Enrique Moyal, is perhaps the best-known example of a phase-space star product: an associative, non-commutative product, , on the functions on , equipped with its Poisson bracket ....

Quantum statistical mechanics approach


Reference needed.

See also

  • Canonical quantization
    Canonical quantization

    In physics, canonical quantization is one of many procedures for quantization a classical theory. Historically, this was the earliest method to be used to build quantum mechanics....
  • Feynman path integral
  • Quantum field theory
    Quantum field theory

    Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
  • Photon polarization
    Photon polarization

    Photon polarization is the Quantum mechanics description of the Classical physics polarized sinusoidal plane wave electromagnetic wave. Individual photons are completely polarized....
  • quantum Hall effect
    Quantum Hall effect

    The quantum Hall effect is a quantum mechanics version of the Hall effect, observed in 2DEG subjected to low temperatures and strong magnetic fields, in which the Hall Electrical conductivity s takes on the quantized values...


External links