In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the procedure for building quantum mechanics from classical mechanics. One also speaks of field quantization, as in the "quantization of the electromagnetic field", where one refers to photons as field "quanta" (for instance as light quanta). This procedure is basic to theories of particle physics, nuclear physics, condensed matter physics, and quantum optics. A Superconductor demonstrating the Meissner Effect. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... In physics, a field is an assignment of a quantity to every point in space (or more generally, spacetime). ... A simple introduction to this subject is provided in Basics of quantum mechanics. ... It has been suggested that this article or section be merged with Newtonian mechanics. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In physics, the photon (from Greek φως, phōs, meaning light) is the quantum of the electromagnetic field; for instance, light. ... In physics, the photon (from Greek φοτος, meaning light) is a quantum of excitation of the quantised electromagnetic field and is one of the elementary particles studied by quantum electrodynamics (QED) which is the oldest part of the Standard Model of particle physics. ... Particles erupt from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... Nuclear physics is the branch of physics concerned with the nucleus of the atom. ... Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ... 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 fields into operators acting on quantum states of the field theory. The lowest energy state is called the vacuum state 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, 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. In physics, a field is an assignment of a quantity to every point in space (or more generally, spacetime). ... Quite literally, quantum state describes the state of a quantum system. ... Field theory (mathematics), the theory of the algebraic concept of field. ... In quantum field theory, the vacuum state, usually denoted , is the element of the Hilbert space with the lowest possible energy, and therefore containing no physical particles. ... Look up Vacuum in Wiktionary, the free dictionary For other uses, see vacuum (disambiguation) A vacuum is a volume of space that is empty of matter, including air, so that gaseous pressure is much less than standard atmospheric pressure. ... In quantum mechanics, a probability amplitude is a complex number-valued function which describes an uncertain or unknown quantity. ... Figure 1. ...

The first method to be developed for quantization of field theories was canonical quantization. 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 has left its mark on the language and interpretation of quantum field theory. In physics, a field is an assignment of a quantity to every point in space. ... In physics, canonical quantization is one of many procedures for quantizing a classical theory. ... In physics, canonical quantization is one of many procedures for quantizing a classical theory. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ...

Canonical quantization

Canonical quantization of a field theory is analogous to the construction of quantum mechanics from classical mechanics. The classical field 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. Technically, one converts the field to an operator, through combinations of creation and annihilation operators. The field operator acts on a quantum states of the theory. The lowest energy state is called the vacuum state. The procedure is also called second quantization. In physics, canonical quantization is one of many procedures for quantizing a classical theory. ... A simple introduction to this subject is provided in Basics of quantum mechanics. ... It has been suggested that this article or section be merged with Newtonian mechanics. ... In physics, a field is an assignment of a quantity to every point in space (or more generally, spacetime). ... In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ... In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ... For an electrical switch that periodically reverses the current see commutator (electric) In mathematics the commutator of two elements g and h of a group G is the element g −1 h −1 gh, often denoted by [ g, h ]. It is equal to the groups identity if and only... A simple introduction to this subject is provided in Basics of quantum mechanics. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In physics, canonical quantization is one of many procedures for quantizing a classical theory. ... A quantum state is any possible state in which a quantum mechanical system can be. ... In quantum field theory, the vacuum state, usually denoted , is the element of the Hilbert space with the lowest possible energy, and therefore containing no physical particles. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ...

This procedure can be applied to the quantization of any field theory: whether of fermions or bosons, and with any internal symmetry. However, it leads to a fairly simple picture of the vacuum state and is not easily amenable to use in a quantum field theory (such as quantum chromodynamics) which is known to have a complicated vacuum characterized by many different condensates. In physics, a field is an assignment of a quantity to every point in space (or more generally, spacetime). ... In particle physics, fermions, (named after Enrico Fermi), are particles with semi-integer spin. ... In physics, bosons, named after Satyendra Nath Bose, are particles with integer spin. ... In physics, a field is an assignment of a quantity to every point in space. ... In quantum field theory, the vacuum state, usually denoted , is the element of the Hilbert space with the lowest possible energy, and therefore containing no physical particles. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... Quantum chromodynamics (QCD) is the theory of the strong interaction, a fundamental force describing the interactions of the quarks and gluons found in nucleons (such as the proton and neutron). ... The QCD vacuum is the vacuum state of quantum chromodynamics (QCD). ... In quantum field theory the vacuum expectation value (also called condensate) of an operator is its average, expected value in the vacuum. ...

For further details consult the article on canonical quantization. In physics, canonical quantization is one of many procedures for quantizing a classical theory. ...

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"). 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. This Poisson algebra is then -deformed in the same way as in canonical quantization. In theoretical physics, an analysis of flows is the study of gauge or gaugelike symmetries (i. ... In physics, the action principle is an assertion about the nature of motion from which the trajectory of an object subject to forces can be determined. ... In theoretical physics, the Peierls bracket is an equivalent description of the Poisson bracket. ...

Actually, there is a way to quantize actions with gauge "flows". It involves the Batalin-Vilkovisky formalism, an extension of the BRST formalism. In theoretical physics, an analysis of flows is the study of gauge or gaugelike symmetries (i. ... In theoretical physics, Batalin-Vilkovisky (BV) formalism was developed as a method for determining the ghost structure for theories, such as gravity and supergravity, whose Hamiltonian formalism has constraints not related to a Lie algebra action. ... In theoretical physics, the BRST formalism is a method of implementing first class constraints. ...

Path integral quantization

The classical theory is given by an action with the permissible configurations being the ones which are extremal with respect to functional variations of the action. The quantum-mechanical counterpart of this is the path integral formulation. This article is about a formulation of quantum mechanics. ... In physics, the action principle is an assertion about the nature of motion, from which the trajectory of an object subject to forces can be determined. ... This article is in need of attention from an expert on the subject. ...

Geometric quantization

See geometric quantization In mathematical physics, geometric quantization is a mathematical approach to define a quantum theory corresponding to a given classical theory 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...

Schwinger's variational approach

See quantum action In Schwingers variational approach to quantum field theory, the quantum action is an operator. ...

Deformation Quantization

See http://idefix.physik.uni-freiburg.de/~star/en/index.html

Quantum statistical mechanics approach

See

• Moyal bracket
• Moyal product

In mathematics, the Moyal product generalizes the idea of the Poisson bracket, with the goal of defining a product of functions on a symplectic manifold that resembles in certain ways the operator product of observables in quantum mechanics. ...

See also

• Canonical quantization
• Feynman path integral
• Quantum field theory

In physics, canonical quantization is one of many procedures for quantizing a classical theory. ... This article is about a formulation of quantum mechanics. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ...

References

• M. Peskin, D. Schroeder, An Introduction to Quantum Field Theory (Westview Press, 1995) [ISBN 0201503972]
• Weinberg, Steven, The Quantum Theory of Fields (3 volumes)

External links

• D. A. Arbatsky, What is "Relativistic Canonical Quantization"?