- Quantization (physics)
In

physics ,**quantization**is a procedure for constructing aquantum field theory starting from a classical field theory. This is a generalization of the procedure for buildingquantum mechanics fromclassical mechanics . One also speaks of**field quantization**, as in the "quantization of theelectromagnetic field ", where one refers tophotons as field "quanta" (for instance as light quanta). This procedure is basic to theories ofparticle physics ,nuclear physics ,condensed matter physics , andquantum optics .**Some quantization methods**Quantization converts classical fields into operators acting on

quantum states of thefield theory . The lowest energy state is called thevacuum state and may be very complicated. The reason for quantizing a theory is to deduce properties of materials, objects or particles through the computation ofquantum amplitude s. Such computations have to deal with certain subtleties calledrenormalization , 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 wascanonical 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 ofcanonical quantization has left its mark on the language and interpretation ofquantum field theory .**Canonical quantization**:"Main article

canonical quantization ".Canonical quantization of a field theory is analogous to the construction of

quantum mechanics fromclassical mechanics . The classical field is treated as a dynamical variable called thecanonical coordinate , and its time-derivative is thecanonical momentum . One introduces acommutation relation between these which is exactly the same as the commutation relation between a particle's position and momentum inquantum mechanics . Technically, one converts the field to an operator, through combinations ofcreation and annihilation operators . Thefield operator acts onquantum state s of the theory. The lowest energy state is called thevacuum state . The procedure is also called second quantization.This procedure can be applied to the quantization of any field theory: whether of

fermion s orboson s, and with anyinternal symmetry . However, it leads to a fairly simple picture of thevacuum state and is not easily amenable to use in some quantum field theories, such asquantum chromodynamics which is known to have a complicated vacuum characterized by many different condensates.**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 equation s. Then, this quotient algebra is converted into a Poisson algebra by introducing a Poisson bracket derivable from the action, called thePeierls bracket . This Poisson algebra is then $hbar$-deformed in the same way as in canonical quantization.Actually, there is a way to quantize actions with gauge "flows". It involves the

Batalin-Vilkovisky formalism , an extension of theBRST formalism .**Path integral quantization**A classical mechanical theory is given by an action with the permissible configurations being the ones which are extremal with respect to

functional variation 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 thepath integral formulation .**Geometric quantization**See

geometric quantization **Schwinger's variational approach**See

quantum action **Deformation Quantization**See

*Weyl quantization

*Moyal bracket

* star product**Quantum statistical mechanics approach**Reference needed.

**See also***

Canonical quantization

*Feynman path integral

*Quantum field theory

*Photon polarization

*quantum Hall effect **References*** Abraham, R. & Marsden (1985): "Foundations of Mechanics", ed. Addison-Wesley, ISBN 0-8053-0102-X.

* M. Peskin, D. Schroeder, "An Introduction to Quantum Field Theory" (Westview Press, 1995) [ISBN 0-201-50397-2]

* Weinberg, Steven, "The Quantum Theory of Fields" (3 volumes)**External links*** [

*http://daarb.narod.ru/wircq-eng.html What is "Relativistic Canonical Quantization"?*]

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