Quantization (physics)

## Contents

* 1 Overview
* 2 Some quantization methods
* 2.1 Canonical quantization
* 2.2 Covariant canonical quantization
* 2.3 Path integral quantization
* 2.4 Geometric quantization
* 2.5 Schwinger's variational approach
* 2.6 Deformation Quantization
* 2.7 Quantum statistical mechanics approach
* 3 See also
* 4 References

## Overview

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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.

## Some quantization methods

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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.

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.

### Canonical quantization

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Main article 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 quantum states of the theory. The lowest energy state is called the vacuum state. The procedure is also called second quantization.

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.

### Covariant canonical quantization

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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 <math>\hbar</math>-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 the BRST formalism.

### Path integral quantization

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Further information: Feynman path integral

A classical mechanical theory is given by an action with the permissible configurations being the ones which are extremal with respect to functional variations 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.

### Geometric quantization

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See geometric quantization

### Schwinger's variational approach

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See quantum action

### Deformation Quantization

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See Weyl quantization

### Quantum statistical mechanics approach

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See

* Moyal bracket
* star product
* phase-space quantization

## See also

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* Canonical quantization
* Feynman path integral
* Quantum field theory
* Photon polarization
* quantum Hall effect

## References

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* 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)

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