In physics, quantization is the process of
transition from a classical understanding
of physical phenomena to a newer understanding
known as quantum mechanics. (It 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.
== 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. The reason for quantizing
a theory is to deduce properties of materials,
objects or particles through the computation
of quantum amplitudes, which may be very complicated.
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 ===
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 some quantum field theories, such as
quantum chromodynamics which is known to have
a complicated vacuum characterized by many
different condensates.
=== 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 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 says 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 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
ℏ
{\displaystyle \hbar }
-deformed in the same way as in canonical
quantization.
There is also a way to quantize actions with
gauge "flows". It involves the Batalin–Vilkovisky
formalism, an extension of the BRST formalism.
=== Deformation quantization ===
=== 
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.
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
(a functor), 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 3ħ2/2. (This
extra term is actually physically significant,
since it accounts for the nonvanishing angular
momentum of the ground-state Bohr orbit in
the hydrogen atom. As a mere representation
change, however, Weyl's map underlies the
alternate Phase space formulation of conventional
quantum mechanics.
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.
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 ===
See Loop quantum gravity.
=== 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 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.
=== Quantum statistical mechanics approach
===
See Uncertainty principle
=== Schwinger's variational approach ===
See Schwinger's quantum action principle
== 
See also ==
First quantization
Feynman path integral
Light front quantization
Photon polarization
Quantum Hall effect
Quantum number
