Canonical quantization

In physics, canonical quantization is one of many procedures for quantizing a classical theory. Historically, this was the earliest method to be used to build quantum mechanics. When applied to a classical field theory it was initially called second quantization. This name has now fallen out of fashion. The word canonical refers actually to a certain structure of the classical theory (called the symplectic structure) which is preserved in the quantum theory. This was first emphasized by Paul Dirac, in his attempt to build quantum field theory.
Contents 
History
Commutators were introduced by Werner Heisenberg; wavefunctions, by Erwin Schrödinger. The connection between the two was discovered by Paul Dirac, who was also the first person to apply this technique to the quantization of the electromagnetic field. Eugene Wigner and Pascual Jordan were the first to quantize the electron field, whose quantum mechanics was first investigated by Dirac. The name canonical quantization may have been first coined by Pascual Jordan.
The exposition here leans heavily on Dirac's influential book on quantum mechanics. This route to quantum mechanics is through uncertainty principle. A later development was the Feynman path integral, a formulation of quantum theory which emphasizes the role of superposition of quantum amplitudes. Needless to say, the two methods give the same results.
Quantum mechanics
In the classical mechanics of a particle one has dynamical variables which are called coordinates (q) and momenta (p). These specify the state of a classical system. The canonical structure (also known as the symplectic structure) of classical mechanics consists of Poisson brackets between these variables. All transformations which keep these brackets unchanged are allowed as canonical transformations in classical mechanics.
In quantum mechanics, these dynamical variables become operators acting on a Hilbert space of quantum states. The Poisson brackets are replaced by commutators, [q,p] = qppq = 1. This readily yields up the uncertainty principle in the form ΔpΔq ≥ 1. This algebraic structure corresponds to a generalization of the canonical structure of classical mechanics.
The states of a quantum system can be labelled by the eigenvalues of any operator. For example, one may write x> for a state which is an eigenvector of q with eigenvalue x. Notationally, one would write this as qx> = xx>. The wavefunction of a state φ> is φ(x)=<xφ>.
In quantum mechanics one deals with the quantum states of a system of a fixed number of particles. This is inadequate for the study of systems in which particles are created and destroyed. Historically, this problem was solved through the introduction of quantum field theory.
Second quantization: field theory
When the canonical quantization procedure is applied to quantum field theory, the classical field variable becomes a quantum operator which acts on a quantum state of the field theory to increase or decrease the number of particles by one. In one way of viewing things, quantizing the classical theory of a fixed number of particles gave rise to a wavefunction. This wavefunction is a field variable which could then be quantized to deal with the theory of many particles. So the process of canonical quantization of a field theory was called second quantization in the early literature.
The rest of this article deals with canonical quantization of field theory. It would be useful to also consult the companion articles on quantum field theory, quantization and the Feynman path integral.
Field operator
One basic notion in this technique is of a vacuum state of a quantum field theory. This is a quantum state containing zero particles. For further elaboration and niceties, see the articles on the quantum mechanical vacuum and the vacuum of quantum chromodynamics. We shall represent this quantum state as 0>.
Then one introduces single particle creation and annihilation operators, a^{+}_{k} and a_{k} respectively, which act on quantum states to increase or decrease the number of particles of the given momentum k. For example—
 a_{k}0>=0, since the vacuum state has no particles, and therefore a state with smaller number of particles cannot exist
 a^{+}_{k}0>=1(k)>, where we have introduced the notation n(k)> to denote the state with n particles of momentum k.
The Hilbert space of states of this kind is called a Fock space and these kinds of states are called Fock states. They are an useful basis with which to discuss quantum field theory, although strictly, their use is limited only to free field theory.
Real scalar field
A classical scalar field can be written as a quantum field operator now by the following simple recipe—
 Make a Fourier transformation of the classical field to find the Fourier coefficients φ(k) and φ^{*}(k). The first corresponds to positive frequencies, and the second, to negative.
 Convert each Fourier coefficient into an operator φ(k)→φ(k) a_{k} and φ^{*}(k)→φ^{*}(k) a^{+}_{k}.
 Reconstruct the field operator by putting together this operator valued Fourier expansion.
Other fields
All other fields can be quantized by a generalization of this procedure. Vector or tensor fields simply have more components, and independent creation and destruction operators must be introduced for each independent component. If a field has any internal symmetry, then creation and destruction operators must be introduced for each component of the field related to this symmetry as well. If there is a gauge symmetry, then the number of independent components of the field must be carefully analyzed. This usually involves gauge fixing.
We have introduced the commutator of two operators, [A,B]. Before producing further we need the anticommutator, which is {A,B} = AB+BA. Note that [A,B]=[B,A], but {A,B}={B,A}.
For all the fields we have named till now, one uses boson creation and annihilation operators. This means that the operators satisfy the commutation relations [a_{k},a^{+}_{k}]=1. All other commutators vanish. To quantize spinor fields, corresponding to fermions, we need to use operators which satisfy the anticommutation relations {a_{k},a^{+}_{k}}=1, and that all other anticommutators vanish.
Condensates
Note that the vacuum expectation value (VEV) <0φ0>=0. Thus, the canonical quantization procedure does not allow for a field condensate in the vacuum state, irrespective of the Lagrangian. The only exception to this is to shift the field by a constant before embarking on the process above, ie, quantize the field φ(x,t)v, where v is a number and not an operator. The quantity v then denotes the condensate of the field φ, and the particle states become the excitations over the new vacuum defined with this condensate. The VEV of any power (or other function) of φ can then be expressed in terms of v. Thus, this procedure allows only a single condensate. This construction is used in the Higgs mechanism which is needed to construct the standard model of particle physics.
Why "canonical"?
Why is this process called canonical quantization? This is because of the strong connection that classical field theory has with classical mechanics, and which is sought to be preserved here. In classical field theory, the field φ(x,t) is the analogue of a dynamical variable, one at each point of spacetime, x,t. Consider this to be the canonical coordinate. Then the canonical momentum is the time derivative of φ. In classical dynamics, the Poisson bracket between these quantities should be unity. In quantum mechanics, the canonical coordinate and momentum become operators, and a Poisson bracket becomes a commutator. This is exactly what happens here.
The one major drawback of this procedure is that Poincare invariance is no longer manifest. That is because to define the time coordinate, one must choose an inertial frame to work with. At the end of the computation one is required to check that relativistic invariance is hidden, but not lost. Field theories used in condensed matter physics are not required to have Poincare invariance, and for them canonical quantization does not suffer from this drawback.
Computing amplitudes
Renormalization
Mathematical quantization
The classical theory is described using a spacelike foliation of spacetime with the state at each slice being described by an element of a symplectic manifold with the time evolution given by the symplectomorphism generated by a Hamiltonian function over the symplectic manifold. The quantum algebra of "operators" is a <math>\hbar<math>deformation of the algebra of smooth functions over the symplectic space such that the leading term in the Taylor expansion over <math>\hbar<math> of the commutator <math>[A,B]<math> is <math>i\hbar\{A,B\}<math>. (Here, the curly braces denote the Poisson bracket.) In general, this <math>\hbar<math>deformation is highly nonunique, which explains the claim that quantization is an art. Now, we look for unitary representations of this quantum algebra. With respect to such a unitary rep, a symplectomorphism in the classical theory would now correspond to a unitary transformation. In particular, the time evolution symplectomorphism generated by the classical Hamiltonian is now a unitary transformation generated by the corresponding quantum Hamiltonian.
We could be more general than this. We can work with a Poisson manifold instead of a symplectic space for the classical theory and perform a <math>\hbar<math> deformation of the corresponding Poisson algebra or even Poisson supermanifolds. (The literal classical interpretation of this, of course, does not exist. This is a purely formal procedure.)
See also
References and external links
Historical
 QED and the men who made it, by S.S.Schweber (http://www.amazon.com/exec/obidos/ASIN/0691033277/qid=1118045494/sr=23/ref=pd_bbs_b_2_3/10232872616119321) [ISBN 0691033277]
Technical
 Principles of quantum mechanics, by P.A.M.Dirac (http://www.amazon.com/exec/obidos/ASIN/0198520115/qid=1118039013/sr=21/ref=pd_bbs_b_2_1/10232872616119321) [ISBN 0198520115]
 An introduction to quantum field theory, by M.E.Peskin and H.D.Schroeder (http://www.amazon.com/exec/obidos/ASIN/0201503972/qid=1117869582/sr=21/ref=pd_bbs_b_2_1/10232872616119321) [ISBN 0201503972]