Inphysics,canonical quantization is a procedure forquantizing aclassical theory, while attempting to preserve the formal structure, such assymmetries, of the classical theory to the greatest extent possible.

Historically, this was not quiteWerner Heisenberg's route to obtainingquantum mechanics, butPaul Dirac introduced it in his 1926 doctoral thesis, the "method of classical analogy" for quantization,^[1] and detailed it in his classic textPrinciples of Quantum Mechanics.^[2] The wordcanonical arises from theHamiltonian approach to classical mechanics, in which a system's dynamics is generated via canonicalPoisson brackets, a structure which isonly partially preserved in canonical quantization.

This method was further used by Paul Dirac in the context ofquantum field theory, in his construction ofquantum electrodynamics. In the field theory context, it is also called thesecond quantization of fields, in contrast to the semi-classicalfirst quantization of single particles.

When it was first developed,quantum physics dealt only with thequantization of themotion of particles, leaving theelectromagnetic fieldclassical, hence the namequantum mechanics.^[3]

Later the electromagnetic field was also quantized, and even the particles themselves became represented through quantized fields, resulting in the development ofquantum electrodynamics (QED) andquantum field theory in general.^[4] Thus, by convention, the original form of particle quantum mechanics is denotedfirst quantization, while quantum field theory is formulated in the language ofsecond quantization.

The following exposition is based onDirac's treatise on quantum mechanics.^[2]In theclassical mechanics of a particle, there are dynamic variables which are called coordinates (x) and momenta (p). These specify thestate of a classical system. Thecanonical structure (also known as thesymplectic structure) ofclassical mechanics consists ofPoisson brackets enclosing these variables, such as{x,p} = 1. All transformations of variables which preserve these brackets are allowed ascanonical transformations in classical mechanics. Motion itself is such a canonical transformation.

By contrast, inquantum mechanics, all significant features of a particle are contained in astate${\displaystyle |\psi ⟩}$, called aquantum state. Observables are represented byoperators acting on aHilbert space of suchquantum states.

The eigenvalue of an operator acting on one of its eigenstates represents the value of a measurement on the particle thus represented. For example, theenergy is read off by theHamiltonian operator${\displaystyle \hat{H}}$ acting on a state${\displaystyle |{\psi }_n⟩}$, yielding$${\displaystyle \hat{H}|{\psi }_n⟩=E_n|{\psi }_n⟩,}$$whereE_n is the characteristic energy associated to this${\displaystyle |{\psi }_n⟩}$eigenstate.

Any state could be represented as alinear combination of eigenstates of energy; for example,$${\displaystyle |\psi ⟩=\sum _{n=0}^{\mathrm{\infty }}{a}_n|{\psi }_n⟩,}$$wherea_n are constant coefficients.

As in classical mechanics, all dynamical operators can be represented by functions of the position and momentum ones,${\displaystyle \hat{X}}$ and${\displaystyle \hat{P}}$, respectively. The connection between this representation and the more usualwavefunction representation is given by the eigenstate of the position operator${\displaystyle \hat{X}}$ representing a particle at position${\displaystyle x}$, which is denoted by an element${\displaystyle |x⟩}$ in the Hilbert space, and which satisfies${\displaystyle \hat{X}|x⟩=x|x⟩}$. Then,${\displaystyle \psi (x)=⟨x|\psi ⟩}$.

Likewise, the eigenstates${\displaystyle |p⟩}$ of the momentum operator${\displaystyle \hat{P}}$ specify themomentum representation:${\displaystyle \psi (p)=⟨p|\psi ⟩}$.

The central relation between these operators is a quantum analog of the abovePoisson bracket of classical mechanics, thecanonical commutation relation,$${\displaystyle [\hat{X},\hat{P}]=\hat{X}\hat{P}-\hat{P}\hat{X}=i\hslash .}$$

This relation encodes (and formally leads to) theuncertainty principle, in the formΔx Δp ≥ħ/2. This algebraic structure may be thus considered as the quantum analog of thecanonical structure of classical mechanics.

When turning to many particle systems, i.e., systems containingNidentical particles (particles characterized by the samequantum numbers such asmass,charge andspin), it is necessary to extend the single-particle state function${\displaystyle \psi (\mathbf{r})}$ to theN-particle state function${\displaystyle \psi ({\mathbf{r}}_1,{\mathbf{r}}_2,\dots ,{\mathbf{r}}_N)}$. A fundamental difference between classical and quantum mechanics concerns the concept ofindistinguishability of identical particles. Only two species of particles are thus possible in quantum physics, the so-calledbosons andfermions which obey the following rules for each kind of particle:

• for bosons:$${\displaystyle \psi ({\mathbf{r}}_1,\dots ,{\mathbf{r}}_j,\dots ,{\mathbf{r}}_k,\dots ,{\mathbf{r}}_N)=+\psi ({\mathbf{r}}_1,\dots ,{\mathbf{r}}_k,\dots ,{\mathbf{r}}_j,\dots ,{\mathbf{r}}_N),}$$
• for fermions:$${\displaystyle \psi ({\mathbf{r}}_1,\dots ,{\mathbf{r}}_j,\dots ,{\mathbf{r}}_k,\dots ,{\mathbf{r}}_N)=-\psi ({\mathbf{r}}_1,\dots ,{\mathbf{r}}_k,\dots ,{\mathbf{r}}_j,\dots ,{\mathbf{r}}_N),}$$

where we have interchanged two coordinates${\displaystyle ({\mathbf{r}}_j,{\mathbf{r}}_k)}$ of thestate function. The usual wave function is obtained using theSlater determinant and theidentical particles theory. Using this basis, it is possible to solve various many-particle problems.

Dirac's book^[2] details his popular rule of supplantingPoisson brackets bycommutators:

One might interpret this proposal as saying that we should seek a "quantization map"${\displaystyle Q}$ mapping a function${\displaystyle f}$ on the classical phase space to an operator${\displaystyle Q_f}$ on the quantum Hilbert space such that$${\displaystyle Q_{\{f,g\}}=\frac{1}{i\hslash }[Q_f,Q_g]}$$It is now known that there is no reasonable such quantization map satisfying the above identity exactly for all functions${\displaystyle f}$ and${\displaystyle g}$.^{[citation needed]}

One concrete version of the above impossibility claim is Groenewold's theorem (after Dutch theoretical physicistHilbrand J. Groenewold), which we describe for a system with one degree of freedom for simplicity. Let us accept the following "ground rules" for the map${\displaystyle Q}$. First,${\displaystyle Q}$ should send the constant function 1 to the identity operator. Second,${\displaystyle Q}$ should take${\displaystyle x}$ and${\displaystyle p}$ to the usual position and momentum operators${\displaystyle X}$ and${\displaystyle P}$. Third,${\displaystyle Q}$ should take a polynomial in${\displaystyle x}$ and${\displaystyle p}$ to a "polynomial" in${\displaystyle X}$ and${\displaystyle P}$, that is, a finite linear combinations of products of${\displaystyle X}$ and${\displaystyle P}$, which may be taken in any desired order. In its simplest form, Groenewold's theorem says that there is no map satisfying the above ground rules and also the bracket condition$${\displaystyle Q_{\{f,g\}}=\frac{1}{i\hslash }[Q_f,Q_g]}$$for all polynomials${\displaystyle f}$ and${\displaystyle g}$.

Actually, the nonexistence of such a map occurs already by the time we reach polynomials of degree four. Note that the Poisson bracket of two polynomials of degree four has degree six, so it does not exactly make sense to require a map on polynomials of degree four to respect the bracket condition. Wecan, however, require that the bracket condition holds when${\displaystyle f}$ and${\displaystyle g}$ have degree three. Groenewold's theorem^[5] can be stated as follows:

The proof can be outlined as follows.^[6][7] Suppose we first try to find a quantization map on polynomials of degree less than or equal to three satisfying the bracket condition whenever${\displaystyle f}$ has degree less than or equal to two and${\displaystyle g}$ has degree less than or equal to two. Then there is precisely one such map, and it is theWeyl quantization. The impossibility result now is obtained by writing the same polynomial of degree four as a Poisson bracket of polynomials of degree threein two different ways. Specifically, we have$${\displaystyle x^2{p}^2=\frac{1}{9}\{x^3,p^3\}=\frac{1}{3}\{x^{2}p,x{p}^2\}}$$On the other hand, we have already seen that if there is going to be a quantization map on polynomials of degree three, it must be the Weyl quantization; that is, we have already determined the only possible quantization of all the cubic polynomials above.

The argument is finished by computing by brute force that$${\displaystyle \frac{1}{9}[Q(x^3),Q(p^3)]}$$does not coincide with$${\displaystyle \frac{1}{3}[Q(x^{2}p),Q(x{p}^2)].}$$Thus, we have two incompatible requirements for the value of${\displaystyle Q(x^2{p}^2)}$.

IfQ represents the quantization map that acts on functionsf in classical phase space, then the following properties are usually considered desirable:^[8]

1. ${\displaystyle Q_x\psi =x\psi }$ and${\displaystyle Q_p\psi =-i\hslash {\mathrm{\partial }}_x\psi }$   (elementary position/momentum operators)
2. ${\displaystyle f⟼Q_f}$   is a linear map
3. ${\displaystyle [Q_f,Q_g]=i\hslash Q_{\{f,g\}}}$   (Poisson bracket)
4. ${\displaystyle Q_{g\circ f}=g(Q_f)}$   (von Neumann rule).

However, not only are these four properties mutually inconsistent,any three of them are also inconsistent!^[9] As it turns out, the only pairs of these properties that lead to self-consistent, nontrivial solutions are 2 & 3, and possibly 1 & 3 or 1 & 4. Accepting properties 1 & 2, along with a weaker condition that 3 be true only asymptotically in the limitħ→0 (seeMoyal bracket), leads todeformation quantization, and some extraneous information must be provided, as in the standard theories utilized in most of physics. Accepting properties 1 & 2 & 3 but restricting the space of quantizable observables to exclude terms such as the cubic ones in the above example amounts togeometric quantization.

Quantum mechanics was successful at describing non-relativistic systems with fixed numbers of particles, but a new framework was needed to describe systems in which particles can be created or destroyed, for example, the electromagnetic field, considered as a collection of photons. It was eventually realized thatspecial relativity was inconsistent with single-particle quantum mechanics, so that all particles are now described relativistically byquantum fields.

When the canonical quantization procedure is applied to a field, such as the electromagnetic field, the classicalfield variables becomequantum operators. Thus, the normal modes comprising the amplitude of the field are simple oscillators, each of which isquantized in standard first quantization, above, without ambiguity. The resulting quanta are identified with individual particles or excitations. For example, the quanta of the electromagnetic field are identified with photons. Unlike first quantization, conventional second quantization is completely unambiguous, in effect afunctor, since the constituent set of its oscillators are quantized unambiguously.

Historically, quantizing the classical theory of a single particle gave rise to a wavefunction. The classical equations of motion of a field are typically identical in form to the (quantum) equations for the wave-function ofone of its quanta. For example, theKlein–Gordon equation is the classical equation of motion for a free scalar field, but also the quantum equation for a scalar particle wave-function. This meant that quantizing a fieldappeared to be similar to quantizing a theory that was already quantized, leading to the fanciful termsecond quantization in the early literature, which is still used to describe field quantization, even though the modern interpretation detailed is different.

One drawback to canonical quantization for a relativistic field is that by relying on the Hamiltonian to determine time dependence,relativistic invariance is no longer manifest. Thus it is necessary to check thatrelativistic invariance is not lost. Alternatively, theFeynman integral approach is available for quantizing relativistic fields, and is manifestly invariant. For non-relativistic field theories, such as those used incondensed matter physics, Lorentz invariance is not an issue.

Quantum mechanically, the variables of a field (such as the field's amplitude at a given point) are represented by operators on aHilbert space. In general, all observables are constructed as operators on the Hilbert space, and the time-evolution of the operators is governed by theHamiltonian, which must be apositive operator. A state${\displaystyle |0⟩}$ annihilated by the Hamiltonian must be identified as thevacuum state, which is the basis for building all other states. In a non-interacting (free) field theory, the vacuum is normally identified as a state containing zero particles. In a theory with interacting particles, identifying the vacuum is more subtle, due tovacuum polarization, which implies that the physical vacuum in quantum field theory is never really empty. For further elaboration, see the articles onthe quantum mechanical vacuum andthe vacuum of quantum chromodynamics. The details of the canonical quantization depend on the field being quantized, and whether it is free or interacting.

Ascalar field theory provides a good example of the canonical quantization procedure.^[10] Classically, a scalar field is a collection of an infinity ofoscillatornormal modes. It suffices to consider a 1+1-dimensional space-time${\displaystyle \mathbb{R}\times S_1,}$ in which the spatial direction iscompactified to a circle of circumference 2π, rendering the momenta discrete.

The classicalLagrangian density describes aninfinity of coupled harmonic oscillators, labelled byx which is now alabel (and not the displacement dynamical variable to be quantized), denoted by the classical fieldφ,$${\displaystyle \mathcal{L}(\varphi )=\frac{1}{2}({\mathrm{\partial }}_t\varphi {)}^2-\frac{1}{2}({\mathrm{\partial }}_x\varphi {)}^2-\frac{1}{2}{m}^2{\varphi }^2-V(\varphi ),}$$whereV(φ) is a potential term, often taken to be a polynomial or monomial of degree 3 or higher. The action functional is$${\displaystyle S(\varphi )=\int \mathcal{L}(\varphi )dxdt=\int L(\varphi ,{\mathrm{\partial }}_t\varphi )dt.}$$The canonical momentum obtained via theLegendre transformation using the actionL is${\displaystyle \pi ={\mathrm{\partial }}_t\varphi }$, and the classicalHamiltonian is found to be$${\displaystyle H(\varphi ,\pi )=\int dx\left[\frac{1}{2}{\pi }^2+\frac{1}{2}({\mathrm{\partial }}_x\varphi {)}^2+\frac{1}{2}{m}^2{\varphi }^2+V(\varphi )\right].}$$

Canonical quantization treats the variablesφ andπ as operators withcanonical commutation relations at timet= 0, given by$${\displaystyle [\varphi (x),\varphi (y)]=0,[\pi (x),\pi (y)]=0,[\varphi (x),\pi (y)]=i\hslash \delta (x-y).}$$Operators constructed fromφ andπ can then formally be defined at other times via the time-evolution generated by the Hamiltonian,$${\displaystyle \mathcal{O}(t)=e^{itH}\mathcal{O}{e}^{-itH}.}$$

However, sinceφ andπ no longer commute, this expression is ambiguous at the quantum level. The problem is to construct a representation of the relevant operators${\displaystyle \mathcal{O}}$ on aHilbert space${\displaystyle \mathcal{H}}$ and to construct a positive operatorH as aquantum operator on this Hilbert space in such a way that it gives this evolution for the operators${\displaystyle \mathcal{O}}$ as given by the preceding equation, and to show that${\displaystyle \mathcal{H}}$ contains a vacuum state${\displaystyle |0⟩}$ on whichH has zero eigenvalue. In practice, this construction is a difficult problem for interacting field theories, and has been solved completely only in a few simple cases via the methods ofconstructive quantum field theory. Many of these issues can be sidestepped using the Feynman integral as described for a particularV(φ) in the article onscalar field theory.

In the case of a free field, withV(φ) = 0, the quantization procedure is relatively straightforward. It is convenient toFourier transform the fields, so that$${\displaystyle {\varphi }_k=\int \varphi (x)e^{-ikx}dx,{\pi }_k=\int \pi (x)e^{-ikx}dx.}$$The reality of the fields implies that$${\displaystyle {\varphi }_{-k}={\varphi }_k^{†},{\pi }_{-k}={\pi }_k^{†}.}$$The classical Hamiltonian may be expanded in Fourier modes as$${\displaystyle H=\frac{1}{2}\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\left[{\pi }_k{\pi }_k^{†}+{\omega }_k^2{\varphi }_k{\varphi }_k^{†}\right],}$$where${\displaystyle {\omega }_k=\sqrt{k^2+m^2}}$.

This Hamiltonian is thus recognizable as an infinite sum of classicalnormal mode oscillator excitationsφ_k, each one of which is quantized in thestandard manner, so the free quantum Hamiltonian looks identical. It is theφ_ks that have become operators obeying the standard commutation relations,[φ_k,π_k^†] = [φ_k^†,π_k] =iħ, with all others vanishing. The collective Hilbert space of all these oscillators is thus constructed using creation and annihilation operators constructed from these modes,$${\displaystyle a_k=\frac{1}{\sqrt{2\hslash {\omega }_k}}\left({\omega }_k{\varphi }_k+i{\pi }_k\right),a_k^{†}=\frac{1}{\sqrt{2\hslash {\omega }_k}}\left({\omega }_k{\varphi }_k^{†}-i{\pi }_k^{†}\right),}$$for which[a_k,a_k^†] = 1 for allk, with all other commutators vanishing.

The vacuum${\displaystyle |0⟩}$ is taken to be annihilated by all of thea_k, and${\displaystyle \mathcal{H}}$ is the Hilbert space constructed by applying any combination of the infinite collection of creation operatorsa_k^† to${\displaystyle |0⟩}$. This Hilbert space is calledFock space. For eachk, this construction is identical to aquantum harmonic oscillator. The quantum field is an infinite array of quantum oscillators. The quantum Hamiltonian then amounts to$${\displaystyle H=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\hslash {\omega }_k{a}_k^{†}{a}_k=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\hslash {\omega }_k{N}_k,}$$whereN_k may be interpreted as thenumber operator giving thenumber of particles in a state with momentumk.

This Hamiltonian differs from the previous expression by the subtraction of the zero-point energyħω_k/2 of each harmonic oscillator. This satisfies the condition thatH must annihilate the vacuum, without affecting the time-evolution of operators via the above exponentiation operation. This subtraction of thezero-point energy may be considered to be a resolution of the quantum operator ordering ambiguity, since it is equivalent to requiring thatall creation operators appear to the left of annihilation operators in the expansion of the Hamiltonian. This procedure is known asWick ordering ornormal ordering.

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 anyinternal symmetry, then creation and destruction operators must be introduced for each component of the field related to this symmetry as well. If there is agauge symmetry, then the number of independent components of the field must be carefully analyzed to avoid over-counting equivalent configurations, andgauge-fixing may be applied if needed.

It turns out that commutation relations are useful only for quantizingbosons, for which the occupancy number of any state is unlimited. To quantizefermions, which satisfy thePauli exclusion principle, anti-commutators are needed. These are defined by{A,B} =AB +BA.

When quantizing fermions, the fields are expanded in creation and annihilation operators,θ_k^†,θ_k, which satisfy$${\displaystyle \{{\theta }_k,{\theta }_l^{†}\}={\delta }_{kl},\{{\theta }_k,{\theta }_l\}=0,\{{\theta }_k^{†},{\theta }_l^{†}\}=0.}$$

The states are constructed on a vacuum${\displaystyle |0⟩}$ annihilated by theθ_k, and theFock space is built by applying all products of creation operatorsθ_k^† to|0⟩. Pauli's exclusion principle is satisfied, because${\displaystyle ({\theta }_k^{†}{)}^2|0⟩=0}$, by virtue of the anti-commutation relations.

The construction of the scalar field states above assumed that the potential was minimized atφ = 0, so that the vacuum minimizing the Hamiltonian satisfies⟨φ⟩ = 0, indicating that thevacuum expectation value (VEV) of the field is zero. In cases involvingspontaneous symmetry breaking, it is possible to have a non-zero VEV, because the potential is minimized for a valueφ =v . This occurs for example, ifV(φ) =gφ⁴ − 2m²φ² withg > 0 andm² > 0, for which the minimum energy is found atv = ±m/√g. The value ofv in one of these vacua may be considered ascondensate of the fieldφ. Canonical quantization then can be carried out for theshifted fieldφ(x,t) −v, and particle states with respect to the shifted vacuum are defined by quantizing the shifted field. This construction is utilized in theHiggs mechanism in theStandard Model ofparticle physics.

The classical theory is described using aspacelikefoliation ofspacetime with the state at each slice being described by an element of asymplectic manifold with the time evolution given by thesymplectomorphism generated by aHamiltonian function over the symplectic manifold. Thequantum algebra of "operators" is anħ-deformation of the algebra of smooth functions over the symplectic space such that theleading term in the Taylor expansion overħ of thecommutator [A,B] expressed in thephase space formulation isiħ{A,B}. (Here, the curly braces denote thePoisson bracket. The subleading terms are all encoded in theMoyal bracket, the suitable quantum deformation of the Poisson bracket.) In general, for the quantities (observables) involved,and providing the arguments of such brackets,ħ-deformations are highly nonunique—quantization is an "art", and is specified by the physical context.(Twodifferent quantum systems may represent two different, inequivalent, deformations of the sameclassical limit,ħ → 0.)

Now, one looks forunitary representations of this quantum algebra. With respect to such a unitary representation, a symplectomorphism in the classical theory would now deform to a (metaplectic)unitary transformation. In particular, the time evolution symplectomorphism generated by the classical Hamiltonian deforms to a unitary transformation generated by the corresponding quantum Hamiltonian.

A further generalization is to consider aPoisson manifold instead of a symplectic space for the classical theory and perform anħ-deformation of the correspondingPoisson algebra or evenPoisson supermanifolds.

In contrast to the theory of deformation quantization described above, geometric quantization seeks to construct an actual Hilbert space and operators on it. Starting with a symplectic manifold${\displaystyle M}$, one first constructs a prequantum Hilbert space consisting of the space of square-integrable sections of an appropriate line bundle over${\displaystyle M}$. On this space, one can mapall classical observables to operators on the prequantum Hilbert space, with the commutator corresponding exactly to the Poisson bracket. The prequantum Hilbert space, however, is clearly too big to describe the quantization of${\displaystyle M}$.

One then proceeds by choosing a polarization, that is (roughly), a choice of${\displaystyle n}$ variables on the${\displaystyle 2n}$-dimensional phase space. Thequantum Hilbert space is then the space of sections that depend only on the${\displaystyle n}$ chosen variables, in the sense that they are covariantly constant in the other${\displaystyle n}$ directions. If the chosen variables are real, we get something like the traditional Schrödinger Hilbert space. If the chosen variables are complex, we get something like theSegal–Bargmann space.

• Correspondence principle
• Creation and annihilation operators
• Dirac bracket
• Moyal bracket
• Phase space formulation (of quantum mechanics)
• Geometric quantization

• Silvan S. Schweber:QED and the men who made it, Princeton Univ. Press, 1994,ISBN 0-691-03327-7

• Alexander Altland, Ben Simons:Condensed matter field theory, Cambridge Univ. Press, 2009,ISBN 978-0-521-84508-3
• James D. Bjorken, Sidney D. Drell:Relativistic quantum mechanics, New York, McGraw-Hill, 1964
• Hall, Brian C. (2013),Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer,Bibcode:2013qtm..book.....H,ISBN 978-1461471158.
• An introduction to quantum field theory, by M.E. Peskin and H.D. Schroeder,ISBN 0-201-50397-2
• Franz Schwabl:Advanced Quantum Mechanics, Berlin and elsewhere, Springer, 2009ISBN 978-3-540-85061-8

• Pedagogic Aides to Quantum Field Theory Click on the links for Chaps. 1 and 2 at this site to find an extensive, simplified introduction to second quantization. See Sect. 1.5.2 in Chap. 1. See Sect. 2.7 and the chapter summary in Chap. 2.

• Quantum field theory
• Mathematical quantization

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