# Canonical quantization

**Canonical quantization** is a procedure for constructing a quantum theory from a classical theory that is already written in Hamiltonian form: the classical dynamical variables are reinterpreted as operators on a [Hilbert space](https://www.edgepedia.org/hilbert-space), and their commutators are fixed by the classical Poisson brackets through the rule \( [A, B] = i \hbar \{ A, B \} \).<sup>[1](https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/d9bac33f6c60b304dc0398e99b327102_MIT8_09F14_full.pdf)</sup> The approach was first studied by [Paul Dirac](https://www.edgepedia.org/paul-dirac), and the replacement of Poisson brackets by commutators is the root of the name "canonical quantization".<sup>[1](https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/d9bac33f6c60b304dc0398e99b327102_MIT8_09F14_full.pdf)</sup> The procedure is called canonical because it follows from the canonical Hamiltonian description of the classical dynamics.<sup>[2](https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf)</sup>

| Key fact | Detail |
|---|---|
| Central rule | \( [A, B] = i \hbar \{ A, B \} \): Poisson brackets of classical variables become commutators of operators, first studied by Dirac<sup>[1](https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/d9bac33f6c60b304dc0398e99b327102_MIT8_09F14_full.pdf)</sup> |
| Fundamental brackets | \( \{ q_i, q_j \} = \{ p_i, p_j \} = 0 \) and \( \{ q_i, p_j \} = \delta_{ij} \), giving \( [\hat{q}_i, \hat{p}_j] = i \hbar \delta_{ij} \)<sup>[1](https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/d9bac33f6c60b304dc0398e99b327102_MIT8_09F14_full.pdf)</sup><sup> • </sup><sup>[5](https://ocw.mit.edu/courses/8-323-relativistic-quantum-field-theory-i-spring-2023/mit8_323_s23_rec02.pdf)</sup> |
| Field version | Equal-time commutators \( [\phi(\mathbf{x}, t), \pi(\mathbf{y}, t)] = i \delta^3(\mathbf{x} - \mathbf{y}) \), with \( [\phi, \phi] = [\pi, \pi] = 0 \); fermionic fields use anticommutators<sup>[3](https://plato.stanford.edu/entries/quantum-field-theory/index.html)</sup> |
| Oscillator result | The commutation relations are equivalent to \( [a_m, a_n^{\dagger}] = \delta_{mn} \) for creation and annihilation operators<sup>[4](https://www.tpi.uni-jena.de/~floerchinger/qft1/lecture10/)</sup> |
| Rigorous status | The prescription breaks down for polynomials of degree 3 or higher in \( q, p \) (Groenewold's theorem); the Stone-von Neumann theorem guarantees existence and uniqueness of representations of the canonical relations<sup>[5](https://ocw.mit.edu/courses/8-323-relativistic-quantum-field-theory-i-spring-2023/mit8_323_s23_rec02.pdf)</sup> |
| Constrained systems | Dirac's program extends the method to Lagrangian-defined theories with constraints and gauge symmetries, quantizing without gauge fixing<sup>[6](https://arxiv.org/abs/quant-ph/9606031)</sup> |

## The classical starting point

The method presupposes a Hamiltonian formulation of the classical theory. For a system with canonical coordinates \( \{ x_j \} \) and conjugate momenta \( \{ p_j \} \), [Hamilton's equations](https://www.edgepedia.org/hamiltons-equations) take the [Poisson bracket](https://www.edgepedia.org/poisson-bracket) form \( \dot{x}_j = \{ x_j, H \} \) and \( \dot{p}_j = \{ p_j, H \} \), and the time development of any dynamical variable is \( \dot{A} = \{ A, H \} \).<sup>[2](https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf)</sup> The Poisson brackets of the coordinates and momenta themselves are remarkably simple: \( \{ x_j, x_k \} = 0 \), \( \{ p_j, p_k \} = 0 \), and \( \{ x_j, p_k \} = \delta_{jk} \).<sup>[2](https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf)</sup>

Comparing these classical equations with the operator equations of motion of quantum mechanics suggests a transcription rule: \( \{ A, B \} \Rightarrow \frac{1}{i \hbar} [A, B] \), where the quantum operators are the same functions of \( \hat{x}_j \) and \( \hat{p}_j \) as the classical variables are of \( x_j \) and \( p_j \).<sup>[2](https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf)</sup> In practice, quantizing a Lagrangian system proceeds in three steps: identify the generalized coordinates \( q_i \), derive the conjugate momenta \( p_i \) from the Lagrangian, and promote \( (q_i, p_i) \) to operators satisfying \( [\hat{q}_i, \hat{p}_j] = i \hbar \{ q_i, p_j \} = i \hbar \delta_{ij} \).<sup>[5](https://ocw.mit.edu/courses/8-323-relativistic-quantum-field-theory-i-spring-2023/mit8_323_s23_rec02.pdf)</sup>

**Operator ordering.** The transcription is not a theorem but a rule with known limits. The prescription breaks down for polynomials of degree 3 or higher in \( q \) and \( p \), a result known as Groenewold's theorem; operator ordering ambiguities generate corrections of higher order in \( \hbar \), though to second order they are not fatal, leaving quantities like the Hamiltonian well defined.<sup>[5](https://ocw.mit.edu/courses/8-323-relativistic-quantum-field-theory-i-spring-2023/mit8_323_s23_rec02.pdf)</sup> In practice one usually restricts quantization to linear, sometimes quadratic, polynomials in \( x \) and \( p \).<sup>[5](https://ocw.mit.edu/courses/8-323-relativistic-quantum-field-theory-i-spring-2023/mit8_323_s23_rec02.pdf)</sup> For the linear sector the Stone-von Neumann theorem supplies existence and uniqueness: there are no finite-dimensional representations of the [Heisenberg group](https://www.edgepedia.org/heisenberg-group), and all irreducible representations of \( \hat{x} \) and \( \hat{p} \) are unitarily equivalent, acting on the Hilbert space of square-integrable functions \( L^2(\mathbb{R}) \).<sup>[5](https://ocw.mit.edu/courses/8-323-relativistic-quantum-field-theory-i-spring-2023/mit8_323_s23_rec02.pdf)</sup>

## Quantizing fields

The same procedure applied to a classical field yields quantum field theory. In classical Lagrangian field theory, each field \( \phi \) is assigned a conjugate field \( \pi = \partial \mathcal{L} / \partial \dot{\phi} \), the direct analog of the canonical coordinate and momentum of particle mechanics.<sup>[3](https://plato.stanford.edu/entries/quantum-field-theory/index.html)</sup> Requiring the canonical variables to satisfy commutation relations makes the basic quantities operator valued. For a bosonic field such as the Klein-Gordon field or the electromagnetic field, the equal-time [canonical commutation relations](https://www.edgepedia.org/canonical-commutation-relation) are

\[ [\phi(\mathbf{x}, t), \pi(\mathbf{y}, t)] = i \delta^3(\mathbf{x} - \mathbf{y}), \qquad [\phi(\mathbf{x}, t), \phi(\mathbf{y}, t)] = [\pi(\mathbf{x}, t), \pi(\mathbf{y}, t)] = 0. \]

For a fermionic field, such as the Dirac field for electrons, one has to use anticommutation relations instead.<sup>[3](https://plato.stanford.edu/entries/quantum-field-theory/index.html)</sup> Unlike the finite set of commutation relations in particle quantum mechanics, the field relations comprise an infinite number of equations, one set for each spacetime point \( (\mathbf{x}, t) \).<sup>[3](https://plato.stanford.edu/entries/quantum-field-theory/index.html)</sup>

**The Klein-Gordon field.** For a free real scalar field of mass \( m \), the equation of motion decouples in momentum space: for each \( \mathbf{k} \in \mathbb{R}^3 \) the mode \( \tilde{\phi}(t, \mathbf{k}) \) obeys the equation of a harmonic oscillator of frequency \( \omega_k \), with \( \omega_k^2 = k^2 + m^2 \).<sup>[5](https://ocw.mit.edu/courses/8-323-relativistic-quantum-field-theory-i-spring-2023/mit8_323_s23_rec02.pdf)</sup> Quantizing each mode as an oscillator gives creation and annihilation operators \( \hat{a}_{\mathbf{k}} \) and \( \hat{a}_{\mathbf{k}}^{\dagger} \), and the field operator takes the mode expansion

\[ \hat{\phi}(t, \mathbf{x}) = \int \frac{d^3 k}{(2 \pi)^3} \frac{1}{\sqrt{2 \omega_k}} \left( \hat{a}_{\mathbf{k}} e^{-i \omega_k t + i \mathbf{k} \cdot \mathbf{x}} + \hat{a}_{\mathbf{k}}^{\dagger} e^{i \omega_k t - i \mathbf{k} \cdot \mathbf{x}} \right). \]

Imposing the canonical commutation relations on \( \hat{\phi} \) and \( \hat{\pi} \) is equivalent to imposing \( [\hat{a}_{\mathbf{k}}, \hat{a}_{\mathbf{k}'}^{\dagger}] = (2 \pi)^3 \delta^3(\mathbf{k} - \mathbf{k}') \) on the mode operators; the two forms of the relations imply each other and serve as a consistency check.<sup>[5](https://ocw.mit.edu/courses/8-323-relativistic-quantum-field-theory-i-spring-2023/mit8_323_s23_rec02.pdf)</sup> The Hilbert space is the [Fock space](https://www.edgepedia.org/fock-space) built by acting with creation operators on a vacuum state \( |0\rangle \), and in field theory a creation operator \( \hat{a}_{\mathbf{k}}^{\dagger} \) creates a bona fide particle of momentum \( \mathbf{k} \), so particle creation and annihilation are intrinsic to the theory rather than added by hand.<sup>[5](https://ocw.mit.edu/courses/8-323-relativistic-quantum-field-theory-i-spring-2023/mit8_323_s23_rec02.pdf)</sup> The reinterpretation of single-particle wave equations as operator-valued fields is sometimes called "second quantization", a term that has been criticized because it blurs the fact that the single-particle wave function and the operator-valued quantum field are fundamentally different kinds of entities.<sup>[3](https://plato.stanford.edu/entries/quantum-field-theory/index.html)</sup>

**Gauge fields.** Quantizing the electromagnetic field requires a gauge-fixed Lagrangian. In Feynman gauge, the canonical momentum density is \( \Pi^{\mu} = - \dot{A}^{\mu} \), and the system is quantized by promoting \( A_{\mu} \) and \( \Pi_{\nu} \) to Heisenberg-picture fields with canonical equal-time commutators \( [A_{\mu}(t, \mathbf{x}), \Pi_{\nu}(t, \mathbf{y})] = i \delta_{\mu \nu} \delta^{(3)}(\mathbf{x} - \mathbf{y}) \).<sup>[7](https://physics.umd.edu/grt/taj/624b/WeigandQFT.pdf)</sup> The gauge-fixed Lagrangian carries a wrong-sign kinetic term for \( \mu = 0 \), and the extra constraint \( \partial \cdot A = 0 \), a consequence of the underlying gauge symmetry, ensures that a consistent quantization is possible despite that sign.<sup>[7](https://physics.umd.edu/grt/taj/624b/WeigandQFT.pdf)</sup>

## Constrained systems and the Dirac program

Many theories of physical interest, including gauge fields and general relativity, do not admit a straightforward Hamiltonian because their Lagrangians are singular: the momenta cannot all be solved for velocities, and the phase space is restricted by constraints. Dirac's general method constructs a quantum theory out of a classical theory defined by a Lagrangian, reviews the Hamiltonian formalism with emphasis on the relation between constraints and gauge symmetries, and carries out quantization without any kind of gauge fixing.<sup>[6](https://arxiv.org/abs/quant-ph/9606031)</sup> The method has been applied to the free electromagnetic field, the relativistic point particle, and the first steps of string theory.<sup>[6](https://arxiv.org/abs/quant-ph/9606031)</sup>

The treatment of constraints matters for what the quantization produces. Analysis of a toy model shows that first-class and second-class constrained systems give physically distinguishable results: second-class systems lead to a well-defined reduced phase space and physical observables, absence of inconsistencies in the closure of the quantum constraint algebra, and consistent promotion of the fundamental Dirac brackets to quantum commutators.<sup>[8](https://arxiv.org/abs/2108.13210)</sup> This bears on the soundness of quantizing first-class systems by simply promoting Poisson, rather than Dirac, brackets to commutators.<sup>[8](https://arxiv.org/abs/2108.13210)</sup>

## Limitations and alternatives

Canonical quantization does not always work. Beyond the Groenewold obstruction on higher-order polynomials, the equal-time commutation relations are not obviously Lorentz invariant, although a manifestly covariant form of the canonical commutation relations can be formulated.<sup>[3](https://plato.stanford.edu/entries/quantum-field-theory/index.html)</sup><sup> • </sup><sup>[5](https://ocw.mit.edu/courses/8-323-relativistic-quantum-field-theory-i-spring-2023/mit8_323_s23_rec02.pdf)</sup> Other quantization methods, including deformation quantization and geometric quantization, exist for systems where the canonical procedure fails or is inconvenient.<sup>[5](https://ocw.mit.edu/courses/8-323-relativistic-quantum-field-theory-i-spring-2023/mit8_323_s23_rec02.pdf)</sup> Even where the fundamental equations of motion can be placed in correspondence by the procedure, the different interpretation of classical and quantum variables leads to very different pictures of phenomena.<sup>[2](https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf)</sup>

## References

1. [8.09 Advanced Classical Mechanics, Lecture notes on Poisson brackets and canonical quantization, MIT OpenCourseWare](https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/d9bac33f6c60b304dc0398e99b327102_MIT8_09F14_full.pdf)
2. [R. L. Jaffe, Supplementary Notes: Canonical Quantization and Application to the Quantum Mechanics of a Charged Particle in a Magnetic Field, MIT 8.06 (2016)](https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf)
3. [Quantum Field Theory, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/quantum-field-theory/index.html)
4. [Quantum Field Theory 1, Lecture 10, Floerchinger Group, University of Jena](https://www.tpi.uni-jena.de/~floerchinger/qft1/lecture10/)
5. [8.323 Relativistic Quantum Field Theory I, Recitation 2, MIT OpenCourseWare (2023)](https://ocw.mit.edu/courses/8-323-relativistic-quantum-field-theory-i-spring-2023/mit8_323_s23_rec02.pdf)
6. [Hans-Juergen Matschull, Dirac's Canonical Quantization Programme, arXiv:quant-ph/9606031 (1996)](https://arxiv.org/abs/quant-ph/9606031)
7. [Timo Weigand, Quantum Field Theory I + II, Lecture notes, University of Hamburg](https://physics.umd.edu/grt/taj/624b/WeigandQFT.pdf)
8. [Eyo Eyo Ita III, Chopin Soo, Abraham Tan, Quantization of constrained systems as Dirac first class versus second class: a toy model and its implications, arXiv:2108.13210 (2021)](https://arxiv.org/abs/2108.13210)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum formalism and states › Operators, observables, and angular momentum › Quantum operators and observables (overview)*

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