Inquantum mechanics, thecanonical commutation relation is the fundamental relation betweencanonical conjugate quantities (quantities which are related by definition such that one is theFourier transform of another). For example,$${\displaystyle [\hat{x},{\hat{p}}_x]=i\hslash \mathbb{I}}$$

between the position operatorx and momentum operatorp_x in thex direction of a point particle in one dimension, where[x ,p_x] =xp_x −p_xx is thecommutator ofx andp_x ,i is theimaginary unit, andℏ is thereduced Planck constanth/2π, and${\displaystyle \mathbb{I}}$ is the unit operator. In general, position and momentum are vectors of operators and their commutation relation between different components of position and momentum can be expressed as$${\displaystyle [{\hat{x}}_i,{\hat{p}}_j]=i\hslash {\delta }_{ij},}$$where${\displaystyle {\delta }_{ij}}$ is theKronecker delta.

This relation is attributed toWerner Heisenberg,Max Born andPascual Jordan (1925),^[1][2] who called it a "quantum condition" serving as a postulate of the theory; it was noted byE. Kennard (1927)^[3] to imply theHeisenberguncertainty principle. TheStone–von Neumann theorem gives a uniqueness result for operators satisfying (an exponentiated form of) the canonical commutation relation.

By contrast, inclassical physics, all observables commute and thecommutator would be zero. However, an analogous relation exists, which is obtained by replacing the commutator with thePoisson bracket multiplied by${\displaystyle i\hslash }$,$${\displaystyle \{x,p\}=1.}$$

This observation ledDirac to propose that the quantum counterparts${\displaystyle \hat{f}}$,${\displaystyle \hat{g}}$ of classical observablesf,g satisfy$${\displaystyle [\hat{f},\hat{g}]=i\hslash \widehat{\{f,g\}}.}$$

In 1946,Hip Groenewold demonstrated that ageneral systematic correspondence between quantum commutators and Poisson brackets could not hold consistently.^[4][5]

However, he further appreciated that such a systematic correspondence does, in fact, exist between the quantum commutator and adeformation of the Poisson bracket, today called theMoyal bracket, and, in general, quantum operators and classical observables and distributions inphase space. He thus finally elucidated the consistent correspondence mechanism, theWigner–Weyl transform, that underlies an alternate equivalent mathematical representation of quantum mechanics known asdeformation quantization.^[4][6]

According to thecorrespondence principle, in certain limits the quantum equations of states must approachHamilton's equations of motion. The latter state the following relation between the generalized coordinateq (e.g. position) and the generalized momentump:$${\displaystyle \{\begin{array}{l}\dot{q}=\frac{\mathrm{\partial }H}{\mathrm{\partial }p}=\{q,H\};\\ \dot{p}=-\frac{\mathrm{\partial }H}{\mathrm{\partial }q}=\{p,H\}.\end{array}}$$

In quantum mechanics the Hamiltonian${\displaystyle \hat{H}}$, (generalized) coordinate${\displaystyle \hat{Q}}$ and (generalized) momentum${\displaystyle \hat{P}}$ are all linear operators.

The time derivative of a quantum state is represented by the operator${\displaystyle -i\hat{H}/\hslash }$ (by theSchrödinger equation). Equivalently, since in the Schrödinger picture the operators are not explicitly time-dependent, the operators can be seen to be evolving in time (for a contrary perspective where the operators are time dependent, seeHeisenberg picture) according to their commutation relation with the Hamiltonian:$${\displaystyle \frac{d\hat{Q}}{dt}=\frac{i}{\hslash }[\hat{H},\hat{Q}]}$$$${\displaystyle \frac{d\hat{P}}{dt}=\frac{i}{\hslash }[\hat{H},\hat{P}].}$$

In order for that to reconcile in the classical limit with Hamilton's equations of motion,${\displaystyle [\hat{H},\hat{Q}]}$ must depend entirely on the appearance of${\displaystyle \hat{P}}$ in the Hamiltonian and${\displaystyle [\hat{H},\hat{P}]}$ must depend entirely on the appearance of${\displaystyle \hat{Q}}$ in the Hamiltonian. Further, since the Hamiltonian operator depends on the (generalized) coordinate and momentum operators, it can be viewed as a functional, and we may write (usingfunctional derivatives):$${\displaystyle [\hat{H},\hat{Q}]=\frac{\delta \hat{H}}{\delta \hat{P}}\cdot [\hat{P},\hat{Q}]}$$$${\displaystyle [\hat{H},\hat{P}]=\frac{\delta \hat{H}}{\delta \hat{Q}}\cdot [\hat{Q},\hat{P}].}$$

In order to obtain the classical limit we must then have$${\displaystyle [\hat{Q},\hat{P}]=i\hslash I.}$$

Thegroup${\displaystyle H_3(\mathbb{R})}$ generated byexponentiation of the 3-dimensionalLie algebra determined by the commutation relation${\displaystyle [\hat{x},\hat{p}]=i\hslash }$ is called theHeisenberg group. This group can be realized as the group of${\displaystyle 3\times 3}$ upper triangular matrices with ones on the diagonal.^[7]

According to the standardmathematical formulation of quantum mechanics, quantum observables such as${\displaystyle \hat{x}}$ and${\displaystyle \hat{p}}$ should be represented asself-adjoint operators on someHilbert space. It is relatively easy to see that twooperators satisfying the above canonical commutation relations cannot both bebounded. Certainly, if${\displaystyle \hat{x}}$ and${\displaystyle \hat{p}}$ weretrace class operators, the relation${\displaystyle \mathrm{Tr}(AB)=\mathrm{Tr}(BA)}$ gives a nonzero number on the right and zero on the left.

Alternately, if${\displaystyle \hat{x}}$ and${\displaystyle \hat{p}}$ were bounded operators, note that${\displaystyle [{\hat{x}}^n,\hat{p}]=i\hslash n{\hat{x}}^{n-1}}$, hence the operator norms would satisfy$${\displaystyle 2\Vert \hat{p}\Vert \Vert {\hat{x}}^{n-1}\Vert \Vert \hat{x}\Vert \ge n\hslash \Vert {\hat{x}}^{n-1}\Vert ,}$$ so that, for anyn,$${\displaystyle 2\Vert \hat{p}\Vert \Vert \hat{x}\Vert \ge n\hslash }$$However,n can be arbitrarily large, so at least one operator cannot be bounded, and the dimension of the underlying Hilbert space cannot be finite. If the operators satisfy the Weyl relations (an exponentiated version of the canonical commutation relations, described below) then as a consequence of theStone–von Neumann theorem,both operators must be unbounded.

Still, these canonical commutation relations can be rendered somewhat "tamer" by writing them in terms of the (bounded)unitary operators${\displaystyle \exp (it\hat{x})}$ and${\displaystyle \exp (is\hat{p})}$. The resulting braiding relations for these operators are the so-calledWeyl relations$${\displaystyle \exp (it\hat{x})\exp (is\hat{p})=\exp (-ist\hslash )\exp (is\hat{p})\exp (it\hat{x}).}$$These relations may be thought of as an exponentiated version of the canonical commutation relations; they reflect that translations in position and translations in momentum do not commute. One can easily reformulate the Weyl relations in terms of therepresentations of the Heisenberg group.

The uniqueness of the canonical commutation relations—in the form of the Weyl relations—is then guaranteed by theStone–von Neumann theorem.

For technical reasons, the Weyl relations are not strictly equivalent to the canonical commutation relation${\displaystyle [\hat{x},\hat{p}]=i\hslash }$. If${\displaystyle \hat{x}}$ and${\displaystyle \hat{p}}$ were bounded operators, then a special case of theBaker–Campbell–Hausdorff formula would allow one to "exponentiate" the canonical commutation relations to the Weyl relations.^[8] Since, as we have noted, any operators satisfying the canonical commutation relations must be unbounded, the Baker–Campbell–Hausdorff formula does not apply without additional domain assumptions. Indeed, counterexamples exist satisfying the canonical commutation relations but not the Weyl relations.^[9] (These same operators give acounterexample to the naive form of the uncertainty principle.) These technical issues are the reason that theStone–von Neumann theorem is formulated in terms of the Weyl relations.

A discrete version of the Weyl relations, in which the parameterss andt range over${\displaystyle \mathbb{Z}/n}$, can be realized on a finite-dimensional Hilbert space by means of theclock and shift matrices.

It can be shown that$${\displaystyle [F(\overrightarrow{x}),p_i]=i\hslash \frac{\mathrm{\partial }F(\overrightarrow{x})}{\mathrm{\partial }{x}_i};[x_i,F(\overrightarrow{p})]=i\hslash \frac{\mathrm{\partial }F(\overrightarrow{p})}{\mathrm{\partial }{p}_i}.}$$

Using${\displaystyle C_{n+1}^k=C_n^k+C_n^{k-1}}$, it can be shown that bymathematical induction$${\displaystyle \left[{\hat{x}}^n,{\hat{p}}^m\right]=\sum _{k=1}^{min\left(m,n\right)}\frac{-{\left(-i\hslash \right)}^{k}n!m!}{k!\left(n-k\right)!\left(m-k\right)!}{\hat{x}}^{n-k}{\hat{p}}^{m-k}=\sum _{k=1}^{min\left(m,n\right)}\frac{{\left(i\hslash \right)}^{k}n!m!}{k!\left(n-k\right)!\left(m-k\right)!}{\hat{p}}^{m-k}{\hat{x}}^{n-k},}$$generally known as McCoy's formula.^[10]

In addition, the simple formula$${\displaystyle [x,p]=i\hslash \mathbb{I},}$$valid for thequantization of the simplest classical system, can be generalized to the case of an arbitraryLagrangian${\displaystyle \mathcal{L}}$.^[11] We identifycanonical coordinates (such asx in the example above, or a fieldΦ(x) in the case ofquantum field theory) andcanonical momentaπ_x (in the example above it isp, or more generally, some functions involving thederivatives of the canonical coordinates with respect to time):$${\displaystyle {\pi }_i\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }(\mathrm{\partial }{x}_i/\mathrm{\partial }t)}.}$$

This definition of the canonical momentum ensures that one of theEuler–Lagrange equations has the form$${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}{\pi }_i=\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{x}_i}.}$$

The canonical commutation relations then amount to$${\displaystyle [x_i,{\pi }_j]=i\hslash {\delta }_{ij}}$$whereδ_ij is theKronecker delta.

Canonical quantization is applied, by definition, oncanonical coordinates. However, in the presence of anelectromagnetic field, the canonical momentump is notgauge invariant. The correct gauge-invariant momentum (or "kinetic momentum") is

${\displaystyle p_{\text{kin}}=p-qA}$ (SI units)    ${\displaystyle p_{\text{kin}}=p-\frac{qA}{c}}$ (cgs units),

whereq is the particle'selectric charge,A is thevector potential, andc is thespeed of light. Although the quantityp_kin is the "physical momentum", in that it is the quantity to be identified with momentum in laboratory experiments, itdoes not satisfy the canonical commutation relations; only the canonical momentum does that. This can be seen as follows.

The non-relativisticHamiltonian for a quantized charged particle of massm in a classical electromagnetic field is (in cgs units)$${\displaystyle H=\frac{1}{2m}{\left(p-\frac{qA}{c}\right)}^2+q\varphi }$$whereA is the three-vector potential andφ is thescalar potential. This form of the Hamiltonian, as well as theSchrödinger equationHψ =iħ∂ψ/∂t, theMaxwell equations and theLorentz force law are invariant under the gauge transformation$${\displaystyle A\to A^{\prime }=A+\mathrm{\nabla }\mathrm{\Lambda }}$$$${\displaystyle \varphi \to {\varphi }^{\prime }=\varphi -\frac{1}{c}\frac{\mathrm{\partial }\mathrm{\Lambda }}{\mathrm{\partial }t}}$$$${\displaystyle \psi \to {\psi }^{\prime }=U\psi }$$$${\displaystyle H\to H^{\prime }=UH{U}^{†},}$$where$${\displaystyle U=\exp \left(\frac{iq\mathrm{\Lambda }}{\hslash c}\right)}$$ andΛ = Λ(x,t) is the gauge function.

Theangular momentum operator is$${\displaystyle L=r\times p}$$and obeys the canonical quantization relations$${\displaystyle [L_i,L_j]=i\hslash {ϵ}_{ijk}{L}_k}$$defining theLie algebra forso(3), where${\displaystyle {ϵ}_{ijk}}$ is theLevi-Civita symbol. Under gauge transformations, the angular momentum transforms as$${\displaystyle ⟨\psi |L|\psi ⟩\to ⟨{\psi }^{\mathrm{\prime }}|L^{\mathrm{\prime }}|{\psi }^{\mathrm{\prime }}⟩=⟨\psi |L|\psi ⟩+\frac{q}{\hslash c}⟨\psi |r\times \mathrm{\nabla }\mathrm{\Lambda }|\psi ⟩.}$$

The gauge-invariant angular momentum (or "kinetic angular momentum") is given by$${\displaystyle K=r\times \left(p-\frac{qA}{c}\right),}$$which has the commutation relations$${\displaystyle [K_i,K_j]=i\hslash {{ϵ}_{ij}}^k\left(K_k+\frac{q\hslash }{c}{x}_k\left(x\cdot B\right)\right)}$$where$${\displaystyle B=\mathrm{\nabla }\times A}$$ is themagnetic field. The inequivalence of these two formulations shows up in theZeeman effect and theAharonov–Bohm effect.

All such nontrivial commutation relations for pairs of operators lead to correspondinguncertainty relations,^[12] involving positive semi-definite expectation contributions by their respective commutators and anticommutators. In general, for twoHermitian operatorsA andB, consider expectation values in a system in the stateψ, the variances around the corresponding expectation values being(ΔA)² ≡ ⟨(A − ⟨A⟩)²⟩, etc.

Then$${\displaystyle \mathrm{\Delta }A\mathrm{\Delta }B\ge \frac{1}{2}\sqrt{{\left|⟨\left[A,B\right]⟩\right|}^2+{\left|⟨\left\{A-⟨A⟩,B-⟨B⟩\right\}⟩\right|}^2},}$$where[A, B] ≡A B −B A is thecommutator ofA andB, and{A, B} ≡A B +B A is theanticommutator.

This follows through use of theCauchy–Schwarz inequality, since|⟨A²⟩| |⟨B²⟩| ≥ |⟨A B⟩|², andA B = ([A, B] + {A, B})/2 ; and similarly for the shifted operatorsA − ⟨A⟩ andB − ⟨B⟩. (Cf.uncertainty principle derivations.)

Substituting forA andB (and taking care with the analysis) yield Heisenberg's familiar uncertainty relation forx andp, as usual.

For the angular momentum operatorsL_x =y p_z −z p_y, etc., one has that$${\displaystyle [L_x,L_y]=i\hslash {ϵ}_{xyz}{L}_z,}$$where${\displaystyle {ϵ}_{xyz}}$ is theLevi-Civita symbol and simply reverses the sign of the answer under pairwise interchange of the indices. An analogous relation holds for thespin operators.

Here, forL_x andL_y ,^[12] in angular momentum multipletsψ = |ℓ,m⟩, one has, for the transverse components of theCasimir invariantL_x² +L_y²+L_z², thez-symmetric relations

⟨L_x²⟩ = ⟨L_y²⟩ = (ℓ (ℓ + 1) −m²) ℏ²/2 ,

as well as⟨L_x⟩ = ⟨L_y⟩ = 0 .

Consequently, the above inequality applied to this commutation relation specifies$${\displaystyle \mathrm{\Delta }{L}_x\mathrm{\Delta }{L}_y\ge \frac{1}{2}\sqrt{{\hslash }^2|⟨L_z⟩{|}^2},}$$hence$${\displaystyle \sqrt{|⟨L_x^2⟩⟨L_y^2⟩|}\ge \frac{{\hslash }^2}{2}|m|}$$and therefore$${\displaystyle \ell (\ell +1)-m^2\ge |m|,}$$so, then, it yields useful constraints such as a lower bound on theCasimir invariant:ℓ (ℓ + 1) ≥ |m| (|m| + 1), and henceℓ ≥ |m|, among others.

• Canonical quantization
• CCR and CAR algebras
• Conformastatic spacetimes
• Lie derivative
• Moyal bracket
• Stone–von Neumann theorem

• Hall, Brian C. (2013),Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer.
• Hall, Brian C. (2015),Lie Groups, Lie Algebras and Representations, An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer.

• Quantum mechanics
• Mathematical physics

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