PlanetPhysics/Canonical Quantization

Canonical quantization is a method of relating, or associating, a classical [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|system]] of the form $(T^{*}X,\omega ,H)$, where $X$ is a [[../NoncommutativeGeometry4/|manifold]], $\omega$ is the canonical symplectic form on $T^{*}X$, with a (more complex) quantum system represented by $H\in C^{\mathrm{\infty }}(X)$, where $H$ is the Hamiltonian operator. Some of the early formulations of [[../QuantumParadox/|quantum mechanics]] used such quantization methods under the umbrella of the \htmladdnormallink{correspondence principle {http://planetphysics.us/encyclopedia/PrincipleOfCorrespondingStates.html} or postulate}. The latter states that a correspondence exists between certain classical and [[../QuantumOperatorAlgebra5/|quantum operators]], (such as the [[../HamiltonianOperator3/|Hamiltonian operators]]) or algebras (such as Lie or Poisson (brackets)), with the classical ones being in the real ($ℝ$) [[../Bijective/|domain]], and the quantum ones being in the complex ($ℂ$) domain. Whereas all classical [[../QuantumOperatorAlgebra4/|Observables and States]] are specified only by real numbers, the [[../TransversalWave/|'wave' amplitudes]] in [[../SpaceTimeQuantizationInQuantumGravityTheories/|quantum theories]] are represented by complex [[../Bijective/|functions]].

Let $(x^i,p_i)$ be a set of Darboux coordinates on $T^{*}X$. Then we may obtain from each coordinate function an [[../QuantumSpinNetworkFunctor2/|operator]] on the [[../NormInducedByInnerProduct/|Hilbert space]] $\mathscr{H}=L^2(X,\mu )$, consisting of functions on $X$ that are square-integrable with respect to some measure $\mu$, by the operator substitution rule:

Failed to parse (unknown function "\pdiff"): {\displaystyle \begin{matrix} x^i \mapsto \hat{x}^i &= x^i \cdot, (1)\\ p_i \mapsto \hat{p}_i &= -i \hbar \pdiff{}{x^i} (2), \end{matrix}}

where $x^i\cdot$ is the "multiplication by $x^i$" operator. Using this rule, we may obtain operators from a larger class of functions. For example,

1. $x^i{x}^j\mapsto {\hat{x}}^i{\hat{x}}^j=x^i{x}^j\cdot$,
2. Failed to parse (unknown function "\pdiff"): {\displaystyle p_i p_j \mapsto \hat{p}_i \hat{p}_j = -\hbar^2 \pdiff{^2}{x^i x^j}} ,
3. if $i\ne j$ then Failed to parse (unknown function "\pdiff"): {\displaystyle x^i p_j \mapsto \hat{x}^i \hat{p}_j = -i \hbar x^i \pdiff{}{x^j}} .

\begin{rmk} The substitution rule creates an ambiguity for the function $x^i{p}_j$ when $i=j$, since $x^i{p}_j=p_j{x}^i$, whereas ${\hat{x}}^i{\hat{p}}_j\ne {\hat{p}}_j{\hat{x}}^i$. This is the operator ordering problem. One possible solution is to choose

$x^i{p}_j\mapsto \frac{1}{2}({\hat{x}}^i{\hat{p}}_j+{\hat{p}}_j{\hat{x}}^i),$

since this choice produces an operator that is self-adjoint and therefore corresponds to a physical [[../QuantumSpinNetworkFunctor2/|observable]]. More generally, there is a construction known as Weyl [[../MoyalDeformation/|quantization]] that uses [[../FourierTransforms/|Fourier transforms]] to extend the substitution rules (2)-(3) to a map

Failed to parse (unknown function "\Op"): {\displaystyle \begin{matrix} C^\infty(T^*X) &\to \Op (\mathcal{H}) \\ f &\mapsto \hat{f}. \end{matrix}}

\end{rmk}

\begin{rmk} This procedure is called "canonical" because it preserves the canonical Poisson brackets. In particular, we have that

$\frac{-i}{\hslash }[{\hat{x}}^i,{\hat{p}}_j]:=\frac{-i}{\hslash }({\hat{x}}^i{\hat{p}}_j-{\hat{p}}_j{\hat{x}}^i)={\delta }_j^i,$

which agrees with the Poisson bracket $\{x^i,p_j\}={\delta }_j^i$. \end{rmk}

\begin{ex} Let $X=ℝ$. The [[../Hamiltonian2/|Hamiltonian]] function for a one-dimensional [[../CenterOfGravity/|point particle]] with [[../CosmologicalConstant/|mass]] $m$ is

$H=\frac{p^2}{2m}+V(x),$

where $V(x)$ is the potential [[../CosmologicalConstant/|energy]]. Then, by operator substitution, we obtain the Hamiltonian operator

$\hat{H}=\frac{-{\hslash }^2}{2m}\frac{d^2}{d{x}^2}+V(x).$

\end{ex}