canonical quantization and quantum symmetry · quantum physics

from poisson brackets to commutators

On the classical phase space $\mathbb{R}^{2n}$ with coordinates $(q_i,p_i)$ the fundamental Poisson brackets are $$ \{q_i,p_j\}=\delta_{ij},\qquad\{q_i,q_j\}=\{p_i,p_j\}=0. $$ Canonical quantization replaces (a suitable subclass of) functions by operators and Poisson brackets by commutators according to the correspondence $$ \{f,g\}\;\longmapsto\;\frac{1}{i\hbar}[\hat f,\hat g]. $$ In particular the canonical commutation relations (CCR) read $$ [Q_i,P_j]=i\hbar\delta_{ij}I,\qquad[Q_i,Q_j]=[P_i,P_j]=0 $$ on a dense domain (for instance the Schwartz space $\mathcal{S}(\mathbb{R}^n)$).

definition (heisenberg lie algebra). the real Lie algebra $\mathfrak{h}_{2n+1}$ spanned by generators $Q_i$, $P_i$, and a central element $C$, with the only non-vanishing brackets $[Q_i,P_j]=C\delta_{ij}$. in quantum mechanics one represents $C$ by $\hbar I$ (or more generally by $\lambda I$ with $\lambda\neq 0$).

The corresponding simply connected Lie group is the Heisenberg group $H_{2n+1}$. The CCR are the infinitesimal form of a unitary representation of that group.

the heisenberg group and the stone-von neumann theorem

The Heisenberg group may be realized as the set of upper-triangular matrices $$ \begin{pmatrix} 1 & \mathbf{a}^T & c\\ 0 & I_n & \mathbf{b}\\ 0 & 0 & 1 \end{pmatrix}, \qquad\mathbf{a},\mathbf{b}\in\mathbb{R}^n,\;c\in\mathbb{R}, $$ with the obvious matrix multiplication. Its irreducible unitary representations in which the center acts by a fixed nontrivial character $e^{i\lambda c/\hbar}$ are classified by the Stone-von Neumann theorem.

theorem (stone-von neumann). for each $\lambda\neq 0$ there is, up to unitary equivalence, a unique irreducible unitary representation of $H_{2n+1}$ on a separable Hilbert space in which the center acts by that character and the representation is strongly continuous (or, equivalently, the integrated Weyl form of the CCR holds).

The Schrodinger representation on $L^2(\mathbb{R}^n)$, $$ \bigl(U(\mathbf{a},\mathbf{b},c)\psi\bigr)(\mathbf{x}) =e^{i\lambda\bigl(c+\mathbf{b}\cdot\mathbf{x}+\tfrac12\mathbf{a}\cdot\mathbf{b}\bigr)/\hbar} \psi(\mathbf{x}+\mathbf{a}), $$ is the standard model. All other regular realizations (momentum representation, Bargmann-Fock representation, and so on) are unitarily equivalent to it. Thus the CCR, together with a mild regularity condition excluding pathological representations of the Weyl relations, determine the kinematics of a quantum particle completely.

example (weyl form). writing $U(\mathbf{a})=e^{-i\mathbf{a}\cdot\mathbf{P}/\hbar}$ and $V(\mathbf{b})=e^{-i\mathbf{b}\cdot\mathbf{Q}/\hbar}$, the integrated CCR become $U(\mathbf{a})V(\mathbf{b})=e^{-i\mathbf{a}\cdot\mathbf{b}/\hbar}V(\mathbf{b})U(\mathbf{a})$. uniqueness theorems are most cleanly stated for these Weyl unitaries rather than for the unbounded generators alone.

the harmonic oscillator and creation/annihilation operators

The one-dimensional harmonic oscillator Hamiltonian $$ H=\frac{P^2}{2m}+\frac12 m\omega^2 Q^2 $$ is most transparent in the complex basis $$ a=\sqrt{\frac{m\omega}{2\hbar}}Q+\frac{i}{\sqrt{2m\omega\hbar}}P,\qquad a^\dagger=\sqrt{\frac{m\omega}{2\hbar}}Q-\frac{i}{\sqrt{2m\omega\hbar}}P. $$ These densely defined operators satisfy $[a,a^\dagger]=1$ on $\mathcal{S}(\mathbb{R})$ and allow $H$ to be rewritten $$ H=\hbar\omega\Bigl(a^\dagger a+\tfrac12\Bigr). $$ The number operator $N=a^\dagger a$ has spectrum $\{0,1,2,\dots\}$. Starting from the unique (up to phase) vacuum vector $|0\rangle$ annihilated by $a$, the excited states $$ |n\rangle=\frac{(a^\dagger)^n}{\sqrt{n!}}|0\rangle $$ form an orthonormal basis of energy eigenvectors with eigenvalues $\hbar\omega(n+\tfrac12)$.

example (matrix elements). one has $a|n\rangle=\sqrt{n}|n-1\rangle$ and $a^\dagger|n\rangle=\sqrt{n+1}|n+1\rangle$. consequently $\langle n|Q|n'\rangle$ and $\langle n|P|n'\rangle$ vanish unless $|n-n'|=1$, recovering the dipole selection rules of the oscillator.

The same construction extends immediately to $n$ degrees of freedom. The resulting bosonic Fock space reappears as the one-particle sector of a free bosonic quantum field.

fourier duality and the metaplectic representation

Position and momentum representations are related by the Fourier transform $$ (\mathcal{F}\psi)(\mathbf{p}) =\frac{1}{(2\pi\hbar)^{n/2}}\int e^{-i\mathbf{p}\cdot\mathbf{q}/\hbar}\psi(\mathbf{q})\,d\mathbf{q}. $$ The Fourier transform intertwines the Schrodinger representation of the Heisenberg group with its dual. More globally, the group of linear canonical transformations $\operatorname{Sp}(2n,\mathbb{R})$ acts on the CCR. This action lifts to a projective unitary representation (the metaplectic representation) on $L^2(\mathbb{R}^n)$.

remark (double cover). the metaplectic group $\operatorname{Mp}(2n,\mathbb{R})$ is a double cover of $\operatorname{Sp}(2n,\mathbb{R})$. the quantum theory therefore sees a projective rather than an ordinary representation of the classical linear symplectic symmetry group. the same projective phenomenon forces half-integer spin for $\operatorname{SO}(3)$.

bargmann-fock space

An especially convenient realization of the CCR employs holomorphic functions. The Bargmann-Fock space is the Hilbert space of entire functions $f:\mathbb{C}^n\to\mathbb{C}$ that are square-integrable with respect to the Gaussian measure $$ d\mu(z)=(\pi\hbar)^{-n}e^{-|z|^2/\hbar}\,d^{2n}z. $$ Creation and annihilation operators become $$ a_i^\dagger=z_i,\qquad a_i=\hbar\frac{\partial}{\partial z_i}, $$ acting on a dense subspace of holomorphic polynomials. The vacuum is the constant function $1$, and the monomials $z^\alpha/\sqrt{\alpha!\,\hbar^{|\alpha|}}$ (with the usual multi-index conventions) furnish the occupation-number basis.

example (coherent states). for $\alpha\in\mathbb{C}$, the coherent state $f_\alpha(z)=e^{-|\alpha|^2/(2\hbar)}e^{\bar\alpha z/\hbar}$ is an eigenvector of $a$ with eigenvalue $\alpha$. coherent states saturate uncertainty relations and provide an overcomplete resolution of the identity, bridging the oscillator algebra to the geometry of $\mathbb{CP}^\infty$.

angular momentum and $\operatorname{SU}(2)$

The commutation relations of angular momentum $$ [J_i,J_j]=i\hbar\epsilon_{ijk}J_k $$ are those of the Lie algebra $\mathfrak{su}(2)\simeq\mathfrak{so}(3)$. Irreducible unitary representations of the compact group $\operatorname{SU}(2)$ are finite-dimensional and labelled by a half-integer or integer spin $j=0,\tfrac12,1,\tfrac32,\dots$. The representation space is $(2j+1)$-dimensional; a standard basis $|j,m\rangle$ diagonalizes $J_z$ with eigenvalues $\hbar m$, $m=-j,\dots,j$.

The orbital angular momentum operators $\mathbf{L}=\mathbf{r}\times\mathbf{p}$ realize only the integer-$j$ representations on $L^2(S^2)$. Half-integer spins require an additional internal Hilbert space and correspond to projective representations of $\operatorname{SO}(3)$, or equivalently to true representations of its double cover $\operatorname{SU}(2)$. The appearance of spin is therefore forced by the topology of the rotation group: $\pi_1(\operatorname{SO}(3))=\mathbb{Z}_2$.

example (spin $1/2$). the fundamental representation has $\mathbf{J}=\tfrac{\hbar}{2}\boldsymbol{\sigma}$. conjugating by a $2\pi$ rotation yields $-I$ on spinors, which is invisible in $\operatorname{SO}(3)$ but nontrivial in $\operatorname{SU}(2)$, matching the double cover.

clifford algebras and fermionic representations

Bosonic canonical commutation relations are replaced, for fermionic degrees of freedom, by canonical anticommutation relations (CAR) $$ \{a_i,a_j^\dagger\}=\delta_{ij},\qquad\{a_i,a_j\}=\{a_i^\dagger,a_j^\dagger\}=0. $$ These relations define a Clifford algebra. The unique (up to equivalence) irreducible representation of the complex Clifford algebra $\operatorname{Cl}(2n,\mathbb{C})$ is $2^n$-dimensional and is realized on the fermionic Fock space generated from a vacuum by the action of the $a_i^\dagger$.

The operators $\gamma_\mu=a_\mu+a_\mu^\dagger$ (and their imaginary companions) satisfy the defining relations of the Clifford algebra and furnish the spin representation of the orthogonal group. The distinction between bosons and fermions is thereby reduced to the distinction between the Heisenberg algebra and the Clifford algebra, or equivalently between symmetric and exterior algebras over the one-particle Hilbert space.

example (one fermionic mode). for a single mode the CAR algebra is represented on $\mathbb{C}^2$ by $a=|0\rangle\langle 1|$ and $a^\dagger=|1\rangle\langle 0|$. the number operator $N=a^\dagger a$ has spectrum $\{0,1\}$, encoding the Pauli exclusion principle for that mode.

symmetry and quantum numbers

A classical symmetry group $G$ that acts by canonical transformations on phase space is expected to act on the quantum Hilbert space by unitary (or projective unitary) operators. Passing to the Lie algebra, one obtains a representation of $\mathfrak{g}$ by (essentially) self-adjoint operators that satisfy the same commutation relations as the classical generators, up to possible central extensions.

Irreducible representations are labelled by a set of discrete or continuous parameters: the quantum numbers. For a compact group the representations are finite-dimensional and the quantum numbers are discrete. For the rotation group they are the familiar angular-momentum labels $j,m$. For the Heisenberg group the Stone-von Neumann theorem supplies a unique kinematics once $\hbar$ (or $\lambda$) is fixed. For the symplectic group the metaplectic representation encodes the quantum theory of linear canonical transformations.

In every case the spectrum of the Casimir operators of the symmetry algebra furnishes the invariants that label physical states. If $C_i$ are independent Casimirs of $\mathfrak{g}$ and $\pi$ is an irreducible representation, the joint eigenvalues of $\pi(C_i)$ are the quantum numbers attached to that representation. In this precise sense, quantum numbers arise as representation-theoretic data rather than as independent phenomenological labels.

summary

Canonical quantization replaces the Poisson algebra of classical observables by a Lie algebra of operators. The Heisenberg algebra and its unique regular irreducible representation fix the kinematics of bosonic particles; Clifford algebras do the same for fermions. Symmetry groups of the classical theory lift to unitary or projective unitary representations on Hilbert space; the irreducible representations of those groups supply the quantum numbers that organize the spectrum.

The harmonic oscillator, the rotation group, and the metaplectic representation of $\operatorname{Sp}(2n,\mathbb{R})$ are the elementary illustrations of this mechanism. All later constructions (geometric quantization, induced representations of the Poincare group, and the operator-algebraic formulation of quantum fields) rest on the same Lie-theoretic foundation.