17.1 Canonical quantization

17.1 Canonical quantization

Concept links · terms present in this machine draft; source roles are unverified: adjoint operator · eigenvalue · Lie algebra · Lie algebra representation · adjoint representation · group representation · unitary representation · Lie group · Lie bracket

Very early on in the history of quantum mechanics, when Dirac first saw the Heisenberg commutation relations, he noticed an analogy with the Poisson bracket. One has

$$\{q,p\}=1\mathrm{and}-\frac{i}{\hslash }[Q,P]=1$$

as well as

$$\frac{df}{dt}=\{f,h\}\mathrm{and}\frac{d}{dt}\mathcal{O}(t)=-\frac{i}{\hslash }[\mathcal{O},H]$$

where the last of these equations is the equation for the time dependence of a Heisenberg picture observable $\mathcal{O}(t)$ in quantum mechanics. Dirac’s suggestion was that given any classical Hamiltonian system, one could “quantize” it by finding a rule that associates to a function $f$ on phase space a self-adjoint operator $O_f$ (in particular $O_h=H)$ , acting on a state space $H$ such that

$$O_{\{f,g\}}=-\frac{i}{\hslash }[O_f,O_g]$$

This is completely equivalent to asking for a unitary representation $({\pi }^{\prime },\mathcal{H})$ of the infinite dimensional Lie algebra of functions on phase space (with the Poisson bracket as Lie bracket). To see this, note that units for momentum $p$ and position $q$ can be chosen such that $\hslash =1$ . Then, as usual getting a skew-adjoint Lie algebra representation operator by multiplying a self-adjoint operator by $-i,$ setting

$${\pi }^{\prime }(f)=-i{O}_f$$

the Lie algebra homomorphism property

$${\pi }^{\prime }(\{f,g\})=[{\pi }^{\prime }(f),{\pi }^{\prime }(g)]$$

corresponds to

$$-i{O}_{\{f,g\}}=[-i{O}_f,-i{O}_g]=-[O_f,O_g]$$

so one has Dirac’s suggested relation.

Recall that the Heisenberg Lie algebra is isomorphic to the three dimensional sub-algebra of functions on phase space given by linear combinations of the constant function, the function $q$ and the function $p.$ The Schr¨odinger representation ${\Gamma }_S^{\prime }$ provides a unitary representation not of the Lie algebra of all functions on phase space, but of these polynomials of degree at most one, as follows

$$O_1=1,O_q=Q,O_p=P$$

so

$${\Gamma }_S^{\prime }(1)=-i1,{\Gamma }_S^{\prime }(q)=-iQ=-iq,{\Gamma }_S^{\prime }(p)=-iP=-\frac{d}{dq}$$

Moving on to quadratic polynomials, these can also be quantized, as follows

$$O_{\frac{p^2}{2}}=\frac{P^2}{2},O_{\frac{q^2}{2}}=\frac{Q^2}{2}$$

For the function pq one can no longer just replace $p$ by $P$ and $q$ by $Q$ since the operators $P$ and $Q$ don’t commute, so the ordering matters. In addition, neither $PQ$ nor $QP$ is self-adjoint. What does work, satisfying all the conditions to give a Lie algebra homomorphism, is the self-adjoint combination

$$O_{pq}=\frac{1}{2}(PQ+QP)$$

This shows that the Schr¨odinger representation ${\Gamma }_S^{\prime }$ that was defined as a representation of the Heisenberg Lie algebra ${\mathfrak{h}}_3$ extends to a unitary Lie algebra representation of a larger Lie algebra, that of all quadratic polynomials on phase space, a representation that we will continue to denote by ${\Gamma }_S^{\prime }$ and refer to as the Schr¨odinger representation. On a basis of homogeneous order two polynomials we have

$${\Gamma }_S^{\prime }\left(\frac{q^2}{2}\right)=-i\frac{Q^2}{2}=-\frac{i}{2}{q}^2$$ $${\Gamma }_S^{\prime }\left(\frac{p^2}{2}\right)=-i\frac{P^2}{2}=\frac{i}{2}\frac{d^2}{d{q}^2}$$ $${\Gamma }_S^{\prime }(pq)=-\frac{i}{2}(PQ+QP)$$

Restricting ${\Gamma }_S^{\prime }$ to linear combinations of these homogeneous order two polynomials (which give the Lie algebra $\mathfrak{sl}(2,\mathbf{R})$ , see theorem 16.1) we get a Lie algebra representation of $\mathfrak{sl}(2,\mathbf{R})$ called the metaplectic representation.

Restricted to the Heisenberg Lie algebra, the Schr¨odinger representation ${\Gamma }_S^{\prime }$ exponentiates to give a representation ${\Gamma }_S$ of the corresponding Heisenberg Lie group (recall section 13.3). As an $\mathfrak{sl}(2,\mathbf{R})$ ) representation however, it turns out that ${\Gamma }_S^{\prime }$ has the same sort of problem as the spinor representation of $\mathfrak{su}(2)=$ $\mathfrak{so}(3)$ , which was not a representation of $SO(3)$ , but only of its double cover $SU(2)=Spin(3)$ . To get a group representation, one must go to a double cover of the group $SL(2,\mathbf{R})$ , which will be called the metaplectic group and denoted $\mathcal{M}p(2,\mathbf{R})$

For an indication of the problem, consider the element

$$\frac{1}{2}(q^2+p^2)\leftrightarrow E-F=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$$

in $\mathfrak{sl}(2,\mathbf{R})$ . Exponentiating this gives a subgroup $SO(2)\subset SL(2,\mathbf{R})$ of clockwise rotations in the qp plane. The Lie algebra representation operator is

$${\Gamma }_S^{\prime }\left(\frac{1}{2}(q^2+p^2)\right)=-\frac{i}{2}(Q^2+P^2)=-\frac{i}{2}\left(q^2-\frac{d^2}{d{q}^2}\right)$$

which is a second-order diferential operator in both the position space and momentum space representations. As a result, it is not obvious how to exponentiate this operator.

One can however see what happens on the state

$${\psi }_0(q)=e^{-\frac{q^2}{2}}\subset \mathcal{H}=L^2(\mathbf{R})$$

where one has

$$-\frac{i}{2}\left(q^2-\frac{d^2}{d{q}^2}\right){\psi }_0(q)=-\frac{i}{2}{\psi }_0(q)$$

so ${\psi }_0(q)$ is an eigenvector of ${\Gamma }_S^{\prime }(\frac{1}{2}(q^2+p^2))$ with eigenvalue $-\frac{i}{2}$ . Exponentiating ${\Gamma }_S^{\prime }(\frac{1}{2}(q^2+p^2))$ , the representation ${\Gamma }_S$ acts on this state by multiplication by a phase. As one goes around the group $SO(2)$ once (rotating the $qp$ plane by an angle from 0 to $2\pi )$ ), the phase angle only goes from 0 to $\pi ,$ demonstrating the same problem that occurs in the case of the spinor representation.

When we study the Schr¨odinger representation using its action on the quantum harmonic oscillator state space $\mathcal{H}$ in chapter 22 we will see that the operator

$$\frac{1}{2}(Q^2+P^2)$$

is the Hamiltonian operator for the quantum harmonic oscillator, and all of its eigenvectors (not just ${\psi }_0(q))$ have half-integer eigenvalues. In chapter 24 we will ${}_{\mathrm{go}}$ on to discuss in more detail the construction of the metaplectic representation, using methods developed to study the harmonic oscillator.

Chapter contents · Next section: 17.2 The Groenewold-van Hove no-go theorem · 中文版本

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来源版本：2025-10-20

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