A.3 Symplectic geometry and quantization

对照

The Lie bracket on the space of functions on phase space $M$ is given by the Poisson bracket, determined by

$$\{q,p\}=1$$

Quantization takes 1, $q,p$ to self-adjoint operators $1,Q,P.$ To make this a unitary representation of the Heisenberg Lie algebra ${\mathfrak{h}}_3$, multiply the self-adjoint operators by $-i,$ so they satisfy

$$[-iQ,-iP]=-i1,\mathrm{or}[Q,P]=i1$$

In other words, our quantization map is the unitary representation of ${\mathfrak{h}}_3$ that satisfies

$${\Gamma }^{\prime }(q)=-iQ,{\Gamma }^{\prime }(p)=-iP,{\Gamma }^{\prime }(1)=-i1$$

Dynamics is determined classically by the Hamiltonian function h as follows

$$\frac{d}{dt}f=\{f,h\}$$

After quantization this becomes the equation

$$\frac{d}{dt}\mathcal{O}(t)=[\mathcal{O},-iH]$$

for the dynamics of Heisenberg picture operators, which implies

$$\mathcal{O}(t)=e^{itH}\mathcal{O}{e}^{-itH}$$

where $\mathcal{O}$ is the Schrödinger picture operator. In the Schrödinger picture, states evolve according to the Schrödinger equation

$$-iH|\psi ⟩=\frac{d}{dt}|\psi ⟩$$

If a group $G$ acts on a space $M$ , the representation one gets on functions on $M$ is given by

$$\pi (g)(f(x))=f(g^{-1}\cdot x)$$

Examples include

• Space translation $(q\to q+a)$ . On states one has

$$|\psi ⟩\to e^{-iaP}|\psi ⟩$$

which in the Schrödinger representation is

$$e^{-ia(-i\frac{d}{dq})}\psi (q)=e^{-a\frac{d}{dq}}\psi (q)=\psi (q-a)$$

So, the Lie algebra action is given by the operator $-iP=-\frac{d}{dq}$ . Note that this has opposite sign to the time translation. On operators one has

$$\mathcal{O}(a)=e^{iaP}\mathcal{O}{e}^{-iaP}$$

or infinitesimally

$$\frac{d}{da}\mathcal{O}(a)=[\mathcal{O},-iP]$$

• The classical expressions for angular momentum quadratic in $q_j,p_j$ , for example

$$l_1=q_2{p}_3-q_3{p}_2$$

under quantization go to the self-adjoint operator

$$L_1=Q_2{P}_3-Q_3{P}_2$$

and $-i{L}_1$ will be the skew-adjoint operator giving a unitary representation of the Lie algebra $\mathfrak{so}(3)$. The three such operators will satisfy the Lie bracket relations of $\mathfrak{so}(3)$ , for instance

$$[-i{L}_1,-i{L}_2]=-i{L}_3$$

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上一节：A.2 Fourier transforms · 下一节：A.4 Complex structures and Bargmann-Fock quantization

附录：约定与记号

来源与版本

正文：英文 · 原著转录

核对状态：AI 辅助转录核对，未作人工审阅

原书 PDF · 印刷页 534、535

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837