22.4 Quantization by annihilation and creation operators

22.4 Quantization by annihilation and creation operators

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · Lie algebra representation · Lie bracket · complexification

The introduction of annihilation and creation operators involves allowing linear combinations of position and momentum operators with complex coeficients. These can be thought of as giving a Lie algebra representation of ${\mathfrak{h}}_3\otimes \mathbf{C}$ , the complexified Heisenberg Lie algebra. This is the Lie algebra of complex polynomials of degree zero and one on phase space $M,$ with a basis $1,z,\overline{z}.$ One has

$${\mathfrak{h}}_3\otimes \mathbf{C}=(\mathcal{M}\otimes \mathbf{C})\oplus \mathbf{C}$$

with

$$z=\frac{1}{\sqrt{2}}(q-ip),\overline{z}=\frac{1}{\sqrt{2}}(q+ip)$$

a basis for the complexified dual phase space $M$⊗C. Note that these coordinates provide a decomposition

$$\mathcal{M}\otimes \mathbf{C}=\mathbf{C}\oplus \mathbf{C}$$

of the complexified dual phase space into subspaces spanned by $z$ and by $z$. The Lie bracket is the Poisson bracket, extended by complex linearity. The only non-zero bracket between basis elements is given by

$$\{z,\overline{z}\}=i$$

Quantization by annihilation and creation operators produces a Lie algebra representation by

$${\Gamma }^{\prime }(1)=-i1,{\Gamma }^{\prime }(z)=-i{a}^{†},{\Gamma }^{\prime }(\overline{z})=-ia\text{\hspace{1em}(22.5)}$$

with the operator relation

$$[a,a^{†}]=1$$

equivalent to the Lie algebra homomorphism property

$$[{\Gamma }^{\prime }(z),{\Gamma }^{\prime }(\overline{z})]={\Gamma }^{\prime }(\{z,\overline{z}\})$$

We have now seen two diferent unitarily equivalent realizations of this Lie algebra representation: the Schr¨odinger version ${\Gamma }_S^{\prime }$ on functions of $q,$ where

$$a=\frac{1}{\sqrt{2}}\left(q+\frac{d}{dq}\right),a^{†}=\frac{1}{\sqrt{2}}\left(q-\frac{d}{dq}\right)$$

and the Bargmann-Fock version ${\Gamma }_{BF}^{\prime }$ on functions of $z,$ where

$$a=\frac{d}{dz},a^{†}=z$$

Note that while annihilation and creation operators $\mathrm{gi}$ ve a representation of the complexified Heisenberg Lie algebra ${\mathfrak{h}}_3\otimes \mathbf{C}$ , this representation is only unitary on the real Lie subalgebra ${\mathfrak{h}}_3$ . This corresponds to the fact that general complex linear combinations of a and $a^{†}$ are not self-adjoint, to get something self-adjoint one must take real linear combinations of

$$a+a^{†}\mathrm{and}i(a-a^{†})$$

Previous section: 22.3 The Bargmann-Fock representation · Next section: 22.5 For further reading · 中文版本

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原书 PDF · 印刷页 244、245、246、247、248、249、250、251、252、253

来源版本：2025-10-20

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