24.2 Intertwining operators in terms of a and a ^

24.2 Intertwining operators in terms of a and a ^

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

Recall from the discussion in chapter 20 that the metaplectic representation of $\mathcal{M}p(2,\mathbf{R})$ can be understood in terms of intertwining operators that arise due to the action of the group $SL(2,\mathbf{R})$ as automorphisms of the Heisenberg group $H_3$ Such intertwining operators can be constructed by exponentiating quadratic operators that have the commutation relations with the $Q,P$ operators that reflect the intertwining relations (see equation 20.3). These quadratic operators provide the Lie algebra version of the metaplectic representation, discussed in section 24.1 using the Lie algebra sl(2, $R$), which is identical to the Lie algebra of $\mathcal{M}p(2,\mathbf{R})$ . In sections 20.3.2 and 20.3.4 these representations were constructed explicitly for $SO(2)$ and R subgroups of $SL(2,\mathbf{R})$ using quadratic combinations of the $Q$ and $P$ operators. Here we’ll do the same thing using annihilation and creation operators instead of $Q$ and $P$ operators.

For the SO(2) subgroup of equation 24.1 (this is the same one discussed in section 20.3.2), in terms of z and $z$ coordinates the moment map will be

$${\mu }_Z=z\overline{z}$$

and one has

$$\{{\mu }_Z,(\genfrac{}{}{0ex}{}{\overline{z}}{z})\}=\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right)\left(\genfrac{}{}{0ex}{}{\overline{z}}{z}\right)\text{\hspace{1em}(24.3)}$$

Quantization by annihilation and creation operators gives (see 24.2)

$${\Gamma }_{BF}^{\prime }(z\overline{z})={\Gamma }_{BF}^{\prime }(Z)=-\frac{i}{2}(a{a}^{†}+a^{†}a)$$

and the quantized analog of 24.3 is

$$\left[-\frac{i}{2}(a{a}^{†}+a^{†}a),\left(\genfrac{}{}{0ex}{}{a}{a^{†}}\right)\right]=\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right)\left(\genfrac{}{}{0ex}{}{a}{a^{†}}\right)\text{\hspace{1em}(24.4)}$$

For group elements, $g_{\theta }=e^{\theta Z}\in SO(2)\subset SL(2,\mathbf{R})$ ) and the representation is given by unitary operators

$$U_{g_{\theta }}={\Gamma }_{BF}(e^{\theta Z})=e^{-i\frac{\theta }{2}(a{a}^{†}+a^{†}a)}$$

which satisfy

$$U_{g_{\theta }}\left(\genfrac{}{}{0ex}{}{a}{a^{†}}\right)U_{g_{\theta }}^{-1}=\left(\begin{array}{cc}{e}^{i\theta }& 0\\ 0& e^{-i\theta }\end{array}\right)\left(\genfrac{}{}{0ex}{}{a}{a^{†}}\right)\text{\hspace{1em}(24.5)}$$

Note that, using equation 5.1

$$\frac{d}{d\theta }{\left(U_{g_{\theta }}\left(\genfrac{}{}{0ex}{}{a}{a^{†}}\right)U_{g_{\theta }}^{-1}\right)}_{|\theta =0}=\left[-\frac{i}{2}(a{a}^{†}+a^{†}a),\left(\genfrac{}{}{0ex}{}{a}{a^{†}}\right)\right]$$

so equation 24.4 is the derivative at $\theta =0$ of equation 24.5. We see that, on operators, conjugation by the action of this $SO(2)$ subgroup of $SL(2,\mathbf{R})$ does not mix creation and annihilation operators. On the distinguished state $|0⟩,U_{g_{\theta }}$ acts as the phase transformation

$$U_{g_{\theta }}|0⟩=e^{-\frac{i}{2}\theta }|0⟩$$

Besides 24.3, there are also the following other Poisson bracket relations between order two and order one polynomials in $z,\overline{z}$

$$\{z^2,\overline{z}\}=2iz,\{z^2,z\}=0,\{{\overline{z}}^2,z\}=-2i\overline{z},\{{\overline{z}}^2,\overline{z}\}=0\text{\hspace{1em}(24.6)}$$

The function

$$\mu =\frac{i}{2}({\overline{z}}^2-z^2)$$

will provide a moment map for the $\mathbf{R}\subset SL(2,\mathbf{R})$ subgroup studied in section 20.3.4. This is the subgroup of elements $g_r$ that for $r\in \mathbf{R}$ act on basis elements $q,p$ by

$$\left(\genfrac{}{}{0ex}{}{q}{p}\right)\to \left(\begin{array}{cc}{e}^r& 0\\ 0& e^{-r}\end{array}\right)\left(\genfrac{}{}{0ex}{}{q}{p}\right)=\left(\genfrac{}{}{0ex}{}{e^{r}q}{e^{-r}p}\right)\text{\hspace{1em}(24.7)}$$

or on basis elements $z,\overline{z}$ by

$$\left(\genfrac{}{}{0ex}{}{\overline{z}}{z}\right)\to \left(\begin{array}{cc}\cosh r& \sinh r\\ \sinh r& \cosh r\end{array}\right)\left(\genfrac{}{}{0ex}{}{\overline{z}}{z}\right)$$

This moment map satisfies the relations

$$\left\{\mu ,\left(\genfrac{}{}{0ex}{}{\overline{z}}{z}\right)\right\}=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\left(\genfrac{}{}{0ex}{}{\overline{z}}{z}\right)\text{\hspace{1em}(24.8)}$$

Quantization $\mathrm{gi}$ ves

$${\Gamma }_{BF}^{\prime }(\mu )=\frac{1}{2}(a^2-(a^{†}{)}^2)$$

which satisfies

$$\left[\frac{1}{2}(a^2-(a^{†}{)}^2),(\genfrac{}{}{0ex}{}{a}{a^{†}})\right]=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\left(\genfrac{}{}{0ex}{}{a}{a^{†}}\right)$$

and intertwining operators

$$U_{g_r}=e^{{\Gamma }_{BF}^{\prime }(r\mu )}=e^{\frac{r}{2}(a^2-(a^{†}{)}^2)}$$

which satisfy

$$U_{g_r}\left(\genfrac{}{}{0ex}{}{a}{a^{†}}\right)U_{g_r}^{-1}=e^{r\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)}\left(\genfrac{}{}{0ex}{}{a}{a^{†}}\right)=\left(\begin{array}{cc}\cosh r& \sinh r\\ \sinh r& \cosh r\end{array}\right)\left(\genfrac{}{}{0ex}{}{a}{a^{†}}\right)\text{\hspace{1em}(24.9)}$$

The operator $\frac{1}{2}(a^2-(a^{†}{)}^2)$ does not commute with the number operator $N_{☉}$ or the harmonic oscillator Hamiltonian $H_{;}$ , so the transformations $U_{g_{\tau }}$ are not “symmetry transformations”, preserving energy eigenspaces. In particular they act non-trivially on the state |0⟩, taking it to a diferent state

$$|0{⟩}_r=e^{\frac{r}{2}(a^2-(a^{†}{)}^2)}|0⟩$$

Previous section: 24.1 The metaplectic representation for d=1 in terms of a and a ^ · Next section: 24.3 Implications of the choice of z, z · 中文版本

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来源版本：2025-10-20

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