20.2 Constructing intertwining operators

20.2 Constructing intertwining operators

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

The method we will use to construct the intertwining operators $U_k$ is to find a solution to the diferentiated version of equation 20.1 and then get $U_k$ by exponentiation. Diferentiating 20.1 for $k=e^{\hat{t}L}$ at $t=0$ gives

$$[U_L^{\prime },{\Gamma }_S^{\prime }(X)]={\Gamma }_S^{\prime }(L\cdot X)\text{\hspace{1em}(20.2)}$$

where

$$L\cdot X=\frac{d}{dt}{\Phi }_{e^{tL}}(X{)}_{|t=0}$$

and we have used equation 5.1 on the left-hand side.

In terms of $Q_j$ and $P_j$ operators, which are i times the ${\Gamma }_S^{\prime }(X)$ for $X$ a basis vector $q_j,p_j$ , equation 20.2 is

$$\left[U_L^{\prime },\left(\genfrac{}{}{0ex}{}{\mathbf{Q}}{\mathbf{P}}\right)\right]=L^T\left(\genfrac{}{}{0ex}{}{\mathbf{Q}}{\mathbf{P}}\right)\text{\hspace{1em}(20.3)}$$

We can find $U_L^{\prime }$ by quantizing the moment map function ${\mu }_L$ , which satisfies

$$\left\{{\mu }_L,\left(\genfrac{}{}{0ex}{}{\mathbf{q}}{\mathbf{p}}\right)\right\}=L^T\left(\genfrac{}{}{0ex}{}{\mathbf{q}}{\mathbf{p}}\right)\text{\hspace{1em}(20.4)}$$

Recall from 16.1.2 that the ${\mu }_L$ are quadratic polynomials in the $q_j,p_j$ . We saw in section 17.3 that the Schr¨odinger representation ${\Gamma }_S^{\prime }$ could be extended from the Heisenberg Lie algebra to the symplectic Lie algebra, by taking a product of $Q_j,P_j$ operators corresponding to the product in ${\mu }_L$ . The ambiguity in ordering for non-commuting operators is resolved by quantizing $q_j{p}_j$ using

$${\Gamma }_S^{\prime }(q_j{p}_j)=-\frac{i}{2}(Q_j{P}_j+P_j{Q}_j)$$

We thus take

$$U_L^{\prime }={\Gamma }_S^{\prime }({\mu }_L)$$

and this will satisfy 20.3 as desired. It will also satisfy the Lie algebra homomorphism property

$$[U_{L_1}^{\prime },U_{L_2}^{\prime }]=U_{[L_1,L_2]}^{\prime }\text{\hspace{1em}(20.5)}$$

If one shifts $U_L^{\prime }$ by a constant operator, it will still satisfy 20.3, but in general will no longer satisfy 20.5. Exponentiating this $U_L^{\prime }$ will give us our $U_k$ , and thus the intertwining operators that we want.

This method will be our fundamental way of producing observable operators. They come from an action of a Lie group on phase space preserving the Poisson bracket. For an element $L$ of the Lie algebra, we first use the moment map to find ${\mu }_L$ , the classical observable, then quantize to get the quantum observable $U_L^{\prime }$

Previous section: 20.1 Intertwining operators and the metaplectic representation · Next section: 20.3 Explicit calculations · 中文版本

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