20.1 Intertwining operators and the metaplectic representation

20.1 Intertwining operators and the metaplectic representation

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · Lie algebra representation · unitary representation · irreducible representation · group action · group homomorphism · Schur’s lemma

For a general semi-direct product $N\times K$ with non-commutative N, the representation theory can be quite complicated. For the Jacobi group case though, it turns out that things simplify dramatically because of the Stone-von Neumann theorem which says that, up to unitary equivalence, we only have one irreducible representation of $N=H_{2d+1}$

In the general case, recall that for each $k\in K$ the definition of the semi-direct product comes with an automorphism ${\Phi }_k:N\to N$ satisfying ${\Phi }_{k_1{k}_2}={\Phi }_{k_1}{\Phi }_{k_2}.$ Given a representation $\pi$ of $N$ , for each k we can define a new representation ${\pi }_k$ of N by first acting with ${\Phi }_k$ :

$${\pi }_k(n)=\pi ({\Phi }_k(n))$$

In the special case of the Heisenberg group and Schr¨odinger representation ${\Gamma }_S$ we can do this for each $k\in K=Sp(2d,\mathbf{R})$ , defining a new representation by

$${\Gamma }_{S,k}(n)={\Gamma }_S({\Phi }_k(n))$$

The Stone-von Neumann theorem assures us that these must all be unitarily equivalent, so there must exist unitary operators $U_k$ satisfying

$${\Gamma }_{S,k}=U_k{\Gamma }_S{U}_k^{-1}={\Gamma }_S({\Phi }_k(n))$$

We will generally work with the Lie algebra version ${\Gamma }_S^{\prime }$ of the Schr¨odinger representation, for which the same argument applies: we expect to be able to find unitary operators $U_k$ relating Lie algebra representations ${\Gamma }_S^{\prime }$ and ${\Gamma }_{S,k}^{\prime }$ by

$${\Gamma }_{S,k}^{\prime }(X)=U_k{\Gamma }_S^{\prime }(X)U_k^{-1}={\Gamma }_S^{\prime }({\Phi }_k(X))\text{\hspace{1em}(20.1)}$$

where $X$ is in the Heisenberg Lie algebra, and $k$ acts by automorphism ${\Phi }_k$ on this Lie algebra.

Operators like $U_k$ that relate two representations are called “intertwining operators”:

Definition (Intertwining operator)

$If\left({\pi }_1,V_1\right),({\pi }_2,V_2)$ are two representations of a group $G,$ an intertwining operator between these two representations is an operator $U$ such that

$${\pi }_2(g)U=U{\pi }_1(g)\forall g\in G$$

In our case $V_1=V_2$ is the Schr¨odinger representation state space $H$ and $U_k$ : $\mathcal{H}$ $H$ is an intertwining operator between ${\Gamma }_S$ and ${\Gamma }_{S,k}$ for each $k\in Sp(2d,\mathbf{R})$ Since

$${\Gamma }_{S,k_1{k}_2}=U_{k_1{k}_2}{\Gamma }_S{U}_{k_1{k}_2}^{-1}$$

one might expect that the $U_k$ should satisfy the group homomorphism property

$$U_{k_1{k}_2}=U_{k_1}{U}_{k_2}$$

and give us a representation of the group $Sp(2d,\mathbf{R})$ on $\mathcal{H}.$ This is what would follow from the general principle that a group action on the classical phase space after quantization becomes a unitary representation on the quantum state space.

The problem with this argument is that the $U_k$ are not uniquely defined. Schur’s lemma tells us that since the representation on $\mathcal{H}$ is irreducible, the operators commuting with the representation operators are just the complex scalars. These give a phase ambiguity in the definition of the unitary operators $U_k$ , which then give a representation of $Sp(2d,\mathbf{R})$ on $\mathcal{H}$ only up to a phase, $i$.e.,

$$U_{k_1{k}_2}=U_{k_1}{U}_{k_2}{e}^{i\phi (k_1,k_2)}$$

for some real-valued function $\phi$ of pairs of group elements. In terms of corresponding Lie algebra representation operators $U_L^{\prime }$ , this ambiguity appears as an unknown constant times the identity operator.

The question then arises whether the phases of the $U_k$ can be chosen so as to satisfy the homomorphism property (i.e., can phases be chosen so that $\phi (k_1,k_2)=N2\pi$ for N integral?). It turns out that this cannot quite be done, since N may have to be half-integral, giving the homomorphism property only up to a sign. Just as in the $SO(d)$ case where a similar sign ambiguity showed the need to go to a double cover $Spin(d)$ to get a true representation, here one needs to go to a double cover of $Sp(2d,\mathbf{R})$ , called the metaplectic group $\mathcal{M}p(2d,\mathbf{R})$ . The nature of this sign ambiguity and double cover is quite subtle, and unlike for the $Spin(d)$ case, we will not provide an actual construction of $\mathcal{M}p(2d,\mathbf{R})$ . For more details on this, see [56] or [37]. In section 20.3.2 we will show by computation one aspect of the double cover.

Since this is just a sign ambiguity, it does not appear infinitesimally: the ambiguous constants in the Lie algebra representation operators can be chosen so that the Lie algebra homomorphism property is satisfied. However, this will no longer necessarily be true for infinite dimensional phase spaces, a situation that is described as an “anoma ${\mathrm{ly}}^{\text{sc},\ast }$ in the symmetry. This phenomenon will be examined in more detail in chapter 39.

Chapter contents · Next section: 20.2 Constructing intertwining operators · 中文版本

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 221、222、223、224、225、226、227、228、229、230、231、232、233

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：d6200b7008ca4a11108f700be118b9c166777e04875cc11a1d08323d8b0f03b6

OCR 产物 SHA-256：d6200b7008ca4a11108f700be118b9c166777e04875cc11a1d08323d8b0f03b6