39.2 The restricted symplectic group

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If one restricts the class of complex structures J to ones not that diferent from the standard one $J_{0},$ then one can recover a version of the Stone-von Neumann theorem and have much the same behavior as in the finite dimensional case. Note that for each invertible linear map $g$ on phase space, $g$ acts on the complex structure (see equation 26.6), taking $J_{0}$ to a complex structure we’ll call $J_{g}$ . One can define subgroups of the infinite dimensional symplectic or orthogonal groups as follows:

Definition (Restricted symplectic and orthogonal groups)

The group of linear transformations g of an infinite dimensional symplectic vector space preserving the symplectic structure and also satisfying the condition
$t r (A^{†} A) < \infty$

on the operator

A = [J_{g}, J_{0}]

is called the restricted symplectic group and denoted $S p_{r es}$ . The group of linear transformations g of an infinite dimensional inner-product space preserving the inner-product and satisfying the same condition as above on $[J_{g}, J_{0}]$ is called the restricted orthogonal group and denoted $S O_{r es}$

An operator $A$ satisfying $t r (A^{†} A) < \infty$ is said to be a Hilbert-Schmidt operator. One then has the following replacement for the Stone-von Neumann theorem:

Theorem

Given two complex structures $J_{1}, J_{2}$ on a Hilbert space such that $[J_{1}, J_{2}]$ is Hilbert-Schmidt, acting on the states
$∣ 0 ⟩_{J_{1}}, ∣ 0 ⟩_{J_{2}}$

by annihilation and creation operators will give unitarily equivalent representations of the Weyl algebra (in the bosonic case), or the Cliford algebra (in the fermionic case).

The standard reference for the proof of this statement is the original papers of Shale [79] and Shale-Stinespring [80]. A detailed discussion of the theorem can be found in [64].

For some motivation for this theorem, consider the finite dimensional case studied in section 25.5 (this is for the symplectic group case, a similar calculation holds in the orthogonal group case). Elements of $sp (2 d, R)$ corresponding to Bogoliubov transformations (i.e., with non-zero commutator with $J_{0})$ were of the form

\frac{1}{2} j k \sum (B_{j k} z_{j} z_{k} + \overline{B}_{j k} \overline{z}_{j} \overline{z}_{k})

for symmetric complex matrices $B .$ . These acted on the metaplectic representation by

- \frac{i}{2} j k \sum (B_{j k} a_{j}^{†} a_{k}^{†} + \overline{B}_{j k} a_{j} a_{k}) (39.1)

and commuting two of them gave a result (equation 25.9) corresponding to quantization of an element of the u(d) subgroup, difering from its normal ordered version by a term

- \frac{1}{2} t r (B \overline{C} - C \overline{B}) 1 = - \frac{1}{2} t r (B C^{†} - C B^{†}) 1

For $d = \infty$ , this trace in general will be infinite and undefined. An alternate characterization of Hilbert-Schmidt operators is that for B and C Hilbert-Schmidt operators, the traces

t r (B C^{†}) and t r (C B^{†})

will be finite and well-defined. So, at least to the extent normal ordered operators quadratic in annihilation and creation operators are well-defined, the Hilbert-Schmidt condition on operators not commuting with the complex structure implies that they will have well-defined commutation relations with each other.

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来源版本：2025-10-20

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39.2 The restricted symplectic group

39.2 The restricted symplectic group

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