an orthogonal state. This introduces the possibility of inequivalent representa-
tions of the commutation relations, built by applying operators to orthogonal
ground states. The physical interpretation again is that such states correspond
to condensates of quanta. For the usual bosonic oscillator case, this phenomenon
occurs in the theory of superfluidity, for fermionic oscillators it occurs in the
theory of superconductivity. It was in the study of such systems that Bogoliubov
discovered the transformations that now bear his name.
39.2 The restricted symplectic group
If one restricts the class of complex structuresJto ones not that different from
the standard oneJ 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 mapgon phase space,gacts on the complex
structure (see equation 26.6), takingJ 0 to a complex structure we’ll callJg. One
can define subgroups of the infinite dimensional symplectic or orthogonal groups
as follows:
Definition(Restricted symplectic and orthogonal groups).The group of linear
transformationsgof an infinite dimensional symplectic vector space preserving
the symplectic structure and also satisfying the condition
tr(A†A)<∞on the operator
A= [Jg,J 0 ]
is called the restricted symplectic group and denotedSpres. The group of linear
transformationsgof an infinite dimensional inner-product space preserving the
inner-product and satisfying the same condition as above on[Jg,J 0 ]is called the
restricted orthogonal group and denotedSOres.
An operatorAsatisfyingtr(A†A)<∞is said to be a Hilbert-Schmidt operator.
One then has the following replacement for the Stone-von Neumann theorem:
Theorem.Given two complex structuresJ 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 2by annihilation and creation operators will give unitarily equivalent representa-
tions of the Weyl algebra (in the bosonic case), or the Clifford 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