Knowlpedia

Symplectic Group Sp(K)

Bounded invertible real-linear maps preserving the symplectic form B

For a (K,B)(K,B), the symplectic group is

Sp(K)={T∈GL(K):B(Tx,Ty)=B(x,y) ∀x,y∈K}.Sp(K)=\{T\in GL(K): B(Tx,Ty)=B(x,y)\ \forall x,y\in K\}.
Remarks

Key properties (paper use):

  • Acts by *-automorphisms on Weyl operators: V(z)↦V(Tz)V(z)\mapsto V(Tz).
  • Only a subgroup is unitarily implementable in Fock space ().
Examples
  • Sp(R2n)Sp(\mathbb R^{2n}) is the classical real symplectic group.