The structure of gauge invariant Gaussian quantum operations on finite Fermion systems

Let H 1 H₁ be a finite dimensional complex Hilbert space. Let ψ ↦ Z ( ψ ) be a canonical anti-commutation relations (CAR) field over H 1 H₁ acting irreducibly on a Hilbert space K K . The ∗-algebra A H 1 A_H₁ generated by { Z ( ψ ) : ψ ∈ H 1 } (ψ ) : ψ ∈ H₁\ , is simply all operators on K K . However, the CAR field endows A H 1 A_H₁ with additional structure, and we are concerned with quantum operations that act in harmony with this structure. In particular, there is a gauge automorphism group generated by “second quantizing” ψ ↦ e it ψ . The fixed point algebra of this group is a sub-algebra of A H 1 A_H₁ ,