抄録
Canonical fields Φφ(f) on the Fock space with an indefinite metric <, >=(, Θ ) and their canonical linear transformations (Bogolyubov transformations) are investigated.
Let T be a bijective real linear operator preserving the φ-symplectic form <, J >r= Real (, φ J ) in one particle Hilbert space \mathscr{H}, where φ is unitary and hermitian and J=√{−1}. It is shown that, under some conditions, T has a decomposition T=V1SV2, where V, are φ-unitary and S is a generalized φ-scaling, namely S(φ)≡φ S*φ=S, JSJ−1=S−1, SK⊂K and SJK⊂JK for a decomposition \mathscr{H}=K⊕JK.
T is called Θ-unitarily implementable if there exists a Θ-unitary (bounded bijective Θ-isometric) operator UT on the Fock space \mathscr{F} such that UTΦφ(f)UT−1=Φφ(Tf). This definition is too restrictive. It is shown that T is Θ-unitarily implementable if and only if [T, φ]=0 and anti-linear part T− of T is of Hilbert-Schmidt class.
We introduce a less restrictive notion: T is called weakly Θ-unitarily implementable if there exist a Θ-isometric operator UT−1 (not necessarily bounded) and a cyclic vector ΩT∈\mathscr{F} such that UT−1Φφ(Tf1)...Φφ(Tfn) Ω=Φφ(f1)...Φφ(fn)ΩT, where Ω is the Fock vacuum. A necessary and a sufficient condition for this implementability are obtained.
As an application, a mass-shift model of the vector field of an indefinite metric formalism (Stückelberg formalism) is discussed. A time-evolution of the system by the model Hamiltonian is investigated.
