5.1 Multiple Systems
77
:=
j∈
ϕ j
(5.1.11)
be a product-vector in H . For any linear densely defined operator A on H (with
domain D(A) ⊂ H) and for ϕ j ∈ H j such that u
−1
j ϕ j ∈ D(A) let π j (A) be the
operator on H determined by
π j (A)) :=
⎛
⎝
k∈\{ j}
ϕ k
⎞
⎠ ⊗ (u j Au
−1
j ϕ j ).
(5.1.12)
Symbolically: π j (A) := I 1 ⊗ I 2 ⊗ · · · ⊗ I j−1 ⊗ A ⊗ I j+1 ⊗ . . . , if = Z + \{0}.
Unitary group action U of G on H is determined by
U (g)) :=
j∈
(u j U (g)u
−1
j ϕ j ).
(5.1.13)
For || (:= the cardinality of ) finite, the representation U is strongly continuous
with generators
X
ξ :=
j∈
π j (X ξ ), ξ ∈ g.
(5.1.14)
U is not weakly continuous in the case of infinite : If ϕ ∈ H is not an eigenvector of X ξ , ϕ j := u j ϕ for all j ∈ , ϕ = 1 and is the corresponding productvector (5.1.11) in H , then = 1 and
((, U (exp(tξ)))) = 0
(5.1.15)
for all sufficiently small |t| = 0, t ∈ R, since
|(ϕ j , u j exp(−it X ξ )u
−1
j ϕ j )| = |(ϕ, exp(−it X ξ )ϕ)| < 1 if e
−it X ξ ϕ = λϕ,
(5.1.16)
for any λ ∈ C, i.e. the function in (5.1.15) is discontinuous at t = 0.
5.1.4 Notes on the structure of CTPS.
We shall not give here a thorough definition of CTPS. We shall assume that the
definitions of (convergence and quasiconvergence of) infinite products and sums
of complex numbers as well as of the scalar product in H according to [227] are
known to the reader. Let z ∈ C
, i.e. z is a function
z : → C, j → z j .
(5.1.17)
Assume that |z j | = 1 for all j ∈ and define a unitary operator U z on H by its
linear action on product vectors (5.1.11) (the set of which is total in H ) given by