I read there are two definitions about [S-operator](https://en.wikipedia.org/wiki/S-matrix#The_S-matrix):
The first one (e.g (8.49) in Greiner's Field Quantization) is:
Sfi≡⟨Ψ−p|Ψ+k⟩
where |Ψ−p⟩ is a state in Heisenberg picture which is |p⟩ at t=+∞ when you calculate the |Ψ−p⟩ in Schrodinger picture , called out state. |Ψ+k⟩ is a state in Heisenberg picture which is |k⟩ at t=−∞, called in state.
SoSfi≡⟨Ψ−p|Ψ+k⟩=⟨p|(Ω−)†Ω+|k⟩
In this case the S-operator ˆS=(Ω−)†Ω+,
where Møller operator
Ω+=limt→−∞U†(t)U0(t)
Ω−=limt→+∞U†(t)U0(t)
So S=UI(∞,−∞)
Another definition (e.g (9.14) (9.17) (9.99) in Greiner's Field Quantization) is :
Sfi≡⟨Ψ−p|Ψ+k⟩≡⟨Ψ−p|ˆS′|Ψ−k⟩=⟨Ψ+p|ˆS′|Ψ+k⟩
where S-operator
ˆS′|Ψ−p⟩=|Ψ+p⟩ that is ˆS′=Ω+(Ω−)†.
It seems that these two definitions are differnt, but many textbook can derive the same dyson formula for these two S-operators.
https://en.wikipedia.org/wiki/S-matrix#The_S-matrix
How to prove: Ω+(Ω−)†=eiα(Ω−)†Ω+
related to this question: http://physics.stackexchange.com/questions/105152/there-are-two-definitions-of-s-operator-or-s-matrix-in-quantum-field-theory-a