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:

$$S_{fi}\equiv \langle \Psi_p^{-}| \Psi_k^{+}\rangle$$

where $|\Psi_p^{-}\rangle$ is a state in Heisenberg picture which is $| p \rangle$ at $t=+\infty$ when you calculate the $|\Psi_p^{-}\rangle$ in Schrodinger picture , called out state. $| \Psi_k^{+}\rangle$ is a state in Heisenberg picture which is $| k \rangle$ at $t=-\infty$, called in state.

So$$S_{fi}\equiv \langle \Psi_p^{-}| \Psi_k^{+}\rangle= \langle p|(\Omega_-)^\dagger\Omega_+|k \rangle$$

In this case the S-operator $\hat S=(\Omega_-)^\dagger\Omega_+$,

where Møller operator

$$\Omega_+ = \lim_{t\rightarrow -\infty} U^\dagger (t) U_0(t)$$

$$\Omega_- = \lim_{t\rightarrow +\infty} U^\dagger (t) U_0(t)$$

So $$S=U_I(\infty,-\infty)$$

Another definition (e.g (9.14) (9.17) (9.99) in Greiner's Field Quantization) is :

$$S_{fi}\equiv \langle \Psi_p^{-}| \Psi_k^{+}\rangle\equiv\langle \Psi_p^{-}| \hat S ^\prime |\Psi_k^{-}\rangle=\langle \Psi_p^{+}| \hat S ^\prime |\Psi_k^{+}\rangle$$

where S-operator

$\hat S ^\prime |\Psi_p^{-}\rangle =|\Psi_p^{+}\rangle$ that is $\hat S^\prime = \Omega_+(\Omega_-)^\dagger$.

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： $$\Omega_+(\Omega_-)^\dagger= e^{i \alpha}(\Omega_-)^\dagger\Omega_+$$

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