I've been trying to solve for scattering amplitudes for 4 graviton scattering in string theory. However, while going through Schwarz, Witten and Green book for string theory, I come across the following formula in Appendix 7.4.
$$\langle :e^{A}: :e^{B}:\rangle=e^{\langle A B \rangle}.\tag{7.A.1}$$
Here $A$ and $B$ are operators, whose commutator $[A,B]$ is proportional to the identity operator. Later this formula is generalized as follows,
$$\langle :e^{A_1}:: e^{A_2}:\ldots:e^{A_n}:\rangle=\prod_{i<j}^n e^{\langle A_i A_j\rangle}.\tag{7.A.12}$$
Can someone guide me on how to prove the above statements in the first place. For example, if I accept the first equation to be true then shouldn't the second equation be
$$\langle e^{A_1} e^{A_2}...e^{A_n}\rangle=e^{\langle A_1 A_2...A_n\rangle}$$
Then using Wick's theorem I could contract pairs of two fields, but that would give me terms like ( for 4 fields ):
$$\langle e^{A_1} e^{A_2}...e^{A_n}\rangle = e^{G_{12}G_{34}}.e^{G_{13}G{24}}.e^{G_{14}G_{23}},$$
here $G$ are the green's functions.
However according to the formula given in the book, the result should be:
$$\langle e^{A_1} e^{A_2}...e^{A_n}\rangle = e^{G_{12}}.e^{G_{13}}.e^{G_{14}}.e^{G_{23}}.e^{G_{24}}.e^{G_{34}}$$
Can someone tell me how to prove the first formula, and then how to generalize it for the case of $n$ fields.
