Unitary transformation in commutator expansion identity

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There is an often used identity* in physics texts that deal with unitary groups. It is an expansion of a unitary transformation of an operator in an infinite series of commutators of the operator and the argument of the unitary transformation, as follows:

exp(A)*B*exp(-A) = B + [A,B] + [A,[A,B]]/2! + [A,[A,[A,B]]]/3! + ...

This identity is not demonstrated in texts I have read, nor is a reference to a proof cited.

Can someone offer an outline of a proof or a proof or a citation to a proof?

Thanks.

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"Quantum Field Theory", by Michio Kaku (OUP, 1991) p39 (2.24)

 

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Solved

There is an often used identity* in physics texts that deal with unitary groups. It is an expansion of a unitary transformation of an operator in an infinite series of commutators of the operator and the argument of the unitary transformation, as follows:

exp(A)*B*exp(-A) = B + [A,B] + [A,[A,B]]/2! + [A,[A,[A,B]]]/3! + ...

This identity is not demonstrated in texts I have read, nor is a reference to a proof cited.

Can someone offer an outline of a proof or a proof or a citation to a proof?

Thanks.

-------------------
"Quantum Field Theory", by Michio Kaku (OUP, 1991) p39 (2.24)

The identity may be proved by collecting the terms on the left side that have sum of power equal to n and applying the next commutator to them. The result is the next commutator on the right side of the identity. In this way, the identity is proved by induction.