The Baker-Campbell-Hausdorff (BCH) formula appears in stochastic analysis and in quantum mechanics. In the context of stochastic analysis, the BCH formula provides a method to calculate the log-signature of the concatenation of two rough paths. In the context of quantum mechanics, the BCH formula allows to compute products of general operators in Hilbert spaces.

The log-signature of the concatenation of two paths in expresses as a sum of Lie brackets of formal power series in the tensor algebra of . To learn how the BCH formula is used for computing log-signatures, the reader is referred to section 2.2.4, p. 37, of the book “Differential equations driven by rough paths” by

This blog post provides a proof of the BCH formula from a quantum perspective, as stated in lemma 34, p. 56, of the lecture notes “Quantum mechanics” by Martin Plenio.

Let be a finite Hilbert space over . Consider two operators and . For the operators , define the so-called commutator . The BCH formula in this setting takes the form

As a side note, the commutator corresponds to a Lie bracket in the rough path case.

To prove the BCH formula, introduce the auxiliary function , where . Take the Taylor series expansion of around , i.e. consider . Setting leads to

By application of definition 27, p. 51, of Plenio’s notes, it holds that . Differentiating gives . Similarly, . The chain rule yields

.

So, . For , it is obvious that .

Working in a similar manner, the second-order derivative of is found to be

.

For , it can be seen that .

The rest of the proof follows inductively.