﻿ The Baker-Campbell-Hausdorff Formula for Exponential Matrices
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The Baker-Campbell-Hausdorff
Formula for Exponential Matrices

The Baker-Campbell-Hausdorff Formula (BCHF) for Exponential Matrices is a beautiful bit of technical mathematics that was worked out separately by John Edward Campbell (1897). It was exteded by Henri Poincaré (1899) and Henry Frederick Baker (1902). It was further elaborated by Felix Hausdorff in 1906. The explicit formula was not formulated until 1947 by Eugene Dynkin.

The BCH ties into the series solution for nonlinear matrix differential equations developed by Wilhelm Magnus in 1954. It also shows why the commutator of two operators plays such a prominent role in physical theory.

The BCHF has various forms, but the one of most use has to do with the transformation of Exp(A+B), where Exp( ) is the matrix exponential function of square matrices and A and B are n×n matrices. If A and B commute (AB=BA) then

#### Exp(A+B) = Exp(A)Exp(B) = Exp(B)Exp(A)

This could be a nice decomposition of a solution in which Exp(A) might represent the oscillatory aspects of the solution and Exp(B) the unoscillatory aspects. For example, consider the graph of the probability density of a harmonic oscillar from Schrödinger's equation in quantum theory.

The heavy line is the spatial average.

The BCHF has to do with what Exp(A+B) is equal to if A and B do not commute. It gives a sequence of products of exponential matrices which asymptotically approach Exp(A+B). The first in the sequence is Exp(A)Exp(B) or Exp(B)Exp(A).

Let [A, B] be the commutator of A and B; i.e., AB−BA. The second in the sequence is

#### Exp(A)Exp(B)Exp(−½[A, B]) = Exp(B)Exp(A)Exp(½[A, B]))

The third in the sequence involves [A, [A, B]) and [B, [A, B]); i.e.,

#### Exp(A)Exp(B)Exp(−½[A, B])Exp(−(1/12){[A, [A, B]) + [B, [A, B])}

Thus if A and B commute with their commutator then

#### Exp(A+B) = Exp(A)Exp(B)Exp(−½[A, B])

The other way the BCHF is represented is as

#### Ln(Exp(A)Exp(B)) = A + B + (1/2)[A, B] + (1/12){[A, [A, B]) + [B, [A, B])} + …

(To be continued.)