Baker–Campbell–Hausdorff formula

The Baker–Campbell–Hausdorff formula expresses the logarithm of a product of exponentials as a series of nested commutators of their exponents. It connects multiplication in a Lie group with the Lie bracket of its Lie algebra.

Lie bracket: A bilinear operation on a Lie algebra that measures the noncommutativity of its elements. The formula’s correction terms are built from this operation.
Lie group: A group that is also a smooth manifold, with smooth multiplication and inversion. Its local multiplication is what the formula translates into Lie-algebra terms.
Henry Frederick Baker: A British mathematician who worked on geometry, Lie groups, and algebraic structures. His work helped establish the formula’s name in the English-language mathematical tradition.
Magnus expansion: A series expressing the solution of a linear differential equation as the exponential of a time-dependent operator series. It adapts commutator expansions to time-dependent evolution.
Commuting matrices: Matrices A and B satisfying AB = BA. When the exponents commute, every correction term disappears and the formula reduces to a sum.
Commutator: An expression measuring the failure of two algebraic operations or elements to commute. The first correction to the sum of exponents is their commutator.
Exponential map: A map from a Lie algebra to its Lie group that sends infinitesimal generators to one-parameter subgroups. It is the bridge between the exponents and group elements in the formula.
John Edward Campbell: A British mathematician whose work included an early form of the exponential-product formula. His 1897 treatment gave the formula one of its namesakes.
Trotter product formula: A limit formula that represents the exponential of a sum of operators through products of their exponentials. It uses repeated exponential products to handle sums of noncommuting operators.
Zassenhaus formula: A factorization expressing the exponential of a sum as a product of exponentials involving nested commutators. It reverses the formula’s direction, splitting one exponential into factors.

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