Lie product formula

A limit formula expressing the exponential of a sum of operators as the limit of repeated products of their exponentials. It extends the scalar identity to settings where the operators may not commute.

Operator exponential: The exponential of a linear operator, defined by a power series or functional calculus. Each short-time factor in the formula is an operator exponential.
Sophus Lie: Norwegian mathematician who developed the theory of continuous transformation groups. The formula bears Lie’s name through its connection to exponentiating sums in Lie theory.
Trotterization: Approximation of an evolution operator by a finite product of simpler exponentials. It applies the formula computationally by replacing a limit with finitely many steps.
Magnus expansion: A series expressing a time-dependent evolution operator as the exponential of a single operator. It handles time-dependent generators through an exponent series rather than repeated products.
Noncommutative algebra: An algebra in which multiplication need not satisfy the commutative law. The formula remains valid when the summands do not commute.
Lie group: A group that is also a smooth manifold, with smooth multiplication and inversion. Exponentials of Lie algebra elements describe local motion within these groups.
Suzuki–Trotter decomposition: A family of product approximations designed to achieve higher-order accuracy. Higher-order variants reduce errors when simulating a sum of noncommuting terms.
Duhamel principle: An integral identity relating solutions of perturbed evolution equations to unperturbed solutions. It treats added generators through an integral equation instead of a product limit.
Baker–Campbell–Hausdorff formula: A series expressing the logarithm of a product of exponentials using commutators. It exposes the commutator corrections behind finite-step product errors.
Lie algebra: A vector space with a bilinear, antisymmetric bracket satisfying the Jacobi identity. The formula relates exponentials of Lie algebra elements to products in a group.

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Golden–Thompson inequalityRelated: Its product approximations connect sums of matrices to exponential products.