TrotterLieTrotter

The Trotter-Lie-Trotter formula is a mathematical decomposition used in the study of Lie groups and Lie algebras, particularly in the context of solving differential equations and approximating group actions. It is a generalization of the Baker–Campbell–Hausdorff formula, which expresses the logarithm of the product of two exponentials in terms of nested commutators.

The formula is named after three mathematicians: William Trotter, Sophus Lie, and another reference to Trotter, though the exact attribution can vary in historical literature. The Trotter-Lie-Trotter formula states that for a Lie algebra **g** and two elements *X* and *Y* in **g**, the following asymptotic equality holds as *t* approaches zero:

exp(t*X*) exp(t*Y*) ≈ exp(t*X*) exp(t*Y*) exp(t[X, Y]) exp(t[X, [X, Y]]) ... (truncated series)

More formally, for a Lie group *G* with Lie algebra **g**, and elements *g₁* and *g₂* in *G* corresponding to *X* and *Y* in **g**, the product *g₁ g₂* can be approximated by the product of exponentials in a nested commutator expansion. The Trotter-Lie-Trotter formula is particularly useful in the context of the Trotter product formula, which approximates the exponential of a sum of Lie algebra elements as the product of exponentials of individual elements.

The formula is widely applied in physics, particularly in quantum mechanics and statistical mechanics, where it helps in approximating time evolution operators. It is also used in numerical analysis for solving ordinary differential equations, especially those arising from Hamiltonian systems. The approximation improves as the time step *t* becomes smaller, making it a valuable tool in both theoretical and computational mathematics.