Ordered exponential
The ordered exponential, also called the path-ordered exponential, is a mathematical operation defined in non-commutative algebras, equivalent to the exponential of the integral in the commutative algebras. In practice the ordered exponential is used in matrix and operator algebras.
Let A be an algebra over a real or complex field K, and a(t) be a parameterized element of A,
The parameter t in a(t) is often referred to as the time parameter in this context.
The ordered exponential of a is denoted
where the term n = 0 is equal to 1 and where is a higher-order operation that ensures the exponential is time-ordered: any product of a(t) that occurs in the expansion of the exponential must be ordered such that the value of t is increasing from right to left of the product; a schematic example:
This restriction is necessary as products in the algebra are not necessarily commutative.
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