Ordered Exponential

The ordered exponential (also called the path-ordered exponential) is a mathematical object, defined in non-commutative algebras, which is equivalent to the exponential function of the integral in the commutative algebras. Therefore it is a function, defined by means of a function from real numbers to a real or complex associative algebra. In practice the values lie in matrix and operator algebras.

For the element A(t) from the algebra (set g with the non-commutative product *), where t is the "time parameter", the ordered exponential

of A can be defined via one of several equivalent approaches:

As the limit of the ordered product of the infinitesimal exponentials:

$OE(t) = \lim_{N \rightarrow \infty} \left\{ e^{\varepsilon A(t_N)}*e^{\varepsilon A(t_{N-1})}* \cdots *e^{\varepsilon A(t_1)}*e^{\varepsilon A(t_0)}\right\}$

where the time moments are defined as for, and .

Via the initial value problem, where the OE(t) is the unique solution of the system of equations:

Via an integral equation:

Via Taylor series expansion:

$\begin{align} OE(t) & = 1 + \int_0^t dt_1 \, A(t_1) + \int_0^t dt_1 \int_0^{t_1} dt_2 \, A(t_1)*A(t_2) \\ & {} \qquad + \int_0^t dt_1 \int_0^{t_1} dt_2 \int_0^{t_2} dt_3 \, A(t_1)*A(t_2)*A(t_3) + \cdots \end{align}$

Famous quotes containing the word ordered:

“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)