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:

“Then he rang the bell and ordered a ham sandwich. When the maid placed the plate on the table, he deliberately looked away but as soon as the door had shut, he grabbed the sandwich with both hands, immediately soiled his fingers and chin with the hanging margin of fat and, grunting greedily, began to much.”
—Vladimir Nabokov (1899–1977)