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:

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—Abraham Lincoln (1809–1865)