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:

    In spite of our worries to the contrary, children are still being born with the innate ability to learn spontaneously, and neither they nor their parents need the sixteen-page instructional manual that came with a rattle ordered for our baby boy!
    Neil Kurshan (20th century)