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 $(g,$
★ ) (set g with the non-commutative product
★ ), where t is the "time parameter", the ordered exponential $OE[A](t):equiv\; left(e^\{int\_0^t\; dt\text{'}\; A(t\text{'})\}$
ight)_+ of A can be defined via one of several equivalent approaches:

★ As the limit of the ordered product of the infinitesimal exponentials:
:$$
OE[A](t) =
lim_{N
ightarrow infty} left{
e^{epsilon A(t_N)}
★ e^{epsilon A(t_{N-1})}
★ cdots

★ e^{epsilon A(t_1)}
★ e^{epsilon A(t_0)}
ight}
where the time moments $\{t\_0,\; t\_1,\; ...\; t\_N\}$ are defined as $t\_j\; =\; j$
★ epsilon for $j=overline\{0,N\}$, and $epsilon\; =\; t/N$.

★ Via the initial value problem, where the OE[A](t) is the unique solution of the system of equations:
:
★ OE[A](t),
:$OE[A](0)\; =\; 1.$

★ Via an integral equation:
:$OE[A](t)\; =\; 1\; +\; int\_0^t\; dt\text{'}\; A(t\text{'})$
★ OE[A](t').

★ Via Taylor series expansion:
:$OE[A](t)\; =\; 1\; +\; int\_0^t\; dt\_1\; A(t\_1)$
+ int_0^t dt_1 int_0^{t_1} dt_2 A(t_1)
★ A(t_2)
:$+\; int\_0^t\; dt\_1\; int\_0^\{t\_1\}\; dt\_2\; int\_0^\{t\_2\}\; dt\_3\; A(t\_1)$
★ A(t_2)
★ A(t_3)
+ cdots
----

★ Related: Path-ordering describes essentially the same concept.