Ordered exponential
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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 :\equiv \left(e^{\int_0^t dt' A(t')}\right)_+](../../../math/4/3/e/43ec8364cd7f71dafb97d5f6feb6febe.png)
of A can be defined via one of several equivalent approaches:
- As the limit of the ordered product of the infinitesimal exponentials:
 = \lim_{N \rightarrow \infty} \left\{ e^{\epsilon A(t_N)}*e^{\epsilon A(t_{N-1})}* \cdots *e^{\epsilon A(t_1)}*e^{\epsilon A(t_0)}\right\}](../../../math/6/0/c/60c9012e556a43241f961c4a854dcaf5.png)
where the time moments {t0,t1,...tN} are defined as tj = j * ε for 
, and ε = t / N.
- Via the initial value problem, where the OE[A](t) is the unique solution of the system of equations:
}{\partial t} = A(t) * OE[A](t),](../../../math/f/e/d/fed7cabf158bb080eac7e1894dce3514.png)
- OE[A](0) = 1.
- Via an integral equation:
 = 1 + \int_0^t dt' A(t') * OE[A](t').](../../../math/4/e/1/4e10d113b3711dba04e7a3060131d151.png)
- Via Taylor series expansion:
 = 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)](../../../math/3/b/7/3b7296f92393a8ea0a1069872aecf8c3.png)
- Related: Path-ordering describes essentially the same concept.
