Ordered exponential

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The ordered exponential (also called 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^\right)_+] 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_ \left\*e^)}* \cdots *e^*e^\right\}]

where the time moments [\] are defined as [t_j = j*\epsilon] for [j=\overline], and [\epsilon = t/N].

Via the initial value problem, where the OE[A](t) is the unique solution of the system of equations:

[\frac = A(t) * OE[A](t),]

[OE[A](0) = 1.]

Via an integral equation:

[OE[A](t) = 1 + \int_0^t dt' A(t') * 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^ dt_2 A(t_1)*A(t_2)]

[ + \int_0^t dt_1 \int_0^ dt_2 \int_0^ dt_3 A(t_1)*A(t_2)*A(t_3) + \cdots]

Related: Path-ordered exponential describes the same concept.

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