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Ordered ExponentialThe ordered exponential is the 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: -
OEA(t) = \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\}
where the time moments are defined as for , and . -
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+ \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
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