Ordered Exponential

Ordered Exponential is the mathematical object, defined in the non-commutative[?] algebras, which is equivalent to the normal exponential function of the integral in the commutative algebras. In practice such objects are observed in the matrix and operator algebras.

For the element A(t) from the algebra <math>(g,*)</math> (set g with the non-commutative product *), where t is the time parameter[?], the ordered exponential <math>OE[A](t):\equiv \left(e^{\int_0^t dt' A(t')}\right)_+</math> of A can be defined via one of several equivalent approaches:

• As the limit of the ordered product[?] of the infinitezimal[?] exponentials:

<math>

OE[A](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\}</math>

where the time moments <math>\{t_0, t_1, ... t_N\}</math> are defined as <math>t_j = j*\epsilon</math> for <math>j=\overline{0,N}</math>, and <math>\epsilon = t/N</math>.

• Via the Initial Value Problem[?], where the OE[A](t) is the unique solution[?] of the system of equations[?]:

<math>\frac{\partial OE[A](t)}{\partial t} = A(t) * OE[A](t),</math>

<math>OE[A](0) = 1.</math>

• Via following integral equation[?]:

<math>OE[A](t) = 1 + \int_0^t dt' A(t') * OE[A](t').</math>

• Via Taylor series expansion:

<math>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</math>