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en>Widr: Reverted edits by 68.191.132.99 (talk) to last version by Kvng

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86.75.143.120 at 01:57, 4 February 2014

2014-02-04T01:57:39Z

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{{Unreferenced|date=March 2008}}
The '''ordered exponential''' (also called the '''path-ordered exponential''') is a [[mathematics|mathematical]] operation defined in [[non-commutative]] [[algebra over a field|algebras]], equivalent to the [[exponential function|exponential]] of the [[integral]] in the [[commutative]] algebras. In practice the ordered exponential is used in [[matrix (math)|matrix]] and [[Operator (mathematics)|operator]] algebras.

== Definition ==

Let ''A'' be an [[algebra over a field|algebra]] over a [[real number|real]] or [[complex number|complex]] [[field (mathematics)|field]] ''K'', and ''a''(''t'') be a [[function (mathematics)|parameterized]] element of ''A'',

: <math>a \mathrel{:} K \to A. \, </math>

The parameter ''t'' in ''a''(''t'') is often referred to as the ''time parameter'' in this context.

The ordered exponential of ''a'' is denoted

: <math>\operatorname{OE}[a](t) \equiv \mathcal{T} \left\{e^{\int_0^t a(t') \, dt'}\right\} \equiv \sum_{n = 0}^\infty \frac{1}{n!} \int_0^t \cdots \int_0^t \mathcal{T} \left\{a(t'_1) \cdots a(t'_n)\right\} \, dt'_1 \cdots dt'_n </math>

where <math>\mathcal{T}</math> is a higher-order operation that ensures the exponential is [[Path-ordering#Time_ordering|time-ordered]]: any product of ''a''(''t'') that occurs in the expansion of the exponential must be ordered such that the value of ''t'' is increasing from right to left of the product; a schematic example:
: <math>\mathcal{T} \left\{a(1.2) a(9.5) a(4.1)\right\} = a(9.5) a(4.1) a(1.2).</math>
This restriction is necessary as products in the algebra are not necessarily commutative.

The operation maps a parameterized element onto another parameterized element, or symbolically,
: <math>\operatorname{OE} \mathrel{:} \left(K \to A\right) \to \left(K \to A\right). </math>

There are various ways to define this integral more rigorously.

=== Product of exponentials ===

The ordered exponential can be defined as the left [[product integral]] of the [[infinitesimal]] exponentials, or equivalently, as an [[ordered product]] of exponentials in the [[Limit (mathematics)|limit]] as the number of terms grows to infinity:

: <math>\operatorname{OE}[a](t) = \prod_0^t e^{a(t') \, dt'} \equiv
\lim_{N \rightarrow \infty} \left(
e^{a(t_N) \Delta t} e^{a(t_{N-1}) \Delta t} \cdots
e^{a(t_1) \Delta t} e^{a(t_0) \Delta t}
\right)
</math>

where the time moments {{nowrap|{''t''0, …, ''t''''N''}}} are defined as {{nowrap|''t''''i'' ≡ ''i'' Δ''t''}} for {{nowrap|''i'' {{=}} 0, …, ''N''}}, and {{nowrap|Δ''t'' ≡ ''t'' / ''N''}}.

=== Solution to a differential equation ===

The ordered exponential is unique solution of the [[initial value problem]]:

: <math>\begin{align}
\frac{d}{d t} \operatorname{OE}[a](t) &= a(t) \operatorname{OE}[a](t) \text{,} \\
\operatorname{OE}[a](0) &= 1 \text{.}
\end{align}</math>

=== Solution to an integral equation ===

The ordered exponential is the solution to the [[integral equation]]:

:: <math>\operatorname{OE}[a](t) = 1 + \int_0^t a(t') \operatorname{OE}[a](t') \, dt'. </math>

This equation is equivalent to the previous initial value problem.

=== Infinite series expansion ===

The ordered exponential can be defined as an infinite sum,

: <math>\operatorname{OE}[a](t) = 1 + \int_0^t a(t_1) \, dt_1+ \int_0^t \int_0^{t_1} a(t_1) a(t_2) \, dt_2 \, dt_1 + \cdots.</math>

This can be derived by recursively substituting the integral equation into itself.

==See also==

* [[Path-ordering]] (essentially the same concept)
* [[Magnus expansion]]
* [[Product integral]]

[[Category:Abstract algebra]]
[[Category:Ordinary differential equations]]

DNA microarray - Revision history

en>Joeinwiki: /* See also */

en>Widr: Reverted edits by 68.191.132.99 (talk) to last version by Kvng

86.75.143.120 at 01:57, 4 February 2014