View source for Ordered exponential

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|integral]] in the commutative algebras.  In practice the ordered exponential is used in [[Matrix (mathematics)|matrix]] and [[Operator (mathematics)|operator]] algebras.

== Definition ==

Let {{math|''A''}} be an [[Algebra over a field|algebra]] over a [[Real number|real]] or [[Complex number|complex]] [[Field (mathematics)|field]] {{math|''K''}}, and {{math|''a''(''t'')}} be a [[Function (mathematics)|parameterized]] element of {{math|''A''}},
<math display="block">a : K \to A. </math>

The parameter {{mvar|t}} in {{math|''a''(''t'')}} is often referred to as the ''time parameter'' in this context.

The ordered exponential of {{math|''a''}} is denoted
<math display="block">\begin{align}
\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 \\
& \equiv \sum_{n = 0}^\infty \int_0^t \int_0^{t'_n} \int_0^{t'_{n-1}} \cdots \int_0^{t'_2} a(t'_n) \cdots a(t'_1) \, dt'_1 \cdots dt'_{n-2} \, dt'_{n-1} \, dt'_n
\end{align}</math>

where the term {{math|1=''n'' = 0}} is equal to 1 and where <math>\mathcal{T}</math> is a higher-order operation that ensures the exponential is [[Physics:Path-ordering#Time ordering|time-ordered]]: any product of {{math|''a''(''t'')}} that occurs in the expansion of the exponential must be ordered such that the value of {{mvar|t}} is increasing from right to left of the product; a schematic example:
<math display="block">\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 display="block">\operatorname{OE} \mathrel{:} (K \to A) \to (K \to A). </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|product integral]] of the [[Infinitesimal|infinitesimal]] exponentials, or equivalently, as an ordered product of exponentials in the [[Limit (mathematics)|limit]] as the number of terms grows to infinity:
<math display="block">\operatorname{OE}[a](t) = \prod_0^t e^{a(t') \, dt'} \equiv
    \lim_{N \to \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 {{math|{{mset|''t''<sub>0</sub>, ..., ''t''<sub>''N''</sub>}}}} are defined as {{math|''t''<sub>''i''</sub> ≡ ''i'' Δ''t''}} for {{math|1=''i'' = 0, ..., ''N''}}, and {{math|Δ''t'' ≡ ''t'' / ''N''}}.

The ordered exponential is in fact a [[Product integral#Type II|geometric integral]].<ref name=nnc>Michael Grossman and Robert Katz. [https://books.google.com/books?q=%22Non-Newtonian+Calculus%22&btnG=Search+Books&as_brr=0 ''Non-Newtonian Calculus''], {{ISBN|0912938013}}, 1972.</ref><ref>A. E. Bashirov, E. M. Kurpınar, A. Özyapıcı. [http://linkinghub.elsevier.com/retrieve/pii/S0022247X07003824 ''Multiplicative calculus and its applications''], Journal of Mathematical Analysis and Applications, 2008.</ref><ref name=FvA>Luc Florack and Hans van Assen.[https://doi.org/10.1007%2Fs10851-011-0275-1 "Multiplicative calculus in biomedical image analysis"], Journal of Mathematical Imaging and Vision, 2011. </ref>

=== Solution to a differential equation ===

The ordered exponential is unique solution of the [[Initial value problem|initial value problem]]:
<math display="block">\begin{align}
    \frac{d}{d t} \operatorname{OE}[a](t) &= a(t) \operatorname{OE}[a](t), \\[5pt]
    \operatorname{OE}[a](0) &= 1.
\end{align}</math>

=== Solution to an integral equation ===

The ordered exponential is the solution to the [[Integral equation|integral equation]]:
<math display="block">\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 display="block">\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.

== Example ==

Given a manifold <math>M</math> where for a <math>e \in TM</math> with [[Group theory|group]] transformation <math>g: e \mapsto g e</math> it holds at a point <math>x \in M</math>:

: <math>de(x) + \operatorname{J}(x)e(x) = 0.</math>

Here, <math>d</math> denotes exterior differentiation and <math>\operatorname{J}(x)</math> is the connection operator (1-form field) acting on <math>e(x)</math>. When integrating above equation it holds (now, <math>\operatorname{J}(x)</math> is the connection operator expressed in a coordinate basis)

: <math>e(y) = \operatorname{P} \exp \left(- \int_x^y J(\gamma (t)) \gamma '(t) \, dt \right) e(x)</math>

with the path-ordering operator <math>\operatorname{P}</math> that orders factors in order of the path <math>\gamma(t) \in M</math>. For the special case that <math>\operatorname{J}(x)</math> is an [[Physics:Antisymmetry|antisymmetric]] operator and <math>\gamma</math> is an infinitesimal rectangle with edge lengths <math>|u|,|v|</math> and corners at points <math>x,x+u,x+u+v,x+v,</math> above expression simplifies as follows :

: <math>
\begin{align}
& \operatorname{OE}[- \operatorname{J}]e(x) \\[5pt]
= {} & \exp [- \operatorname{J}(x+v) (-v)] \exp [- \operatorname{J}(x+u+v) (-u)] \exp [- \operatorname{J}(x+u) v] \exp [- \operatorname{J}(x) u] e(x) \\[5pt]
= {} & [1 - \operatorname{J}(x+v) (-v)][1 - \operatorname{J}(x+u+v) (-u)][1 - \operatorname{J}(x+u) v][1 - \operatorname{J}(x) u] e(x).
\end{align}
</math>

Hence, it holds the group transformation identity <math>\operatorname{OE}[- \operatorname{J}] \mapsto g \operatorname{OE}[\operatorname{J}] g^{-1}</math>. If <math>- \operatorname{J}(x)</math> is a smooth connection, expanding above quantity to second order in infinitesimal quantities <math>|u|,|v|</math> one obtains for the ordered exponential the identity with a correction term that is proportional to the [[Riemann curvature tensor|curvature tensor]].

==See also==

* [[Physics:Path-ordering|Path-ordering]] (essentially the same concept)
* [[Physics:Magnus expansion|Magnus expansion]]
* [[Product integral]]
* List of derivatives and integrals in alternative calculi
* [[Indefinite product]]
*[[Fractal derivative]]

==References==

<references />

==External links==
* [https://sites.google.com/site/nonnewtoniancalculus/ Non-Newtonian calculus website]

[[Category:Abstract algebra]]
[[Category:Ordinary differential equations]]
[[Category:Non-Newtonian calculus]]

{{Sourceattribution|Ordered exponential}}