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Ordered exponential

  (Redirected from Path-ordered exponential)

The ordered exponential, also called the path-ordered exponential, is a mathematical operation defined in non-commutative algebras, equivalent to the exponential of the integral in the commutative algebras. In practice the ordered exponential is used in matrix and operator algebras.

DefinitionEdit

Let A be an algebra over a real or complex field K, and a(t) be a parameterized element of A,

 

The parameter t in a(t) is often referred to as the time parameter in this context.

The ordered exponential of a is denoted

 

where the term n = 0 is equal to 1 and where   is a higher-order operation that ensures the exponential is 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:

 

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,

 

There are various ways to define this integral more rigorously.

Product of exponentialsEdit

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 as the number of terms grows to infinity:

 

where the time moments {t0, …, tN} are defined as tii Δt for i = 0, …, N, and Δtt / N.

The ordered exponential is in fact a geometric integral.[1][2] [3]

Solution to a differential equationEdit

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

 

Solution to an integral equationEdit

The ordered exponential is the solution to the integral equation:

 

This equation is equivalent to the previous initial value problem.

Infinite series expansionEdit

The ordered exponential can be defined as an infinite sum,

 

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

ExampleEdit

Given a manifold   where for a   with group transformation   it holds at a point  :

 

Here,   denotes exterior differentiation and   is the connection operator (1-form field) acting on  . When integrating above equation it holds (now,   is the connection operator expressed in a coordinate basis)

 

with the path-ordering operator   that orders factors in order of the path  . For the special case that   is an antisymmetric operator and   is an infinitesimal rectangle with edge lengths   and corners at points   above expression simplifies as follows :

 

Hence, it holds the group transformation identity  . If   is a smooth connection, expanding above quantity to second order in infinitesimal quantities   one obtains for the ordered exponential the identity with a correction term that is proportional to the curvature tensor.

See alsoEdit

ReferencesEdit

  1. ^ Michael Grossman and Robert Katz. Non-Newtonian Calculus, ISBN 0912938013, 1972.
  2. ^ A. E. Bashirov, E. M. Kurpınar, A. Özyapıcı. Multiplicative calculus and its applications, Journal of Mathematical Analysis and Applications, 2008.
  3. ^ Luc Florack and Hans van Assen."Multiplicative calculus in biomedical image analysis", Journal of Mathematical Imaging and Vision, 2011.

External linksEdit