If $f: [a,b] \to V$ is a (nice) function taking values in a vector space, one can define the definite integral $\int_a^b f(t)\ dt \in V$ as the limit of Riemann sums $\sum_{i=1}^n f(t_i^*) dt_i$, or as the final value $F(b)$ of the solution $F: [a,b] \to V$ to the ODE problem $F'(t) = f(t); F(a) = 0$.

In a similar spirit, given a (nice) function $f: [a,b] \to {\mathfrak g}$ taking values in a Lie algebra $\mathfrak g$ of a Lie group $G$, one can define the multiplicative definite integral, which for sake of discussion I will denote $\Pi_a^b \exp(f(t)\ dt) \in G$, either as the limit of Riemann products $\prod_{i=1}^n \exp(f(t_i^*) dt_i)$ (with the product read from left to right), or as the final value $F(b)$ of the solution $F: [a,b] \to G$ of the ODE $F'(t) = F(t) f(t); F(a) = 1$.

Thus, for instance, when the Lie algebra is abelian, the multiplicative integral is just the exponential of the ordinary integral,

$$\Pi_a^b \exp(f(t)\ dt) = \exp( \int_a^b f(t)\ dt)$$

but in general the two are a little bit different, though still quite analogous.

This notion arises implicitly in many places (solving ODE, integrating connections along curves, dynamics and random walks on Lie groups (e.g. in the work of Terry Lyons), the "noncommutative Fourier transform" from scattering theory, etc.), but I am sure that it must be studied explicitly in some body of literature (and even vaguely recall seeing such at some point in the past). But I am having difficulty locating this literature because I am not sure I have the correct terminology for this concept. So my questions are:

  1. What is the accepted name and notation for this concept in the literature? (Perhaps there is more than one such notation, coming from separate bodies of literature.)

  2. What are the references for the theory of this concept?

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    $\begingroup$ There are wikipedia articles "Product Integral" and "Multiplicative calculus". $\endgroup$ – Roy Maclean Jul 20 '10 at 22:26
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    $\begingroup$ Tha main ref appears to be: J. D. Dollard, C. N. Friedman, Product integration with applications to differential equations, $\endgroup$ – Roy Maclean Jul 20 '10 at 22:27
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    $\begingroup$ I think you want the "Time Ordered Exponential", as used in quantum physics. Alternatively, the "Path Ordered Exponential". $\endgroup$ – George Lowther Jul 20 '10 at 22:49
  • $\begingroup$ This also appears in the study of time-inhomogeneous Markov processes. As I recall you have to be careful about the global time-ordering convention. $\endgroup$ – Steve Huntsman Jul 21 '10 at 0:24

This type of construction also arises in topology and algebraic geometry as "iterated integrals" or "Chen's iterated integrals". There are many sources of which a famous one by Chen himself is: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183539443 .

Path-ordered (or time-ordered) exponential, as suggested in the other answer, is the most common term, or at least would get the most hits in a search, but this is due to the usage in physics.

ADDED: this paper by Hain (Chen's student) calls the construction "iterated line integrals". http://arxiv.org/abs/math.AG/0109204 . Another paper calls it "iterated integrals" in a more specific context matching that of the question: p.21 of http://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf .


As I mentioned in my comment, this concept is the time-ordered exponential (as I remember from Quantum Field Theory lectures, long ago). Alternatively, the path-ordered exponential. I'm no expert here, so I can't really point you to any definitive sources that you couldn't find with a bit of searching yourself anyway.

The function f(t) does not have to be particularly 'nice', just locally integrable should be enough. You can go even further and replace f(t)dt by dF(t) for a continuous finite-variation function F. In fact, F can be any continuous semimartingale, as long as you use Stratonovich integration in the associated SDE (as Ito integration is not coordinate independent). This is the method used by Rogers & Williams (Diffusions, Markov Processes and Martingales) to construct Brownian motions on Lie groups and referred to there as the product-integral injection. Actually, it's a bijection from the continuous semimartingales X in the Lie algebra starting at $X_0=0$ to the continuous semimartingales Y in the Lie group starting at the point $Y_0=1$ satisfying the Stratonovich SDE $$ \partial Y = Y\,\partial X. $$

Also, why restrict to Lie groups/algebras? Any manifold with an affine connection will do, where f maps to the tangent space at some base point, and is moved along the generated curve by parallel transport. There is the possibility of the solution exploding in finite time though. In the case of Lie groups, there is a standard invariant connection which gives you the time-ordered exponential.


In the physics world, the notation is $P\exp(\int_a^b f(t)\,dt)$ or $T\exp(\int_a^b f(t)\,dt)$, where the "$P$" and "$T$" stand for "path ordered" and "time ordered". The idea of time-ordered arithmetic I think is originally due to Feynman:

  • R.P. Feynman (1951). An operator calculus having applications in quantum electrodynamics. Physical Review. vol. 84 (1) pp. 108-128.

In the UC Berkeley 2008 course on Lie theory by Mark Haiman (my edited lecture notes are available as a PDF), we called it just $\int$, which was a bit of an abuse of notation. Or rather, for any ODE, we referred to the corresponding "flow" as $\int$: $\int_p(\vec x)(t)$ was the point that you get to by starting at a point $p$ and flowing via the vector field $\vec x$ by $t$ seconds. I'm not a fan of this notation, myself, whereas I'm reasonably happy with "$T\exp$".

  • $\begingroup$ Now I see that George Lowther has given the same answer above. $\endgroup$ – Theo Johnson-Freyd Jul 21 '10 at 2:15
  • $\begingroup$ As far as I can make out, physicists have a habit of simply slapping a capital T in front of an expression and declaring it to be "time-ordered". Not just exponentials, but whatever takes their fancy really. I guess, this assumes that everything can be expanded out into a sum of products of terms associated with a time index which are then ordered. Or, into fields associated with space-time points where it is assumed that terms associated to space-like separated points commute. $\endgroup$ – George Lowther Jul 21 '10 at 2:43
  • $\begingroup$ "terms associated to space-like separated points commute"... or anticommute! Remember there's a spin-statistics theorem, at least in 4-dimensional relativistic quantum field theory. $\endgroup$ – José Figueroa-O'Farrill Jul 21 '10 at 2:48
  • $\begingroup$ Yes, but you have even numbers of anticommuting terms in the products at each point. So grouped together they will commute. (I think) $\endgroup$ – George Lowther Jul 21 '10 at 2:56
  • $\begingroup$ I've seen both TO and PO used as well. $\endgroup$ – Steve Huntsman Jul 21 '10 at 12:46

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