I am working on a theory of quantum information and am unsure on some of the mathematical formalism I need.
I have learned that integration can be thought of as summing up infinitely thin slices.
My question is, is there an equivalent idea for multiplication? Id est, is there an integral-like operator for multiplying infinitely thin slices? If so, what is the mathematics behind it?
Thank you in advance!
If you restrict yourself to positive functions, you can "convert from addition to multiplication" using the exponential map $$\exp : (\mathbb R, +) \to (\mathbb R^+, \times) : x \mapsto e^x.$$
If you wanted to, you could build up a complete theory of "multiplicative integration" by mimicking standard integration, just replacing addition with multiplication and multiplication with exponentiation. For example, the naive additive integral formula $$\int_0^1 f(x) dx = \sum_{i=0}^N \frac{f(i)}{N}$$ would be rewritten as $$\Pi_0^1 \;g(x)^{dx} = \prod_{i=0}^N g(i)^{1/N}.$$
There isn't much point in actually developing this as a separate theory, though - since $\exp$ is a smooth isomorphism, it preserves all algebraic and analytic properties, and thus the resulting integration theory is entirely isomorphic to the standard one. Given a positive function $g,$ its "multiplicative integral" can be written as a standard integral: $$\Pi_a^b \;g(x)^{dx} = \exp\left(\int_a^b \ln g(x) dx\right).$$