In the context of Feynman integrals, certain values of the zeta function appear at certain loop orders. Given that the tree-level amplitudes are rational, and **assuming** the zeta values to be transcendental and algebraically independent, this means that the loop amplitudes live in extensions of $\mathbb{Q}$ of increasing transcendence degree. There is a notion in physics that associates a "[degree of transcendentality](https://mathoverflow.net/questions/50465/degree-of-transcendentality-and-feynman-diagrams)" to the occurring zeta values, and which appears to be related to the weight of the numbers seen as multiple zeta values.


There is of course also a mathematically well-defined degree for an algebraic number.


All this leads me to wonder if one can make the physics notion of "degree of transcendality" more precise, and extend it from multiple zeta values to periods in general by defining the degree of a period as the lowest dimension of an integral in which it appears. Is a definition like this being used anywhere? And if not, would such a definition make sense?


(I apologize if this is all hopelessly naive — the number-theoretical aspects of this are way outside my area of expertise.)



Can one define a degree of a period?
Jianming Wan's preprint on "Degrees of Periods" should answer your question, using the notion of periods in the sense of Kontsevich-Zagier: <https://arxiv.org/abs/1102.2273>


