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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" 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.)

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    $\begingroup$ I don't have a reference but I think it's conjectured that the ring of periods is graded by the weight (maybe after inverting $2\pi i$). The weight corresponds to the weight of the Hodge structure or motive defining the period (in some sense it is a motivic avatar of periods, which enables to do more things like the action of motivic Galois group). In my view this is a more natural notion than the dimension of the domain of integration. $\endgroup$
    – François Brunault
    Commented Feb 11, 2021 at 8:26

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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

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    $\begingroup$ The paper presents it as a problem to give a concrete period whose degree is $\ge 3$ -- it would be more satisfying to have a notion of degree with more interesting examples. $\endgroup$
    – user44143
    Commented Feb 10, 2021 at 11:43
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    $\begingroup$ @MattF. IMO one should either (1) assume Grothendieck's period conjecture or (2) define a "period" not as a number but as an integral representing a number, and only allow "algebraic" manipulations, so that each period defines a motive, and we have a reasonable theory. $\endgroup$
    – Will Sawin
    Commented Feb 11, 2021 at 0:57
  • $\begingroup$ This is indeed more or less answers my question: one can define a degree in terms of the dimension of the region of integration, but this degree is indeed not terribly satisfying, and it doesn't look like the physicists' notion of degree at all (it appears to equal two for all of the periods commonly occurring from Feynman integrals, no matter the loop order). $\endgroup$
    – gmvh
    Commented Feb 11, 2021 at 15:12
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    $\begingroup$ I was indeed a bit doubtful of the relevance of my answer but it might be a first step towards further investigations, and as such I thought it might be somewhat useful to you (and maybe other people as well). $\endgroup$
    – Sylvain JULIEN
    Commented Feb 11, 2021 at 15:37

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