Degree of Transcendentality and Feynman Diagrams - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T05:36:34Z http://mathoverflow.net/feeds/question/50465 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/50465/degree-of-transcendentality-and-feynman-diagrams Degree of Transcendentality and Feynman Diagrams Jeff Harvey 2010-12-27T02:55:58Z 2010-12-27T11:51:58Z <p>Physicists computing multiloop Feynman diagrams have introduced various techniques and conjectures that involve the notion of Degree of Transcendentality (DoT). From what I understand one defines</p> <p>1) $DoT(r)=0$, r rational </p> <p>2) $DoT(\pi^k)=k$, $k \in {\mathbb N}$,</p> <p>3) $DoT(\zeta(k))=k$,</p> <p>4) $DoT( a \cdot b)= DoT(a)+DoT(b)$</p> <p>One then proves for example that the $\ell$-loop contribution to a certain scaling function in $N=4$ Supersymmetric gauge theory consists of a sum of terms all of which have DoT equal to $2 \ell-2$. </p> <p>This can't be rigorous mathematically, since it is not even known that $\zeta(2n+1)$ is transcendental, but is there some circle of ideas, or conjecture in mathematics that if true would give a precise definition to DoT?</p> http://mathoverflow.net/questions/50465/degree-of-transcendentality-and-feynman-diagrams/50489#50489 Answer by Wadim Zudilin for Degree of Transcendentality and Feynman Diagrams Wadim Zudilin 2010-12-27T11:51:58Z 2010-12-27T11:51:58Z <p>Your <em>transcendentality</em> reminds me about the Institute of Algebraic Meditation at H&ouml;&ouml;r (Sweden). To be honest, your definition corresponds to what is known as the <em>weight</em> of a (multiple) zeta value (see Michael Hoffman's <a href="http://www.usna.edu/Users/math/meh/mult.html" rel="nofollow">http://www.usna.edu/Users/math/meh/mult.html</a>, especially the <a href="http://www.usna.edu/Users/math/meh/biblio.html" rel="nofollow">references on MZVs</a>). These indeed occur in the computation of Feynman's diagrams. As for conjectures related to the transcendental number theory tag, a belief is that $\pi$ and odd zeta values $\zeta(3)$, $\zeta(5)$, etc, are algebraically independent over the rationals.</p>