Transcendence
Recall that the spectrum of the MZV algebra is somehow embedded in the Grothendieck-Teichmuller group, which Bar-Natan has defined in terms of non-associative braids. Today God Plays Dice discusses MZVs and the conjecture that all zeta values are transcendental numbers.
This suggests a real dichotomy between algebraic (in fact integral) arguments and non-algebraic values. On the other hand, only $\pi$ seems to crop up, which is not surprising given the appearance of volumes of spherical polytopes. If $\pi$ represents the sum of angles of a Euclidean triangle
$\pi = \theta_1 + \theta_2 + \theta_3$
then integral powers of $\pi$ look like homogeneous symmetric polynomials in three variables. For even $n$, the basic zeta values always contain a factor of $\pi^n$. (Do non-commutative zeta algebras contain a concept of non-commutative angle?)
This suggests a real dichotomy between algebraic (in fact integral) arguments and non-algebraic values. On the other hand, only $\pi$ seems to crop up, which is not surprising given the appearance of volumes of spherical polytopes. If $\pi$ represents the sum of angles of a Euclidean triangle
$\pi = \theta_1 + \theta_2 + \theta_3$
then integral powers of $\pi$ look like homogeneous symmetric polynomials in three variables. For even $n$, the basic zeta values always contain a factor of $\pi^n$. (Do non-commutative zeta algebras contain a concept of non-commutative angle?)
posted by Kea | 10:21 AM
4 Comments:
Funny thing that these mathematicians are playing with these things which I thought to be cognitive side tracks of a theoretical physicist with very primitive mathematical skills and lost into the number theoretical idea jungle.
For few years ago I played similar games with multiple zetas: the physical idea was that if zeros of zeta describe complex conformal weights of single particle state then the zero distributions of multiple zeta describe the values of conformal weights assignable to many-particle state. Also I identified what might be called bound state conformal weights from the continuum of zeros of multiplet zetas.
If the complexity of conformal weight disturbs, one might find better to identify zeros as analogs of complex vacuum expectations of Higgs and identify conformal weight with its modulus squared. This identification makes sense for complex generalized eigenvalues of Dirac operator and they are even continuous functions as Higgs also but points of number theoretic braid select discrete value set.
pi is very challenging number from the point of view of p-adicization programs. e^p is a p-adic number so that it can define extension of p-adics. But Pi is really very strongly transcendental.
*Should one accept the extension of p-adics containing pi and all its powers: is this still mathematically completely ok?
*Or should one use only the phases exp(i2pi/n) and very "quantal" algebraic numbers to define of p-adics. pi obviously appears when on takes derivative of Lie-group element exp(i2pit) with respect to t at t=0. For quantum groups derivatives are discretized: could this be interpreted by saying that in p-adic worlds you can measure only discrete phases. You cannot weigh thoughts;-)! Thoughts can be however characterized by their discrete symmetries (cyclic symmetry groups Z_n) and symmetries define geometry;-).
*You end up with pi and logarithms of integers when you integrate rational functions. And definite integrals are indeed the problem for p-adic calculus. In TGD this finally led to the recent picture where the action principle does not make sense separately for p-adic counterparts of p-adic 3-surfaces and they obey same *algebraic* equations as their real counterparts. Almost topological QFT character of TGD and p-adicization are ideas forcing each other.
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If one does not accept pi what this means physically, that is from the point of S-matrix?
*Zeta(n) pops up in Feynman graphs in perturbation theory. Does this mean that radiative corrections vanish. They indeed must vanish in TGD framework for many reasons: one is that otherwise the p-adicization by algebraization program would fail.
*In TGD radiative corrections are coded into Kahler action via the proportionality of M^4 covariant metric factor on hbar^2 so that Kahler function K plays the role of effective action summing up all perturbative corrections in its extremely nonlinear dependence on induced metric. At leat for maxima of K further radiative corrections vanish. Note that radiative corrections are essentially gravitational.
Be it as it may, I believe that a genuine understanding of this pi issue probably would mean a very profound further understanding of also TGD.
Yes, pi is annoying, but I'm hopeful that a better geometric characterisation of pi will come from the n-operad heirarchy.
I spent some time looking for the form of zeta(3). I got the computer to show that it isn't a rational multiple of pi^3 with the rational fraction having a denominator with less than 100 digits.
I also tried various other stuff, like ln(2) pi^3, but always the result was a denominator > 100 digits.
All this was as class project for a C++ class.
It's possible that there is some form for the number that is simple, but I just didn't guess it.
LOL, Carl. Guessing might not be a great strategy in this game! I was thinking of playing with properties of triangulated polygons, since these label associahedra.
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