Are

there hidden relations between mathematical and physical constants such

as

$\frac{e^2}{4 \pi \epsilon_0 h c} \sim \frac{1}{137} $

or are these numerical relations mere accidents? A couple of years

ago, Pierre Cartier proposed in his paper A mad day’s work : from Grothendieck to Connes and

Kontsevich : the evolution of concepts of space and symmetry that

there are many reasons to believe in a cosmic Galois group acting on the

fundamental constants of physical theories and responsible for relations

such as the one above.

The Euler-Zagier numbers are infinite

sums over $n_1 > n_2 > ! > n_r \geq 1 $ of the form

$\zeta(k_1,\dots,k_r) = \sum n_1^{-k_1} \dots n_r^{-k_r} $

and there are polynomial relations with rational coefficients between

these such as the product relation

$\zeta(a)\zeta(b)=\zeta(a+b)+\zeta(a,b)+\zeta(b,a) $

It is

conjectured that all polynomial relations among Euler-Zagier numbers are

consequences of these product relations and similar explicitly known

formulas. A consequence of this conjecture would be that

$\zeta(3),\zeta(5),\dots $ are all trancendental!

Drinfeld

introduced the Grothendieck-Teichmuller group-scheme over $\mathbb{Q} $

whose Lie algebra $\mathfrak{grt}_1 $ is conjectured to be the free Lie

algebra on infinitely many generators which correspond in a natural way

to the numbers $\zeta(3),\zeta(5),\dots $. The Grothendieck-Teichmuller

group itself plays the role of the Galois group for the Euler-Zagier

numbers as it is conjectured to act by automorphisms on the graded

$\mathbb{Q} $-algebra whose degree $d $-term are the linear combinations

of the numbers $\zeta(k_1,\dots,k_r) $ with rational coefficients and

such that $k_1+\dots+k_r=d $.

The Grothendieck-Teichmuller

group also appears mysteriously in non-commutative geometry. For

example, the set of all Kontsevich deformation quantizations has a

symmetry group which Kontsevich conjectures to be isomorphic to the

Grothendieck-Teichmuller group. See section 4 of his paper Operads and motives in

deformation quantzation for more details.

It also appears

in the renormalization results of Alain Connes and Dirk Kreimer. A very

readable introduction to this is given by Alain Connes himself in Symmetries Galoisiennes

et renormalisation. Perhaps the latest news on Cartier’s dream of a

cosmic Galois group is the paper by Alain Connes and Matilde Marcolli posted

last month on the arXiv : Renormalization and

motivic Galois theory. A good web-page on all of this, including

references, can be found here.

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