Connection between Hurwitz zetas, quantum groups, and hierarchy of Planck constants?
Kea mentioned in her blog the action of modular group SL(2,Z) on Riemann Zeta induced by its action on theta function. In particular, she mentioned the action of the translation τ→ τ+1. Recall that the action of the generator τ→ -1/τ on theta function is essential in providing the functional equation for Riemann Zeta.
1. Hurwitz zetas form n-plets closed under the action of fractional modular group
I checked the formula for Riemann zeta in terms of theta function in Wikipedia and found that τ→τ+1 transforms θ(0,τ); to θ(1/2,τ). When applied to the representation of zeta in terms of θ this gives Hurwitz zeta ζ(s,z), with z=1/2. Riemann Zeta corresponds to ζ(s,n)=ζ(s,0) by the periodicity of theta with respect to first argument.
Thus ζ(s,0) and ζ(s,1/2) behave like a doublet under modular transformations. Under the subgroup of modular group obtained by replacing τ→τ+1 with τ→τ+2 Riemann Zeta forms a singlet. The functional equation for Hurwitz zeta relates ζ(1-s,1/2) to ζ(s,1/2) and ζ(s,1)=ζ(s,0) so that also now one obtains a doublet. This doublet might be the proper object to study instead of singlet if one considers full modular invariance.
Hurwitz zeta is obtained by replacing integers m with fractionals m/n in the defining sum formula for Riemann Zeta:
ζ(s,z)= ∑m (m+z)-s.
Obviously Riemann zeta results for z=n. The inspection of the functional equation for ζ(s, m/n), m=0,1,...,m-1, demonstrates that form n-plets under fractional modular transformations obtained by using generators τ→-1/τ and τ→τ+2/n. The latter corresponds to the unimodular matrix (a,b;c,d)= (1, 2/n;0,1). These matrices obviously form a group. Note that Riemann zeta is always one member of the multiplet containing n Hurwitz zetas.
These observations bring in mind fractionization of quantum numbers, quantum groups corresponding to the quantum phase q=exp(i2π/n), and the inclusions for hyper-finite factors of type II1 partially characterized by these quantum phases. Fractional modular group obtained using generator τ→τ+2/n and Hurwitz zetas ζ(s,k/n) would very naturally relate to these and related structures.
2. Hurwitz zetas and TGD
I realized that these observations might have a direct application to quantum TGD.
- In TGD framework inclusions of HFFs are directly related to the hierarchy of Planck constants involving a generalization of the notion of imbedding space obtained by gluing together copies of 8-D H=M4×CP2 with a discrete bundle structure H→ H/Zna×Znb together along the 4-D intersections of the associated base spaces (see this). A book like structure results and various levels of dark matter correspond to the pages of this book. One can say that elementary particles proper are maximally quantum critical and live in the 4-D intersection of these imbedding spaces whereas they field bodies are on the pages of the Big Book.
- The integers na and nb give Planck constant as hbar/hbar0=na/nb, whose most general value is rational number (I hope that Lubos forgives the presence of hbar0: I know that I could put it equal to 1;-)). In Platonic spirit one can argue that number theoretically simple integers involving only powers of 2 and Fermat primes are favored physically. Phase transitions between different matters occur at the intersection.
- The inclusions N subset M of HFFs relate also to quantum measurement theory with finite measurement resolution with N defining measurement resolution so that N-rays replace complex rays in the projection postulate and quantum space M/N having fractional dimension effectively replaces M.
- The basic hypothesis is that the inverses of zeta function or of more general variants of zeta coding information about the algebraic structure of the partonic 2-surface appear in the admittedly speculative fundamental formula for the generalized eigenvalues of modified Dirac operator D (see this). This formula is consistent with the generalized eigenvalue equation for D but is not the only one that one can imagine.
- The generalized eigen spectrum of D should code information both about the p-adic prime p characterizing particle and about quantum phases q=exp(i2π/n) assignable to the particle in M4 and CP2 degrees of freedom. I understand how p-adic primes appear in the spectrum of D and therefore how coupling constant evolution emerges at the level of free field theory so that radiative corrections can vanish without the loss of coupling constant evolution (see this). The problem has been to understand how the quantum phase characterizing the sector of imbedding space could make itself visible in these formulas and therefore in quantum dynamics at the level of free spinor fields. The replacement of Riemann zeta with an n-plet of Hurwitz zetas would resolve this problem.
- Geometrically the fractional modular invariance would naturally relate to the fact that Riemann surface (partonic 2-surface) can be seen as an na× nb-fold covering of its projection to the base space of H: fractional modular transformations corresponding to naand nb would relate points at different sheets of the covering of M4 and CP2. This suggests that the fractionization could be a completely general phenomenon happening also for more general zeta functions.
Also n=1 and n=2 are present in the hierarchy of Hurwitz zetas (singlet and doublet). They do not correspond to allowed Jones inclusion since one has n>2 for them. What could this mean?
- It would seem that the fractionization of modular group relates to Jones inclusions (n>2) giving rise to fractional statistics. n=2 corresponding to the full modular group Sl(2,Z) could relate to the very special role of 2-valued logic, to the degeneracy of n=2 polygon in plane, to the very special role played by 2-component spinors playing exceptional role in Riemann geometry with spinor structure, and to the canonical representation of HFFs of type II1 as fermionic Fock space (spinors in the world of classical worlds). Note also that SU(2) defines the building block of compact non-commutative Lie groups and one can obtain Lie-algebra generators of Lie groups from n copies of SU(2) triplets and posing relations which distinguish the resulting algebra from a direct sum of SU(2) algebras.
- Also n=2-fold coverings M4→ M4/Z2 and CP2 → CP2/Z2 seem to make sense. By quantum classical correspondence also the spin half property of imbedding space spinors should have space-time correlate. Could n=2 coverings allow to define the space-time correlates for particles having half odd integer spin or weak isospin? If so, bosons would correspond to n=1 and fermions to n=2.