Light Speed III
Carl Brannen has a new blog post refuting Mottle's opinion piece on variable speed of light theories. Meanwhile Matti Pitkanen has been posting interesting updates on TGD and the zero energy ontology. As good old Einstein once said, "We can't solve problems by using the same kind of thinking we used when we created them."
In Lesson 59 we saw how the functional equation for the Riemann zeta function depended on a theta series at two different arguments, $\tau$ and $\frac{-1}{\tau}$. The interchange of these two complex parameters is familiar as an order 2 operation $S$ for the modular group. Perhaps the order 3 operation, $TS$, should give us a triality relation for the zeta function, analogous to the triality satisfied by the j-invariant. Recall that the operation $T$ sends $\tau$ to $\tau + 1$, which is why a fundamental region in the upper half plane has width 1. $TS$ fixes the cubed root of unity $\omega$. Complex numbers that get sent to the unit circle under $TS$ take the form $(\frac{1}{2} + iy)$.
But from M Theory one would guess that a triality relation arises from a triple of moduli, rather than the 1-punctured torus case. Is there a natural way to view the modular generators in terms of higher dimensional Belyi maps? This is presumably the sort of thing that nice Calabi-Yau 3-folds can do.
In Lesson 59 we saw how the functional equation for the Riemann zeta function depended on a theta series at two different arguments, $\tau$ and $\frac{-1}{\tau}$. The interchange of these two complex parameters is familiar as an order 2 operation $S$ for the modular group. Perhaps the order 3 operation, $TS$, should give us a triality relation for the zeta function, analogous to the triality satisfied by the j-invariant. Recall that the operation $T$ sends $\tau$ to $\tau + 1$, which is why a fundamental region in the upper half plane has width 1. $TS$ fixes the cubed root of unity $\omega$. Complex numbers that get sent to the unit circle under $TS$ take the form $(\frac{1}{2} + iy)$.
But from M Theory one would guess that a triality relation arises from a triple of moduli, rather than the 1-punctured torus case. Is there a natural way to view the modular generators in terms of higher dimensional Belyi maps? This is presumably the sort of thing that nice Calabi-Yau 3-folds can do.
posted by Kea | 11:21 AM
7 Comments:
Dear Kea,
I checked the formula for Riemann zeta in terms of theta function in Wikipedia. tau-->tau+1 seems to transform theta(0,tau) to theta(1/2,tau). When applied to the representation of zeta in terms of zeta seems to give Hurwitz zeta zeta(s,z), with z=1/2. Riemann Zeta corresponds to zeta(s,n)=zeta(s,0) by periodicity of theta with respect to first argument.
It would seem that zeta and Hurwitz zeta can be said to behave like a doublet under modular transformations.
The functional equation for Hurwitz zeta relates zeta(1-s,1/2) to zeta(s,1/2) and zeta(s,1)=zeta(s,0) so that also now one obtains a doublet. This this doublet might be the proper object to study instead of singlet.
I would guess that Hurwitz zetas for fractional arguments m/n form n-plets under modular transformations and in functional equation. Could these n-plets relate interestingly to quantum groups characterized by phase q=exp(ipi/n), inclusions of HFFs of type II_1, and to dark matter hierarchy in quantum TGD? Objection: n=2 does not correspond to allowed Jones inclusion (n>2 for them). Neither does n=2 correspond to genuine polygon in plane. And n=2-valued logic is in preferred position.
It would seem that zeta and Hurwitz zeta can be said to behave like a doublet under modular transformations.
Thanks, Matti! This is really nice. I had not thought about Hurwitz at all.
Actually, my attention has been diverted yet again because I've just come across the amazingly original Vepstas and his work on binary trees and the modular group. And now I recall that Lieven Le Bruyn was discussing this .... will post more soon.
Dear Kea,
still a comment about Hurwitz zetas. Riemann zeta defines a singlet if one takes subgroup obtained using the generator tau-->tau+2 instead of tau-->tau+1.
The definition of Hurwitz zeta zeta(s,k/n) as a sum is obtained by replacing the integer m in the term m^(-s) with m+k/n. This is very much analogous to the fractionization of an integer valued quantum number so that the proposed connection with quantum groups with quantum phase exp(i2pi/n) and inclusions of HFFs of type II_1 and quantization of Planck constant looks natural and one could indeed speak of n-multiplets under modular group.
Modular group generates however only the doublet corresponding to n=2. Fractionized modular group would be obtained by using generators tau-->-1/tau and tau-->tau+2/n. The latter corresponds to unimodular matrix (a,b;c,d)= (1, 2/n;0,1). These matrices form obviously a group.
It would seem that the fractionization of modular group correspond to Jones inclusions (n>2), to 2-valued logic, to degenerate n=2 polygon in plane, and to 2-component spinors playing exceptional role in case of HFFs of type II_1, and SU(2) defining the building block of compact non-commutative Lie groups. One can also 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.
I added to my blog a little posting about the possible role of Hurwitz zetas in TGD.
The message is that Hurwitz zetas of more general fractional zetas suggested by the covering space structure of sectors of the generalized imbedding space would allow to code quantum phase q to the structure of eigenmodes of the modified Dirac operator: how to achieve this has indeed been an open problem.
Calabi Yau manifolds may also be inherently electromagnetic.
Consider the 1890s work of CP Steinmetz on phasor equations [from Grassmann Algebra] and the association of the imaginary unit "i" with electromagnetism.
Perhaps electromagnetism is ubiquitous in our universe because of the presence of Calabi Yau manifolds?
(Being optimistic) Perhaps LM and Ellis are complaining about the ad hoc nature of some VSL theories. It is a sign of success that a "VSL industry" is growing. Perhaps, like inflation and "dark energy," it will lead to a number of theories before settling on one.
Hi all. Matti, that's great! I'm going to dig up Schwinger's old paper with Hurwitz zetas (er, maybe tomorrow, since today is the Queen's Birthday holiday and the library is closed) ... the n=2 doublet really makes sense for 2-logic.
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