### Hilbert p-adics, hierarchy of Planck constants, and finite measurement resolution

The hierarchy of Planck constants assigns to the N-fold coverings of the imbedding space points N-dimensional Hilbert spaces. The natural identification of these Hilbert spaces would be as Hilbert spaces assignable to space-time points or with points of partonic 2-surfaces. There is however an objection against this identification.

- The dimension of the local covering of imbedding space for the hierarchy of Planck constants is constant for a given region of the space-time surface. The dimensions of the Hilbert space assignable to the coordinate values of a given point of the imbedding space are defined by the points themselves. The values of the 8 coordinates define the algebraic Hilbert space dimensions for the factors of an 8-fold Cartesian product, which can be integer, rational, algebraic numbers or even transcendentals and therefore they vary as one moves along space-time surface.
- This dimension can correspond to the locally constant dimension for the hierarchy of Planck constants only if one brings in finite measurement resolution as a pinary cutoff to the pinary expansion of the coordinate so that one obtains ordinary integer-dimensional Hilbert space. Space-time surface decomposes into regions for which the points have same pinary digits up to p
^{N}in the p-adic case and down to p^{-N}in the real context. The points for which the cutoff is equal to the point itself would naturally define the ends of braid strands at partonic 2-surfaces at the boundaries of CD:s. - At the level of quantum states pinary cutoff means that quantum states have vanishing projections to the direct summands of the Hilbert spaces assigned with pinary digits p
^{n}, n>N. For this interpretation the hierarchy of Planck constants would realize physically pinary digit representations for number with pinary cutoff and would relate to the physics of cognition.

One of the basic challenges of quantum TGD is to find an elegant realization for the notion of finite measurement resolution. The notion of resolution involves observer in an essential manner and this suggests that cognition is involved. If p-adic physics is indeed physics of cognition, the natural guess is that p-adic physics should provide the primary realization of this notion.

The simplest realization of finite measurement resolution would be just what one would expect it to be except that this realization is most natural in the p-adic context. One can however define this notion also in real context by using canonical identification to map p-adic geometric objets to real ones.

**Does discretization define an analog of homology theory?**

Discretization in dimension D in terms of pinary cutoff means division of the manifold to cube-like objects. What suggests itself is homology theory defined by the measurement resolution and by the fluxes assigned to the induced Kähler form.

- One can introduce the decomposition of n-D sub-manifold of the imbedding space to n-cubes by n-1-planes for which one of the coordinates equals to its pinary cutoff. The construction works in both real and p-adic context. The hyperplanes in turn can be decomposed to n-1-cubes by n-2-planes assuming that an additional coordinate equals to its pinary cutoff. One can continue this decomposition until one obtains only points as those points for which all coordinates are their own pinary cutoffs. In the case of partonic 2-surfaces these points define in a natural manner the ends of braid strands. Braid strands themselves could correspond to the curves for which two coordinates of a light-like 3-surface are their own pinary cutoffs.
- The analogy of homology theory defined by the decomposition of the space-time surface to cells of various dimensions is suggestive. In the p-adic context the identification of the boundaries of the regions corresponding to given pinary digits is not possible in purely topological sense since p-adic numbers do not allow well-ordering. One could however identify the boundaries sub-manifolds for which some number of coordinates are equal to their pinary cutoffs or as inverse images of real boundaries. This might allow to formulate homology theory to the p-adic context.
- The construction is especially interesting for the partonic 2-surfaces. There is hierarchy in the sense that a square like region with given first values of pinary digits decompose to p square like regions labelled by the value 0,...,p-1 of the next pinary digit. The lines defining the boundaries of the 2-D square like regions with fixed pinary digits in a given resolution correspond to the situation in which either coordinate equals to its pinary cutoff. These lines define naturally edges of a graph having as its nodes the points for which pinary cutoff for both coordinates equals to the actual point.
- I have proposed earlier what I have called symplectic QFT involving a triangulation of the partonic 2-surface. The fluxes of the induced Kähler form over the triangles of the triangulation and the areas of these triangles define symplectic invariants, which are zero modes in the sense that they do not contribute to the line element of WCW although the WCW metric depends on these zero modes as parameters. The physical interpretation is as non-quantum fluctuating classical variables. The triangulation generalizes in an obvious manner to quadrangulation defined by the pinary digits. This quadrangulation is fixed once internal coordinates and measurement accuracy are fixed. If one can identify physically preferred coordinates - say by requiring that coordinates transform in simple manner under isometries - the quadrangulation is highly unique.
- For 3-surfaces one obtains a decomposition to cube like regions bounded by regions consisting of square like regions and Kähler magnetic fluxes over the squares define symplectic invariants. Also Kähler Chern-Simons invariant for the 3-cube defines an interesting almost symplectic invariant. 4-surface decomposes in a similar manner to 4-cube like regions and now instanton density for the 4-cube reducing to Chern-Simons term at the boundaries of the 4-cube defines symplectic invariant. For 4-surfaces symplectic invariants reduce to Chern-Simons terms over 3-cubes so that in this sense one would have holography. The resulting structure brings in mind lattice gauge theory and effective 2-dimensionality suggests that partonic 2-surfaces are enough.

**Does the notion of manifold in finite measurement resolution make sense?**

A modification of the notion of manifold taking into account finite measurement resolution might be useful for the purposes of TGD.

- The chart pages of the manifold would be characterized by a finite measurement resolution and effectively reduce to discrete point sets. Discretization using a finite pinary cutoff would be the basic notion. Notions like topology, differential structure, complex structure, and metric should be defined only modulo finite measurement resolution. The precise realization of this notion is not quite obvious.
- Should one assume metric and introduce geodesic coordinates as preferred local coordinates in order to achieve general coordinate invariance? Pinary cutoff would be posed for the geodesic coordinates. Or could one use a subset of geodesic coordinates for δ CD× CP
_{2}as preferred coordinates for partonic 2-surfaces? Should one require that isometries leave distances invariant only in the resolution used? - A rather natural approach to the notion of manifold is suggested by the p-adic variants of symplectic spaces based on the discretization of angle variables by phases in an algebraic extension of p-adic numbers containing n
^{th}root of unity and its powers. One can also assign p-adic continuum to each root of unity (see this). This approach is natural for compact symmetric Kähler manifolds such as S^{2}and CP_{2}. For instance, CP_{2}allows a coordinatization in terms of two pairs (P^{k},Q^{k}) of Darboux coordinates or using two pairs (ξ^{k},ξ^{k*}), k=1,2, of complex coordinates. The magnitudes of complex coordinates would be treated in the manner already described and their phases would be described as roots of unity. In the natural quadrangulation defined by the pinary cutoff for |ξ^{k}| and by roots of unity assigned with their phases, Kähler fluxes would be well-defined within measurement resolution. For light-cone boundary metrically equivalent with S^{2}similar coordinatization using complex coordinates (z,z^{*}) is possible. Light-like radial coordinate r would appear only as a parameter in the induced metric and pinary cutoff would apply to it.

**Hierachy of finite measurement resolutions and hierarchy of p-adic normal Lie groups**

The formulation of quantum TGD is almost completely in terms of various symmetry group and it would be highly desirable to formulate the notion of finite measurement resolution in terms of symmetries.

- In p-adic context any Lie-algebra g with p-adic integers as coefficients has a natural grading based on the p-adic norm of the coefficient just like p-adic numbers have grading in terms of their norm. The sub-algebra g
_{N}with the norm of coefficients not larger than p^{-N}is an ideal of the algebra since one has [g_{M},g_{N}]⊂ g_{M+N}: this has of course direct counterpart at the level of p-adic integers. g_{N}is a normal sub-algebra in the sense that one has [g,g_{N}]⊂ g_{N}. The standard expansion of the adjoint action gg_{N}g^{-1}in terms of exponentials and commutators gives that the p-adic Lie group G_{N}=exp(tpg_{N}), where t is p-adic integer, is a normal subgroup of G=exp(tpg). If indeed so then also G/G_{N}is group, and could perhaps be interpreted as a Lie group of symmetries in finite measurement resolution. G_{N}in turn would represent the degrees of freedom not visible in the measurement resolution used and would have the role of a gauge group. - The notion of finite measurement resolution would have rather elegant and universal representation in terms of various symmetries such as isometries of imbedding space, Kac-Moody symmetries assignable to light-like wormhole throats, symplectic symmetries of δCD× CP
_{2}, the non-local Yangian symmetry, and also general coordinate transformations. This representation would have a counterpart in real context via canonical identification I in the sense that A→ B for p-adic geometric objects would correspond to I(A)→ I(B) for their images under canonical identification. It is rather remarkable that in purely real context this kind of hierarchy of symmetries modulo finite measurement resolution does not exist. The interpretation would be that finite measurement resolution relates to cognition and therefore to p-adic physics. - Matrix group G contains only elements of form g=1+O(p
^{m}), m≥ 1 and does not therefore involve matrices with elements expressible in terms roots of unity. These can be included, by writing the elements of the p-adic Lie-group as products of elements of above mentioned G with the elements of a discrete group for which the elements are expressible in terms of roots of unity in an algebraic extension of p-adic numbers. For p-adic prime p p:th roots of unity are natural and suggested strongly by quantum arithmetics.

For background see the chapter Quantum Adeles of "Physics as Generalized Number Theory".

## 9 Comments:

Thanks Matti. Earlier today I finished reading 'Goldilocks Enigma' by Paul Davies, book about Anthropic Principle dedicated to Wheeler - and Wheeler's Delayed Choice Experiment (http://www.bottomlayer.com/bottom/basic_delayed_choice.htm) was news to me.

In the end of the book Davies shortly discussed his old idea of "Ground Hog Day"-universe, a rather special form of Hell. And I thought that Matti has found the math for the way out of that hell of finite information content and self-referential universe of causal loop repeating the same experiences ad infinitum. As also Davies suggested, information content can be finite, but there is no upper limit.

Transcendentals like pi contain (potentially) infinite amount of information, and pinary cut-of giving finite amount of information can grow ad infinitum with increasing value of p.

What might be the significance for finite measurement resolution and information content of observable universe if any, of the largest Mersenne Primes found so far by human mathematicians and their computers, as well as of the biggest number of pi digits computed so far, given Wheeler's causal loop cosmos->life->mind->cosmos.

And no, I still don't get how trascendental number like pi that is not constructible with compass and ruler can end up as inner structure of a point in euclidean Hilbert space, as it defines the ratio between compass and ruler. I guess I'm still asking the kinds of questions that lead to Gödelian answer's like "I cannot be played by record player 1".

The name Zurek popped up and lead to this: http://en.wikipedia.org/wiki/No-cloning_theorem

I do not know what Wheeler thought about anthropic principle. I am not able to take AP seriously because it raises just this particular form of consciousness and life to a special position and gives humans completely special role.

I see observer participancy as something much more general than AP. OP does not raise humans in any special position but makes conscious entities - whatever they are- re-creators of the Universe: correct if I am wrong;-).

As mathematician I would not call this self-referential world hell! For a mathematician this world would be heaven making possible mathematical Nirvana since it would be possible to achieve arbitrary high levels of reflective consciousness. For us even a statement about statements about statements representing N=3 level overcomes our intellectual abilities and N=10 is absolutely impossible for us. First order logic involving existential quantors "for every" and "there exists" is what we mostly use. Maybe, the number of active abstraction levels could serve as a measure for the mathematical evolutionary level allowing to dimly imagine what kind of math ETs are doing and applying.

p-Adic arithmetics could provide a better approach to numerical calculations than ordinary real number based mathematics. Canonical identification would mediate between reals and p-adics. To achieve this effectively p-adic aritmetics should be realized at the level of computer hardware. Maybe living system a has adopted this option for computation.

AP is historical misnomer for biofriendly universe and OP is better and more general, but the AP-term is now hard to get rid of.

And I expressed myself poorly, hell didn't refer to self-referential world but the causal loop that reminds on cosmic scale the movie Ground Hog Day, where main character lives the same day over and over again.

This self loop brings in mind Stanislav Lem's story about getting stuck in time loop which must be a hellish experience. And Douglas Adam's story about a fly which re-incarnation after re-incarnation gets killed by the same person. And of course this TGD fellow whose work gets rejected again and again by irritated colleagues but who still continues to irritate. A good hell needs both the fly and fly killer!

http://arxiv.org/pdf/1203.5557v1.pdf

John P. Ralston

Planck’s constant was introduced as a fundamental scale in the early history of quantum mechanics. We find a modern approach where Planck’s constant is absent: it is unobservable except as a constant of human convention. Despite long reference to experiment, review shows that Planck’s constant cannot be obtained from the data of Ryberg, Davisson and Germer, Compton, or that used by Planck himself. In the new approach Planck’s constant is tied to macroscopic conventions of Newtonian origin, which are dispensable. The precision of other fundamental constants is substantially improved by eliminating Planck’s constant. The electron mass is determined about 67 times more precisely, and the unit of electric charge determined 139 times more precisely. Improvement in the experimental value of the fine structure constant allows new types of experiment to be compared towards finding “new physics.” The long-standing goal of eliminating reliance on the artifact known as the International Prototype Kilogram can be accomplished to assist progress in fundamental physics……

http://physicsforme.wordpress.com/2012/03/28/can-gps-find-variations-in-plancks-constant/

http://physics.aps.org/synopsis-for/10.1103/PhysRevLett.108.110801

Kentosh and Mohageg looked through a year’s worth of GPS data and found that the corrections depended in an unexpected way on a satellite’s distance above the Earth. This small discrepancy could be due to atmospheric effects or random errors, but it could also arise from a position-dependent Planck’s constant.

I looked at the article. At least particle physicists have been using hbar=1 convention as dimensional convention for more than half century. Also I have used to think about mass and frequency as inverse length because it is so convenient.

The situation becomes non-trivial when Planck constant is varying. In TGD hbar is not a continuous variable but integer multiple of its basic value. In this situation the basic value ban be taken just hbar_0=1 still and other values are integers: hbar= n_hbar_0= n=1,2,....

When experimentalist tells that Planck constant is known to be constant with so and so many digits he actually says that hbar_0 is constant in excellent approximation. TGD predicts the constancy of hbar_0. TGD does not predict position dependent Planck constant so that in TGD Universe the claimed observation mentioned in latter comment of Ulla must be due to some other effect.

The existing experimental data are completely consistent with integer multiples of ordinary Planck constant since multiples are predicted to be associated with dark matter about which we know in practice only its existence and something about its distribution.

Even the information about galactic dark matter is grossly misinterpreted! The rotation spectra of distant stars around galaxies can be explained by dark matter associated with cosmic strings- magnetic flux tubes of cosmic size scale- perpendicular to the galactic plane: galaxies would be like pearls in a necklace. The motion along the direction of cosmic string would be almost free and this the testable prediction and one can quite well imagine large scale motions in which galaxies- the pearls- move along the necklace. Candidates for this kind of motions exist.

The standard interpretation assumes however a 3-D halo of dark matter around galaxy rather than cosmic string. This kind of halo might be present but need not make any significant contribution to the gravitational field explaining the rotation spectrum. So the frustrating construction of more and more complex models that fail to work continues!

The development of ideas is painstakingly slow! There is the communication barrier created by the fact that career building scientists behave very much like a flock of sheep. It also seems that the process selecting the career builders does not favor innovative persons. I have been waiting for years that some cosmologists would make this simple observation about cosmic strings but in vain.

Similar incredible slow-mindedness prevails in the case of p-adic mass calculations. It is becoming clear that 125 GeV particle is not Higgs but they have begun to talk about fermion-phobic or top-phobic Higgses, which - apart from name- have nothing common with the Higgs and mean giving up the original motivations for Higgs. The exceptional inability to see the things as they are and to stick to the recipes of past is what surprises me again and again.

Well, I have very much been accused to be a stupid sheep in a flock.

The difference between hbar hierarchy and p-adic hierarchy is still dim, as for the brain. Then there are also different primes... ZEO ...

The hbar hierarchy in the body for color or chakras are due to primes (pearls in a necklace) or resonances, that is p-adics? Consciousness grows with use of a bigger diamond? Not necessarily intelligence or ethics grows at the same time (they are independant), as Murphy showed. In what way does the periodic table reflect this hbar/p-adics hierarchy? Or should it be contrarily?

If quantum mechanics is behind everything, then also behind astrophysics. This was the reason for my controverse with Keith Ashman et co. He went furious when I pointed out it was due to interpretation alone.

Once I had a vision of diamonds in a giant wheel, an Ouroborus :) One small sector was enlighted at a time, like it was time that circulated as in a clock. Ye, I know :) I am very ignorant. The comfort is that I know of it.

Lubos has a ralling on the Ralstone post http://motls.blogspot.com/2012/04/lets-fix-value-of-plancks-constant.html

What he doesn't realize is that he actually say the constant is varying in reality :) Just a little... But this is the effect of something else?

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