### Quantum Arithmetics and the Relationship between Real and p-Adic Physics

p-Adic physics involves two only partially understood questions.

- Is there a duality between real and p-adic physics? What is its precice mathematic formulation? In particular, what is the concrete map p-adic physics in long scales (in real sense) to real physics in short scales? Can one find a rigorous mathematical formulation of canonical identification induced by the map p→ 1/p in pinary expansion of p-adic number such that it is both continuous and respects symmetries.
- What is the origin of the p-adic length scale hypothesis suggesting that primes near power of two are physically preferred? Why Mersenne primes are especially important?

A possible answer to these questions relies on the following ideas inspired by the model of Shnoll effect. The first piece of the puzzle is the notion of quantum arithmetics formulated in non-rigorous manner already in the model of Shnoll effect.

- Quantum arithmetics is induced by the map of primes to quantum primes by the standard formula. Quantum integer is obtained by mapping the primes in the prime decomposition of integer to quantum primes. Quantum sum is induced by the ordinary sum by requiring that also sum commutes with the quantization.
- The construction is especially interesting if the integer defining the quantum phase is prime. One can introduce the notion of quantum rational defined as series in powers of the preferred prime defining quantum phase. The coefficients of the series are quantum rationals for which neither numerator and denominator is divisible by the preferred prime.
- p-Adic--real duality can be identified as the analog of canonical identification induced by the map p→ 1/p in the pinary expansion of quantum rational. This maps maps p-adic and real physics to each other and real long distances to short ones and vice versa. This map is especially interesting as a map defining cognitive representations.

Quantum arithmetics inspires the notion of quantum matrix group as counterpart of quantum group for which matrix elements are ordinary numbers. Quantum classical correspondence and the notion of finite measurement resolution realized at classical level in terms of discretization suggest that these two views about quantum groups are closely related. The preferred prime p defining the quantum matrix group is identified as p-adic prime and canonical identification p→ 1/p is group homomorphism so that symmetries are respected.

- The quantum counterparts of special linear groups SL(n,F) exists always. For the covering group SL(2,C) of SO(3,1) this is the case so that 4-dimensional Minkowski space is in a very special position. For orthogonal, unitary, and orthogonal groups the quantum counterpart exists only if quantum arithmetics is characterized by a prime rather than general integer and when the number of powers of p for the generating elements of the quantum matrix group satisfies an upper bound characterizing the matrix group.
- For the quantum counterparts of SO(3) (SU(2)/ SU(3)) the orthogonality conditions state that at least some multiples of the prime characterizing quantum arithmetics is sum of three (four/six) squares. For SO(3) this condition is strongest and satisfied for all integers, which are not of form n= 2
^{2r}(8k+7)). The number r_{3}(n) of representations as sum of squares is known and r_{3}(n) is invariant under the scalings n→ 2^{2r}n. This means scaling by 2 for the integers appearing in the square sum representation. - r
_{3}(n) is proportional to the so called class number function h(-n) telling how many non-equivalent decompositions algebraic integers have in the quadratic algebraic extension generated by (-n)^{1/2}.

The findings about quantum SO(3) suggest a possible explanation for p-adic length scale hypothesis and preferred p-adic primes.

- The basic idea is that the quantum matrix group which is discrete is very large for preferred p-adic primes. If cognitive representations correspond to the representations of quantum matrix group, the representational capacity of cognitive representations is high and this kind of primes are survivorsâ€Ť in the algebraic evolution leading to
algebraic extensions with increasing dimension.
- The preferred primes correspond to a large value of r
_{3}(n). It is enough that some of their multiples do so (the 2^{2r}multiples of these do so automatically). Indeed, for Mersenne primes and integers one has r_{3}(n)=0, which was in conflict with the original expectations. For integers n=2M_{m}however r_{3}(n) is a local maximum at least for the small integers studied numerically. - The requirement that the notion of quantum integer applies also to algebraic integers in quadratic extensions of rationals requires that the preferred primes (p-adic primes) satisfy p=8k+7. Quite generally, for the integers n=2
^{2r}(8k+7) not representable as sum of three integers the decomposition of ordinary integers to algebraic primes in the quadratic extensions defined by (-n)^{1/2}is unique. Therefore also the corresponding quantum algebraic integers are unique for preferred ordinary prime if it is prime also in the algebraic extension. If this were not the case two different decompositions of one and same integer would be mapped to different quantum integers. Therefore the generalization of quantum arithmetics defined by any preferred ordinary prime, which does not split to a product of algebraic primes, is well-defined for p=2^{2r}(8k+7). - This argument was for quadratic extensions but also more complex extensions defined by higher polynomials exist. The allowed extensions should allow unique decomposition of integers to algebraic primes. The prime defining the quantum arithmetics should not decompose to algebraic primes. If the algebraic evolution leadis to algebraic extensions of increasing dimension it gradually selects preferred primes as survivors.

## 10 Comments:

Happy birthday.

http://arxiv.org/abs/1101.3619

I post my first lesson tonight as a 'gift'. Hope you enjoy it.

Matti,

several orders of magnitude simpler, hmmmmm if that is an advantage.

At last, a surprisingly recent paper on a simpler way and more advanced way to view the plane (as I have posted all along- the quasic plane) and it is much simpler and more general than the link supplied above by Ulla on things like Clifford space.

Now that some theorists realize there can be combinations of what seems complete physics- to ask as you did the difference in physics a and physics p, it is high time we brought it all together.

The PeSla

Thank you for the birthday gift. As become clear from the posting, the main challenges in the fusion of real and p-adic physics reduce to purely mathematical problems besides deep interpretational problems. One "knows" or better to say -"feels"- what the outcome roughly is: how to understand it mathematically is the challenge.

Again the basic boost came from experimental anomaly: Shnoll's strange findings. These findings where made by an outsider as so often: Shnoll was a biologists who made a discovery which - according to my almot-belief (professionals never believe;-)- will modify throughly physicists views about what the physical world is.

To Ulla:

Thank you for the link. A couple of comments from my own point of view.

The proposal of the article relies on 3-D Clifford algebra. I like the idea. This notion is central also in TGD and generalized so that infinite-D Clifford algebra of fermionic oscillator operators corresponds to Clifford algebra of "world of classical worlds":

The approach relies also on quaternions and is restricted to *flat* 3-space (or Euclidian 4-space). This does *not* allow a description of gravitation. Already Riemann realized that one needs the notion of curved space to understand everyday macroscopic world and a mathematization of the notion of length measurement, and performed the needed generalization.

The article as such does not seem to make any wrong statements. All these notions have been however discovered and developed for a long time ago by both mathematicians and physicists. The problem that every physicists encounters is that already the classical mathematics is so huge a discipline that it is almost hopeless to gain an overall view about what is already done.

I like also the idea of quaternionicity of space-time: it is one of the cornerstones of TGD. The notion must be however refined to allow curved space-time and gravitation. This requires the identification of space-time as 4-surface (neutrino super-luminality would be more sexy justification;-).

The notion of quaternionic space-time surface in octonionic space-time and quaternionicity as a local property of the tangent space determining the classical dynamics. To this I will would put my money if I had money;-)!

Hi Matti. I have a question motivated by the desire to make a bridge between TGD and physical models where people already know how to write and solve equations of motion, etc.

Basically I'm asking: what in your opinion is an existing tractable model of mathematical physics which comes closest to TGD?

I can look at this description of TGD from 2003 and it's very comprehensible. But also it's mostly just conceptual. Many-sheeted 3-surfaces moving around in M^4+ x CP2 - OK, that's a well-defined concept. But what's the equation of motion? Let's say it's the extremization of some geometric quantity - I'm sure the details, or various hypotheses, can be found in your work. But even then, this still might not be enough to calculate anything.

And then there are all the extra details, once you introduce quantum mechanics, p-adics, etc.

On the other hand, a while back we had this discussion about similarity and difference between TGD and the Type IIA description of AdS/CFT for N=4 Yang-Mills. I would like to find something that is even closer to the true TGD - something which resembles the informal description from 2003 - but which has had some mathematical development. Can you suggest anything? Any clues about what to look for?

Mitchell

Hi,

thank you for asking. I will respond in two parts.

The first thing to notice is that path integral approach fails in TGD. Therefore the stationary phase approximation giving classical theory in the usual sense fails. This leads to the geometrization program for "the world of classical worlds".

Preferred extremals analogous to Bohr orbits. But what "preferred" means? This is the question and I have proposed several identifications during years. There is a lot of information about them but no strict recipe for constructing them. I have not been able to fuse all the parallel threads to single coherent overall view giving a collection of formulas.

The basic variational principle is defined by Kahler action, this has been clear for ore than two decades. Kahler action is Maxwell action but for induced Kahler form of CP_2 so that only 4 coordinate variables of M^4xCP_2 serve as dynamical variables.

I know a lot about extremals of Kahler action. Huge solution families.

*Vacuum extremals with cP_" projection which is Lagrangian manifold with at most 2-D CP_2 projection. 4-D spin glass degeneracy is the basic implication and allows a lot of intuitive understanding about the theory.

Solutions are of Einstein's equation correspond to the vacuum extremals: the energy-momentum currents asssociated with the small deformations would be expressible in terms of Einstein tensor.

* CP_2 type vacuum with 1-D light-like curve as M^4 projection. The deformations of these Euclidian space-time regions have interpretation as lines of generalized Feynman graphs.

*String like objects of form X^2xY^2: with Y^2 complex manifold of CP_2. String like objects prevail in primordial cosmology and during cosmic expansion transform to magnetic flux tubes with 4-D M^4 projection. They are fundamental for understanding dark energy and dark matter and birth of galaxies and stars. Also in biology.

*Massless extremals about which simplest cases are graphs of arbitrary maps from M^4 to CP_2 with CP_2 coordinates depending on arbitrary manner on light-like coordinate and linear coordinate orthogonal to it. These serve as correlates for massless radiation fields carrying photons etc as topologically condensed CP_2 type extremals.

All phenology and applications rely on these bits of information.

To be continued....

One extremely interesting conjecture is that the dynamics of preferred extremals reduces to associativity. Imbedding space is endowed with octonionic Clifford algebra structure and space-time surfaces have tangent spaces which are quaternionic.

A closely related conjecture is that octonion real-analytic functions in H defined space-time surfaces. Associative/co-associative space-time region would be defined via the vanishing of the "imaginary" /"real" part of octonion-real analytic function in the representation of octonion as sum of quaternion ad I*quaternion. The space of this kind of functions would be closed under local arithmetic operations and form function field. Also it would be closed under composition of functions. Space-time surfaces would form a field in well-defined sense. This is big idea- so big that it is very difficult to conclude what it implies!

At this moment it is not possible to give Feynman rules allowing to calculate. Nor a general recipe for the construction of preferred extremals. Just general very abstract results such as the almost topological QFT picture, weak form of electric magnetic duality, effective 2-dimensionality, etc.

Concrete pictures -like those appearing in 2003 representation- would be certainly helpful but unfortunately I do not have time to start fighting with drawing programs. This would require a professional with nerves of a work horse.

You asked also about my opinion about tractable models of mathematical physics coming close to TGD. The 3-branes in M-theory or string theory would be the analog of space-time surfaces in M^4xCP_2. But can one say that string theory or M-theory is tractable?! And does it really help at all? I am a little bit skeptic. String theorists miss quite too much important physics. The analogy also breaks down however because the induced metric and spinor structure play a minor role in these theories: I hope that the apparent neutrino super-luminality could teach realism for the colleagues.

I believe that TGD is integrable model in some sense: this means among other things integrability of classical field equations.

Also the value of K\"ahler action must be calculable without knowing all the details about the 4-D solution. Reduction to 3-D Chern-Simons terms with boundary conditions given by teh weak form of electric-magnetic duality and bringing in dependence on metric is an enormous step in this respect and is well-understood.

Something even better might happen. The latest conjecture inspired by the effective 2-dimensionality is that real Euclidian contribution reduces to the total area of partonic 2-surfaces and the purely imaginary contribution from Minkoskian regions to that for string world sheets. String world-sheet-- partonic 2-surface duality would suggest that the magnitudes of these contributions are actually identical. For details see this:

http://matpitka.blogspot.com/2011/10/is-k-action-expressible-in-terms-of.html

Mitchell

The complex square root was introduced by Hermann Weyl (Space, Time, Matter p. 212) in a Lagrangian not for Maxwell's electrodynamics but for the theory of Gustav Mie, which extends within the electron or proton with a "pressure" balancing electrostatic self-repulsion.

Remember also that a quaternion is just the (not nec. parallel) displacement of a vector in Eucuidean space. Let the vector be any electrodynamic quantity, and the displacement involve gravitational curvature.

Mesons remain a worthy target, much where standard nuclear engineering fails us.

Could you elaborate a little?

a quaternion is just the (not nec. parallel) displacement of a vector in Eucuidean space. Let the vector be any electrodynamic quantity, and the displacement involve gravitational curvature.Post a Comment

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