### Progress in understanding of quantum p-adics

Quantum arithmetics is a notion which emerged as a possible resolution of long-lived challenge of finding mathematical justification for the canonical identification mapping p-adics to reals playing key role in p-adic mass calculations. The model for Shnoll effect was the bridge leading to the discovery of quantum arithmetics.

I have been gradually developing the notion of quantum p-adics and during the weekend made quite a step of progress in understanding the concept and dare say that the notion now rests on a sound basis.

- What quantum arithmetics suggests is a modification of p-adic numbers by replacing p-adic pinary expansions with their quantum counterparts allowing the coefficients of prime powers to be integers not divisible by p. A further important constraint is that the factors of coefficients are primes smaller than p. If the coefficients are smaller than p, one obtains something reducing effectively to ordinary p-adic number field.
- A further constraint is that quantum integers respect the decomposition of integer to powers of prime. Quantum p-adic integers are to p-adic integers what the integers in the extension of number field are for the number field and one can indeed identify Galois group G
_{p}for each prime p and form adelic counterpart of this group as Cartesian product of all G_{p}:s. - After various trials it turned out (this is what motivated this posting!) that quantum p-adics are indeed quantal in the sense that one can assign to given quantum p-adic integer n a wave function at the orbit of corresponding Galois group decomposing to Galois groups of its prime factors of n.
- The basic conditions are that ×
_{q}and +_{q}satisfy the basic associativity and distributivity laws. These conditions are extremely powerful and can be formulated in terms of number theoretic Feynman diagrams assignable to sequences of arithmetical operations and their co-algebra counterparts. This brings in physical insight. - One can interpret ×
_{q}and +_{q}and their co-algebra operations as 3-vertices for number theoretical Feynman diagrams describing algebraic identities X=Y having natural interpretation in zero energy ontology. The two vertices have direct counterparts as two kinds of basic topological vertices in quantum TGD, namely stringy vertices in which 3-surface splits and vertices analogous to those of Feynman diagrams in which lines join along their 3-D ends. Only the latter vertices correspond to particle decays and fusions whereas stringy vertices correspond to decay of particle to path and simultaneous propagation along both paths: this is by the way one of the fundamental differences between quantum TGD and string models. This plus the assumption that Galois groups associated with primes define symmetries of the vertices allows to deduce very precise information about the symmetries of the two kinds of vertices needed to satisfy the associativity and distributivity and actually fix them highly uniquely, and therefore determine corresponding zero energy states having collections of integers as counterparts of incoming positive energy (or negative energy) particles. - Zero energy ontology leads naturally zero energy states for which time reversal symmetry is broken in the sense that either positive or negative energy part corresponds to a single collection of integers as incoming lines. What is fascinating is the the prime decomposition of integer corresponds to a decomposition of braid to strands. C and P have interpretation as formations of multiplicative and additive inverses of quantum integers and CP=T changes the positive and negative energy parts of the number theoretic zero energy states.
- This gives strong support for the old conjecture that generalized Feynman diagrams have number theoretic interpretation and allow moves transforming them to tree diagrams - also this generalization of old-fashioned string duality is old romantic idea of quantum TGD, which I however gave up as too "romantic". I noticed the analogy of Feynman diagrams with the algebraic expressions but failed to realize how extremely concrete the connection could be. What was left from the idea were some brief comments in Appendix A: Quantum Groups and Related Structures to one of the chapters of "Towards M-matrix".
The moves for generalized Feynman diagrams would code for associativity and distributivity of quantum arithmetics and we have actually learned them in elementary school as a process simplifying algebraic expressions! Also braidings with strands labeled by the primes dividing the integer emerge naturally so that the connection with quantum TGD proper becomes very strong and consistent with the earlier conjecture inspired by the construction of infinite primes stating that transition amplitudes have purely number theoretic meaning in ZEO.

- The basic conditions are that ×
- Canonical identification finds a fundamental role in the definition of the norm for both quantum p-adics and quantum adeles. The construction is also consistent with the notion of number theoretic entropy which can have also negative values (this is what makes living systems living!).
- There are arguments suggesting that quantum p-adics form a field - one might say "quantum field" - so that also differential calculus and even integral calculus would make sense since quantum p-adics inherit almost well-ordering from reals via canonical identification.
- One can also generalize the construction to algebraic extensions of rationals. In this case the coefficients of quantum adeles are replaced by rationals in the extension and only those p-adic number fields for which the p-adic prime does not split into a product of primes of algebraic extension are kept in the quantum adele associated with rationals. This construction gives first argument in favor of the crazy conjecture that the Absolute Galois group (AGG) is isomorphic with the Galois group of quantum adeles.

To sum up, the vision abut "Physics as generalized number theory" can be also transformed to "Number theory as quantum physics"!

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

## 5 Comments:

Dear Matti,

some questions:

-What is difference between x^3-l and other light like 3-surfaces in the H?

-In many sheeted space time, when we concentrate on two of them, condensing space time is in the H=M4*cp2 and other space time is in the other H’=m’4*cp’2, then both of space time sheet with the wormhole between them are in the space H*H’? (Cartesian product or tensor product of imbedding spaces H and H’?)

-Is the wormhole represented by D3*cp2? Then D3 has Euclidian signature of induced metric?

-If a piece of Generalized Feynman diagram represented by a cylinder, then what is the space of side? And what is the space of plane perpendicular to axis of cylinder?

Best regards

Matti:

Cymatics/Sacred Geometry

http://www.youtube.com/watch?v=9R4Bkwh9h9c

Are the patterns being formed reflecting a dimensional reality?

Rotating wave motion at:

http://www.youtube.com/watch?feature=player_profilepage&v=9R4Bkwh9h9c#t=790s

Twin opposing vertices - a fundamental phenomenon - a scaled-down variant of macroscopic motion.

Regards.

Dear Hamed,

thank you for good answers. Here are my answers to your questions.

a) There are general light-like surfaces in general and wormhole throats which are light-like surfaces at which the signature of the induced metric changes. This means that both 4-metric and 3-metric are degenerate (determinant of the metric vanishes). If I remember correctly I use X^3_l as a symbol for the latter.

b) Imbedding space is just one and unique if one forgets p-adic generalizations and the use of covering of imbedding spaces as effective tool in the description of hierarchy of Planck constants. These generalization might easily lead to confusion.

*All space-time sheets are therefore region of 4-D space-time in M^4xCP_2. There is complete analogy with strings and branes.

*Think first in terms of 2-D surfaces in 3-space E^3 since this is very concrete and familiar. Take two planar pieces of surface which are parallel and connect by wormhole contacts. After this generalize: 3-D space to 8-D spacetime and 2-D surface to 4-D space-time surface.

*The illustrations at my homepage should help to get an idea what the topological condensation and wormhole throats mean. There is no Cartesian product or tensor product involved. See

http://tgdtheory.com/figu.html

If you understand the low-D situation, you easily understand the higher-D situation.

c) Wormhole contact is a 4-D object since it is a small regions of space-time surface.

* A simple model of wormhole contact is as a small deformation of a piece of what I call CP_2 type vacuum extremal.

*Take CP_2 coordinates as local space-time coordinates so that the roles of M^4 and CP_2 reversed and assume that M^4 coordinates are of form m^k= f^k(s), s some function of CP_2 coordinates and f^k some functions. This gives 1-D M^4 projection.

*If you assume also that the condition

m_kl df^k/dsdf^l/ds=0 holds true,

the contribution of M^4 to induced metric vanishes so that surface is just piece of CP_2 in induced metric.

* M^4 projection is light-like random curve since f^k can be chosen otherwise arbitrarily as well as function s. This is vacuum extremal of Kaehler action and the educated guess is that small deformations of these correspond to wormhole contacts defining analogs of lines of generalized Feynman diagram: the M^4 projections of lines would be light-like curves in this approximation.

*The line of generalized Feynman diagrams correspond to wormhole contact and in reasonable approximation piece for the deformation of CP_2 type vacuum extremal. Piece is in question. If you regard the line as cylinder in space-time then the analog for the boundary of cylinder is light-like wormhole throat at which induced 4-metric changes its signature. Blackhole horizon would be a reasonable but not a complete analogy (at Schwartshild blackhole horizon is light-like but the determinant of 4-metric determinant is non-vanishing whereas it vanishing for wormhole throat in TGD).

"Thank you very much Matti, it Gave me a fresh perspective, the question about space time sheets returned to when I asked this by model of a lot of boxes in my mind, then in really there is one box and all the space time sheets are in it. But in my mind there is some misunderstanding yet. If all the space time sheets are in the one H=M4*CP2 then distance between them should be smaller than CP2 radios. but in TGD, CP2 Radios is smallest length scale."

To be continued....

Dear Hamed,

for some reason my blog did not show your question so that I glued it here.

The answer is completely trivial but there is some misunderstanding. I hope the following helps.

a) Space-time sheets have typically finite size. They are inside CDxCP_2 where CD is the intersection of future and past directed light-cones. [I often all CDxCP_" just CD - this kind of laziness is criminal: I am sorry for my sloppiness;-)].

The space-time surface and also space-time sheets inside CD - not usually filling it- have some* finite-sized M^4-projection*. These projections can intersect - sheets are "on top of each other" - but not necessarily. If they are on top of each other, the distance between them is indeed of the orfer of CP_2 size: very very small- about 10^4 Planck lengths: you are completely right in this case. Second possibility is that the first space-time sheet of finite size is here and second in Andromeda. Their distance is essentially the distance using M^4 metric and very very large.

Illustration: think of two very thin glass plates - thinner than CP_2 - in Euclidian 3-space. Imagine that z-direction is analogous to the directions along CP_2 in 8-D space. If the second plate is here and second in Andromeda, the distance between them is very large. If both plates are here at Earth at the same position and parallel to x-y plane, their distance can be arbitrary small as orthonormal distance along z-axis between them. Also wormhole contacts can form between them.

b) The fact that sheets "on top of each other" are so incredibly near each other, makes very probable the formation of wormhole contacts between them and as a special case wormhole contacts gives rise to particles like gauge bosons. If you have fermion modellable as piece of CP_2 type vacuum extremal topologically condensed to another space-time sheet it can touch also second space-time sheet and very probably does so and it feels its presence as external field. About this I talked in some earlier blog postings. This is very important and rather new realization concerning the physical interpretation of the theory.

c) One cannot say that CP_2 radius is the smallest possible length since there is no discretization. Think in terms of the analogy with Kauza-Klein theories. CP_2 length scale only defines the natural geometrically defined length unit which could quite well be smallest possible fundamentla *length scale* but not distance!! If one goes to energies, at which CP_2 radius is the natural length scale defined by Uncertainty Principle, CP_2 begins to be visible directly in physics. This is of course impossible in accelerates that human kind can build during next millenium;-) so that CP_2 makes itself visible only indirectly. CP_2 makes itself indirectly visible via color interactions but QCD is expected to be a good first approximation since CP_2 is so small.

Continuation of the response to Hamed..,

What about Planck length which is about 10^(-4) times smaller than CP_2 size? Does it define fundamental length scale in TGD Universe? I do not actually know. It might emerge from dynamics of quantum TGD as genuine fundamental scale or be just a parameter popping up only at the long length scale limit of the theory so that the whole Planck length scale heuristics would be bad mysticism;-). I however have a nice formula for G in terms of CP_2 radius and exponent of Kahler action for CP_2 type vacuum extremal.

What is clear that Planck length L_Pl propto sqrt(hbar G) and if one takes the hierarchy of Planck constants seriously then for astrophysical values of Planck constant given by hbar_gr= GMm/v_0 Planck length L_Pl is of the order of black hole radius 2GM as you see just by substituting!

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