### About the basic assumptions behind p-adic mass calculations

The motivation for this piece of text was the basic horror experience of theoretician waking him up at early morning hours. Is s there something wrong with basic assumptions of some particular piece of theory? At this time it was p-adic thermodynamics. Theoretician tries to figure this out in a drowsy state between wake-up and sleep, fails repeatedly, and blames the mighties of the Universe for his miserable fate as eternal doubter. Eventually merciful sleep arrives and theoretician wakes up in the morning, recalls the problem and feels that nothing is wrong. But theoretician knows that it is better to check everything once again.

So that this is what I am doing in the sequel: listing and challenging the basic assumptions and philosophy behind p-adic mass calculations. As always in this kind of situation, I prefer to think it allover again rather than finding what I have written earlier: reader can check whether the recent me agrees with the earlier me. This list is not the only one that I have made during these years and other, possibly different, lists can be found in the chapters of various books. Although the results of calculations are unique and involve only very general assumptions, the guessing of the detailed physical picture behind them is difficult.

**Why p-adic thermodynamics?**

p-Adic thermodynamics is a fundamental assumption behind the p-adic mass calculations: p-adic mass squared is identified as a thermal average of mass squared for super-conformal representation with p-adic mass squared given essentially by the conformal weight.

Zero energy ontology (ZEO) has gradually gained a status of second fundamental assumption. In fact, ZEO strongly suggests the replacement of p-adic thermodynamics with its "complex square root" so that one would be actually considering genuine quantum states squaring to thermodynamical states. This idea looks highly satisfactory for anyone used to think that elementary particles cannot be thermodynamical objects. The square root of p-adic thermodynamics raises delicate number theoretical issues since the p-adic square root of the conformal weight having value p does not exist without a proper algebraic extension of p-adic numbers leading to algebraic integers and generalized notion of primeness.

**Q**: Why p-adic thermodynamics, which predicts the thermal expectation of p-adic mass squared and requires the mapping of p-adic valued mass squared to real mass squared by some variant of canonical identification?

**A**: Number theoretical universality requires fusion of real and p-adic number based physics for various primes so that p-adic thermodynamics becomes natural.

- The answer inspired by TGD inspired theory of consciousness would be that the interaction of p-adic space-time sheets serving as correlates of cognition with real space-time sheets representing matter makes p-adic topology effective topology in some length scale range also for real space-time sheets (as an effective topology for discretization). One could even speak about cognitive representations of elementary particles using the rational or algebraic intersections of real and p-adic space-time sheets. These cognitive representations are very simple in p-adic topology and it is easy to calculate the masses of the particles using p-adic thermodynamics. Since representation is in question, the result should characterize also real particle.

- The pragmatic answer would be that p-adic thermodynamics gives extremely powerful number theoretical constraints leading to the quantization of mass scales and masses with p-adic temperature T=1/n and p-adic prime appearing as free parameters. Also conformal invariance is strongly favored since the counterpart of Hamiltonian must be integer valued as the super-conformal scaling generator indeed is.

- By number theoretical universality one can require that the p-adic mass thermodynamics is equivalent with real thermodynamics for real mass squared. This is the case if partition function has cutoff so that conformal weights only up to some maximum value N are allowed. This has no practical consequences since the real-valued contribution from the conformal weight n is proportional to p
^{-n+1/2}and for n>2 is completely negligible since the primes involved are so large (p=M_{127}=2^{127}-1 for electron for instance).

**Q**: Is the canonical identification mapping the p-adic mass squared to real mass squared unique?

**A**: This is not the case. One can imagine a family of identification for which integers n<p^{N}, N=1,2,... are mapped to itself. This however has no practical implications for the calculations since the values of primes involved are so large.

The calculations themselves assume only p-adic thermodynamics and super-conformal invariance. The most important thing that matters is the number of tensor factors in the tensor product of representations of conformal algebra, which must be *five*.

**Q**: What are the fundamental conformal algebras giving rise to the super conformal symmetries?

**A**: There are two conformal algebras involved.

- The symplectic algebra of δ M
^{4}_{+/-}× CP_{2}has the formal structure of Kac-Moody algebra with the light-like radial coordinate r of the light-cone boundary δ M^{4}_{+/-}taking the role of complex coordinate z. It has symplectic algebras of CP_{2}and sphere S^{2}of light-cone boundary as building blocks taking the role of the finite-dimensional Lie group defining Kac-Moody algebra. This algebra has not in string models.

- There is also the Kac-Moody algebra assignable to the light-like wormhole throats and assignable to the isometries of the imbedding space havingM
^{4}and CP_{2}isometries as factors. There are also electroweak symmetries acting on spinor fields. In fact, the construction of the solutions of the modified Dirac equation suggests that electroweak and color gauge symmetries become Kac-Moody symmetries in TGD framework. In practice this means that only the generators with positive conformal weight annihilate the physical states. For gauge symmetry also those with negative conformal weight annihilate the physical states.

One can of course ask whether also SU(2) sub-algebra of SL(2,C) acting on spinors should be counted. One could argue that this is not the case since spin does correspond to gauge or Kac-Moody symmetry as electroweak quantum numbers do.

**Q**: One must have five tensor factors. How should one count the number of tensor factors, in other words what is the basic building brick to which one identifieds as a tensor factor of Super-Virasoro algebra?

**A**: One can imagine two options.

- The most general option is that one takes the CP
_{2}and S^{2}symplectic algebras as factors in the symplectic sector. In Kac-Moody sector one has E^{2}⊂ M^{4}isometries (longitudinal degrees of freedom of string world sheet carrying induce spinors fields are not physical) and SU(3). Besides this one has electroweak algebra U(2), which almost but quite not decomposes to SU(2)_{L}× U(1) (there are correlations between SU(2)_{L}and U(1) quantum numbers and the existence of spinor structure of CP_{2}makes also these correlations manifest). This would give 5 tensor factors as required.

- I have also considered Cartan algebras as separate tensor factors. I must confess, that at this moment I am unable to rediscover what my motivation for this actually has been. This would give a larger number of tensor factors: 1+2 factors in symplectic sector from Cartan algebras of SO(3)× SU(3) defining subgroup of symplectic group, 2+2 for isometries in Kac-Moody sector from E
^{2}and SU(3), and 1+1 in the electroweak sector with spin giving a possible further factor. This means 9 (or possibly 10) factors so that thermalization is not possible for all Cartan algebra factors. Symplectic sectors are certainly a natural candidate in this respect so that one would have 5 as required (or 6 if spin is allowed to have Kac-Moody structure) sectors.

**How to understand the conformal weight of the ground state?**

Ground state conformal weight which is non-positive can receives various contributions. One contribution is negative and therefore corresponds to a tachyonic mass squared, second contribution corresponds to CP_{2} cm degrees of freedom and together with the momentum squared boils down to an eigenvalue of the square of spinor d'Alembertian for H=M^{4}× CP_{2} (by bosonic emergence). Third one comes from the conformal moduli of the partonic 2-surface at the end of the space-time sheet at light-like boundary of causal diamond and distinguishes between different fermion families.

**Q**: Tachyonic ground state mass does not look physical and is quite generally seen as a serious - if not lethal - problem also in string models. What is the origin of the tachyonic contribution to the mass squared in TGD framework?

**A**: The recent picture about elementary particles is as lines of generalized Feynman diagram identified as space-time regions with Euclidian signature of the induced metric. In this regions mass squared is naturally negative and it is natural to think that ground state mass squared receives contributions from both Euclidian and Mionkowskian regions. If so, the necessary tachyonic contribution would be a direct signal for the presence of the Euclidian regions, which have actually turned out to define a generalization of blackhole interior and be assignable to any system as a space-time sheet characterizing the system geometrically. For instance, my own body as I experience it would correspond to my personal Euclidian space-time seet as a line of generalized Feynman diagram.

**Q**: Where does the H=M^{4}× CP_{2} contribution to the scaling generator L_{0} assignable to spinor partial waves in H come from?

**A**: Zero energy ontology (ZEO) allows to assign to each particle a causal diamond CD and according to the recent view emerging from the analysis of the relationship between subjective (experienced) time and geometric time, particle is characterized by a quantum superposition of CDs. Every state function reduction means localization of the upper of lower tip of all CDs in the superposition and delocalization of the other tip. The position of the upper tip has wave function in H_{+/-}=M^{4}_{+/-}× CP_{2} and there is a great temptation to identify the wave function as being induced from a partial wave in H=M^{4}× CP_{2}. As a matter fact, number theoretic arguments and arguments related to finite measurement resolution strongly suggest discretization of H_{+/-}. M^{4}_{+/-} would be replaced with a union of hyperboloids with a distance from the tip of M^{4}_{+/-} which is quantized as a multiple of CP_{2} radius. Furthermore at each hyperboloid the allowed points would correspond to the orbit of some discrete subgroup of SL(2,C). CP_{2} would be also discretized.

**What about Lorentz invariance?**

The square root of p-adic thermodynamics implies quantum superposition of states with different values of mass squared and hence four-momenta. In ZEO this does not mean obvious breaking of Lorentz invariance since physical states have vanishing total energy. Note that coherent states of Cooper pairs, which in ordinary ontology would have both ill-defined energy and fermion number, have a natural interpretation in ZEO.

- A natural assumption is that the state in the rest system involves only a superposition of states with vanishing three-momentum. For Lorentz boosts the state would be a superposition of states with different three-momenta but same velocity. Classically the assumption about same 3-velocity is natural.

**Q**: Could Lorentz invariance break down by the presence of the superposition of different momenta?

**A**: This is not the case if only the average four-momentum is observable. The reason is that average four-momentum transforms linearly under Lorentz boosts. I have earlier considered the possibility of replacing momentum squared with conformal weight but this option looks somewhat artificial and even wrong to me now.

- The decomposition M
^{4}=M^{2}× E^{2}is fundamental in the formulation of quantum TGD, in the number theoretical vision about TGD, in the construction of preferred extremals, and for the vision about generalized Feynman diagrams. It is also fundamental in the decomposition of the degrees of string to longitudinal and transversal ones. An additional item to the list is that also the states appearing in thermodynamical ensemble in p-adic thermodynamics correspond to four-momenta in M^{2}fixed by the direction of the Lorentz boost.

**Q**: In parton model of hadrons it is assumed that the partons have a distribution with respect to longitudinal momentum, which means that the velocities of partons are same along the direction of motion of hadron. Could one have p-adic thermodynamics for hadrons?

**A**: For hadronic p-adic thermodynamics the value of the string tension parameter would be much smaller and the thermal contributions from n>0 states would be completely negligible so that the idea does not look good. In p-adic thermodynamics for elementary particles one would have distribution coming from different values of p-adic mass squared which is integer valued apart from ground state configuration.

**What are the fundamental dynamical objects?**

The original assumption was that elementary particles correspond to wormhole throats. With the discovery of the weak form of electric-magnetic duality came the realization that wormhole throat is homological magnetic monopole (rather than Dirac monopole) and must therefore have (Kähler) magnetic charge. Magnetic flux lines must be however closed so that the wormhole throat must be associated with closed flux loop.

The most natural assumption is that this loop connects two wormhole throats at the first space-time sheet, that the flux goes through a second wormhole contact to another sheet, returns back along second flux tube, and eventually is transferred to the original throat along the first wormhole contact.

The solutions of the Modified Dirac equation assign to this flux tube string like curve as a boundary of string world sheet carrying the induced fermion field. This closed string has "short" portions assignable to wormhole contacts and "long" portions corresponding to the flux tubes connecting the two wormhole contacts. One can assign a string tension defined by CP_2 scale with the "short" portions of the string and string tension defined by the primary or perhaps secondary p-adic length scale to the "long" portions of the closed string.

Also the "long" portion of the string can contribute to the mass of the elementary particle as a contribution to the vacuum conformal weight. In the case of weak gauge bosons this would be the case and since the contribution is naturally proportional to gauge couplings strength of W/Z boson one could understand Q/Z mass ratio if the p-adic thermodynamics gives a very small contribution from the "short" piece of string (also photon would receive this small contributionin ZEO): this is the case if one must have T=1/2 for gauge bosons. Note that "long" portion of string can contribute also to fermion masses a small shift. Hence no Higgs vacuum expectation value or coherent state of Higgs would be needed. There are two options for the interpretation of recent results about Higgs and Option II in which Higgs mechanism emerges as an ffective description of particle massivation at QFT limit of the theory and both gauge fields and Higgs fields and its vacuum expectation exist only as constructs making sense at QFT limit. Higgs like particles do of course exist. At WCW limit they are replaced by WCW spinor fields as fundamental object.

**Q**: One can consider several identifications of the fundamental dynamical object of p-adic mass calculations. Either as a wormhole throat (in the case of fermions for which either wormhole throat carries the fermion quantum number this looks natural), as entire wormhole contact, or as the entire flux tube having two wormhole contacts. Which one of these options is correct?

**A**: The strong analogy with string model implied by the presence of fermionic string world sheet would support that the identification as entire flux tube in which case the large masses for higher conformal excitations could be interpreted in terms of string tension. Note that this is the only possibility in case of gauge bosons.

**Q**: What about p-adic thermodynamics or its square root in hadronic scale?

**A**: As noticed the contributions from n>0 conformal excitations would be extremely small in p-adic thermodynamics for "long" portions. It would seem that this contribution is non-thermal and comes from each value of n labelling states in Regge trajectory separately just as in old-fashioned string model. Even weak bosons would have Regge trajectories. The dominant contribution to the hadron mass can be assigned to the magnetic body of the hadron consisting of Kähler magnetic flux tubes. The Kähler-magnetic (or equivalently color-magnetic) flux tubes connecting valence quarks can contribute to the mass squared of hadron. I have also considered the possibility that symplectic conformal symmetries distinguishing between TGD and superstring models could be responsible for a contribution identifiable as color magnetic energy of hadron classically.

## 14 Comments:

Sometimes I get comment to blog arriving to my email but not visible in blog. The following comment is example of this.

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The zero-energy ontology has one large implication: it cuts you out of higher number systems. There you loose factorial identity and prime number

theory, which lead Dedekind and Dirichlet to construct ideal numbers, but with a generic inverse energy and Kahler norm you have no non-trivial

ideals. It looks like the mathematics/physics divide persists here.

[MP] Sorry. I have absolutely no idea about how ZEO or the dynamics of Kahler action would cut off higher number systems. The notion of inverse energy is also totally mysterious to me.

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Peirce remarked that if you pursue relations far enough, you loose all logical structure, as also in higher number systems: they merge into pure

character, the free creative space that draws ThePesla. I don't see the children of the Information Age foregoing that horizon.

[MP] Same again. Not a slightest idea about connection with Peirce. I would like to see something more concrete to be able to answer.

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Phenomenology is supposed ot mediate this tension, but I'm afraid Heidegger was mad (or psychopathic), and presented as such to his psychiatry seminar

by Jaques Lacan.

[MP] Same problem again. How Heidegger and his possible madness might possibly relate to TGD and description of particle massivation by p-adic thermodynamics? Something more concrete is needed.

http://physicsworld.com/cws/article/news/2012/nov/21/babar-makes-first-direct-measurement-of-time-reversal-violation

On the Vixra blog you say: In TGD framework this is even more the case since at microscopic level bosons consist of fermion antifermion pairs meaning that all bosons emerge.

Can bosons then be seen as a BEC? In biology the carbon (can be a boson) and light interaction is a basic signal, and the interaction is in the gap, as a superposition?

Light interaction and the quantum jump as a 'particle' for a cognate can then be the same thing? Also in the other interactions is light emitted as radiation, as Popp has explained. The DNA as a quantum computer has also difficulties with the epigenetic (light-interactive sheet around genes? Light overrules carbon regulation sometimes.

This also links light and time, as matter and antimatter are linked.

Are y ok?

Space-time sheets with negative time orientation carry negative energies.

Is it any difference in energies between light as a particle or light as a wave? When light (and information entangled with light) forms A BEC and is teleportated it is dissipationless, as is much the superconduction in nerves? The dissipationless state makes time impossible (instant), or then the negative time is very compressed, almost presens? (Or very diluted like the diluted energy in 'lower than Planck scale', labelled with primes?)

http://phys.org/news/2012-11-super-atoms-rydberg-quantum-gas.html

Dear Matti,

I want to learn Quantum TGD step by step. I think it is more useful for me to learn it as like TGD evolved from first years of your thesis to now. This process helps me to understand the motivations behind it and how the concepts generalized step by step as TGD evolved. I know it will be long process, but I try to be patient ;).

Therefore I am concentrating on the space of 3-metrics of Wheeler precisely. I will ask the questions about it in following days.

Yes, I also think that is the way to go. This is why all those old texts also should be saved, although there are mistakes and wrong guesses. Mark Williams does a good servise in saving them too.

Dear Matti,

If it is possible, please write a program step by step for studying Quantum TGD in the first of your thesis!! As Ulla pointed, it would be very good if it contains mistakes and wrong guesses. I hope you remember them after more than 30 years ;-). Thanks a lot.

If you want, it would be useful for other readers, if you devote a new posting about it.

Dear Hamed,

sounds like an excellent idea. I could try to write this after I get my recent project done: an extremely nervestaking programming task at this age. Takes still few days probably.

I have been worked hardly to learn Unix command level language (Bash) from web. This just to rationalize the hopelessly complex latexing procedure of tex files (MacLatex does not work properly so that I need both command Level Unix and Python as auxiliary tools). This would make possible batch operations: all chapters of all books latexed simultaneously.

To Ulla about bosonic emergence and BEC:

Bose Einstein condensate is quantum field theoretic notion. I have a proposal for the microscopic counterpart of BEC in TGD developed during my long Higgs Odysseia , which I have summarized as a chapter http://tgdtheory.com/public_html/paddark/paddark.html#higgs .

If TGD has QFT limit (I suggest a concrete construction of this limit at the end of Higgs chapter), BEC makes sense also in TGD as an approximate concept at least. Of course, microscopic description could bring in daylight some new aspects about this process.

A basic objection to BEC in TGD is that for members of fermion-antifermion pair exclusion principle holds true. Can one really put two such pairs at same position (at single point at QFT limit)?

Maybe many-sheeted space-time allows to solve the problem. There is room in CP_2 degrees of freedom. One can put the bosons as pairs of space-time sheet on top of each other as parallel space-time sheets so that only their M^4 projections overlap.

The space-time interpretation for a generation of large value of Planck constant is as a multifurcation generating N-branched space-time. Could also this have something to do with BEC like process?

Thanks, I wish you do these tasks in the days well with no problem.

Quantum theory of continuous feedback. Thoughts? could dN(t) be a geiger counter?

https://docs.google.com/open?id=0B5kp8BrW_9rdZl9mVEExMUZoLVU

Hi Matti,

I managed to download this page on a flash drive and read it carefully. I made some general comments inspired by it concerning the nature of doubts including a rather strange idea upon awakening with some rather unusual for me images and dreams. I work off line mostly with no connections but the library. This nature of doubts and its philosophy.

But you may find it quite entertaining when from the view The Tarot of Physics what if they will look back on our theories like many now look back on the tarot deck for these standard and string physicists?

when does the magic become the science anyway?

The PeSla

To Stephen,

the article is rather technical and about quantum effects to feedback if I recall correctly. Unfortunately I do not have time to read it properly to comment it.

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