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.

1. 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.
   
   
2. 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.
   
   
3. 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=M₁₂₇=2¹²⁷-1 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.

1. The symplectic algebra of δ M⁴_+/-× CP₂ has the formal structure of Kac-Moody algebra with the light-like radial coordinate r of the light-cone boundary δ M⁴_+/- taking the role of complex coordinate z. It has symplectic algebras of CP₂ and sphere S² 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.
   

2. There is also the Kac-Moody algebra assignable to the light-like wormhole throats and assignable to the isometries of the imbedding space havingM⁴ and CP₂ 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.

1. The most general option is that one takes the CP₂ and S² symplectic algebras as factors in the symplectic sector. In Kac-Moody sector one has E²⊂ M⁴ 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 CP₂ makes also these correlations manifest). This would give 5 tensor factors as required.
   
   
2. 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 E² 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.
   
   

The first option looks more convincing to me.

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 CP₂ cm degrees of freedom and together with the momentum squared boils down to an eigenvalue of the square of spinor d'Alembertian for H=M⁴× CP₂ (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=M⁴× CP₂ contribution to the scaling generator L₀ 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_+/-=M⁴_+/-× CP₂ and there is a great temptation to identify the wave function as being induced from a partial wave in H=M⁴× CP₂. As a matter fact, number theoretic arguments and arguments related to finite measurement resolution strongly suggest discretization of H_+/-. M⁴_+/- would be replaced with a union of hyperboloids with a distance from the tip of M⁴_+/- which is quantized as a multiple of CP₂ radius. Furthermore at each hyperboloid the allowed points would correspond to the orbit of some discrete subgroup of SL(2,C). CP₂ 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.

1. 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.
   
   

2. The decomposition M⁴=M²× E² 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 M² 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.