Indeed! Is it really Higgs?

Jester comments the latest release of results from LHC relating to the signal interpreted by all fashionable and well-informed physics bloggers as Higgs.

Additional support for a resonance at 125 GeV is emerging. What is new are two events which are interpreted as fusion of two W bosons to Higgs. This is very nice. The only problem is that the predicted rate for these events is so small for standard model Higgs that they would not have been observed. Second not anymore pleasant surprise is that Higgs candidates are indeed produced but with a rate twice than the predicted rate.

Hitherto these signals which are too strong to allow interpretation as standard model Higgs have been interpreted by saying that both CMS and ATLAS have been "lucky". I warned already in the previous Higgs posting that if this good luck continues, it turns to a serious problem. And as Jester mentions, already now people are beginning to suspect that this Higgs is not quite the standard model Higgs. The next step will come sooner or later and will be a cautious proposal spoiling the euphoric mood of co-bloggers: perhaps it is not Higgs at all!

But things go slowly. Colleagues are rather conformistic and remarkably slow as thinkers. There are even those who are still making bets for standard SUSY;-)! I can however hope that after this step colleagures would be finally psychologically mature to consider the TGD prediction for M₈₉ hadron physics as an alternative to Higgs. Accepting this hypothesis as something worth of testing would mean enormous progress on both the theoretical and the experimental side.

One more good reason for p-adic cognition

One can present several justifications for why p-adic numbers are natural correlates of cognition and why p-adic topology is tailor-made for computation. One possible justification derives from the ultrametricity of p-adic norm stating that the p-adic norm is never larger than the maximum of the norms of summands.

If one forms functions of real arguments, a cutoff in decimal or more general expansion of arguments introduces a cumulating error, and in principle one must perform calculation assuming that the number of digits for the arguments of function is higher than the number digits required by the cutoff, and drop the surplus digits at the end of the calculations.

In p-adic case the situation is different. The sum for the errors resulting from cutoffs is never p-adically larger than the largest individual error so that there is no cumulation of errors , and therefore no need for surplus pinary digits for the arguments of the function. In practical computations this need not have great significance unless they involve very many steps but in cognitive processing the situation might be different.

Teasing as a national disease

We have a presidential election in Finland. The two main candidates are Sauli Niinistö and Pekka Haavisto. Niinistö can be said to represent the old world order in which economical values dictate everything. Haavisto is a representative of the new world order in which humanity, freedom, equality, and environment represent the most important values. For me the choice between these options is easy although I have nothing against Niinistö personally.

Haavisto crystallized something very essential about Finland as a nation as he said that teasing is the national disease of Finland. Teasing begins already in elementary schools and continues in various educational establishments and eventually it continues at working places. Web has become also an arena of teasing providing completely new opportunities. Now and then some-one gets enough. The two mass murders that took place in educational establishments for few years ago are just two sad examples of what "enough is enough" really means.

Personally I belong to the victims of academic teasing. The terror began for 34 years ago and has continued since then. I have lost my academic human rights and have been unemployed most of the time after I began to write my thesis 1977. I will remain so until I get to the age of 63 (only two years of this humiliation anymore!) and start to receive a minimal pension. I have done impressive life work: 15 books making about 12 thousand pages and a lot of articles. This does not means anything since so called "evaluation by equals" (direct translation for "vertaisarviointi"), a scientific equivalent of inquisition, can be used to label my work as crackpottery. There are many people out-broad and also in Finland who appreciate my work and they have made attempts to inform about my work in Wikipedia but (very probably finnish) censors have reacted immediately and vandalized the attempts.

I have tried to understand what drives people to this kind of sadistic behaviors in which human life is literally destroyed. As far as I know, these people are quite descent human beings as individuals. But as members of collective they become sadistic beasts. Or some of them. The others remain completely passive and this is probably the core problem. We do not have the courage to say no when some sociopath starts the cruel game. I am of course just one of the many victims of this national hobby and I sincerely hope that Finland as a nation could heal from it. Haavisto is certainly experienced as a symbol of this healing and my sincere hope is that he wins.

Quantum p-adic deformations of space-time surfaces as a representation of finite measurement resolution?

A mathematically - and also physically - fascinating question is whether one could use quantum arithmetics as a tool to build quantum deformations of partonic 2-surfaces or even of space-time surfaces and how could one achieve this. These quantum space-times would be commutative and therefore not like non-commutative geometries assigned with quantum groups. Perhaps one could see them as commutative semiclassical counterparts of non-commutative quantum geometries just as the commutative quantum groups (see this) could be seen commutative counterparts of quantum groups.

As one tries to develop a new mathematical notion and interpret it, one tends to forget the motivations for the notion. It is however extremely important to remember why the new notion is needed.

1. In the case of quantum arithmetics Shnoll effect is one excellent experimental motivation. The understanding of canonical identification and realization of number theoretical universality are also good motivations coming already from p-adic mass calculations. A further motivation comes from a need to solve a mathematical problem: canonical identification for ordinary p-adic numbers does not commute with symmetries.

2. There are also good e motivations for p-adic numbers? p-Adic numbers and quantum phases can be assigned to finite measurement resolution in length measurement and in angle measurement. This with a good reason since finite measurement resolution means the loss of ordering of points of real axis in short scales and this is certainly one outcome of a finite measurement resolution. This is also assumed to relate to the fact that cognition organizes the world to objects defined by clumps of matter and with the lumps ordering of points does not matter.

3. Why quantum deformations of partonic 2-surfaces (or more ambitiously: space-time surfaces) would be needed? Could they represent convenient representatives for partonic 2-surfaces (space-time surfaces) within finite measurement resolution?
   1. If this is accepted there is not compelling need to assume that this kind of space-time surfaces are preferred extremals of Kähler action.

   2. The notion of quantum arithmetics and the interpretation of p-adic topology in terms of finite measurement resolution however suggest that they might obey field equations in preferred coordinates but not in the real differentiable structure but in what might be called quantum p-adic differentiable structure associated with prime p.

   3. Canonical identification would map these quantum p-adic partonic (space-time surfaces) to their real counterparts in a unique a continuous manner and the image would be real space-time surface in finite measurement resolution. It would be continuous but not differentiable and would not of course satisfy field equations for Kähler action anymore. What is nice is that the inverse of the canonical identification which is two-valued for finite number of pinary digits would not be needed in the correspondence.

   4. This description might be relevant also to quantum field theories (QFTs). One usually assumes that minima obey partial differential equations although the local interactions in QFTs are highly singular so that the quantum average field configuration might not even possess differentiable structure in the ordinary sense! Therefore quantum p-adicity might be more appropriate for the minima of effective action.

   The conclusion would be that commutative quantum deformations of space-time surfaces indeed have a useful function in TGD Universe.

Consider now in more detail the identification of the quantum deformations of space-time surfaces.

1. Rationals are in the intersection of real and p-adic number fields and the representation of numbers as rationals r=m/n is the essence of quantum arithmetics. This means that m and n are expanded to series in powers of p and coefficients of the powers of p which are smaller than p are replaced by the quantum counterparts. They are quantum quantum counterparts of integers smaller than p. This restriction is essential for the uniqueness of the map assigning to a give rational quantum rationals.

2. One must get also quantum p-adics and the idea is simple: if the pinary expansions of m and n in positive powers of p are allowed o become infinite, one obtains a continuum very much analogous to that of ordinary p-adic integers with exactly the same arithmetics. This continuum can be mapped to reals by canonical identification. The possibility to work with numbers which are formally rationals is utmost importance for achieving the correct map to reals. It is possible to use the counterparts of ordinary pinary expansions in p-adic arithmetics.

3. One can defined quantum p-adic derivatives and the rules are familiar to anyone. Quantum p-adic variants of field equations for Kähler action make sense.
   1. One can take a solution of p-adic field equations and by the commutativity of the map r=m/n→ r_q=m_q/n_q and of arithmetic operations replace p-adic rationals with their quantum counterparts in the expressions of quantum p-adic imbedding space coordinates h^k in terms of space-time coordinates x^α.

   2. After this one can map the quantum p-adic surface to a continuous real surface by using the replacement p→ 1/p for every quantum rational. This space-time surface does not anymore satisfy the field equations since canonical identification is not even differentiable. This surface - or rather its quantum p-adic pre-image - would represent a space-time surface within measurement resolution. One can however map the induced metric and induced gauge fields to their real counterparts using canonical identification to get something which is continuous but non-differentiable.

4. This construction works nicely if in the preferred coordinates for imbedding space and partonic (space-time) surface itself the imbedding space coordinates are rational functions of space-time coordinates with rational coefficients of polynomials (also Taylor and Laurent series with rational coefficients could be considered as limits). This kind of assumption is very restrictive but in accordance with the fact that the measurement resolution is finite and that the representative for the space-time surface in finite measurement resolution is to some extent a convention. The use of rational coefficients for the polynomials involved implies that for polynomials of finite degree WCW reduces to a discrete set so that finite measurement resolution has been indeed realized quite concretely!

Consider now how the notion of finite measurement resolution allows to circumvent the objections against the construction.

1. Manifest GCI is lost because the expression for space-time coordinates as quantum rationals is not general coordinate invariant notion unless one restricts the consideration to rational maps and because the real counterpart of the quantum p-adic space-time surface depends on the choice of coordinates. The condition that the space-time surface is represented in terms of rational functions is a strong constraint but not enough to fix the choice of coordinates. Rational maps of both imbedding space and space-time produce new coordinates similar to these provided the coefficients are rational.

2. Different choices for imbedding space and space-time surface lead to different quantum p-adic space-time surface and its real counterpart. This is an outcome of finite measurement resolution. Since one cannot order the space-time points below the measurement resolution, one cannot fix uniquely the space-time surface nor uniquely fix the coordinates used. This implies the loss of manifest general coordinate invariance and also the non-uniqueness of quantum real space-time surface. The choice of coordinates is analogous to gauge choice and quantum real space-time surface preserves the information about the gauge.

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

The anatomy of quantum jump in zero energy ontology

The understanding of the anatomy of quantum jump identified as a moment of consciousness in the framework of Zero energy ontology (ZEO) is gradually getting more detailed and the following is the summary of the recent understanding. The general vision about quantum jump in zero energy ontology generalizes the ordinary auantum measurement theory bringing in also the selection of maximal set of mutually commuting set of observables. Also the connection with the breaking of time reversal invariance at the level of zero energy states as a necessary condition for the non-triviality of U-matrix is new.

1. Quantum jump begins with unitary process U described by unitary matrix assigning to a given zero energy state a quantum superposition of zero energy states. This would represent the creative aspect of quantum jump - generation of superposition of alternatives.

2. The next step is a cascade of state function reductions proceeding from long to short scales. It starts from some CD and proceeds downwards to sub-CDs to their sub-CDs to ...... At a given step it induces a measurement of the quantum numbers of either positive or negative energy part of the quantum state. This step would represent the measurement aspect of quantum jump - selection among alternatives.

3. The basic variational principle is Negentropy Maximization Principle(NMP) stating that the reduction of entanglement entropy in given quantum jump between two subsystems of CD assigned to sub-CDs is maximal. Mathematically NMP is very similar to the second law although states just the opposite but for individual quantum system rather than ensemble. NMP actually implies second law at the level of ensembles as a trivial consequence of the fact that the outcome of quantum jump is not deterministic.

   For ordinary definition of entanglement entropy this leads to a pure state resulting in the measurement of the density matrix assignable to the pair of CDs. For hyper-finite factors of type II₁ (HFFs) state function reduction cannot give rise to a pure state and in this case one can speak about quantum states defined modulo finite measurement resolution and the notion of quantum spinor emerges naturally. One can assign a number theoretic entanglement entropy to entanglement characterized by rational (or even algebraic) entanglement probabilities and this entropy can be negative. Negentropic entanglement can be stable and even more negentropic entanglement can be generated in the state function reduction cascade.

The irreversibility is realized as a property of zero energy states (for ordinary positive energy ontology it is realized at the level of dynamics) and is necessary in order to obtain non-trivial U-matrix. State function reduction should involve several parts. First of all it should select the density matrix or rather its Hermitian square root. After this choice it should lead to a state which prepared either at the upper or lower boundary of CD but not both since this would be in conflict with the counterpart for the determinism of quantum time evolution.

Generalization of S-matrix

ZEO forces the generalization of S-matrix with a triplet formed by U-matrix, M-matrix, and S-matrix. The basic vision is that quantum theory is at mathematical level a complex square roots of thermodynamics. What happens in quantum jump was already discussed.

1. U-matrix as has its rows M-matrices , which are matrices between positive and negative energy parts of the zero energy state and correspond to the ordinary S-matrix. M-matrix is a product of a hermitian square root - call it H - of density matrix ρ and universal S-matrix S commuting with H: [S,H]=0. There is infinite number of different Hermitian square roots H_i of density matrices which are assumed to define orthogonal matrices with respect to the inner product defined by the trace: Tr(H_iH_j)=0. Also the columns of U-matrix are orthogonal. One can interpret square roots of the density matrices as a Lie algebra acting as symmetries of the S-matrix.

2. One can consider generalization of M-matrices so that they would be analogous to the elements of Kac-Moody algebra. These M-matrices would involve all powers of S.

   1. The orthogonality with respect to the inner product defined by < A| B> = Tr(AB) requires the conditions Tr(H₁H₂Sⁿ)=0 for n≠ 0 and H_i are Hermitian matrices appearing as square root of density matrix. H₁H₂ is hermitian if the commutator [H₁,H₂] vanishes. It would be natural to assign n:th power of S to the CD for which the scale is n times the CP₂ scale.

   2. Trace - possibly quantum trace for hyper-finite factors of type II₁) is the analog of integration and the formula would be a non-commutative analog of the identity ∈t_S¹ exp(inφ) dφ=0 and pose an additional condition to the algebra of M-matrices. Since H=H₁H₂ commutes with S-matrix the trace can be expressed as the sum

      ∑_i,jh_is_j(i)= ∑_i,j h_i(j)s_j

      of products of correspondence eigenvalues and the simplest condition is that one has either ∑_j s_j(i)=0 for each i or ∑_i h_i(j)=0 for each j.

   3. It might be that one must restrict M matrices to a Cartan algebra for a given U-matrix and also this choice would be a process analogous to state function reduction. Since density matrix becomes an observable in TGD Universe, this choice could be seen as a direct counterpart for the choice of a maximal number of commuting observables which would be now hermitian square roots of density matrices. Therefore ZEO gives good hopes of reducing basic quantum measurement theory to infinite-dimensional Lie-algebra.

Unitary process and choice of the density matrix

Consider first unitary process followed by the choice of the density matrix.

1. There are two natural state basis for zero energy states. The states of these state basis are prepared at the upper or lower boundary of CD respectively and correspond to various M-matrices M_K⁺ and M_L^-. U-process is simply a change of state basis meaning a representation of the zero energy state M_K^+/- in zero energy basis M_K^-/+ followed by a state preparation to zero energy state M^+/-_K with the state at second end fixed in turn followed by a reduction to M_L^-/+ to its time reverse, which is of same type as the initial zero energy state.

   The state function reduction to a given M-matrix M_K^+/- produces a state for the state is superposition of states which are prepared at either lower or upper boundary of CD. It does not yet produce a prepared state on the ordinary sense since it only selects the density matrix.

2. The matrix elements of U-matrix are obtained by acting with the representation of identity matrix in the space of zero energy states as

   I= ∑_K | K⁺> < K⁺|

   on the zero energy state | K^-> (the action on | K⁺> is trivial!) and gives

   U⁺_KL= Tr(M⁺_KM⁺_L) .

   In the similar manner one has

   U^-_KL=(U^+†)_KL= Tr(M^-_LM^-_K) = (U⁺_LK)^* .

   These matrices are Hermitian conjugates of each other as matrices between states labelled by positive or negative energy states. The interpretation is that two unitary processes are possible and are time reversals of each other. The unitary process produces a new state only if its time arrow is different from that for the initial state. The probabilities for transitions |K₊> → |K_-> are given by

   p_mn= |Tr(M_K⁺ M_L⁺)|².

State function preparation

Consider next the counterpart of the ordinary state preparation process.

1. The ordinary state function process can act either at the upper or lower boundary of CD and its action is thus on positive or negative energy part of the zero energy state. At the lower boundary of CD this process selects one particular prepared states. At the upper boundary it selects one particular final state of the scattering process.

2. Restrict for definiteness the consideration to the lower boundary of CD. Denote also M_K by M. At the lower boundary of CD the selection of prepared state - that is preparation process- means the reduction

   ∑_m⁺n^-M^+/-_m⁺n^-| m⁺> | n^-> → ∑_n^-M^+/-_m⁺n^-| m⁺> | n^-> .

   The reduction probability is given by

   p_{m= ∑n^- | Mm⁺n^-|² = ρm⁺m⁺ .}

   For this state the lower boundary carries a prepared state with the quantum numbers of state | m₊> . For density matrix which is unit matrix (this option giving pure state might not be possible) one has p_m=1.

State function reduction process

The process which is the analog of measuring the final state of the scattering process is also needed and would mean state function reduction at the upper end of CD - to state | n^-> now.

1. It is impossible to reduce to arbitrary state | m₊> | n_-> and the reduction must at the upper end of CD must mean a loss of preparation at the lower end of CD so that one would have kind of time flip-flop!

2. The reduction probability for the process

   | m₊ >== ∑_n^-M_m⁺n^-| m⁺> | n^-> → n_->= ∑_m⁺M_m⁺n^-| m⁺> | n^->

   would be

   p_{mn =| Mmn|² .}

   This is just what one would expect. The final outcome would be therefore a state of type | n^-> and - this is very important- of the same type as the state from which the process began so that the next process is also of type U⁺ and one can say that a definite arrow of time prevails.

3. Both the preparation and reduction process involves also a cascade of state function reductions leading to a choice of state basis corresponding to eigenstates of density matrices between subsystems.

Can the arrow of geometric time change?

A highly interesting question is what happens if the first state preparation leading to a state | K⁺> is followed by a U-process of type U^- rather than by the state function reduction process |K⁺> → |L^->. Does this mean that the arrow of geometric time changes? Could this change of the arrow of geometric time take place in living matter? Could processes like molecular self assembly be entropy producing processes but with non-standard arrow of geometric time? Or are they processes in which negentropy increases by the fusion of negentropic parts to larger ones? Could the variability relate to sleep-awake cycle and to the fact that during dreams we are often in our childhood and youth. Old people are often said to return to their childhood. Could this have more than a metaphoric meaning? Could biological death mean return to childhood at the level of conscious experience? I have explained the recent views about the arrow of time here .

For background see new chapter Construction of Quantum Theory: More About Matrices of "Towards M-matrix" .

How it went?

Mark McWilliams requested some kind of summary about the development of TGD, and I decided to write an article about the the history of TGD. I could not avoid telling also about turning points of my personal life since my work and life are to high extent one and the same thing.

I have tried to represent the development chronologically but I must confess that I have forgotten precise dates so that the chronology is not exact. Very probably I have also forgotten many important ideas and many side tracks which led nowhere. Indeed, the study of the tables of contents of books and old blog postings and What's New articles at the homepage forces me to wonder how I can forget something so totally.

The article should help a novice to get an overall view about the basic ideas of TGD are their evolution during these 34 years. To myself a real surprise was to see how many deep ideas have emerged after 2005: one can really speak about a burst of new ideas. Most of them relate to the evolution of the mathematical aspects of TGD and to their physical interpretation but also the experimental input from LHC, Fermilab, and elsewhere has played a decisive role in stimulating ideas about the interpretation of the theory.

Unavoidably the emphasis is on the latest ideas and there is of course the risk that some of them are not here to stay. Even during writing process some ideas developed into more concrete form. A good example is the vision about what happens in quantum jump and what the unitarity of U-matrix really means, how M-matrices generalize to form Kac-Moody type algebra, and how the notion of quantum jump in zero energy ontology (ZEO) reproduces the basic aspects of quantum measurement theory. Also a slight generalization of quantum arithmetics suggested itself during the preparation of the article.

I gave to the article a title which is easy to guess: "Evolution of TGD". It can be found at my homepage which is now living at webhotel with address http://tgdtheory.com/.

Note: The links of old postings to my homepage do not work anymore. Apologies. To get to the link work one can replace "http://tgd.wippiespace.com/" with "http://tgdtheory.com/", and if this does not work, with "http://tgdtheory.com/public_html/".

Does 2-adic quantum arithmetics explain p-adic length scale hypothesis?

For p=2 quantum arithmetics looks singular at the first glance. This is actually not the case since odd quantum integers are equal to their ordinary counterparts in this case. This applies also to powers of two interpreted as 2-adic integers. The real counterparts of these are mapped to their inverses in canonical identification.

Clearly, odd 2-adic quantum quantum rationals are very special mathematically since they correspond to ordinary rationals. It is fair to call them "classical" rationals. This special role might relate to the fact that primes near powers of 2 are physically preferred. CDs with n=2^k would be in a unique position number theoretically. This would conform with the original - and as such wrong - hypothesis that only these time scales are possible for CDs. The preferred role of powers of two supports also p-adic length scale hypothesis.

The discussion of the role of quantum arithmetics in the construction of generalized Feynman diagrams allows to understand how for a quantum arithmetics based on particular prime p particle mass squared - equal to conformal weight in suitable mass units - divisible by p appears as an effective propagator pole for large values of p. In p-adic mass calculations real mass squared is obtained by canonical identification from the p-adic one. The construction of generalized Feynman diagrams allows to understand this strange sounding rule as a direct implication of the number theoretical universality realized in terms of quantum arithmetics.

Witten about mass gap

Witten has a nice talk about mass gap problem in 3-D (mostly) and 4-D gauge theories demonstrating how enormous his understanding and knowledge about mathematical physics is. Both Peter Woit and Kea have commented it.

In 3-D case the coupling strength g² has the dimension of inverse length and therefore it would not be surprising if mass gap would emerge. Witten argues that by adding to the theory a Chern-Simons term the theory could reduce in long length scales to non-trivial topological QFT at the IR limit. This would be also a nice manner to resolve the IR difficulties of 3-D gauge theories. Could one imagine effective reduction to topological QFT in long length scales also in 4-D case as a solution to IR divergences?

In D=4 the situation is much more difficult since the gauge coupling is dimensionless. My un-educated is opinion is that the proper question is whether the theory actually exists mathematically and my equally un-educated guess is that it does not - unless one brings in the mass scale somehow by hand. The standard Muenchausen trick to bring the scales in perturbation theory is via UV and IR cutoffs. This is going outside what one means with gauge theory strictly mathematically. In order to make progress, one must bring in the new physics and mathematics. A rigorous mathematical formulation of 4-D gauge theory is not enough: it simply does not exist since something very important is missing.

TGD view about the mass gap problem

TGD is one proposal for what this new physics and mathematics could be. I do not try to re-explain in any detail what this new physics and mathematics might be since I have done this explaining for 6 years in this blog. The basic statement is however that the fundamental UV length scale must be present explicitly in the definition of the theory and must have concrete geometric interpretation rather than being a dimensional number like string tension. In TGD framework it corresponds to the "radius" of CP₂, which is fixed from simple symmetry arguments as the only possible choice. This scale is not an outcome of some conceptually highly questionable procedure like spontaneous compactification, which has paralyzed theoretical physics for more than two decades and led to the landscape problem and the proposal to bring anthropic principle to physics - something extremely uninviting for anyone who has spent few minutes by trying to understand what one can say about consciousness as a physicist and mathematician.

To my rebellionary view the mathematics of standard gauge theories is not enough.

1. Quite a far reaching generalization is needed besides the replacement of the recent view about space-time with the identification of space-times as 4-surfaces.

2. The usual positive energy ontology having its roots in Newtonian mechanics based on absolute time (Hamiltonian approach especially) must be replaced with zero energy ontology which is natural in the relativistic context.

3. A further generalization is number theoretical universality requiring that the physics in different number fields must be unified to single coherent whole.

4. In some of the latest postings I have explained how the number theoretical universality would be realized in terms of quantum arithmetics - something also missing from ordinary gauge theory but for the existence of which there exist indications (quantum groups, inclusions of hyper-finite factors, and Shnoll effect on experimental side). In particular, the size scales of CDs coming as powers of 2 correspond to p=2 quantum arithmetics, which is very special in the sense that for odd integers it is just the ordinary arithmetics. For other primes p the corresponding p-adic length scale is in preferred position since the states with mass squared proportional to p as almost massless states giving an almost pole to the propagator which is given in terms of M² momentum. I dare to hope that these observations finally answer the question why p-adic primes near powers of two are favored physically.

Further comments about mass gap

I cannot avoid the temptation to represent some further comments related to how the mass gap - or rather a hierarchy of mass gaps defined by p-adic mass scales in turn expressible in terms of p-adic prime and the fundamental mass scale defined by CP₂ mass - emerges in TGD.

1. One of the surprises of zero energy ontology was that that all - including those associated with virtual particles - braid strands carrying fermion numbers are on massless on mass shell states with possibly negative sign of energy so that wormhole contact can have space-like virtual net momentum. This leads to extremely powerful restrictions of loops integrals and guarantees finiteness and with certain additional natural assumptions deriving from ZEO the number of contributing diagrams is finite (discussed in recent postings: see this and this), and therefore guarantees algebraic universality (sum of infinite number of rationals (rational functions) need not be rational (rational function)!).

2. Quite recently I have learned finally to accept that for generalized Feynman diagrams the presence of preferred M²⊂M⁴ having interpretation in terms of quantization axis of energy and spin is unavoidable (see for instance this). Of course, also propagators for on mass shell massless states are literally infinite unless one restricts the momentum in propagator to its M² projection. There is integral over different choices of M² so that Poincare invariance is not lost. Also number theoretic vision forces M² with an interpretation as commutative subspace of complexified octonions. The last posting about Very Special Relativity and TGD gave one additional justification M².

3. Witten talks about 3-D gauge theories with emphasis on Chern-Simons term and the idea that in long length scales one obtains a non-trivial topological QFT for non-trivial mass gap. In TGD framework effective 2-dimensionality - or strong form of holography - follows from strong form of General Coordinate Invariance and for preferred extremals of Kähler action the action reduces to Chern-Simons terms if weak form of electric-magnetic duality holds true at the space-like 3-D surfaces at the ends of space-time sheet and at wormhole throats.

   The special features of light-like 3-surfaces and boundaries of CDs is that they allow an extension of 2-D conformal invariance by their metric 2-dimensionality: this actually raises 4-D space-time and 4-D Minkowski space in completely unique position mathematically. An extremely simple and profound discovery, whose communication has turned out to be impossible- I think that even my cat is able to understand its significance-: what is wrong with these bright-minded colleagues in their academies;-)? An interesting question raised by Witten's talk is whether also TGD as almost topological TGD in some sense reduces to topological QFT at long length scales for given p-adic length scale. Exponential decrease of correlation function as function of distance might imply this but what happens on light-like boundaries of CDs?

4. There are also open questions. For instance, should one assign different M² to each sub-CD of CD or to each propagator line connecting the 3-vertices? One can be even more general and also consider a local choices of M² defined by an integrable distribution of M^{2⊂ M4 defining the analog of string world sheet.}

The special role of M²⊂M⁴ in relation to mass gap

The special role of M²⊂M⁴ in the construction of generalized Feynman diagrams deserves additional comments.

1. What is remarkable that gauge conditions generalize in the sense that it is M² momentum that appears in gauge conditions so that also the third polarization for gauge bosons creeps into the spectrum and even photon, gluons, and gravitons would receive small mass given in terms of IR cutoff given by the largest causal diamond in the hierarchy of causal diamonds defining the experimental range of length scales about which experimentalist can gain information. This is of course extremely natural from the view point of experimentalist.

2. Physical particles are bound states of massless states with parallel M² momenta assigned with the wormhole throats of the same wormhole contact. Also this bring in the overall important IR cutoff - and thus mass gap - not present in gauge theories. UV cutoff given by the size of the smallest CD gives UV cutoff. As already mentioned, there is no breaking of Poincare invariance.

3. The highly non-trivial question is how the p-adic mass calculations can be consistent with the massless of braid strands. How can wormhole throat satisfy stringy mass formula if it is massless? One of the latest realizations is that that it is not the full M⁴mass squared but the longitudinal M² mass squared, which is quantized by the stringy mass formula! The modified Dirac equation indeed strongly suggests that M² momenta have integer valued components. I could not however decide whether only hyper-complex primes should be accepted: it now seems that integers coming as multiple of given hyper-complex integer, whose modulus square is prime, must be allowed. Particles would get longitudinal mass squared by p-adic thermodynamics and this mass would be the observed mass. Mass gap again but only in longitudinal degrees of freedom

4. There is also experimental support for the necessity of introducing M². In QCD one characterizes partons with M² momentum and this again brings to gauge theory as a purely mathematical construct - something which really is not there! The great experimental question is whether Higgs exists or not. In TGD Higgs mechanism is replaced by a microscopic mechanism based on p-adic thermodynamics and identification of the mass squared as longitudinal mass squared (in Lorentz invariance manner since one averages over different M²:s). The natural prediction is that instead of Higgs there is entire M⁸⁹ hadron physics to be discovered. If Higgs really is there as some bloggers have already revealed to us;-), profound re-interpretation of TGD is necessary.

New physics in non-perturbative sector

Asymptotically free gauge theories can handle the UV divergences by using renormalization group approach bringing in a scale analogous to QCD Lambda. Lambda defines the IR scale identified as the length scale associated with hadronization and confinement. One gets rid of UV scale altogether (but not from the mathematically tedious and ugly procedures removing UV infinities). In perturbative gauge theories IR remains however a source of difficulties since one really does not know how to calculate anything since the proposed expression for IR scale is non-analytic function of coupling constant strength (expressible in terms of exp(-8π²ℏ/g²), I hope I remember correctly). Also twistor approach is plagued by IR divergences. These difficulties are of course the reason for arranging a conference about mass gap problem! I do not however believe that the mass gap is a mathematical problem within the framework of 4-D gauge theories. One must go outside the system.

In TGD framework magnetic flux tubes are the concrete classical space-time correlate for non-perturbative aspects of quantum theory and appear in all applications from primordial cosmology to biology to elementary particle physics. They are not present in gauge theories. They are obtained as deformations of what I call cosmic strings, which are Cartesian products of string world sheets in M² with 2-D complex sub-manifolds of CP². In this case one cannot anymore speak about space-time as a small deformation of Minkowski space. The quantized size scale of the complex sub-manifolds brings in the scale via string tension. Wormhole contacts themselves are magnetic monopoles and thus homologically non-trivial surfaces of CP₂ with quantized area so that again the fundamental mass scale creeps in. Note that the Kähler action for the magnetic flux tubes and also for deformations CP₂ vacuum extremals contains a power of exp(-8π²ℏ/g²) giving rise to non-analyticity in g. In gauge theories classical action for instantons would give rise kind of factor.

Note: The address of my homepage has changed and the links to my homepage from the earlier postings will fail. The cure of the problem is the replacement of tgd.wippiespace or tgdq.wippiespace in the address with tgdtheory.