Does muon's anomalous anomalous magnetic moment imply new physics?

Lepton universality predicts that the magnetic moments of leptons should be the same apart from the corrections due to different masses. Leptons have besides the magnetic moment predicted by Dirac equation also anomalous magnetic moment which is predicted to come from various radiative corrections.

The standard model predictions for the anomalous magnetic moments of the electron are a_e= (g_e-2)/2= .00115965218091 and a_μ =(g_μ-2)/2= .00116591804.

The anomalous magnetic moments of electron and muon differ by .1 per cent. This breaking of universality is however due to the different masses of electron and muon rather than different interactions.

1. The finding of the Fermilab experiment

The breaking of universality could also come from interactions and the Fermilab experiment (see this) and earlier experiments suggest this. The experiment shows that in the case of muon the magnetic moment differs by from the predicted: the deviation from the standard model prediction is 2.5×10^-4 per cent. This indicates that there might be interactions violating the lepton universality. Besides the problem with the muon's magnetic moment, which differs from that of the electron, there is also a second problem. The decays of B mesons seem to break universality of fermion interactions: indications for the breaking of universality have emerged during years so that this is not new.

The measurement result involves various sources of error and one can estimate the probability that the measurement outcome is due to this kind of random fluctuations. The number of standard deviations tells how far the measurement result is from the maximum of the probability distribution. The deviation is expressed using standard deviation as a unit. Standard deviation is essentially the width of the distribution. For instance, 4 standard deviations tells that the probability that the result is random fluctuation is .6 per cent. For 5 standard deviations from predicted is .0001 per cent and is regarded as the discovery limit.

2. Theoretical uncertainties

There are also theoretical uncertainties related to the calculation of magnetic moment. There are 3 contributions: electroweak, QCD, and hadronic contributions. The electroweak and QCD corrections are "easily" calculable. The hadronic contributions are difficult to estimate since perturbative QCD does not apply at the hadronic energies. There are groups which claim that their estimation of hadronic contributions produces a prediction consistent with the Fermilab finding and the earlier findings consistent with the Fermilab finding.

The prediction based on experimentally deduced R ratio characterizing the rate for the decay of a virtual photon  to  a qquark pair allows to estimate the hadronic contribution and gives a prediction for hadronic contributions which is in conflict with experimental findings. On the other hand, the calculations based on lattice QCD give a result consistent with the experimental value (see this). Should one trust experiment or theory?

3. Is a wider perspective needed?

To my opinion, one should see the problem from a bigger perspective than a question about how accurate the standard model is.

1. Standard Model does not explain fermion families. Also GUTs fail in this respect: the mass ratios of fermions vary in the range spanned by 11 orders of magnitude. This is not a small gauge symmetry breaking but something totally different: mass scale is the appropriate notion and p-adic length scale hypothesis provides it.
2. One must also challenge the belief that lattice QCD can describe low energy hadron physics. There might be much deeper problems than the inability to compute hadronic contributions to g-2. Perturbative QCD describes only high energy interactions and QCD might exist only in the perturbative sense.The fact is that low energy hadron physics is virtually existent. Saying this aloud of course irritates lattice QCD professionals but the reduction of QCD to thermodynamics in the Euclidian space-time looks to me implausible. There are deep problems with Wick rotation.

   For instance, massless dispersion relation E²-p²= 0 in M⁴ translates to E²+p² =0 in E⁴: massless fields disappear completely since one has only E=0,p=0 zero mode. There are similar problems with the massless Dirac equation. For the massive case the situation is not so bad as this. There is the strong CP problem caused by instantons and a problem with multiplication of spinor degrees of freedom since the 4-D cube has the topology of 4-torus and allows 16 spinor structures.

   Quarks explain only a few per cent of hadron mass just as ordinary matter explains only a few percent of mass in cosmology. Hadron physics might therefore involve something totally new and color interaction could differ from a genuine gauge interaction.

   4. What TGD can say about family replication phenomenon?

   In TGD framework, the topological explanation of family replication phenomenon identifying partonic 2-surfaces as fundamental building blocks of elementary particles provides the needed understanding and predicts 3 different fermion generations corresponding to 3 lowest general: sphere, torus, and sphere with two handles (see this).

   Conformal Z₂ symmetry for partonic 2-surfaces is present for the lowest 3 genera but not for the higher ones for which one must talk about many handle states with continuous mass spectrum. p-Adic thermodynamics allows to estimate the masses of new boson by simple scaling arguments and Mersenne prime hypothesis.

   In the TGD framework the two findings can be seen as indications for the failure of lepton universality. Besides 3 light fermion generations TGD also predicts 3 light generations for electroweak bosons, gluons, and Higgs. These generations are more massive than weak bosons and p-adic length scale hypothesis also allows to estimate their masses.

   The couplings of the lightest generations to the gauge bosons obey fermion universality (are identical) but the couplings of the 2 higher generations cannot do so since the charge matrices of 3 generations must be orthogonal to each other. This predicts breaking of fermion universality which in quantum field theory approximation comes from the loops coupling fermions to the 2 higher boson generations.

   This prediction is a test for TGD based topological view about family replication phenomenon in terms of the genus of partonic 2-surface: partonic 2-surface can be sphere, torus or sphere with two handles. TGD also explains why higher generations are experimentally absent.

   5. What does TGD say about low energy hadron physics?

   There is also the question about whether QCD catches all aspects of strong interactions. In TGD color magnetic flux tubes carry Kaehler magnetic energy and volume energy parametrized by length scale dependent cosmological constant so that a connection with cosmology indeed emerges. The reconnections of U-shaped flux tubes give rise to the TGD counterparts of meson exchanges of old-fashioned hadron physics. See this .

   Color group need not be a gauge group but analogous to a Kac-Moody group or Yangian group (only non-negative conformal weights). In TGD framework SU(3) at the level of M⁴xCP₂ is not a gauge symmetry but acts as isometries of CP₂ and fermions do not carry color as analog of spin but as angular momentum like quantum number. At the level of compelexified M₈ SU(3) is a subgroup of G₂ acting as octonion automorphism and defines Yangian replacing the local gauge group.

   For the TGD based model see this and this.

   For a summary of earlier postings see Latest progress in TGD.

   Articles and other material related to TGD.

EEG and the structure of magnetosphere

Roughly 15 years  ago I proposed the idea that Earth's  magnetosphere (MS) could serve as a sensory canvas in the sense that biological systems, in particular the vertebrate brain, could have sensory representations realized at the "personal" magnetic body (MB)  closely associated with the MS of the Earth. EEG would make communications to  and control by MB possible. 

 At that time I did not yet  have  the idea about number theoretical realization of the  hierarchy of Planck constants h_eff=nh₀ in the framework of adelic physics fusing the physics of sensory experience and cognition. This hierarchy is crucial for understanding the basic aspects of living matter such as metabolism, coherence in long scales, correlates of cognition, and even evolution.

Also the concept of zero energy ontology (ZEO) forming now the basis of the quantum TGD was missing although there was already the about communication to past using negative energy signals. ZEO is now in a central role in the understanding of self-organization - not only the biological one. The new view about time predicting that time reversal occurs in ordinary state function reductions (SFRs) allows to understand homeostasis as self-organized quantum criticality. 

For these reasons it is interesting to consider the notion of sensory canvas from the new perspective. This article discusses besides  the earlier ideas about the MS  also the proposal that it is possible to associate EEG bands to the regions of MS via the correspondence between EEG   frequency with the distance of the region from Earth.   Also the idea  that the structure of MS could be a  fractal analog of the vertebrate body is tested quantitatively by comparing various scales involved.

See the article EEG and the structure of magnetosphere or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD . 


Three alternative generalizations of Nottale's hypothesis in TGD framework

Nottale's gravitational Planc constant ℏ_gr= GMm/v₀ contains  the velocity parameter v₀ as the only parameter. In the perturbative expansion  of  the scattering amplitudes β₀=v₀/c appears  in the role of fine structure constant.    

There is however a problem.

1. The model  for the effects of ELF radiation on vertebrate brain  inspired by  a generalization of Nottale's hypothesis by replacing the total mass M in the case of Earth by M_D≈ 10^-4M_E suggests that in this case the dark particles involved couple only to a part of mass identifiable as dark mass M_D.
2.   Since only GM appears in the basic formulas, the  alternative option is that the value of G is reduced to G_D. This conforms with the fact that in the  TGD framework CP₂ length is the fundamental parameter  G is a prediction of the theory and therefore can vary. 
3. A further option is that the parameter β₀=v₀/c≤ 1 is variable and equals to β₀=1 or to a value not much smaller than 1, say β₀=1/2.

These three options are  critically discussed and compared. The cautious conclusion is that the the third option is the most plausible one.

See the article Three alternative generalizations of Nottale's hypothesis in TGD framework or the chapter About the Nottale's formula for h_gr and the relation between Planck length and CP₂ length.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD. 


Does Goedel's incompleteness theorem hold true for reals?

I have many times wondered whether the incompleteness theorem extends to real numbers, which are usually the stuff used by physics. There is a very nice discussion of this point here. Strongly recommended.

Real numbers and all algebraically closed number fields such as complex numbers and algebraic numbers are complete. All truths are provable. If physics is based on complex numbers or algebraic numbers, Goedel's theorem has no direct implications for physics.This however implies that integers cannot be characterized using the axiomatics of these number fields since if this were the case, Gdel's incompleteness theorem would not hold true for integer arithmetics. One can also say that Goedel numbers for unprovable theorems are not expressible as a natural number but are more general reals or complex numbers.

Since algebraic numbers are complete, a good guess is that algebraic numbers label all true statements about integer arithmetics and also about arithmetics of algebraic integers for extensions of rationals.

In TGD adelic physics definescorrelates for cognition. Adeles for the hierarchy labelled by algebraic extensions (perhaps also extensions involving roots of e since e^p is p-adic number). These are not complete and Goedel's incompleteness theorem applies to them. Only at the never achievable limit of algebraic numbers the system becomes complete. This would strongly suggest a generalization of Turing's view about computation by replacing integer arithmetics with a hierarchy of arithmetics of algebraic integers associated with extensions of rationals. See this article .

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Does the quantal gravitational force vanish below critical radius in average sense?

Nottale's gravitational constant hbar_gr= GM_Dm/v₀ contains dark mass M_D as a parameter. At the surface of Earth M_D much smaller than M_D and for the planets  one has M_D=M_Sun. It turns out that in the  average sense  M_D must grow to M.   This is  required by  the condition that  Bohr radii correspond to the classical radii in the average sense. The actual dependence of M_D on r  is expected to  be a staircase like function.

At the quantum level, this   effectively eliminates  the average  gravitational force in the scales below the critical radius r_cr above  which M_D=M is true.   Indeed, due to the average M_D∝ r dependence,  gravitational potential would be constant on the average. 

 Could one regard this   effective elimination of  the gravitational force as a kind of    Quantum Equivalence Principle or   as an analog of asymptotic freedom?

See the article Two alternative generalizations of Nottale's hypothesis or the chapter About the Nottale's formula for h_gr and the relation between Planck length and CP₂ length.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

The idea of Connes about inherent time evolution of certain algebraic structures from TGD point of view

Alan Connes has proposed that certain mathematical structures known as hyperfinite factors contain in their structure inherent time evolution.This time evolution is determined only modulo unitary automorphism analogous to a time evolution determined by Hamiltonian so that this time evolution seems to be too general for the purposes of a physicist.

Zero energy ontology of TGD combined with adelic physics leads to a vision that the sequences of state function reductions implies a mathematical evolution in the sense that the extensions of rationals characterizing the space-time region increases gradually. This induces the increase of algebraic complexity implying time evolution as the analog of biological evolution.

The dimension of extension corresponds to an effective Planck constant assumed to label dark matter as phases of ordinary matter. Therefore quantum coherence lengths increase in this evolution.

This generalization of the idea of Connes is discussed in the framework provided by the recent view about TGD. In particular, the inclusion hierarchies of hyper-finite factors, the extension hierarchies of rationals, and fractal inclusion hierarchies of subalgebras of supersymplectic algebra isomorphic with the entire algebra are proposed to be more or less one and the same thing in TGD framework.

See the article The idea of Connes about inherent time evolution of certain algebraic structures from TGD point of view.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Pomeron and Odderon as direct support for the notion of color magnetic body

The following comments were inspired by a popular article"> telling about the empirical support for a particle christened Odderon. As the name tells, Odderon is not well-understood in QCD framework.

Odderon is a cousin of Pomeron which emerged already about half century ago in the so called Regge theory to explain the logarithmically rising (rather than decreasing) cross sections in proton-proton and proton-antiproton collisions. Pomeron is part of low energy phenomenology and perturbative QCD cannot say much about it. Since the charge parity is C=1 for Pomeron C=-1 for Odderon, these states are analogous to pion with spin 0 and ρ meson with spin 1.

Pomeron and Odderon have not been in the interests of the frontier of theoretical physics: they represent for an M-theorist a totally uninteresting and primitive low energy phenomenology - as all that we used to call physics before the first superstring revolution -, and does not therefore deserve the attention of an ambitious superstring theorist more interested in the marvels of brane words, landscape, swampland, and multiverse.

I have written about Pomeron for years ago. The following is something different since the view about low energy strong interactions according to TGD (see this) has developed considerably (see for instance this and this)

One can go first to Wikipedia to learn about Pomeron.

1. Pomeron exchange in the t-channel was postulated to explain the slowly (logarithmically) rising scattering cross sections in proton-proton and proton-antiproton collisions. For quarks and gluons the scattering cross sections fall down rather rapidly with energy (by dimensional argument like inverse 1/s of cm energy squared) so that something else would be in question.
2. The cross sections did not depend on the charges of the colliding baryons. The usual shower of Cerenkov radiation was missing from Pomeron exchange events. The absence of pions usually present was interpreted as absence of color charge and therefore. This suggests that quarks and gluons do not participate the Pomeron events. There is often also a large rapidity gap in which no outgoing particles are observed.
3. In the Regge theory which later was concretized in terms of the hadronic string model. Pomeron would correspond to a Regge trajectory for which the Reggeon would have quantum numbers of vacuum except for mass and angular momentum. Regge trajectory would satisfy the formula M²= M₀² =α(s) J, M mass, J angular momentum. Odderon would be Pomeron like state with an odd charge parity C=-1 instead of C=1.
4. In the QCD picture Pomeron and Odderon are assumed to be associated with the gluonic exchanges. Pomeron would be a many-gluon state.

In the many-sheeted space-time of TGD, hadrons are many-sheeted objects.

1. There is a hadronic space-time sheet and quark and gluon space-time sheets are glued to this. There is a magnetic body (MB) of hadron having a layered structure. In particular, there are em/color/weak MBs consisting of flux tubes and "tentacles", which are U-shaped flux tubes.

   Low energy hadron physics would be described in terms of reconnections of these tentacles. This is a rather new element in the picture. In a reasonable approximation, flux tubes are strings and the reconnection of closed strings appears as a basic reaction vertex for closed strings. This gives a connection with the hadronic string model. TGD indeed emerged as a generalization of the hadronic string model 43 years ago (and also as a solution of the energy problem of GRT).

2. Most of the energy of hadron is assumed to be carried by color MB: quarks and gluons carry only a small part of energy. In QCD space-time dynamics is not present and the analog of hadron as space-time surfaces would be a gluon condensate of some kind.
3. Low energy hadron reactions would consist of reconnections of the U-shaped flux tubes of the colliding color MBs. Besides this there are also the collisions of quarks and gluons having approximate description in terms of QCD. The already mentioned connection with hadronic string model suggests a connection with Regge and string model descriptions of Pomeron/Odderon.
4. Hadrons have U-shaped flux tubes acting like tentacles and reconnect to form a bridge of two flux tubes between colliding hadrons. This topological interaction mechanism would be universal and occur in all scales. In biology the ability of reacting biomolecules to magically find each other in the dense molecular soup would rely on this mechanism. It would be also a mechanism of high Tc - and biological superconductivity.

Could this explain the basic properties of the Pomeron?

1. Charge independence and absence of pion emission assignable to quark-gluon reactions can be understood. Gluons and quarks of colliding hadrons would not meet each other at all. The two colliding hadrons would just touch each other with their "tentacles" which would transfer some momentum between them in elastic collisions. This would explain the rapidity gap.
2. What about the slow dependence on collision energy? Why the cross section describing the probability of the formation of reconnection would not depend on collision energy?
   1. One could visualize the cross section in cm frame geometrically as the area of a 2-D surface cylinder parallel to the line connecting the colliding particles. The area of this cylinder would tell the probability for the formations of reconnection. If I try to touch some object in darkness, its area tells how probable the success is.
   2. In elastic scattering the t-channel momentum exchange would be orthogonal to this cylinder and have vanishing energy component. It would not change in Lorentz boosts increasing the cm collision energy. If the contribution to the cross section depends only on t, it would be independent of collision energy.

The TGD view about this finding is described in the article Some unexpected findings in hadron and nuclear physics from TGD point of view and in a chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

MeshCODE theory from TGD point of view

Benjamin Goult has made an interesting proposal in an article The Mechanical Basis of Memory the MeshCODE Theory published in Frontiers of Molecular Neuroscience (see this).

The proposal is that the cell - or at least synaptic contacts - realize mechanical computation in terms of adhesive structures consisting of hundreds of proteins known as talins, which act as force sensors. Talins are connected to integrins in the extracellular matrix, to each other, and to the actins in the cell interior. This proposal has far reaching consequences for understanding formation of memomies as behaviors at the synaptic level.

This proposal does not conform with the TGD vision but inspires a series of questions leading to a rather detailed general vision for how magnetic body (MB) receives sensory input from biological body (BB) coded into dark 3N-photons a representing genes with N codons and as a response activates corresponding genes, RNA or proteins as a reaction. Sensory input and the response to it would be coded by the same dark genes.

See the article MeshCODE theory from TGD point of view or the chapter An Overall View about Models of Genetic Code and Bio-harmony.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD. 


Zero energy states as scattering amplitudes and subjective time evolution as sequence of SSFRs

Zero energy states code for the ordinary time evolution in the QFT sense described by the S-matrix. Construction of zero energy is reasonably well understood (see this, this, and this ).

This is not yet the whole story. One should also understand the subjective time evolution defined by a sequence of "small" state function reductions (SSFRs) as analogs of "weak" measurements followed now and then by BSFRs. How does the subjective time evolution fit with the QFT picture in which single particle zero energy states are planewaves associated with a fixed CD?

1. The size of CD increases at least in statistical sense during the sequence of SSFRs. This increase cannot correspond to M⁴ time translation in the sense of QFTs. Single unitary step followed by SSFR can be identified asa scaling of CD leaving the passive boundary of the CD invariant. One can assume a formation of an intermediate state which is quantum superposition over different size scales of CD: SSFR means localization selecting single size for CD. The subjective time evolution would correspond to a sequence of scalings of CD.
2. The view about subjective time evolution conforms with the picture of string models in which the Lorentz invariant scaling generator L₀ takes the role of Hamiltonian identifiable in terms of mass squared operator allowing to overcome the problems with Poincare invariance. This view about subjective time evolution also conforms with super-symplectic and Kac-Moody symmetries of TGD.

   One could perhaps say that the Minkowski time T as distance between the tips of CDs corresponds to exponentiated scaling: T= exp(L₀t). If t has constant ticks, the ticks of T increase exponentially.

The precise dynamics of the unitary time evolutions preceding SSFRs has remained open.

1. The intuitive picture that the scalings of CDs gradually reveal the entire 4-surface determined by polynomial P in M⁸: the roots of P as "very special moments in the life of self" would correspond to the values of time coordinate for which SSFRs occur as one new root emerges. These moments as roots of the polynomialdefining the space-time surface would correspond to scalings of the size of both half-cones for which the space-time surfaces are mirror images. Only the upper half-cone would be dynamical in the sense that mental images as sub-CDs appear at "geometric now" and drift to the geometric future.
2. The scaling for the size of CD does not affect the momenta associated with fermions at the points of cognitive representation in X⁴⊂ M⁸ so that the scaling is not a genuine scaling of M⁴ coordinates which does not commute with momenta. Also the fact that L₀ for super symplectic representations corresponds to mass squared operator means that it commutes with Poincare algebra so that M⁴ scaling cannot be in question.
3. The Hamiltonian defining the time evolution preceding SSFR could correspond to an exponentiation of the sum of the generators L₀ for super-symplectic and super-Kac Moody representations and the parameter t in exponential corresponds to the scaling of CD assignable to the replaced of root r_n with root r_n+1 as value of M⁴ linear time (or energy in M⁸). L₀ has a natural representation at light cone boundaries of CD as scalings of light-like radial coordinate.
4. Does the unitary evolution create a superposition over all over all scalings of CD and does SSFR measure the scale parameter and select just a single CD?

   Ordoes the time evolution correspond to scaling? Is it perhaps determined by the increase of CD from the size determinedby the root r_n as "geometric now" to the root r_n+1 so that one would have a complete analogy with Hamiltonian evolution? The scaling would be the ratio r_n+1/r_n which is an algebraic number.

   Hamiltonian time evolution is certainly the simplest option and predicts a fixed arrow of time during SSFR sequence. L₀ identifiable essentially as a mass squared operator acts like conjugate for the logarithm of the logarithm of light-cone proper time for a given half-cone.

   One can assume that L₀ as the sum of generators associated with upper and lower half-cones if the fixed state at the lower half-cone is eigenstate of L₀ not affect in time evolution by SSFRs.

How does this picture relate to p-adic thermodynamics in which thermodynamics is determined by partition function which would in real sector be regarded as a vacuum expectation value of an exponential exp(iL₀t) of a Hamiltonian for imaginary time t=iβ β=1/T defined by temperature? Here L₀ is proportional to mass squared operator.

1. In p-adic thermodynamics temperature T is dimensionless parameter and β=1/T is integer valued. The partition function as exponential exp(-H/T) is replaced with p^{β L₀)}, β=n, which has the desired behavior if L₀ has integer spectrum. The exponential form e^L₀/T_R), β_R= nlog(p) equivalent in the real sector does not make sense p-adically since the p-adic exponential function has p-adic norm 1 if it exists p-adically.
2. The time evolution operator exp(-iL₀t) for SSFRs (t would be the scaling parameter) makes sense for the extensions of p-adic numbers if the phase factors for eigenstates are roots of unity belonging to the extension. t= 2π k/n since L₀ has integer spectrum. SSFRs would define a clock. The scalingexp(t)= exp(2π k/n) is however not consistent with the scaling by r_n-1/r_n.

   Both the temperature and scaling parameter for time evolution by SSFRs would be quantized by number theoretical universality. p-Adic thermodynamics could have its origins in the subjective time evolution by SSFRs.

3. In the standard thermodynamics it is possible to unify temperature and time by introducing a complex time variable \tau = t+iβ, where β=1/T is inverse temperature. For the space-time surface in complexified M⁸, M⁴ time is complex and the real projection defines the 4-surface mapped to H. Could thermodynamics correspond to the imaginary part of the time coordinate?

   Could one unify thermodynamics and quantum theory as I have indeed proposed: this proposal states that quantum TGD can be seen as a "complex square root" of thermodynamics. The exponentials U=exp(\tau L₀/2) would define this complex square root and thermo-dynamical partition function would be given by UU^†= exp(-β L₀).

See the article Is M⁸-H duality consistent with Fourier analysis at the level of M⁴× CP₂?.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.