Sunday, August 05, 2012

Pseudo-scalar Higgs as Euclidian pion?

The observations of previous postings (see this and this) and earlier work suggest that pion field in TGD framework is analogous to Higgs field.

This raises questions. Assuming that QFT in M4 is a reasonable approximation, does a modification of standard model Higgs mechanism allow to approximate TGD description? What aspects of Higgs mechanism remain intact when Higgs is replaced with pseudo-scalar? Those assignable to electro-weak bosons? The key idea allowing to answer these questions is that "Higgsy" pion and ordinary M89 pion are not one and the same thing: the first one corresponds to Euclidian flux tube and the latter one to Minkowskian flux tube. Hegel would say that one begins with thesis about Higgs, represents anti-thesis replacing Higgs with pion, and ends up with a synthesis in which Higgs is transformed to pseudo-scalar Higgs, "Higgsy" pion, or Higgs like state if you wish! Higgs certainly loses its key role in the massivation of fermions.

Can one assume that M4 QFT limit exists?

The above approach assumes implicitly - as all comparisons of TGD with experiment - that M4 QFT limit of TGD exists. The analysis of the assumptions involved with this limit helps also to understand what happens in generation of "Higgsy" pions.

  1. QFT limit involves the assumption that quantum fields and also classical fields superpose in linear approximation. This is certainly not true at given space-time sheet since the number of field like is only four by General Coordinate Invariance. The resolution of the problem is simple: only the effects of fields carried by space-time sheets superpose and this takes place in multiple topological condensation of the particle on several space-time sheets simultaneously. Therefore M4 QFT limit can make sense only for many-sheeted space-time.

  2. The light-like 3-surfaces representing lines of Feynman graphs effectively reduce to braid strands and are just at the light-like boundary between Minkowskian and Euclidian regions so that the fermions at braid strands can experience the presence of the instanton density also in the more fundamental description. The constancy of the instanton density can hold true in a good approximation at braid strands. Certainly the M4 QFT limit treats Euclidian regions as 1-dimensional lines so that instanton density is replaced with its average.

  3. In particular, the instanton density can be non-vanishing for M4 limit since E and B at different space-time sheets can superpose at QFT limit although only their effects superpose in the microscopic theory. At given space-time sheet I can be non-vanishing only in Euclidian regions representing lines of generalized Feynman graphs.

  4. The mechanism leading to the creation of pion like states is assumed to be the presence of strong non-orthogonal electric and magnetic fields accompanying colliding charged particles (see this): this of course in M4 QFT approximation. Microscopically this corresponds to the presence of separate space-time sheets for the colliding particles. The generation of "Higgsy" pion condensate or pion like states must involve formation of wormhole contacts representing the "Higgsy" pions. These wormhole contacts must connect the space-time sheets containing strong electric and magnetic fields.

Higgs like pseudo-scalar as Euclidian pion?

The recent view about the construction of preferred extremals predicts that in Minkowskian space-time regions the CP2 projection is at most 3-D. In Euclidian regions M4 projection satisfies similar condition. As a consequence, the instanton density vanishes in Minkowskian regions and pion can generate vacuum expectation only in Euclidian regions. Long Minkowskian flux tubes connecting wormhole contacts would correspond to pion like states and short Euclidian flux tubes connecting opposite wormhole throats to "Higgsy" pions.

  1. If pseudo-scalar pion like state develops a vacuum expectation value the QFT limit, it provides weak gauge bosons with longitudinal components just as in the case of ordinary Higgs mechanism. Pseudo-scalar boson vacuum expectation contributes to the masses of weak bosons and predicts correctly the ratio of W and Z masses. If p-adic thermodynamics gives a contribution to weak boson masses it must be small as observed already earlier. Higgs like pion cannot give dominant contributions to fermion masses but small radiative correction to fermion masses are possible.

    Photon would be massless in 4-D sense unlike weak bosons. If ZEO picture is correct, photon would have small longitudinal mass and should have a third polarization. One must of course remain critical concerning the proposal that longitudinal M2 momentum replaces momentum in gauge conditions. Certainly only longitudinal momentum can appear in propagators.

  2. If 3 components of Euclidian pion are eaten by weak gauge bosons, only single neutral pion-like state remain. This is not a problem if ordinary pion corresponds to Minkowskian flux tube. Accordingly, the 126 GeV boson would correspond to the remaining component Higgs like Euclidian pion and the boson with mass around 140 GeV for which CDF has provided some evidence to the Minkowskian M89 pion (see this) and which might have also shown itself in dark matter searches (see this and this).

  3. By the previous construction one can consider two candidates for pion like pseudo-scalars as states whose form apart from parallel translation factor is ‾Ψ1 jAkγkΨ2. Here jA is generator of color isometry either in U(2) sub-algebra or its complement. The state in U(2) algebra transforms as 3+1 under U(2) and the state in its complement like 2+2* under U(2).

    These states are analogous of CP2 polarizations, whose number can be at most four. One must select either of these polarization basis. 2+2* is an unique candidate for the Higgs like pion and can be be naturally assigned with the Euclidian regions having Hermitian structure. 3+1 in turn can be assigned naturally to Minkowskian regions having Hamilton-Jacobi structure.

    Ordinary pion has however only three components. If one takes seriously the construction of preferred extremals the solution of the problem is simple: CP2 projection is at most 3-dimensional so that only 3 polarizations in CP2 direction are possible and only the triplet remains. This corresponds exactly to what happens in sigma model combining describing pion field as field having values at 3-sphere.

  4. Minkowskian and Euclidian signatures correspond naturally to the decompositions 3+1 and 2+2*, which could be assigned to quaternionic and co-quaternionic subspaces of SU(3) Lie algebra or imbedding space with tangent vectors realized in terms of the octonionic representation of gamma matrices.
One can proceed further by making objections.
  1. What about kaon, which has a natural 2+2* composition but can be also understood as 3+1 state? Is kaon is Euclidian pion which has not suffered Higgs mechanism? Kaons consists of usbar, dsbar and their antiparticles. Could this non-diagonal character of kaon states explain why all four states are possible? Or could kaon corresponds to Minkowskian triplet plus singlet remaining from the Euclidian variant of kaon? If so, then neutral kaons having very nearly the same mass - so called short lived and long lived kaons - would correspond to Minkowskian and Euclidian variants of kaon. Why the masses if these states should be so near each other? Could this relate closely to CP breaking for non-diagonal mesons involving mixing of Euclidian and Minkowskian neutral kaons? Why CP symmetry requires mass degeneracy?

  2. Are also M107 electroweak gauge bosons? Could they correspond to dark variant of electro-weak bosons with non-standard value of Planck constant? This would predict the existence of additional - possibly dark - pion-like state lighter than ordinary pion. The Euclidian neutral pion would have mass about (125/140)× 135 ∼ 125 MeV from scaling argument. Interestingly, there is evidence for satellites of pion: they include also state which are lighter than pion. The reported masses of these states would be M = 62, 80, 100, 181, 198, 215, 227.5, and 235 MeV. 125 MeV state is not included. The interpretation of these states is as IR Regge trajectories in TGD framework.

How the vacuum expectation of the pseudo-scalar pion is generated?

Euclidian regions have 4-D CP2 projection so that the instanton density is non-vanishing and Euclidian pion generates vacuum expectation. In the following an attempt to understand details of this process is made using the unique Higgs potential consistent with conformal invariance.

  1. One should realize the linear coupling of Higgs like pion to instanton density. The problem is that Tr(F^Fπ) since π does not make sense as such since π is defined in terms of gamma matrices of CP2 and F in terms of sigma matrices. One can however map gamma matrices to sigma matrices in a natural manner by using the quaternionic structure of CP2. γ0 corresponding to e0 is mapped to unit matrix and γi to the corresponding sigma matrix: γi→ εijkσjk. This map is natural for the quaternionic representation of gamma matrices. What is crucial is the dimension D=4 of CP2 and the fact that it has U(2) holonomy.


  2. Vacuum expectation value derives from the linear coupling of pion to instanton density. If instanton density is purely electromagnetic, one obtains correct pseudo-scalar Higgs vacuum expectation commuting with photon.


  3. If the action density contains only the mass term m2π/2 plus instanton term 1/32π2fππ I,
    where I is the instanton density, one obtains the standard PCAC relation between the vacuum expectation of the pion field and instanton density.

    π0= (1/32π2fπmπ2) I .

    This relation appears also in the model for leptopion production. In the standard model the mass term must be tachyonic. This leads to so called hierarchy problem hierarchy problem. The source of the problem are the couplings of Higgs to fermions proportional to the mass of fermion. The radiative corrections to Higgs mass squared are positive and proportional to fermion mass so that top quark gives the dominating contribution. This implies that the sign of the mass squared can become positive and the state with vanishing vacuum expectation value of Higgs field becomes the ground state. In the recent situation this is not a problem since fermions couple to the pion like state only radiatively.

  4. When one adds to the fields appearing in the classical instanton term the quantum counter parts of electroweak gauge fields, one obtains an action giving rise to the anomaly term inducing the anomalous decays to gamma pairs and also other weak boson pairs. The relative phase between the instanton term and kinetic term of pion like state is highly relevant to the decay rate. If the relative phase corresponds to imaginary unit then the rate is just the sum of the anomalous and non-anomalous rates since interference is absent.

What is the window to M89 hadron physics?

Concerning the experimental testing of the theory one should have a clear answer to the question concerning the window to M89 hadron physics. One can imagine several alternative windows.

  1. Two gluon states transforming to M89 gluons could be one possibility proposed earlier. The model contains a dimensional parameter characterizing the amplitude for the transformation of M107 gluon to M89 gluon. Dimensional parameters are not however well-come.

  2. Instanton density as the portal to new hadron physics would be second option but works only in the Euclidian signature. One can however argue that M89 Euclidian pions represent just electroweak physics and cannot act as a portal.

  3. Electroweak gauge bosons correspond to closed flux tubes decomposing to long and short parts. Two short flux tubes associated with the two wormhole contacts connecting the opposite throats define the "Higgsy" pions. Two long flux tubes connect two wormhole contacts at distance of order weak length scale and define M89 pions and mesons in the more general case. In the case of weak bosons the second end of long flux tube contains neutrino pair neutralizing the weak isospin so that the range of weak interactions is given the length of the long flux tube. For M89 the weak isospins at the ends need not sum up to zero and also other states that neutrino pair are allowed, in particular single fermion states. This allows an interpretation as electroweak "de-confinement" transition producing M89 mesons and possibly also baryons. This kind of transition would be rather natural and would not requite any specific mechanisms.

For a TGD based discussion of the general theoretical background for Higgs and possible TGD inspired interpretation of the new particle as pionlike state of scaled variant of hadron physics see Is it really Higgs?. See also the chapter " New particle physics predicted by TGD: Part I of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".

19 Comments:

At 5:25 AM, Blogger hamed said...

Dear Matti,
Sorry for delaying, I must be more here in the blog. These months is not repeated anytime!
I congratulate you for your new understandings on M89 physics. I have best wishes for this.
I have a methodological question! : My world view as I deduce from TGD! say to me that it is not a good method when most of physicists rely on the huge laboratories like as LHC for their theories. But for the current theories, it is only method! Every time that a theory become much richer, it’s need to experiments is lesser. Finally it is any need to experiment at all and in really direct observation is enough?(what is your opinion?)
In the days I am struggling with testi.pdf. The process as you noted in the testi.pdf containing some preliminaries like understanding homotopy and covering group and after it spinor bundle and Stiefel-Whitney class and some basic things about relations between octonions and dirac spinors. I am between the process and it would be very sweet for me to understanding them ;-).
Until now, Despite TGD concepts, also learning the preliminaries of TGD was very useful for me to improve my physical and mathematical intuitions :-) and I really thank you for this.

 
At 8:16 AM, Blogger Ulla said...

http://arxiv.org/abs/0705.4160
http://www3.imperial.ac.uk/ccm/research/edm/overview

and my modest contribution http://zone-reflex.blogspot.fi/2011/02/edm-reveals-antimatter-without-mssm-and.html

There are other ways into the problem, but LHC has got so much attention that the other are forgotten?
Take also a look at the muonic hydrogen.

http://physicsforme.wordpress.com/2012/08/05/the-beauty-of-the-higgs-boson/ Where did the asymmetry go? the action principle? the mass? The Higgs boson should explain the massivation and skewness of 4,5% of total? It is quite peculiar that the massivation is exactly that percent too.

 
At 8:56 AM, Anonymous Matti Pitkanen said...

Dear Hamed,

I do not believe on the possibility to build develop theory without a close contact with the experimental world. It is of course impossible to specialize to what people in LHC modelling the collisions are doing. Just the basic understanding about what we know is enough. One must become aware about the big problems. A course about basic problems and anomalies of modern physics should basic stuff in universities.

Every step of progress in TGD has is due to taking seriously some aspect of theory related to the experiment. For instance, construction of quantum TGD involved the realization that the definition of Kahler geometry must define classical physics as exact part of quantum theory. It took five years to realize this;-).

Now it was the realization that the Higgs like behavior of the new particle in decays to ew boson pairs must be taken seriously. This led to as such trivial realization that pseudoscalar in Higgs mechanism behaves in many respects just like scalar but in in some very improtant aspects differently. It does not explain fermion masses but p-adic mass calculations take care of this. Higgs mechanism is also realized differently: in terms of linear instanton density coupling. This allows avoid instability of vacuum due to the loss of conformal invariance caused by the tachyonic mass of Higgs: this instability is the fundamental problem of standard model and the hope was that standard SUSY could solve it.

Experimental thinking is also kind of dual to mathematical thinking. Together they give very powerful tools. Theory alone leads to dead formalisms and physics problems are replaced with theory problems. Landscape difficulty in string theory is a school example about this.

My worst fear is that some-one takes TGD too seriously;-). Critical attitude is the most fruitful one. TGD is just one attempt to build an overview and work of single person forced to work in isolation and therefore must contain flaws and mis-interpretations. It cannot yet provide mechanical calculational recipes and there are very many open interpretational questions and conjectures to be proved or disproved.

 
At 11:34 AM, Blogger Ulla said...

http://www.optical-lattice.com/index.php?lattice-site=cooper-pairs

bosonic cooper pairs?

http://greiner.physics.harvard.edu/QGM.html

 
At 11:47 AM, Blogger hamed said...

Thanks. You are right, so maybe i should repair my worldview ;-)

 
At 8:18 PM, Anonymous Matti Pitkanen said...

Dear Hamed,

I hope that I would not sound like a preacher;-). It is wonderful that we have gigantic experiments like LHC and that there are colleagues who have enormous knowledge about some special area. The information about new particle is precious also for me.

There is however a lot of overall important knowledge which does not require LHC. For a theoretician trying to build overall view also this knowledge is very important. One must also know what one does not know! These white areas tell in which direction one can deform existing world view.

Proton stability is an example about enormously important data bit, which is forgotten. If one takes it seriously, one can forget entire GUT philosophy, standard SUSY and and superstrings. TGD is a candidate for this kind of new particle theory. There are also various anomalies forgotten during years: for instance the anomalies to which I assign leptohadron physics.

Inflation is one example: here theoretical difficulties have been present from the beginning and the problem is that theoreticians assume quite too much. What is really known is flatness of 3-space! Quantum criticality makes the same prediction and leads to quite different view. The generation of ordinary matter as decay products of inflaton field is second key assumption: Idea is beautiful but if Higgs is not there, we can forget also inflaton. In TGD this mechanism it is replaced with decay of magnetic fields associated with magnetic flux quanta: this also explains dark energy.

Biology is of course one of the most important anomalies. No one can claim of really understanding biochemistry- it is a compete mystery for standard physics. And there is also dark matter, which could relate to biology. Again unnecessary strong assumptions are made about its character. For instance, the identification as lightest supersymmetric particle has turned out to be wrong. Just the assumption that dark and ordinary particles cannot appear in the same vertex of Feynman graph might be enough.

There is also belief that quantum theory is something final although von Neuman already at thirties demonstrated that quantum theory allows 3 different mathematical frameworks. Quantum measurement theory is one the saddest examples about areas of theoretical physics, where thinking is explicitly forbidden.

To my view this kind of simple knowledge about physical and also knowledge about lack of knowledge are important for a theoretician who really wants to understand.

There is of course the question about career: we do not live in a world were truth would be the value number number one! Just few days ago I was censored out from a finnish discussion group relating to physics - the only one in Finland actually and in principle allowed to any-one. The reason was that the young fellow moderating the messages knows quite well that certain influental colleagues do not like at all that I am allowed to talk publicly! He must think of his career!

 
At 6:42 AM, Blogger hamed said...

Dear Matti,

It would not sound like a preacher but I wish that see you as a preacher sometimes ;-), you are more than enough humble!
I become very sad when I see your position in Finland. If I was there, don’t let them these behaviors ;-)
It is very hard imaginable for me that they without reading your theory, censored you. Scientific world is very sick for this and i am sorry for that.

 
At 10:39 AM, Blogger Ulla said...

Anyons are one example of a particle with one end in the dark, undulating between both worlds? Anyons are abelian and non-abelian, the latter hypothethical. In high energetic conditions anyons also can gain energy, but I have not succeeded to find references of how much mass, and Wilczek don't say anything either. Some also talked of Higgs as dark. Have you any opinion?

In the Higgs discussion the 'thinking is explicitly forbidden' has been seen easily.

 
At 11:52 AM, Blogger Ulla said...

http://physicsworld.com/cws/article/news/2012/aug/07/physicists-see-hints-of-majorana-fermions
interesting.
"Majorana fermions" – theoretically proposed particles that are also their own anti-particles – could be found by studying the behaviour of an electrical device known as a Josephson junction. That is the view of physicists at Stanford University in the US, who have examined the properties of a Josephson junction that incorporates material calle
d a "topological insulator" sandwiched between two superconducting contacts. The researchers found significant deviations from what is seen in conventional Josephson junctions – differences that they believe could be explained in terms of Majorana-like quasiparticles.

 
At 7:43 PM, Anonymous Matti Pitkanen said...

To Hamed:

The attitude of colleagues brings in my mind a story about the awakening of Buddha. Buddha realized that the court is sleeping, and left left the court to see the world and realize that it is full of suffering.

Same applies to colleagues receiving monthly salaries and enjoying the social status of professor: they are sleeping. In this state they refuse to open their minds to see the enormous gaps and contradictions in their thinking. I do not know whether to blame this young and opportunistic career builder for this cruel and stupid behavior. Sleeping child is beautiful;-).

 
At 7:48 PM, Anonymous Matti Pitkanen said...

To Ulla:

I have written about anyons and hierarchy of Planck constants: http://tgdtheory.com/public_html/paddark/paddark.html#anyontgd .

There has been a lot of hype about Majorana fermions. They stubbornly speak about Majorana fermions: this is simply wrong. Majorana like quasiparticle would be the correct term.

 
At 3:49 AM, Blogger Ulla said...

Actually they talked about that, the quasiparticle.
"Although definite proof of the existence of Majorana fermions has not yet been obtained, theorists have calculated that particle-like excitations, or quasiparticles, which look like Majorana fermions could exist at the interface where a topological insulator – a material that only conducts electricity on its surface – is placed next to an ordinary superconductor. These quasiparticles are called “zero-energy modes” because they lie along the Fermi energy of the material."

In this way Josephson junction would be directly linked to vacuum or zero energy conditions, maybe even zero magnetism?

"In the case of a Josephson junction containing a topological insulator as the “weak link” between two superconductors, there are actually two superconductor–topological insulator interfaces back-to-back, and the Majoranas are expected to couple to each other and depart from zero energy. However, if a tiny magnetic field – even as small as half a superconducting flux quantum – is applied to the junction, the two Majorana modes decouple and both reside at zero energy."

There are a new way of doing research, in open 'collaborations' like the Human genome Project. LHC tried something similar too? This way makes the individ unimportant? No singular fonds, nor singular CV notes?

 
At 5:00 AM, Blogger Ulla said...

http://sandwalk.blogspot.fi/2012/08/changing-ideas-about-origin-of-life.html this you like :)

 
At 5:16 AM, Anonymous Matti Pitkanen said...

When the two young finnish professors doomed about 15 years ago years of my work to be trash (11 years earlier Wheeler had regarded as brilliant), one of the justifications was that the age of individuals is over in physics. Critical mass was the buzz word.

This is pure nonsense. The failure to introduce even single really great working idea to theoretical physics during last four decades (Penrose - not a group!- introduced twistors 45 years ago!) convincingly demonstrates that Big Sciene does not is not good idea in theory front: individuals are needed.

Theoretical physics performs a suicide by censoring bottleneck ideas and by forcing theoreticians to generate CV by producing papers from fashionable topics.

 
At 8:32 AM, Blogger Ulla said...

Ye, I could wait for that response of yours, but things change when the names don't come forth as much (ego not so important). It is the plethora of ideas that make it. There are always something that attatch individuals, a bit like flux tubes. This is a chaotic process, and fast, very creative.

Today the critical mass effect is used instead to conserve old attitudes and inhibit creative processes. Peer review is the uttermost tool, that can fail. Is it really necessary? I don't think so. The experiments are the one collapsing wrong ideas, not peer reviews.

I know you prefer to be the lonely thinker, and maybe there needs to be both. You are very talented and intuitive.

One question: happen entanglement easier in 2D than in 3D? I think it would be more natural with a yes? The dark matter interference on DNA happen in 2D (it must be so?)? Some think the dimensional discussion is a big mess, and I am bond to agree.

http://www.nature.com/ncomms/journal/v3/n8/full/ncomms1988.html
Imaging high-dimensional spatial entanglement with a camera.

modern electron-multiplying charge-coupled device cameras can measure correlations in both position and momentum across a multi-pixel field of view. This capability allows us to observe entanglement of around 2,500 spatial states and demonstrate Einstein–Podolsky–Rosen type correlations by more than two orders of magnitude.

 
At 9:20 PM, Anonymous Matti Pitkanen said...

Entanglement in 2-D case is special in the sense that braiding -called often entanglement - is a good candidate for a geometric correlate of entanglement.
One can obtain entangled states by permuting particles in many-particle state. Permutation corresponds to braiding in 2 dimensions.

This has quite concrete realization in the model of the flux tubes assumed to emanated from DNA in the model of DNA as topological quantum computer.

 
At 10:51 PM, Blogger Ulla said...

http://physics.aps.org/articles/v5/87

a DNA hairpin (folded: left; folding transition state: middle; coil: right). The molecule spends the vast majority of time in one or the other basin, with very rare but very rapid transitions between them. The transition path is the (yellow thick) segment of the unfolding trajectory (continuous white), see fig.

The hairpin is very important for the regulation in general. It works as a dipole, due to the pi-stacks charachteristics of DNA?

Transition path projected on schematic two-dimensional free-energy surface. The surface is indicated by contour lines and coloring, with the red and yellow areas bounded by blue contours corresponding to low (favorable) free energies.

What give the low energy surfaces?

measured average transition path times that are within a factor of 5 for proteins whose folding rates differ by 4 orders of magnitude. Given the complexity of the structural rearrangements involved in folding, this finding is surprising but, interestingly, can be readily explained by the theory for diffusion of a single particle on a one-dimensional free-energy surface..
an analytic expression relates the duration of the transition path between two states on a one-dimensional free-energy surface to the barrier height ΔG≠ and diffusivity D [4]. An accurate approximation derived by Szabo [8] shows that the average transition path time τTP−−− grows only logarithmically with ΔG≠ for a high and parabolic free-energy barrier,... The molecular extensions are a reasonable choice of reaction coordinate for the folding by zipping and unfolding by unzipping of nucleic acid hairpins and similar structures ... consistent transition path times for the DNA hairpins... consistent with the rough estimate of the bead relaxation time above.

Flux tubes at work together with 'homeostasis' like the quantum well? Creation of borders are essential? And exactly the fact that 1D DNA folds to 3D proteins, which is seen as problematic, but it indeed is obligatory? But this 'classic homeostasis' lies a step above the intermediate anyonic 'quantum homeostasis'?

If we could say that living matter is an expression for the 'Higgs field'???? This is why we can 'die' and be 'born'?

Tzymanski: "one brain is equally efficient as all the quantum computers in the world, but it use only 25 W energy."

That sentence is very important. Today LHC eats enormous amounts of energy, not to talk about the follower LEC. Is the thinking really right?

 
At 1:11 PM, Blogger Ulla said...

http://phys.org/news/2012-08-magnetic-monopoles-dipoles-vice-versa.html

Magnetic monopoles, entities with isolated north or south magnetic poles, weren't supposed to exist. If you try to saw a bar magnet in half, all you succeed in getting are two magnets, each with a south and north pole. In recent years, however, the existence of monopoles, at least in the form of "quasiparticles" consisting of collective excitations among many atoms, has been predicted and demonstrated in the lab. Now Stephen Powell, a scientist at the Joint Quantum Institute (JQI) and the University of Maryland, has sharpened the theoretical framework under which monopoles can operate.

 
At 12:08 PM, Blogger Ulla said...

http://www.journal-of-nuclear-physics.com/?p=695
Is proton the harmonic mean of up and down quark fermi-gluons!

 

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