### Could N=2 or N=4 SUSY have something to do with TGD?

N=4 SYM has been the theoretical laboratory of Nima and others. The article Scattering Amplitudes and the Positive Grassmannian by Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, and Trnka summarizes the recent situation in a form meant to be should be accessible to ordinary physicist.

N=4 SYM is definitely a completely exceptional theory and one cannot avoid the question whether it could in some sense be part of fundamental physics. In TGD framework right handed neutrinos have remained a mystery: whether one should assign space-time SUSY to them or not. Could they give rise to N=2 or N=4 SUSY with fermion number conservation?

** Earlier results**

My latest view is that * fully* covariantly constant right-handed neutrinos decouple from the dynamics completely. I will repeat first the earlier arguments which consider only fully covariantly constant right-handed neutrinos.

- N=1 SUSY is certainly excluded since it would require Majorana property not possible in TGD framework since it would require superposition of left and right handed neutrinos and lead to a breaking of lepton number conservation. Could one imagine SUSY in which both MEs between which particle wormhole contacts reside have N=2 SUSY which combine to form an N=4 SUSY?

- Right-handed neutrinos which are covariantly constant right-handed neutrinos in both M
^{4}degrees of freedom cannot define a non-trivial theory as shown already earlier. They have no electroweak nor gravitational couplings and carry no momentum, only spin.

The fully covariantly constant right-handed neutrinos with two possible helicities at given ME would define representation of SUSY at the limit of vanishing light-like momentum. At this limit the creation and annihilation operators creating the states would have vanishing anticommutator so that the oscillator operators would generate Grassmann algebra. Since creation and annihilation operators are hermitian conjugates, the states would have zero norm and the states generated by oscillator operators would be pure gauge and decouple from physics. This is the core of the earlier argument demonstrating that N=1 SUSY is not possible in TGD framework: LHC has given convincing experimental support for this belief.

** Could massless right-handed neutrinos covariantly constant in CP _{2} degrees of freedom define N=2 or N=4 SUSY?**

Consider next right-handed neutrinos, which are covariantly constant in CP_{2} degrees of freedom but have a light-like four-momentum. In this case fermion number is conserved but this is consistent with N=2 SUSY at both MEs with fermion number conservation. N=2 SUSYs could emerge from N=4 SUSY when one half of SUSY generators annihilate the states, which is a basic phenomenon in supersymmetric theories.

- At space-time level right-handed neutrinos couple to the space-time geometry - gravitation - although weak and color interactions are absent. One can say that this coupling forces them to move with light-like momentum parallel to that of ME. At the level of space-time surface right-handed neutrinos have a spectrum of excitations of four-dimensional analogs of conformal spinors at string world sheet (Hamilton-Jacobi structure).

For MEs one indeed obtains massless solutions depending on longitudinal M

^{2}coordinates only since the induced metric in M^{2}differs from the light-like metric only by a contribution which is light-like and contracts to zero with light-like momentum in the same direction. These solutions are analogs of (say) left movers of string theory. The dependence on E^{2}degrees of freedom is holomorphic. That left movers are only possible would suggest that one has only single helicity and conservation of fermion number at given space-time sheet rather than 2 helicities and non-conserved fermion number: two real Majorana spinors combine to single complex Weyl spinor.

- At imbedding space level one obtains a tensor product of ordinary representations of N=2 SUSY consisting of Weyl spinors with opposite helicities assigned with the ME. The state content is same as for a reduced N=4 SUSY with four N=1 Majorana spinors replaced by two complex N=2 spinors with fermion number conservation. This gives 4 states at both space-time sheets constructed from ν
_{R}and its antiparticle. Altogether the two MEs give 8 states, which is one half of the 16 states of N=4 SUSY so that a degeneration of this symmetry forced by non-Majorana property is in question.

** Is the dynamics of N=2 or N=4 SYM possible in right-handed neutrino sector? **

Could N=2 or N=4 SYM be a part of quantum TGD? Could TGD be seen a fusion of a degenerate N=4 SYM describing the right-handed neutrino sector and string theory like theory describing the contribution of string world sheets carrying other leptonic and quark spinors? Or could one imagine even something simpler?

What is interesting that the net momenta assigned to the right handed neutrinos associated with a pair of MEs would correspond to the momenta assignable to the particles and obtained by p-adic mass calculations. It would seem that right-handed neutrinos provide a representation of the momenta of the elementary particles represented by wormhole contact structures. Does this mimircry generalize to a full duality so that all quantum numbers and even microscopic dynamics of defined by generalized Feynman diagrams (Euclidian space-time regions) would be represented by right-handed neutrinos and MEs? Could a generalization of N=4 SYM with non-trivial gauge group with proper choices of the ground states helicities allow to represent the entire microscopic dynamics?

Irrespective of the answer to this question one can compare the TGD based view about supersymmetric dynamics with what I have understood about N=4 SYM.

- In the scattering of MEs induced by the dynamics of Kähler action the right-handed neutrinos play a passive role. Modified Dirac equation forces them to adopt the same direction of four-momentum as the MEs so that the scattering reduces to the geometric scattering for MEs as one indeed expects on basic of quantum classical correspondence. In ν
_{R}sector the basic scattering vertex involves four MEs and could be a re-sharing of the right-handed neutrino content of the incoming two MEs between outgoing two MEs respecting fermion number conservation. Therefore N=4 SYM with fermion number conservation would represent the scattering of MEs at quantum level.

- N=4 SUSY would suggest that also in the degenerate case one obtains the full scattering amplitude as a sum of permutations of external particles followed by projections to the directions of light-like momenta and that BCFW bridge represents the analog of fundamental braiding operation. The decoration of permutations means that each external line is effectively doubled. Could the scattering of MEs can be interpreted in terms of these decorated permutations? Could the doubling of permutations by decoration relate to the occurrence of pairs of MEs?

One can also revert these questions. Could one construct massive states in N=4 SYM using pairs of momenta associated with particle with label k and its decorated copy with label k+n? Massive external particles obtained in this manner as bound states of massless ones could solve the IR divergence problem of N=4 SYM.

- The description of amplitudes in terms of leading singularities means picking up of the singular contribution by putting the fermionic propagators on mass shell. In the recent case it would give the inverse of massless Dirac propagator acting on the spinor at the end of the internal line annihilating it if it is a solution of Dirac equation.

The only way out is a kind of cohomology theory in which solutions of Dirac equation represent exact forms. Dirac operator defines the exterior derivative d and virtual lines correspond to non-physical helicities with dΨ ≠ 0. Virtual fermions would be on mass-shell fermions with non-physical polarization satisfying d

^{2}Ψ=0. External particles would be those with physical polarization satisfying dΨ=0, and one can say that the Feynman diagrams containing physical helicities split into products of Feynman diagrams containing only non-physical helicities in internal lines.

- The fermionic states at wormhole contacts should define the ground states of SUSY representation with helicity +1/2 and -1/2 rather than spin 1 or -1 as in standard realization of N=4 SYM used in the article. This would modify the theory but the twistorial and Grassmannian description would remain more or less as such since it depends on light-likeneness and momentum conservation only.

** 3-vertices for sparticles are replaced with 4-vertices for MEs**

In N=4 SYM the basic vertex is on mass-shell 3-vertex which requires that for real light-like momenta all 3 states are parallel. One must allow complex momenta in order to satisfy energy conservation and light-likeness conditions. This is strange from the point of view of physics although number theoretically oriented person might argue that the extensions of rationals involving also imaginary unit are rather natural.

The complex momenta can be expressed in terms of two light-like momenta in 3-vertex with one real momentum. For instance, the three light-like momenta can be taken to be p, k, p-ka, k= ap_{R}. Here p (incoming momentum) and p_{R} are real light-like momenta satisfying p⋅ p_{R}=0 with opposite sign of energy, and a is complex number. What is remarkable that also the negative sign of energy is necessary also now.

Should one allow complex light-like momenta in TGD framework? One can imagine two options.

- Option I: no complex momenta. In zero energy ontology the situation is different due to the presence of a pair of MEs meaning replaced of 3-vertices with 4-vertices or 6-vertices, the allowance of negative energies in internal lines, and the fact that scattering is of sparticles is induced by that of MEs. In the simplest vertex a massive external particle with non-parallel MEs carrying non-parallel light-like momenta can decay to a pair of MEs with light-like momenta. This can be interpreted as 4-ME-vertex rather than 3-vertex (say) BFF so that complex momenta are not needed. For an incoming boson identified as wormhole contact the vertex can be seen as BFF vertex.

To obtain space-like momentum exchanges one must allow negative sign of energy and one has strong conditions coming from momentum conservation and light-likeness which allow non-trivial solutions (real momenta in the vertex are not parallel) since basically the vertices are 4-vertices. This reduces dramatically the number of graphs. Note that one can also consider vertices in which three pairs of MEs join along their ends so that 6 MEs (analog of 3-boson vertex) would be involved.

- Option II: complex momenta are allowed. Proceeding just formally, the (g
_{4})^{1/2}factor in Kähler action density is imaginary in Minkowskian and real in Euclidian regions. It is now clear that the formal approach is correct: Euclidian regions give rise to Kähler function and Minkowskian regions to the analog of Morse function. TGD as almost topological QFT inspires the conjecture about the reduction of Kähler action to boundary terms proportional to Chern-Simons term. This is guaranteed if the condition j_{K}^{μ}A_{μ}=0 holds true: for the known extremals this is the case since Kähler current j_{K}is light-like or vanishing for them. This would seem that Minkowskian and Euclidian regions provide dual descriptions of physics. If so, it would not be surprising if the real and complex parts of the four-momentum were parallel and in constant proportion to each other.

This argument suggests that also the conserved quantities implied by the Noether theorem have the same structure so that charges would receive an imaginary contribution from Minkowskian regions and a real contribution from Euclidian regions (or vice versa). Four-momentum would be complex number of form P= P

_{M}+ iP_{E}. Generalized light-likeness condition would give P_{M}^{2}=P_{E}^{2}and P_{M}⋅P_{E}=0. Complexified momentum would have 6 free components. A stronger condition would be P_{M}^{2}=0=P_{E}^{2}so that one would have two light-like momenta "orthogonal" to each other. For both relative signs energy P_{M}and P_{E}would be actually parallel: parametrization would be in terms of light-like momentum and scaling factor. This would suggest that complex momenta do not bring in anything new and Option II reduces effectively to Option I. If one wants a complete analogy with the usual twistor approach then P_{M}^{2}=P_{E}^{2}≠ 0 must be allowed.

** Is SUSY breaking possible or needed?**

It is difficult to imagine the breaking of the proposed kind of SUSY in TGD framework, and the first guess is that all these 4 super-partners of particle have identical masses. p-Adic thermodynamics does not distinguish between these states and the only possibility is that the p-adic primes differ for the spartners. But is the breaking of SUSY really necessary? Can one really distinguish between the 8 different states of a given elementary particle using the recent day experimental methods?

- In electroweak and color interactions the spartners behave in an identical manner classically. The coupling of right-handed neutrinos to space-time geometry however forces the right-handed neutrinos to adopt the same direction of four-momentum as MEs has. Could some gravitational effect allow to distinguish between spartners? This would be trivially the case if the p-adic mass scales of spartners would be different. Why this should be the case remains however an open question.

- In the case of unbroken SUSY only spin distinguishes between spartners. Spin determines statistics and the first naive guess would be that bosonic spartners obey totally different atomic physics allowing condensation of selectrons to the ground state. Very probably this is not true: the right-handed neutrinos are delocalized to 4-D MEs and other fermions correspond to wormhole contact structures and 2-D string world sheets.

The coupling of the spin to the space-time geometry seems to provide the only possible manner to distinguish between spartners. Could one imagine a gravimagnetic effect with energy splitting proportional to the product of gravimagnetic moment and external gravimagnetic field B? If gravimagnetic moment is proportional to spin projection in the direction of B, a non-trivial effect would be possible. Needless to say this kind of effect is extremely small so that the unbroken SUSY might remain undetected.

- If the spin of sparticle be seen in the classical angular momentum of ME as quantum classical correspondence would suggest then the value of the angular momentum might allow to distinguish between spartners. Also now the effect is extremely small.

## 26 Comments:

Dear Matti,

I am logged on yet superposions of the effects in TGD;-).

About my last question, I asked from you: ” if we have some distribution of matters on the space-time, In GR one can calculate energy momentum tensor from the distribution of matters and Einstein equation gives us metric of space-time. How do you the same thing in TGD?”

You answered me: “In the case of the gravitation the effective deviation of metric experienced by particle would be estimated as follows. Use common (say linear) M^4 coordinates for the space-time sheets. One sums up the deviations of the corresponding induced metrics from flat M^4 metric at various space-time sheet. This defines the effective metric giving rise to the gravitational acceleration experienced by the test particle idealized to point particle. Particle experiences sum of gravitational accelerations just as in standard description. Similar prescription applies to classical gauge forces.”

For doing the superposions of the effects, one must have induced metrics of various space-time sheets. Hence one must write independently for every space-time sheets a field equation to get induced metric?? (Multiplying energy momentum tensor by second fundamental form and equal the result to zero)

When you say in GRT there are superpositions of gravitational fields themselves. I don’t understand it clearly. In GRT we only sum up the energy momentum tensor for dust and electromagnetism. After it, one can derive Einstein tensor from Einstein equation and from it, equations of motion are deduced. Where is the “superposition of gravitational fields themselves”?

For deriving induced metric, because it is deduced from second fundamental form (? As I think!) One must have energy momentum tensor. In GRT energy momentum tensor is sum of the stress energy tensors for the dust (all of the macroscopic objects are in the form of dust) and electromagnetism. But in TGD it is energy momentum tensor of kahler field. There is a contribution of energy momentum tensor of electromagnetism in U(1) part of kahler field. As I understand it is not needed to take account contribution of dust, because dusts are in form of space-time sheets and only effects of the space time sheets are contributes as sum of deviations of metric fields from M4 for every space-time sheets.

Hence when an object is on a larger space-time sheet, it’s gravity doesn’t contribute to curvature of the larger space time sheet? Because it doesn’t contribute in energy momentum tensor of kahler field of it.

Now I think that I don’t regard anything from boundary conditions of every space-time sheets. If it is important in your answers, please make it clear.

Dear Hamed,

thank you for good questions again. And good new year!

[Hamed]

"Multiplying energy momentum tensor by second fundamental form and equal the result to zero"

[MP] This is indeed true for preferred extremals for which Einstein-Maxwell equations with cosmological term guarantee the vanishing of second term proportional to contraction of Kahler current in Kahler field: physically this means that 4-D Lorentz force vanishes (equilibrium condition) so that Kahler energy momentum tensor has vanishing divergence just like Einstein tensor and metric. Cosmological constant Lambda and Newton's constant are predictions.

Every space-time sheet satisfied the field equations independently in regions far from wormhole contacts. Near wormhole contacts and in their Euclidian interior situation of course changes.

[Hamed] For deriving induced metric, because it is deduced from second fundamental

form (? As I think!)

[MP] No! You ask about the relationship between induced metric and second fundamental form.

a) Induced metric is deduced just as a projection: you have metric tensor h_kl of the imbedding space. You have coordinate gradients partial_alpha h^k defining tangent vectors for the space-time sheet. The induced metric is the projection of h_kl to space-time surface: g_{alphabeta}= h_{kl} partial_alpha h^k partial_beta h^l. Same recipe applies to components of spinor connection, and gauge fields.

b) Second fundamental form is formed by the covariant derivatives of tangent vectors and is orthogonal to space-time surface as H-vector. H^k_{alphabeta}= D_beta h^k_alpha. It contains second derivatives of imbedding space coordinates. Minimal surface equations g^{alphabeta} H^k_{alphabeta}=0 are geometric counterpart for a massless wave equation and are also satisfied by preferred extremals: that 3 kinds of different field equations are true simultaneously is due to the generalization of complex structure to Hamilton-Jacobi structure in Minkowskian regions and to 4-D complex structure in Euclidian regions. It is this structure which codes the entire dynamics. A still open question is whether it implies or/and is implied by the quaternionicity (associativity) of tangent spaces with integrable distribution of M^2:s (hypercomplex plane) in tangent spaces. For preferred extremals the local distribution of M^2:s code for a distribution of local planes of light-like momentum and of polarization directions so that physically the equivalence is highly suggestive.

[Hamed] Where is the superposition of gravitational fields themselves??

[MP] In GRT as in all interacting theories the superposition of fields is approximation. In the case of gravitation the extreme weakness of interaction makes this approximation excellent and one can develop a classical perturbation theory and attempt to quantize GRT are based on this expansion. In low energy QCD situation is different.

In TGD linear superposition is replaced with a set theoretic union of space-time sheets and the formation of wormhole contacts between test particle space-time sheet and the space-time sheeets in the union gives to a superposition of various effects: gravitational, electgromagnetic,....

To be continued...

Dear Hamed,

answers to your remaining questions.

[Hamed] Hence when an object is on a larger space-time sheet, it's gravity doesn't contribute to curvature of the larger space time sheet? Because it doesn't contribute in energy momentum tensor of kahler field of it.

[MP] This is true only in the first approximation. In GRT applied in macroscales the energy momentum tensor - I will call it just T - is what you say. In QFT approach to GRT T is deduced as energy momentum tensor of the matter part of the action, which is ad hoc: Noether theorem would give vanishing energy momentum currents associated with Noether currents reflecting the fact that general coordinate invariance is gauge symmetry. One could perhaps cautiously say that in TGD Kahler action codes for the matter classically (note however that induced spinors are present also and that one does not have Einstein-Maxwell action!).

Condensing space-time sheet deforms the larger space-time sheet and vice versa!! Wormhole contacts are formed and larger space-time sheet it is deformed near the wormhole throats. Is a small gravitational deformation created also also in faraway regions? Or is gravitational interaction mediated by MEs connecting the masses condensed at spce-time sheet. String world sheets having their boundaries at the light-like wormhol contacts associated with the two subsystems would be a natural candidate for a microscopic description very much analogous to that provided by AdS_5 correspondence. Are both views correct and correspond to descriptions in different scales?

The two space-time sheets can form also form a bound states. Magnetic charges are needed to make wormhole contacts stable: these occur in pairs and have interpretation as elementary particles particles. Closed magnetic flux tube defines a string like object indeed containg closed string world sheet carrying spinors (right-handed neutinos form an exception). When bound states are formed in this manner simple cylindrical MEs with constant light-like momentum and fixed polarization representing free propagation are deformed so that a bound state of them results. The motion assignable to MEs is not linear anymore and momentum exchange takes place so that only the total momentum which is timelike is conserved. The massless momenta of MEs sum up to the massive momentum of particle. The Minkowskian preferred extremals (apart for cosmic string like objects) are generalizations of MEs. There are also string like objects with 2-D M^4 and CP_2 projections. This leads to a a concrete picture about how twistorial description (for which masslessness is absolutely essential) and possibly the analog of N=2 or N=4 SYM as part of TGD emerges in TGD framewor.

The interaction of particle on background geometry corresponds to the direct action on space-time sheet and in the vicinity of wormhole contact and in its interior of course there is a new contribution also to the curvature of the large space-time sheet. A simple idealized model for the condensation of massive space-time sheet to canonically imbedded M^4 (CP_2 coordinates are constant) deforming it to an imbedding of Schwartschild or Reissner-Nordstoem metric as extremal. I have studied these imbeddings in http://tgdtheory.com/public_html/tgdclass/tgdclass.html#tgdgrt. The ansatz is same as for the imbedding of AdS_4 metric blog in the recent blog posting at http://matpitka.blogspot.fi/2012/12/is-there-connection-between-preferred.html.

So thanks, Happy new year too. i remembered last year at this time, it was starting of my process for learning TGD with the questions on the blog :-).

Hou have made good progress!

http://www.nature.com/news/2011/110405/full/news.2011.210.html

Researchers in Austria have made what they call the "fattest Schrödinger cats realized to date". They have demonstrated quantum superposition – in which an object exists in two or more states simultaneously – for molecules composed of up to 430 atoms each, several times larger than molecules used in previous such experiments

Dear Matti,

“[Matti] Induced metric is deduced just as a projection”

Yes, Induced metric is deduced just as a projection but this is in theory. My question was in practical sense when we have some distribution of matters in some region and we want to deduce the value of components of the metric of the larger space-time sheet containing these matters.

For example in GRT about Schwarzschild solution, we are concerned with a solution of the vacuum field equations. Therefore we request T_munu=0 and from G_munu=0 we can deduce the value of components of metric tensor.

Another my practical question is how one can deduce the value of components of energy momentum tensor of kahler field from distribution of matters?

http://www.nature.com/news/quantum-gas-goes-below-absolute-zero-1.12146

Schneider and his colleagues reached such sub-absolute-zero temperatures with an ultracold quantum gas made up of potassium atoms. Using lasers and magnetic fields, they kept the individual atoms in a lattice arrangement. At positive temperatures, the atoms repel, making the configuration stable. The team then quickly adjusted the magnetic fields, causing the atoms to attract rather than repel each other. “This suddenly shifts the atoms from their most stable, lowest-energy state to the highest possible energy state, before they can react,” says Schneider. Another peculiarity of the sub-absolute-zero gas is that it mimics 'dark energy'

To Hamed:

Thanks for good questions. Here is the first one.

[Hamed] My question was in practical sense when we have some distribution of matters in some region and we want to deduce the value of components of the metric of the larger space-time sheet containing these matters.

[MP] This question involves hidden assumptions.In GRT the perturbations would indeed add as deformations of metric in single space-time. Now the perturbations are associated with different space-time sheets, sum is replaced with union of space-time sheets, and only their effects on test particle add.

I summarize what I believe I know.

a) Preferred extremals (PEs) are known in general form. The challenge is to find a description for their interaction with a test particle - small space-time sheet - which is expected to be via touching. I will return to this.

a) In Minkowskian regions PEs can be seen as radiation with local light-like wave vector and transversal local polarization. Massless extremals (MEs) for which massless momentum vector and polarization are in constant direction are simplest example. From more complex ones one can build elementary particles as bound states by putting between them wormhole contact pairs with magnetic charges. Thus one can say that elementary particles and thus all systems are accompanied by preferred extremals representing radiation to which test particle responds. Basically all interactions are via radiation as already in classical ED ( this also makes twistor approach possible).

b) Only the effects of external sources on test particle add. One must describe the situation using only the preferred extremals associated with the matter - ultimately elementary particles. This microscopic picture is basically that provided by generalized Feynman diagrams.

You can imagine of having a tests particle, a small space-time sheet and space-time sheets assignable to the various systems generating gravitational fields at their own sheets or more generally, at space-time sheets assignable to them. The "space-time sheets assignable to them" would correspond to radiation fields represented by preferred extremals having same general structure than massless extremals (MEs). To calculate the acceleration/ associated 4-force one would add the effects of various space-time sheets- basically MEs and get effective metric for which deformations from various space-time sheets would add.

d) This kind of effective metric very rarely allows imbedding in M^4xCP_2. The space-time with this effective metric - General Relativity - becomes an effective theory to describe the situation. The Universe of GRT is therefore only a fiction albeit very practical fiction in astrophysics.

Only in some idealized cases such as Sun describable as a static point like mass and planets describable as test masses GRT description using Schwarthild or Reissner-Nordstrom metric, the metric is imbeddable. When one must take into account the response of planets, this description fails in TGD framework. Although GRT is a convenient fiction, a huge amount of information and predictivity is lost since the constraints due to imbeddability condition and preferred extremal property are forgotten. There is no hope about microscopic quantal description if one starts from this picture directly and as we know, the quantization of GRT indeed fails also technically. This is like trying to describe atoms using classical hydrodynamics.

To be continued....

To Hamed:

[Hamed] Another my practical question is how one can deduce the value of components of energy momentum tensor of kahler field from distribution of matters?

[MP] Also this question involves the hidden assumption that single space-time sheet is enough. This is not true in TGD framework, where one can speak only about superposition of effects and forces from material objecs.

If one introduces the effective metric in which deformations of metric add up then one is using General Relativity as an *effective* and fictive description which could be quite practical. Einstein-Maxwell system with cosmological term would be a good guess for the action. Reissner-Nordstroem solution and Scrwartschild solution would be basic building bricks.

Although induced Kahler field is fundamental at the level of TGD in GRT approximation it is the four-momentum assigned to effectively point like sources which matters in astrophysics and is described phenomenologically without any attempt to do it microscopically.

The attempt to describe matter microscopically in terms of Kahler field would lead to TGD and generalized Feynman diagrammatics.

a) This attempt would lead to the discovery of Euclidian space-time regions: CP_2 type extremals carrying very strong Kahler fields and behaving like instantons.

b) Also the discovery that elementary particles correspond to closed wormhole flux tubes represent would emerge sooner or later.

c) The realization that the interiors of black holes are actually Euclidian space-time regions assignable to any objects in the scale of object would emerge in the process.

d) Sooner or later would come the discovery of "cosmic strings" and the realization that dark energy is identifiable as Kahler magnetic energy of magnetic flux tubes of cosmological size scale.

A general vision about galaxies as string like objects and about their organization along cosmic strings would develop, and one would eliminate galactic dark matter and replace it with long range gravitational fields created by the dark energy associated with these strings.

A further positive surprise would be explanation for the magnetic fields filling the cosmos but having no explanation in terms of standard cosmology.

But all this will take one century if the colleagues continue their discoveries with the recent pace...;-)

To Hamed:

Just a couple of random remarks about topological sum (touching) of space-time sheets) and the detailed description for the interaction of test particle and preferred extremals representing radiation.

a) One could argue that touching is the mechanism of interaction. Touching certainly creates region of Euclidian signature and thus something which could correspond to a line of generalized Feynman diagram. Is this something which much be added to the general Feynman diagrammatics? Note that also string world sheets can touch each other. And what about string world sheets carrying the fermion fields (all except right handed neutrino) which are present in TGD?

b) The touching occurs at single point for two 4-D space-time surfaces in 8-D imbedding space occurs unavoidably at discrete points. When you have two two 2 slightly deformed pieces of Minkowski space which are "parallel" and within distance about CP2 size scale, one expects touchings to occur with very high probability. This is like two threads in extremely narrow tube: they will touch with very high probability.

c) The description for the formation of wormhole contacts must be statistical. Quantal description by generalized Feynman diagrams could be the only reasonable description. From this

one would deduce QFT limit and finally get to the practical level. Since Einstein-Maxwell equations hold true classically one expects that Einstein-Hilbert action with a small cosmological term is a reasonable approximation for the effective action at QFT limit.

Matti,

If you have time I would like your take on the articles of this week on the idea of negative temperature below (but hotter than at minus infinity) from the TGD perspective. I also continue to mention you in my blog in relation to recent generalizations of systems theory.

If you follow, from a philosophic view at least, how I use groups as you seem to do here and from a wider than the limited standard viewpoint, how close is it to what is actually such views as physics?

Anyway, may you and yours have a very great new and creative era that begins for us this year!

ThePeSla

Here is the last link I saw on it yesterday...

http://www.sciencedaily.com/releases/2013/01/130104143516.htm

To Ulla and Pesla:

Thank you for the link.

In thermodynamical equilibrium for a closed system

entropy is maximized subject to the constraint that average energy per particle is fixed. The probabilities for single particle energies E

are proportional to exp(-E/kT). Temperature T must be positive in order that probabilities stay finite as E becomes infinite. E is sum of kinetic energy and interaction energy which can be also negative. Usually the kinetic energy grows without limit.

Negative temperatures are possible when the energy spectrum changes from the usual spectrum extending to plus infinity to a spectrum which extends to - infinity or to very high energies.

This requires the presence of strong negative interaction energy between particles (attractive force) and some kind of cutoff on kinetic energies. Cutoff can be effectively realized by starting from a situation in which one has very low but positive temperature.

In the experimental situation an attractive interaction is generated between particles: this corresponds to negative interaction energy. Thermal equilibrium exists only below some time scale and researchers compare the situation to a pyramid which stands on its top. The negative interaction energy stabilizes the situation.

Some further comments on negative temperatures.

a) Researchers mention also antigravity.

Effective antigravity is possible if negative gravitational interaction energy overcomes the kinetic energies and there is cutoff for kinetic energies.

For a gas in a box at the surface of Earth the smallest magnitude for E corresponds to largest height so that particles tend to be near the top of the box for negative temperatures. This would be the effective antigravity.

Could this have something to do with Podkletnov effect. Superconductors at low temperature are present. Could the temperature of Cooper pairs become negative in the gravitational field of Earth and could "antigravity" inside rotating superconductor cause the Cooper pairs to leak to the air above the super-conductor and induce the claimed flow of air upwards? Just asking;-).

b) Question: could the ability of superfluid flow to apparently resist gravitational force is due to negative temperature?

This situation is realized only if one can pose an effective cutoff on kinetic energies: this requires very low starting temperatures.

c) Electron holes in condensed matter systems have in natural manner negative energy and on can imagine that they are in thermal equilibrium with negative temperature.

d) In zero energy ontology the ends of CD can be said to correspond to opposite signs of energy and therefore also of temperature in thermal equilibrium. Phase conjugate laser rays would be standard example of negative sign of energy and they obey second law in reverse time direction. One can assign to them negative temperature naturally.

And the dark energy perspective? zero temp is analog to cosmological constant?

Matti, can you discuss a TGD based explanation?

http://phys.org/news/2013-01-non-causal-quantum-eraser.html#jCp

This is cool!

In the new experiment, this second photon is so far separated from the first photon that no transfer of information whatsoever (the velocity of which can never exceed the speed of light) would be fast enough. Yet, the first photon behaves like a wave or like a particle, still depending on the measurement performed on the second.

While the results of such experiments are fully consistent with quantum physics, a clear explanation in terms of causality is impossible, as, according to Einstein's relativity theory, any transfer of information is limited to the speed of light.

To Ulla:

This dark energy speculation is hype. Also the claim that negative temperatures are something totally new, is not true. It was mentioned already in my book of statistical physics which I studied 1970+ few years.

To Stephen:

The result is consistent with quantum theory. Quantum entanglement implies correlations which are non-causal and there is no need to send signals.

Quantum classical correspondence however requires that also this phenomenon has classical space-time correlate. Standard physics with point like particles does not allow this.

Generalization of particle to space-time sheet leads to the view that the entanglement of distant particles has flux tubes/massless extremals/.. connecting the two systems as space-time correlates so that they effectively form a single particle.

The replacement of point like particle with 3-surface which can have arbitrarily large size that any system has elementary particle like aspects at large space-time sheets resolves many of the mysteries of quantum theory basically due to the lack of quantum classical correspondence. How simple and elegant and how hopelessly difficult to communicate!

Dear Matti,

First I have a humble request!!: when you answer the questions, please answer at this level, the concepts of quantum TGD or recently views of field equations in TGD makes me confuse and makes my process slow. Although for you different concepts of TGD aren’t separable but in the process of teaching TGD, you can for simplification at each step concentrate on some concepts and assign other concepts to next steps. So let’s go in logical way step by step without intermixing steps together :-).

****** I think learning TGD (in principle not practical!) is simpler than M-theory or TQFT or any physical theories with advanced mathematics, because intuitions in TGD makes learning TGD simple so that advanced mathematics in TGD are very intuitional. But this is only possible practically, if the process of teaching be logically step by step! *******

At TGDHistory.pdf:

“At this period I did not have yet the alternative view idea TGD as a generalization of string

Model: I wanted Einstein's equations and conservation laws! Curvature scalar was therefore the

Obvious first guess for the action density.”

At this period, did you know about space-time sheets? If yes, first lets we don’t know about it!! It makes easier for me to understand it. Although description of gravity without space-time sheets is not possible, but let to me to start without space-time sheets, and when I understand basic views correctly, you can show me that without space-time sheets these views lead to contradiction.

Therefore I explain a review of what I know and ask my questions:

There is a global space-time in M4*S^2. The metric of the space-time is induced from M4*S^2. Curvature scalar was the first guess for the action density in TGD.(Although it is a wrong action) This is like derivation of Einstein equation from the Einstein–Hilbert action in general relativity. Only seeming difference is the metric is induced from imbedding space.

This leads to the field equation as G time to H equal zero. Energy momentum tensor of the action is a constant multiple of Einstein tensor.

- What is physical meaning of this energy momentum tensor of the action? Is it gravitational energy?

In this framework that is without space-time sheets, as like GRT, physical fields make the global space-time curve? Therefore the correct formula for field equation is G time to H equal to sum of energy momentum tensors of these physical fields like in GRT?

At TGDHistory.pdf:

?At this period I did not have yet the alternative view idea TGD as a

generalization of string Model: I wanted Einstein's equations and conservation laws! Curvature

scalar was therefore themObvious first guess for the action density.?

At this period, did you know about space-time sheets? If yes, first lets we

don't know about it!! It makes easier for me to understand it. Although

description of gravity without space-time sheets is not possible, but let

to me to start without space-time sheets, and when I understand basic views

correctly, you can show me that without space-time sheets these views lead

to contradiction.

[MP] I learned about space-time sheets, massless extremals, and string like objects and cosmic strings as call the, during the first years by finding solutions of field equations which are highly symmetries. For instance, space-time sheets emerge when one tries to imbed constant magnetic field as surface. Imbedding exists only in finite volume and the question is what happens at resulting boundaries. Are they real boundaries or causal boundarires at which the signature of metric changes?

Therefore I explain a review of what I know and ask my questions:

There is a global space-time in M4*S^2. The metric of the space-time is

induced from M4*S^2. Curvature scalar was the first guess for the action

density in TGD.(Although it is a wrong action) This is like derivation of

Einstein equation from the Einstein?Hilbert action in general relativity.

Only seeming difference is the metric is induced from imbedding space.

This leads to the field equation as G time to H equal zero. Energy momentum

tensor of the action is a constant multiple of Einstein tensor.

General Relativity based on single space-time sheet is utterly unphysical: typical solution of Einstein's equations requires rather high-dimensional imbedding space.

[MP] Second very essential difference is that the geometry of surface dictates gravitational dynamics and U(1) gauge dynamics as submanifold dynamics. This is very essential element and manysheeted space saves the situation concerning the failure of linear superposition even approximately.

[Hamed] What is physical meaning of this energy momentum tensor of the action? Is

it gravitational energy?

[MP] It is essentially energy momentum tensor of matter which for preferred extremals coincides with what you obtain from Einstein-Maxwell. To answer this detail I need to have a model for particle in the Minkowskian and Eucdian region. By quantum classical correspondence I *assume* that the sum of lightlike moment associated with Minkowskian preferred extremals (in first approximation MEs) is the momentum of particle which is identified as wormhole contacts structure between them, All momenta are light-like at the level single space-time sheet.

[Hamed] In this framework that is without space-time sheets, as like GRT, physical

fields make the global space-time curve? Therefore the correct formula for

field equation is G time to H equal to sum of energy momentum tensors of

these physical fields like in GRT?

[MP] One cannot obtain Einstein-Maxwell system in any natural manner without long and tedious path to the recent situation. Hilbert's approach is certainly purely ad hoc since Noether theorem gives identically vanishing conserved charges infinitesimal general coordinate transformation.

I am not sure would you mean with G time to H. Curvature scalar as action give Tr(GH^k)=0. This is equivalent with conservation of isometry currents so that his part of dream would be achieved.

Most of the text in the chapter http://tgdtheory.com/public_html/tgdclass/tgdclass.html#class is about single space-time sheet. The challenge would be to understand how wormhole contatcs and massless preferred extremals are combine to more complex structures. Here only physical intuition can help at recent stage.

Dear Hamed,

still a little comment about Einstein's equations.

*One thing is sure: they do not follow from a variational principle in TGD. In TGD framework the only possible conclusion is that attempts to quantize general relativity by starting from curvature scalar as action have been a wrong track.

*One natural guess would be that Einstein's equations are predicted only at the long length scale limit of the theory. This the view provided also by string theory. This was my belief for a long time but it seems that this was wrong.

*That Einstein-Maxwell with cosmological constant equations follow from the condition that the Maxwellian energy momentum tensor has vanishing divergence (, which in turn is necessary to satisfy the field equations for preferred extremals) was a total surprise (this condition is also the manner to end up with Einstein's equation without mentioning Hilbert action). Notice however that G and Lambda are now predictions of the theory rather than input parameters, and one can consider the possiblity that they have a spectrum. Lambda certainly has: Lambda should decrease with cosmic time roughly as 1/a^2 from the estimate following from p-adic length scale hypothesis.

Dear Matti,

So thanks. I gradually taste the beauty of gravity in this sense of TGD :-).

Question about extremals of kahler action:

Solutions of field equations are extremals of kahler action. One can classify them as dimension of CP2 projection in one way(in respect to phases of matter) and in other way are classified as vacuum extremals and non-vacuum extremals.

Vacuum extremals are in two basic types: CP2 type vacuum extremals for which the induced Kahler

Field and Kahler action are non-vanishing and the extremals for which the induced Kahler Field vanishes.

non-vacuum extremals are in 3 basic type “ string like objects, ME and Maxwell’s phase

Now ordinary macroscopic objects are in which of them?

One can say that these solutions aren’t classical and we need some preferred ones. These preferred extremals can be obtained from Quantum TGD. In other word quantum TGD can say what are classical extremals!

Dear Hamed,

it was good that you listed the basic type of extremals as I have given them in the chapter I gave link to. I found that about Maxwell phase my views have become obsolete.

I try so summarize the situation as it as between 1980-1990 when worked with extremals of Kahler action.

a) The idea about Maxwell phase relates to MEs and the fact that they allow extremely restricted linear superposition. That is only for modes moving in the same direction and all wave-vectors in 4-D sense must be parallel or antiparallel. ME is like wave tube inside which excitations with arbitrary transversal dependence move to single direction with light-velocity without change in the shape of pulse. This like having left- (or right-) movers but not both. MEs are very much like space-time correlates for field quanta which conforms with Bohr orbitology.

b) Guided by simple common sense I however hoped linear superposition in a more general sense - albeit only approximately. I called this desired phase Maxwell phase. I proposed the identification of Maxwell phase as small deformations of vacuum extremals with vanishing Kahler form and assumed that they are separate from MEs and their generalizations. Something very natural at least when one consider Kahler form. My sincere hope was that this would give the analog of Maxwell's ED for single space-time sheet. I was wrong.

At that time (1980-1990) I did not have a slightest idea about the replacement of linear superposition for fields with linear superposition for their effects realized in terms of set theoretic union of space-time sheets representing field modes. This is like geometric counterpart of Fourier decomposition with each mode represented as space-time sheet. Geometric form of quantization.

Does Maxwell phase exist at all? Are extremals known at that time and their generalization to the ansatz for preferred extremals relying on Hamilton-Jacobi structure and complex structure in Minkowskian/Euclidian regions enough?

a) The existence of Hamilton Jacobi-structure implies that all Minkowskian pieces of preferred extremals represent a space-time region which could be called space-time correlate for a massless quantum. Polarization vector and light-like wave-vector are however local rather than constant. By adding wormhole contacts one can engineer "bound states" of these objects. This transforms simplest linear MEs to "curved" MEs and more general preferred extremals.

This does not support the view that Maxwell phase could be realized at the level of single space-time sheet.

b) Does one then obtain the analog of Maxwell phase at all? Yes. But it is indeed many particle system in the sense that many space-time sheets are involved and as the term "phase" would suggest. My choice of the terminology happened to be correct!

If you however have large number of preferred extremals in single region of Minkowki space, the summation of their effects on particle having topological sum contacts with them implies that effectively one has superposition of Fourier components of fields with different wave vectors. Hence this many-sheeted situation would correspond to Maxwell phase as I thought it.

Quite generally, the very-many-sheeted situation quite generally corresponds to the classical limit of the the theory in accordance with the basic philosophy that classicality emerges at the limit of large quantum numbers.

Dear Hamed,

I continue to answer your questions. They are very useful since they help me to see things from a new view point.

[Hamed] Now ordinary macroscopic objects are in which of them?

[MP] I dare cautiously believe;-) that I have now a reduction of the dynamics of Kahler action at basic level to Minkowskian and Eucldiian preferred extremals with first ones representing spacetime correlates for radiation in very general sense. All elementary particles could effectively consist of massless quanta in the sense that massive four-momentum of the particle would be sum of massless but non-parallel momenta.

What about macroscopic objects? One identification for the space-time representative of macroscopic objects is as the analog of line of Feynman diagram, that is space-time region with Euclidian signature. Basically a deformation of CP_2 vacuum extremals which can increase the E

3 projection from that for a point to arbitrarily large would be in question.

Line of Feynman diagram is something very quantal so that this is consistent with the vision that quantal aspects are present even at macroscopic level. One way to end up with this vision was a view about blackhole interiors as Euclidian regions. Then I realized that blackhole property in the sense that there very strong gravitational fields present is not essential. As a matter this brings in mind with the present tendency to describe everything in terms of blackhole via Ads/CFT and wormhole contacts are indeed blackhole like objects.

[Hamed] One can say that these solutions aren’t classical and we need some preferred ones. These preferred extremals can be obtained from Quantum TGD. In other word quantum TGD can say what are classical extremals!

[MP] Here I think that we think differently. The motivations for preferred extremals are different.

a) The starting point is construction Kahler geometry of WCW. It consists of 3-surfaces but one must realized 4-D general coordinate invariance.

The very definition of Kahler function should be such that it assigns a unique space-time surface to a given 3-surface for general coordinate transformations to act on.

One must also have direct connection with classical physics which must exact part of quantum physics. This is achieved if Kahler function is identified as Kahler action associated with a preferred extremal containing a given 3-surface X^3. Preferred extremals are like Bohr-orbits since instead of infinite number of 4-surfaces solving field equations and going through X^3 one accepts only single one, the "preferred one". General Coordinate Invariance implies obviously also holography since knowing the 3-surface means that one knows space-time surface. It is GCI which implies automatically very quantal classical behavior as Bohr orbitology. This is something totally unexpected and shows how powerful GCI is in the case of sub-manifold gravity.

b) The question has of course been what "preferred" means. My first guess was that it could mean absolute minimum for Kahler action. Number theoretical universality does not conform with this idea. In p-adic context the absolute minimum is however not a sensical notion since p-adic numbers are not well-ordered.

Number theoretical universality suggests that one should be able to completely algebraize the "preferred". Hamilton-Jacobi structure/4-D complex structure does this in Minkowskian/Euclidian regions and leads to deep analogies with string theory. What is so nice that all tensor contractions in field equations vanish automatically thanks to this structure. In string models holomorphy leads to the same outcome. The hypothesis that space-time surfaces are quatermionic in some sense does the same and I hope that the identifications are actually equivalent.

It is amusing that Equivalence Principle, the second great principle of GRT, holds true for preferred extremals!

So thanks. Because of my course examinations at the next days, i will try to continue the discussion at some days later.

http://phys.org/news/2013-01-dark-energy-alternatives-einstein-room.html#ajTabs

Post a Comment

<< Home