Considerable progress in the understanding of M8-H duality

The idea of M⁸-H duality (H=M⁴× CP₂) has progressed through frustratingly many several twists and turns. The evolution of the ideas Consider first the development of the key ideas and the related problems.

1. The first key idea (see this, this and this) was that one can interpret octonions O as Minkowski space M⁸ (see this) by using the number theoretic inner product defined by the real part Re(o₁o₂) of the octonion product. Later I gave up this assumption and considered complexified octonions, which do not form a number field, but finally found that the original option is the only sensible option.
2. The second key idea was that if either the tangent spaces T or normal spaces N of Y⁴ ⊂ M⁸ are quaternionic and therefore associative, and also contain a commutative subspace C, they can be parameterized by points of CP₂ and mapped to H=M⁴× CP₂. This would be the first half or M⁸-H duality.
3. How to map the M⁴ ⊂ M⁸ projection to M⁴× CP₂? This question did not have an obvious answer. The simplest map is direct identification whereas the inversion with respect to the cm or tip of causal diamond cd⊂ M⁴⊂ H is strongly suggested by Uncertainty Principle and the interpretation of M⁸ coordinates as components of 8-momentum (see this. Note that one can considerably generalize the simplest view by replacing the fixed commutative subspace of quaternion space M⁴ with an integrable distribution of them in M⁸.
4. I considered first the T option in which T was assumed to be associative. The cold shower was that there might be very few integrable distributions of associative tangent spaces (see this and this). As a matter of fact, M⁴ and E⁴ were the only examples of associative 4-surfaces that I knew of. On the other hand, any distribution of quaternionic normal spaces is integrable and defines an associative surface Y⁴. This led to a too hasty conclusion that only the N option might work.
5. If M⁸ is not complexified, the surfaces Y⁴ in M⁸ are necessarily Euclidean with respect to the number theoretic metric (see this). This is in sharp conflict with the original intuitive idea that Y⁴ has a number theoretic Minkowski signature. Is it really the normal space N , which must have a Minkowskian signature? Is also T possible.
6. The minimal option in which M⁸-H duality determines only the 3-D holographic data as 3-surfaces Y³⊂ M⁸ mapped by M⁸-H duality to H. The images of Y³ could define holographic data consistent with the holography = holomorphy (H-H) vision (see this, this, this, this and this). Both M⁸ and H sides of the duality would be necessary.

Interpretational problems

There are also interpretational problems.

1. The proposed physical interpretation for the 4-surface Y⁴⊂ M⁸ was as the analog of momentum space for a particle identified as a 3-D surface. In this interpretation the Y⁴ would be an analog of time evolution with time replaced with energy. A more concrete interpretation of the 3-D holographic data would be as a dispersion relation and Y⁴ could also represent off-mass shell states. Momentum space description indeed relies on dispersion relations and space-time description to the solutions of classical field equations.
2. For N option Y⁴ must be Euclidean in the number theoretic metric. Therefore the momenta defined in terms of the tangent space metric are space-like. What does this mean physically?

   Could the problem be solved if the momentum assignable to a given point of Y⁴ is identified as a point of its quaternionic normal space as proposed in (see this).

   Or should one accept both T and N options and interpret the Euclidean Y⁴ as as a counterpart of a virtual particle with space-like momenta and of CP₂ type extremals at the level of H. At vertices belonging to Y⁴₁∩ Y⁴₂ me, quaternionic N(Y⁴₁) would contain points of quaternionic T(Y⁴₂) so that the earlier proposal would not be completely wrong.

3. A further criticism against the M⁸-H duality is that its explicit realization has been missing. For N option, the distributions of the quaternionic normal spaces N are always integrable but their explicit identification has been the problem. For T option even the existence of integrable distributions of T has remained open.

A possible solution of the problems of the earlier view

Consider now how the view to be described could solve the listed problems.

1. There are two options, which could be called T and N: either the local tangent space T or normal space N is quaternionic. Which one is correct or are both correct?

   Any integrable distribution of quaternionic normal spaces is allowed whereas for tangent spaces this is not the case. This does not mean that non-trivial solutions would not exist. Perhaps the rejection of the T option was too hasty.

   Furthermore, for X⁴⊂ H both Minkowskian and Euclidean signatures of the induced metric are possible: could T and N option be their M⁸ counterparts?

2. Holography= holomorphy vision (H-H) allows an explicit construction of the space-time surfaces X⁴⊂ H. For Y⁴⊂ M⁸ the situation has been different. The very nature of duality concept suggests that the explicit construction must be possible also at the level of H.
3. Is M⁸-H duality between 4-D surfaces Y⁴⊂ M⁸ and space-time surfaces X⁴⊂ H or only between the 3-D holographic data Y³⊂ H and X³⊂ M⁸-H?

It turns out that a modification of the original form of the M⁸-H duality, formulated in terms of a real octonion analytic functions f(o):O→ O, leads to a possible solution of these problems.

1. All the conditions f(o)=0, f(o)=1 and Imf)(o)=0, and Ref)(o)=0 are invariant under local G₂ acting as as a dynamical spectrum generating symmetry group since f_º g₂= g_2º f holds true. The task reduces to that of finding the 4-surfaces with a constant quaternionic T or N.
2. In particular, M⁴⊂ M⁸ has been hitherto the only known Y⁴ of type T and the action of local G₂ generates a huge number of Y⁴ of type T. Both T and N option are possible after all! G₂ symmetry applies also to the N option for which E⁴ is the simplest representative!
3. The roots of Im(f)(o)=0 resp. Re(f)(o)=0 are unions ∪_o0S⁶(o₀) of 6-spheres, where o₀ is octonionic real coordinate o₀. The 3-D union Y⁴= ∪_o0S⁶(o₀) ∩ M⁴=S²(o₀)⊂ M⁴ has quaternionic tangent space T=M⁴. The interpretation as holographic data and the M⁸ counterpart of a partonic orbit is suggestive.
4. The Euclidean 3-surface Y³= S⁶(o₀) ∩ E⁴(o₀)=S³(o₀) could serve as a holographic data for Y⁴ with quaternionic normal spaces and with an Euclidean number theoretic signature of the metric. Obviously, the option Y⁴=∪_o0 Y³(o₀) fails to satisfy this condition. The interpretation would be as the M⁸ counterpart of CP₂ type extremal with a Euclidean signature of the induced metric. The identification as the M⁸ counterpart of a virtual particle with space-like momentum is suggestive.

   If T resp. N contains a commutative hyper-complex subspace, it corresponds to a point of CP₂. Hence Y⁴ can be mapped to X⁴⊂ H=M⁴× CP₂ as M⁸-H duality requires.

5. What could be the counterpart of H-H vision in M⁸? One can choose the function f(o) to be an analytic function of a hypercomplex coordinate u or v of M⁴ and 3 complex coordinates of M⁸. The natural conjecture is that the image X⁴ of Y⁴ has the same property and satisfies H-H.

This view solves the interpretational problems.

1. The proposed physical interpretation for the 4-surface Y⁴⊂ M⁸ was as the analog of momentum space for a particle identified as a 3-D surface. The interpretation the Y⁴ as the analog of time evolution with time replaced with energy looks range. A more concrete interpretation of the 3-D holographic would be as a dispersion relation emerges and Y⁴ could also represent off-mass shell states. Momentum space description indeed relies on dispersion relations and space-time description to the solutions of classical field equations.

   Number theoretic discretization as a selection of points as elements of the extensions of rationals defining the coefficient field for f(o) and the replacement of fermions to the "active" points of discretization would realize many fermion states at the level of H. Galois confinement (see this and this) stating that the total momenta are rational numbers would provide a universal mechanism for the formation of bound states.

2. For N option Y⁴ must be Euclidean in the number theoretic metric. Therefore the momenta defined in terms of the tangent space metric are space-like. What does this mean physically?

   Could the problem be solved if the momentum assignable to a given point of Y⁴ is identified as a point of its quaternionic normal space as proposed in (see this).

   Or should one accept both T and N options and interpret the Euclidean Y⁴ as as a counterpart of a virtual particle with space-like momenta and of CP₂ type extremals at the level of H.

   At 2-D vertices belonging to the intersection Y⁴₁∩ Y⁴₂, quaternionic N(Y⁴₁) would contain points of quaternionic T(Y⁴₂) so that the first proposal would not be completely wrong.

3. Could the TGD analogs of Feynman diagrams be built by gluing together T and N type surfaces Y⁴ along 3-surfaces Y³ defining analogs of vertices. In the role of consciousness theorist, I have called them "very special moments in the life of self" (see this) at which the non-determinism of the classical field equations in H-H vision is localized. At these 3-surfaces the smoothness of Y⁴ fails and could give a connection to the notion of exotic smooth manifold (see this, this, and this), conjectured to make possible particle vertices and fermion pair creation in TGD despite the fact that fermions in H are free (see this, this and this).

See the article Still about M⁸-H duality or the chapter chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Did Google quantum computer change the arrow of time?

The latest FB news is that Google quantum computer has changed the arrow of time for a period of about 1 second. There is no publication of this yet but IBM made a similar announcement in 2019 (see this). It remains unclear to me what they have achieved without knowing what they mean with the time reversal.

1. Time reversal can be interpreted as a time reflection, which is a slightly broken discrete symmetry in fundamental physics. Do they construct a time revered time evolution in which initial and final states are permuted? Since complex conjugation is associated with the time reversal, a positive Hamiltonian can induce the time-reversed time evolution and no new physics would be involved.
2. Time reversal can be also interpreted as a thermodynamical time reversal. The reversal of the thermodynamical arrow of time is thought to be impossible in standard thermodynamics. There are however anomalies suggesting that this is possible. Phase conjugate laser rays are a basic example. In biology Pollack effect suggests the reversal of the arrow of time in the negatively charged exclusion zone. The inpurities are cleaned out of the system whereas thermodynamics suggests the reversal. Dissipation with a reversed arrow of time.
3. In TGD, time reversal occurs in "big" state function reductions (BSFRs) occurring in quantum measurement and induce a thermodynamical time reversal. In "small" SFRs (SSFRs), which have interpretation as internal quantum measurements of the system involving no external observer and assumed to give rise to cognition and consciousness, this does not happen. These measurements would take place in the discrete degrees of freedom predicted to be associated with the non-determism, of the classical dynamics. The sequence of SSFRs defines a conscious entity self, which would die and reincarnate with an opposite arrow of time in BSFR. Falling asleep or biological death would be familiar examples of this.

By writing "Google's quantum computer reversed the arrow of time" to Google, one learns more. AI claims that the thermodynamic arrow was not really changed. A quantum computational feat would be in question. But what does it really mean? The computer is externally controlled and the time evolution A→B is continued so that one has A→B→A. This means that a quantum measurement by an external system takes place at time T when the B→A starts.

In the TGD framework this would mean that the control inducing what looks B→A corresponds to a BSFR at time T, which reverses the arrow of geometric time. A time evolution to the geometric past by SSFRs begins at time T-ΔT₁ and eventually at time T-ΔT₂ a second BSFR occurs and the evolution with the standard arrow of time time by SSFRs begins, not at T but, at T+ΔT. ΔT would be about 1 second. The same happens as we fall asleep: we wake-up after, say, ΔT=12 hours but make a time travel to the geometric past during sleep lasting for say, 12 hours. If this interpretation is correct, the experiment could provide a direct support for the zero energy ontology of TGD.

However, a more precise TGD view of what happened is needed. In TGD cognition is predicted to be present in all scales so that I will approach the question from the point of view of TGD inspired theory of consciousness.

1. There are very delicate details involved. Second law says that the entropy S(S) of a closed system S increases. Now S is the quantum computer. There is also the entanglement entropy S(S-O) between S and that observer O. It is reported that the entropy of the quantum computer decreased during the period B→ A. It is not clear to me whether this entropy was S(S) or S(O,S)? If the system was closed during this period, the decrease of S(S) would allow us to conclude that the arrow of time was effectively changed.
2. What does the period B→ A does correspond to in TGD? Suppose that it corresponds to [T-Δ T₁, T-Δ T₂ when geometric time decreases. What does the entropy correspond to. Does it correspond to the entanglement entropy of the system + observer or to the sum of this entropy and system's internal entropy? Or is also the system s internal entropy basically entanglement entropy?
3. Intuitively it looks obvious that the first BSFR increases the information of an external observer of the system and reduces S(S-O) to zero. If the internal degrees of freedom are not entangled with external degrees of freedom, BSFR leaves S(S) unaffected. The time reversed time evolution would increase S(S) but in the opposite time direction. If S(S-O) is generated, it is reduced to zero in the second BSFR. S(S) would increase during the period [T-Δ T₁, T-Δ T₂. If an observer O with a standard arrow of time were able to observe S during this period, it would see this as a decrease of S(S).

   Also in the second BSFR the entropy would have decreased and the evolution in standard direction would have started at T+Δ T. It is difficult to say whether the entropy increased or decreased. If this was the case, the experiment would provide direct support for the zero energy ontology of TGD.

4. To gain some insight, one can compare the situation to what happens during sleep. Sleep has positive effects, perhaps due to the fact that it reduces the entropy of the sleeper. These positive effects are however felt also subjectively, rather than perceived by the external observer. Should one identify O as the field body of the system observing the biological body or is some other interpretation more appropriate?
5. There is a further delicacy involved. In TGD, the fundamental objects are 4-surfaces as slightly non-deterministic analogs of Bohr orbits for particles identified as 3-surfaces. The discrete degrees of freedom associated with the non-determinism are identified as cognitive degrees of freedom. Therefore it might make sense to speak of cognitive entropy S(cogn) associated with them.

   S(cogn) would increase as we get tired and would be reduced during sleep. Could S(cogn) correspond to entanglement entropy between cognitive degrees of freedom and those of the external world? If so, S(cogn) would contribute also to S(S-O). In BSFR also S(cogn) would be reduced to zero. If this view holds true, then the second BSFR would be responsible for the decrease of entanglement entropy.

Note that there are also other kinds of entanglements involved. Consider two systems A and B.

1. The IDF of A need not entangle with the ODF of A although the ODF of A and B can entangle. Could this relate to sensory perception?
2. The IDF of A can entangle with the ODF B. Could this make possible psychokinesis and hypnosis?
3. The IDF of A and B can entangle. Could this relate to telepathy?
4. Entanglement can also occur between the IDF and ODF of A. Could this relate to the realization of intentions in motor degrees of freedom?

See the article Are Conscious Computers Possible in TGD Universe? or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

The most distant known galaxy JADES-GS-z14-0 has high metallicity and 10 times higher oxygen content than expected

The popular article with title "Oxygen Has Been Discovered In The Most Distant Known Galaxy JADES-GS-z14-0" (see this) tells that the most distant known Galaxy JADES-GS-z14-0 existed about 300 million years after Big Bang. It was discovered by James Webb and ALMA telescope has continued the observations. What is totally surprising is that the stars of JADES-GS-z14-0 have high metallicity. In particular, the oxygen content is 10 times higher than expected. In the standard view of stellar evolution, this requires that the stars have suffered several supernova explosions. This is not possible.

There is an article by Schouws et al in arXiv with title "Detection of [OIII]88μm in JADES-GS-z14-0 at z=14.1793" (see this). Here is the abstract of the article.

We report the first successful ALMA follow-up observations of a secure z>10 JWST-selected galaxy, by robustly detecting (6.6σ) the [OIII]88μm line in JADES-GS-z14-0 (hereafter GS-z14). The ALMA detection yields a spectroscopic redshift of z=14.1793+/- 0.0007, and increases the precision on the prior redshift measurement of z=14.32^+0.08_-0.20 from NIRSpec by ≥ 180×. Moreover, the redshift is consistent with that previously determined from a tentative detection (3.6σ) of CIII]1907,1909 (z=14.178+/- 0.013), solidifying the redshift determination via multiple line detections. We measure a line luminosity of L[OIII]88=(2.1+/- 0.5)× 10⁸L_Sun, placing GS-z14 at the lower end, but within the scatter of, the local L[OIII]88-star formation rate relation.

No dust continuum from GS-z14 is detected, suggesting an upper limit on the dust-to-stellar mass ratio of < 2× 10^-3, consistent with dust production from supernovae with a yield y_d<0.3M_Sun. Combining a previous JWST/MIRI photometric measurement of the [OIII]λλ 4959,5007 Angstrom and Hβ lines with Cloudy models, we find GS-z14 to be surprisingly metal-enriched (Z∈ [0.05,0.2]Z_Sun), a mere 300 Myr after the Big Bang. The detection of a bright oxygen line in GS-z14 thus reinforces the notion that galaxies in the early Universe undergo rapid evolution.

This finding conforms with the general TGD based view of the formation of galaxies and stars (see this and this). Galaxies would not be formed by gravitational condensation but by the thickening of tangles of a cosmic string leading to the liberation of energy giving rise to ordinary matter, somewhat like inflation theory. Also the intersections of two cosmic strings and self intersections could be involved and generate galactic nuclei. The model explains the flat galactic rotation curves in terms of dark energy assignable to cosmic strings.

Also stars would be formed by a similar mechanism (see this). That no dust continuum created by supernova explosions was observed, is consistent with the assumption that no supernova explosions have occurred as standard model requires in order to explaing the high metallicity.

This explosive process can be considerably faster than the formation by gravitational condensation and dominate in the very early cosmology. TGD leads also to a model of stars based on new physics predicted by TGD and differing dramatically from the standard view (see this) and could change profoundly the views about stellar evolution.

See the article ANITA anomaly, JWST observation challenging the interpretation of CMB, star formation in the remnant of a star, and strange super nova explosion or the chapter About the recent TGD based view concerning cosmology and astrophysics.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

The internal degrees of freedom related to the classical non-determinism and consciousness

In TGD, consciousness and cognition are assigned with the internal degrees of freedom (IDF) assignable with the classical non-determinism. Ordinary SFRs (BSFRs) are assigned with the ordinary degrees of freedom (ODF) assignable to the entire Bohr orbits and measured in the ordinary quantum measurements. Several questions related to the relationship of IDF and ODF) come to mind.

Consider two systems A and B.

1. The IDF of A need not entangle with the ODF of A although the ODF of A and B can entangle. Could this relate to sensory perception?
2. The IDF of A can entangle with the ODF B. Could this make possible psychokinesis and hypnosis?
3. The IDF of A and B can entangle. Could this relate to telepathy?
4. Entanglement can also occur between the IDF and ODF of A. Could this relate to the realization of intentions in motor degrees of freedom?

The precise role of quantum criticality should be understood. Conservation laws pose strong restrictions here: a stable particle like proton or electron serves as an example. The intuitive idea is that a perturbation is needed to trigger a BSFR, which transforms intention realized as entanglement to a motor action or to an action affecting the external world. Is the quantum entanglement of IDF with its ODF enough to trigger a BSFR of the system: a spontaneous decay of an unstable particle would be an example now.

See the article The problem of time and the TGD counterpart of F= ma or the chapter Comparing the S-matrix descriptions of fundamental interactions provided by standard model and TGD.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Is cognition present at elementary particle level and are particle reactions intentional actions?

An innocent looking question "What F=ma means in TGD?", posed by Lawrence B. Crowell in a FB discussion, can be abstracted to the question how the transfers of conserved isometry charges of H=M⁴× CP₂ are realized at the level of fundamental interactions. At this level, the question is about how the conserved charges associated with the initial state particles are redistributed between the final state particles.

Somewhat surprisingly, TGD based quantum ontology implies that quantum non-determinism is an essential part of the answer to the question. Equally surprisingly, also a connection with the theory of consciousness and cognition emerges at the fundamental elementary particle level.

In the TGD view, the weak violation of the classical non-determinism in holography = holomorphy vision of TGD leads to the identification of self as a sequence of "small" state function reductions (SSFRs) identified as TGD counterparts of repeated measurements of the same observables: now however the observables related to the non-determinism are measured in SSFRs and give rise to the correlates of cognition. By quantum criticality, a "big" state function reduction (BSFR) as the TGD counterpart of what occurs in quantum measurement, can take place. BSFR means the death of self and its reincarnation with an opposite arrow of time.

Quantum criticality of the TGD Universe, realized in terms of holography = holomorphy principle, would be essential for this. For instance, particle decay involving topological changes could correspond to this process and all particle interactions would basically be due to the quantum criticality so that instead of SSFR, BSFR takes place. Intention is transformed to action at fundamental level. Cognition is dangerous in the TGD Universe!

See the article The problem of time and the TGD counterpart of F= ma or the chapter Comparing the S-matrix descriptions of fundamental interactions provided by standard model and TGD.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

From the problem of time in general relativity to the TGD counterpart of F= ma

This contribution was inspired by a posting of Lawrence B. Crowell related to one particular problem related to the notion of time in general relativity: the general coordinate invariance implies interpretational problems since it is difficult to identify any preferred time coordinates used by observers. The identification of spatial coordinates is problematic for the same reason.

There are also other deep problems. In curved space-time, the classical conservation laws are lost. Einstein's equations are formulated in terms of energy momentum tensor but Newton's equations (F=ma) expressing the conservation of momentum and energy cannot be formulated in a general coordinate invariant way since the notions of energy and momentum fluxes are lost. No F=ma!

The TGD view of the problem of time inspired Lawrence B. Crowell to ask about the TGD counterpart of $F=ma$. This question can be abstracted to the question about how the transfers of conserved isometry charges of H=M⁴× CP₂ are realized at the level of fundamental interactions. At this level the question is about how the conserved charges associated with the initial state particles are redistributed between the final state particles.

Somewhat surprisingly, TGD based quantum ontology implies that quantum non-determinism is an essential part of the answer to the question. Equally surprisingly, also a connection with the theory of consciousness and cognition emerges at the fundamental elementary particle level.

A detailed consideration of the question of Crowell led to a considerable clarification of the previous views. In the TGD view, the weak violation of the classical non-determinism in holography = holomorphy vision of TGD leads to the identification of self as a sequence of "small" state function reductions (SSFRs) identified as TGD counterparts of repeated measurements of the same observables: now however the observables related to the non-determinism are measured in SSFRs and give rise to the correlates of cognition. By quantum criticality, a "big" state function reduction (BSFR) as the TGD counterpart of what occurs in quantum measurement, can take place. BSFR means the death of self and its reincarnation with an opposite arrow of time.

Quantum criticality of the TGD Universe, realized in terms of holography = holomorphy principle, would be essential for this. For instance, particle decay could correspond to this process and all particle interactions would basically be due to the quantum criticality so that instead of SSFR, BSFR takes place. Intention is transformed to action but a self at some level of the hierarchy dies! Thinking is dangerous in the TGD Universe!

The 3 problems related to the notion of time

It is good to beging with my first comment to the posting of Lawrence Crowell.

1. In the materialistic ontology subjective and geometric time are identified and this leads to deep problems.

   In TGD zero energy ontology (see this and this) allows both times and solves both the measurement problem and the problem of free will. Subjective time corresponds to a sequence of "small" state function reductions (SSFRs) replacing the Zeno effect, in which nothing happens, with the notion of self.

   Zero energy ontology replaces space-time surfaces as analogs of Bohr orbits for 3-surfaces having an interpretation as a geometric representation of particles. The classical dynamics is slightly non-deterministic although field equations are satisfied. This has crucial implications for the description of fundamental interactions (see this) and this). This non-determinism also makes possible the description of physical correlates of cognition.

2. The second problem relates to the geometric time. General coordinate invariance allows an endless number of identifications of the time coordinate. In TGD, space-times are 4-surfaces in H=M⁴× CP₂ and M⁴ provides linear Minkowskian time or light-cone proper time (cosmic time) as a preferred time coordinate for the space-time surfaces.
3. The loss of Poincare invariance in General Relativity is the third problem and led to TGD. In TGD one obtains classical conservation laws due to the Poincare invariance of M⁴ factor of H.

What does F= ma mean in TGD?

Lawrence Crowell asked about how the counterpart of F= ma emerges in TGD. At the general level the answer is as follows.

1. F= ma states momentum conservation for a particle plus its environment by characterizing the momentum is exchanged between the particle and environment. In TGD, momentum conservation generalizes to field equations for the space-time surfaces as analogs of Bohr orbits of particles identified as 3-surfaces. The field equations state the conservation of Poincare and color charges classically.

   Newton's equations are not given up as in General Relativity and their generalization defines the dynamics of space-time surfaces. Formally TGD is therefore like hydrodynamics. Einstein's equations follow naturally at the QFT limit as a remnant of Poincare invariance when the sheets of the many-sheeted space-time are replaced with a slightly curved region of M⁴ carrying the sum of induced gauge fields and gravitational fields defined as deviation from M⁴ metric.

2. This can be made more precise holography = holomorphy (H-H) hypothesis ( (see this, (see this, and this) reduces the field equations to local algebraic equations in terms of generalized holomorphy irrespective of action as long as it general coordinate invariant and expressible in terms of the induced geometry. This generalizes the role of holomorphy in string models.

   In conformal field theories holomorphy is a correlate for criticality. In TGD it would be a correlate for quantum criticality in the 4-D sense. This principle is extremely powerful since various dynamical parameters are analogous to a critical temperature.

   Space-time surfaces as minimal surfaces become analogs of Bohr orbits. Minimal surface equations generalize massless field equations and the TGD counterparts of field equations of gauge theories follow automatically.

3. Holomorphy is violated as 3-D surfaces which are analog for the singularities of analytic functions. This can be seen as a generalization of the fact that analytic functions can be expressed in terms of the holographic data given at poles and cuts.
4. What is crucial is that there is a light failure of determinism (but not of field equations). This occurs also for soap films, modellable as 2-D minimal surfaces: the frames do not uniquely determine the soap film. In TGD, the identification of this non-determinism as a p-adic non-determinism is attractive and leads to a generalization of the notion of p-adic number field to a function field (this and this ).

   H-H vision leads to a long-sought-for understanding of the origin of p-adic length scales hypothesis which for 30 years ago led to a surprisingly successful particle mass calculations based on p-adic thermodynamics (see this, this and this). The most recent article is about the application to the calculation of the mass spectrum of quarks and hadrons this). The p-adic non-determinism and p-adic length scale hypothesis would have origin in the iterations of polynomials defining dynamical symmetries in H-H vision and also giving a connection to the Mandelbrot fractals and Julia sets becomes possible.

Geometric and fermionic counterparts of F= ma in TGD

The next question concerns the counterpart of F= ma at geometric and fermionic levels respectively.

F= ma is a simple model for interactions. How are the interactions of two space-time surfaces A and B as analogs of Bohr orbits described geometrically?

1. Geometric vision suggests that a generalization of a contact interaction is in question. The intersection A and B defines the contact points. Without additional assumptions the intersection of A and B would be a discrete set of points. One can argue that this is not enough.
2. The intuitive idea is that there must exist additional prerequisites for the formation of a quantum coherent structure, at least in the interaction region. The proposal is that A and B share a common generalized complex structure, which I call Hamilton-Jacobi structure (see this) so that the conformal moduli defining the H-J structure would identical.

   The common H-J coordinates involve hypercomplex coordinate pair u,v with light-like coordinate curves, common complex M⁴ coordinate w and common complex CP₂ coordinates. The analytic functions defining A and B as their root must be generalized analytic functions of the same H-J coordinates.

3. The solution of field equations in Minkowskian space-time regions implies that either u or v is a passive coordinate since it cannot appear in the generalized analytic functions (the real hypercomplex coordinates u and v are analogous to z and z). The 2-D surfaces at which u and v vary, are generalizations of straight strings of M⁴ and dynamically very simple. Vibrational string degrees of freedom are frozen but the string ends at the partonic orbits are dynamical and can carry fermion numbers.

   In this case, the intersection of A and B consists of 2-D string world sheets connecting light-like partonic orbits. This gives a connection with string model type description.

4. Self-interactions of the space-time surface correspond to self-intersections consisting of string world sheets. For instance, the description of the internal dynamics of hadrons (see this and this) is realized in terms of self-intersections.

In the fermionic sector modified/induced Dirac equation at the space-time level holds true for the induced spinor fields and can be solved exactly by the holomorphy just as in the case of string models. At the level of scattering amplitudes the dynamics reduces to the fermionic N-point functions.

Propagators are free propagators in H and the hard problem is to understand how fermion pair creation is possible when fermions are free in H. The notion of exotic smooth structure, possible only for 4-D space-time surfaces, solves the problem.

How to translate F=ma to a view about the transfer of isometry charges between initial and final state particles?

Let us return to the original questions. How can one understand the generalization of F= ma in terms of a transfer of isometry charges of A (momenta color charges) from the initial state particles A and B to the final state particles? The classical field equations state the local conservation of isometry currents. How can this give rise to a transfer of total charges?

1. In the interaction region the Hamilton-Jacobi structures for A and B must be identical. Intersection consists of string world sheets. The interacting state therefore differs from the non-interacting state. Intuitively it is clear that the incoming states in the distant geometric past approach disjoint Bohr orbits. This is true also in the remote future except that the scattering need not be elastic and the particles identifiable bremsstrahlung can be emitted in the interactions. The interaction can also induce the decay of A and B. What happens in hadronic reactions gives a good idea of what happens.
2. The key notion is the mild failure of classical determinism for the Bohr orbits, which also characterizes criticality. The minimal surfaces describing the space-time surfaces have 3-D loci of non-determinism at which the classical determinism fails. These loci are analogous to the 1-D frames spanning 2-D soap films, which are also slightly non-deterministic minimal surfaces. There are several soap films spanned by the same collection of frames. The non-determinism gives rise to a sequence of small state function reductions (SFRs) generalizing the Zeno effect of standard quantum mechanics.

The description of the scattering in space-time degrees of freedom

Consider first the situation for a single particle as a 4-D Bohr orbit.

1. The non-determinism gives rise to internal interactions assignable to the self intersection as string world sheets. In the TGD inspired theory of consciousness, they can be identified in terms of geometric correlates of cognition.
2. Thinking is however dangerous also at the elementary particle level! The non-determinism is associated with quantum criticality and can lead to the decay of a partonic orbit to two or even more pieces. It can also change the topology of the partonic 2-surface characterized by genus g (CKM mixing). The partonic decay can in turn induce the decay of the space-time surface itself. The outcome would be a particle decay.

   This also leads to an emission of virtual particles as Bohr orbits, which appear as exchanges in 2-particle interactions. Massless extremals/topological light rays (see this) as counterparts of massless modes of gauge fields can be emitted. Closed 2-sheeted monopole flux tubes as geometric particles can be created by a splitting of a single monopole flux pair by a reconnection: this would be involved with a particle decay and emission of a virtual particle.

3. The scattering of two interacting particles A and B reduces to self interaction in the interaction region behaving like a single particle. The slight non-determinism makes possible a geometric generalization of the Feynman diagram type description. Now however all would be discrete and finite. There would be no path integral and therefore no divergences.

   The outcome would be a classical description and the generalization of F= ma would code for the transfers of isometry charges from A and B to the final state particles generated in the scattering. The slight classical non-determinism would make this possible.

The description of the scattering in fermionic degrees of freedom

The description of the scattering in fermionic degrees of freedom involves highly non-trivial aspects.

1. In the fermionic degrees of freedom, the fermionic propagators between points of the singular 3-surfaces defining the interaction vertices as ends of string world sheets of the intersection would describe the situation.
2. The crucial point is the possibility of fermion pair creation only for 4-D space-time surfaces due to the existence of the exotic smooth structures (see this, this). In other space-time dimensions the fermions would be free.
3. Fermion pair creation corresponds intuitively to the turning of a fermion line in time direction. At the 3-D holomorphic singularities X³ (analogous to cuts of analytic functions) the minimal surface equations fail and the additional pieces of the classical action, in particular 4-volume as the analog of cosmological constant term, become relevant. The twistor lift of TGD (see this and this) implies that the action is sum of K\"ahler action and volume term.
4. X³ represents a defect of the standard smooth structure and the turning of a fermion line at X³ at which standard smoothness fails corresponds to the transformation of u-type coordinate curve to a v-type coordinate curve takes place in the pair creations. u and v are associated with the parallel Minkowskian space-time sheets of the 2-sheeted space-time region. The creation of fermion occurs at the boundaries of two string world sheets at different monopole flux tubes so that a decay of monopole flux tube to a pair of them occurs.

   The pair creation would take place in the interaction regions and lead to the generation of final state fermions as the decay to 3-surfaces takes place. Closed 2-sheeted monopole flux tubes carrying fermion lines at their ends defined by Euclidean wormhole contacts connecting two sheets are generated.

5. At the 3-D defects minimal surface property fails and the trace of the second fundamental form, call it H^k, which vanishes almost everywhere, has a delta function singularity. By its group theoretical properties H^k has an interpretation as a generalized Higgs field. What is new is its M⁴ part, which has an interpretation as a local acceleration for a 4-D Bohr orbit. The same interpretation applies to the TGD counterpart of the ordinary Higgs.
6. The vertices at 3-D singularities are analogous to points at which the direction of the motion for a Brownian particle changes. Conformal invariance suggests that the 8-D Higgs vector H^k is light-like so that one has H_kH^k=0. Higgs has the dimension of ℏ/length and its vanishing gives rise to the analog of 8-D massless fixing M⁴ mass squared in terms of CP₂ mass squared. A reasonable guess is that the square of M⁴ part of H^k is proportional to particle mass squared. This would give quantum-classical correspondence.

See the article The problem of time and the TGD counterpart of F= ma or the chapter Comparing the S-matrix descriptions of fundamental interactions provided by standard model and TGD.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

The surprise of the year: a Nobel prize for discovering macrosopic quantum tunnelling!

I must confess  that the work of Nobel prize winners John Clarke, Michel H. Devoret, and John M. Martinis "for the discovery of macroscopic quantum mechanical tunnelling and energy quantisation in an electric circuit" (see this) came as a total surprise for me.

Macroscopic quantum coherence is what makes possible quantum tunnelling in macroscopic scales but the value of Planck constant h is too small to make it plausible.

I have been talking about macroscopic quantum coherence for decades as a basic prediction of Topological Geometrodynamics (TGD). TGD forces us to generalize the standard quantum theory. One prediction is the spectrum h_eff=nh₀ of effective Planck constants as multiples of the minimal value h₀ of Planck constant which is smaller than h. It however seems that the multiples h_eff=nh are the most  important ones. Quantum coherence becomes possible in even astrophysical scales and would play a central role in the TGD inspired quantum biology.    

In the work of Nobel laureates, the quantization of energy and and macroscpic tunnelling were discovered in Josephson junctions.  An interesting question is whether the macroscopic quantum coherence observed in the quantum tunnelling can be really  explained in terms of ordinary quantum mechanics or whether TGD is required. One can indeed  argue that the extremely low temperatures presumably use in the experiments make it possible to have macroscopic quantum coherence in superconducting systems even in the framework of standard quantum mechanics.

The basic mystery of biological systems is their coherence. Could it be induced by macroscopic quantum coherence?  The quantum coherence at room  temperature is  definitely in conflict with standard quantum theory assuming a single value of Planck constant.  Macroscopic quantum tunnelling in high temperature superconductors would  obviously challenge the standard quantum mechanics.

TGD  could explain the quantum coherence at room temperature.  The new view of space-time leads to the notion of a field body, having  magnetic, electric and gravitational field bodies as special cases. These field bodies  carry  phases of ordinary matter with large h_eff. The quantum coherence at the level of  the field body, serving as a controller of the biological body,  would induce the macroscopic coherence (not quantum-) of  the ordinary biomatter.  

For instance, EEG would   relate  to the communications to and the  control by  the field body, which can have layers with the size of the Earth and even larger. In TGD quantum coherence at room  temperature superconductivity would make possible cell membranes as Josephson junctions. They would  play a central role in TGD inspired  quantum biology and neuroscience. The model of nerve pulse and EEG (see this) is one example.  

See for instance https://tgdtheory.fi/public_html/articles/TGD2024I.pdf and https://tgdtheory.fi/public_html/articles/TGD2024II.pdf .

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

A refined view of the phenomenology of hadron physics and p-adic mass calculations

In the recent article with title "A refined view of the phenomenology of hadron physics and p-adic mass calculations" (see this) the implications of the updated vision of standard model physics and hadron physics are considered. The goal is to develop a phenomenological picture of hadrons based on the general mathematical framework of TGD and on the interpretation of strong and weak interactions as different aspects of color interaction.

The additivity of the mass squared values identified as conformal weights at the level of the embedding space $H$ is a crucial assumption made also in the p-adic mass calculations. One must check whether this assumption is physically sensical and how it relates to the additivity of masses assumed in the constituent quark model and understand the relation between the notions of current quark mass and constituent quark mass. One should also identify various contributions to the hadron mass squared in the new picture and understand the hadronic mass splittings.

The results of the simple calculations deducing the p-adic mass scales of hadrons and quarks mean a breakthrough in the quantitative understanding of the hadronic mass spectrum. In particular, the identification of color interactions in fermionic isospin degrees of freedom as weak interactions with a p-adically scaled up range explains the mass splittings due to isospin. The smallness of the Weinberg angle for scaled up weak interactions can explain how the interactions become strong and why the parity violation for strong interactions is small.

A refined view of the phenomenology of hadron physics and p-adic mass calculations or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.