About some unclear technical issues of TGD

TGD has been in the middle of palace revolution during last two years and it is almost impossible to keep the chapters of the books updated. Adelic vision and twistor lift of TGD are the newest developments and there are still many details to be understood and errors to be corrected. The description of fermions in TGD framework has contained some unclear issues. Hence the motivation for the following brief comments.

Adelic vision and symmetries

In the adelic TGD SH is weakened: also the points of the space-time surface having imbedding space coordinates in an extension of rationals (cognitive representation) are needed so that data are not precisely 2-D. I have believed hitherto that one must use preferred coordinates for the imbedding space H - a subset of these coordinates would define space-time coordinates. These coordinates are determined apart from isometries. Does the number theoretic discretization imply loss of general coordinate invariance and also other symmetries?

The reduction of symmetry groups to their subgroups (not only algebraic since powers of e define finite-dimensional extension of p-adic numbers since e^p is ordinary p-adic number) is genuine loss of symmetry and reflects finite cognitive resolution. The physics itself has the symmetries of real physics.

The assumption about preferred imbedding space coordinates is actually not necessary. Different choices of H-coordinates means only different and non-equivalent cognitive representations. Spherical and linear coordinates in finite accuracy do not provide equivalent representations.

Quantum-classical correspondence for fermions

Quantum-classical correspondence (QCC) for fermions is rather well-understood but deserves to be mentioned also here.

QCC for fermions means that the space-time surface as preferred extremal should depend on fermionic quantum numbers. This is indeed the case if one requires QCC in the sense that the fermionic representations of Noether charges in the Cartan algebras of symmetry algebras are equal to those to the classical Noether charges for preferred extremals.

Second aspect of QCC becomes visible in the representation of fermionic states as point like particles moving along the light-like curves at the light-like orbits of the partonic 2-surfaces (curve at the orbit can be locally only light-like or space-like). The number of fermions and antifermions dictates the number of string world sheets carrying the data
needed to fix the preferred extremal by SH. The complexity of the space-time surface increases as the number of fermions increases.

Strong form of holography for fermions

It seems that scattering amplitudes can be formulated by assigning fermions with the boundaries of strings defining the lines of twistor diagrams. This information theoretic dimensional reduction from D=4 to D=2 for the scattering amplitudes can be partially understood in terms of strong form of holography (SH): one can construct the theory by using the data at string worlds sheets and/or partonic 2-surfaces at the ends of the space-time surface at the opposite boundaries of causal diamond (CD).

4-D modified Dirac action would appear at fundamental level as supersymmetry demands but would be reduced for preferred extremals to its 2-D stringy variant serving as effective action. Also the value of the 4-D action determining the space-time dynamics would reduce to effective stringy action containing area term, 2-D Kähler action, and topological Kähler magnetic flux term. This reduction would be due to the huge gauge symmetries of preferred extremals. Sub-algebra of super-symplectic algebra with conformal weigths coming as n-multiples of those for the entire algebra and the commutators of this algebra with the entire algebra would annihilate the physical states, and thecorresponding classical Noether charges would vanish.

One still has the question why not the data at the entire string world sheets is not needed to construct scattering amplitudes. Scattering amplitudes of course need not code for the entire physics. QCC is indeed motivated by the fact that quantum experiments are always interpreted in terms of classical physics, which in TGD framework reduces to that for space-time surface.

The relationship between spinors in space-time interior and at boundaries between Euclidian and Minkoskian regions

Space-time surface decomposes to interiors of Minkowskian and Euclidian regions. At light-like 3-surfaces at which the four-metric changes, the 4-metric is degenerate. These metrically singular 3-surfaces - partonic orbits- carry the boundaries of string world sheets identified as carriers of fermionic quantum numbers. The boundaries define fermion lines in the twistor lift of TGD. The relationship between fermions at the partonic orbits and interior of the space-time surface has however remained somewhat enigmatic.

So: What is the precise relationship between induced spinors Ψ_B at light-like partonic 3-surfaces and Ψ_I in the interior of Minkowskian and Euclidian regions? Same question can be made for the spinors Ψ_B at the boundaries of string world sheets and Ψ_I in interior of the string world sheets. There are two options to consider:

• Option I: Ψ_B is the limiting value of Ψ_I .
  
  
• Option II: Ψ_B serves as a source of Ψ_I .
  
  

For Opion I it is difficult to understand the preferred role of Ψ_B.

I have considered Option II already years ago but have not been able to decide.

1. That scattering amplitudes could be formulated only in terms of sources only, would fit nicely with SH, twistorial amplitude construction, and also with the idea that scattering amplitudes in gauge theories can be formulated in terms of sources of boson fields assignable to vertices and propagators. Now the sources would become fermionic.
   
   
2. One can take gauge theory as a guideline. One adds to free Dirac equation source term γ^kkA_kΨ. Therefore the natural boundary term in the action would be of the form (forgetting overall scale factor)
   

   S_B=Ψ_IΓ^α(C-S)A_αΨ_B+ c.c .
   

   Here the modified gamma matrix is Γ^α(C-S) (contravariant form is natural for light-like 3-surfaces) is most naturally defined by the boundary part of the action - naturally Chern-Simons term for Kähler action. A denotes the Kähler gauge potential.
   
   

3. The variation with respect to Ψ_B gives
   

   G^α(C-S)A_αΨ_I=0
   

   at the boundary so that the C-S term and interaction term vanish. This does not however imply vanishing of the source term! This condition can be seen as a boundary condition.
   
   

The same argument applies also to string world sheets.

About second quantization of the induced spinor fields

The anti-commutation relations for the induced spinors have been a long-standing issue and during years I have considered several options. The solution of the problem looks however stupifuingly simple. The conserved fermion currents are accompanied by super-currents obtained by replacing Ψ with a mode of the induced spinor field to get u_nΓ^αΨ or ΨΓ^αu_n with the conjugate of the mode. One obtains infinite number of conserved super currents. One can also replace both Ψ and Ψ in this manner to get purely bosonic conserved currents Ψ_mΓ^αu_n to which one can assign a conserved bosonic charges Q_mn.

I noticed this years ago but did not realize that these bosonic charges define naturally anti-commutators of fermionic creation and annihilation operators! The ordinary anti-commutators of quantum field theory follow as a special case! By a suitable unitary transformation of the spinor basis one can diagonalize the hermitian matrix defined by Q_mn and by performing suitable scalings one can transform anti-commutation relations to the standard form. An interesting question is whether the diagonalization is needed, and whether the deviation of the diagonal elements from unity could have some meaning and possibly relate to the hierarchy h_eff=n× h of Planck constants - probably not.

Is statistical entanglement "real" entanglement?

The question about the "reality" of statistical entanglement has bothered me for years. This entanglement is maximal and it cannot be reduced by measurement so that one can argue that it is not "real". Quite recently I learned that there has been a longstanding debate about the statistical entanglement and that the issue still remains unresolved.

The idea that all electrons of the Universe are maximally entangled looks crazy. TGD provides several variants for solutions of this problem. It could be that only the fermionic oscillator operators at partonic 2-surfaces associated with the space-time surface (or its connected component) inside given CD anti-commute and the fermions are thus indistinguishable. The extremist option is that the fermionic oscillator operators belonging to a network of partonic 2-surfaces connected by string world sheets anti-commute: only the oscillator operators assignable to the same scattering diagram would anti-commute.

What about QCC in the case of entanglement. ER-EPR correspondence introduced by Maldacena and Susskind for 4 years ago proposes that blackholes (maybe even elementary particles) are connected by wormholes. In TGD the analogous statement emerged for more than decade ago - magnetic flux tubes take the role of wormholes in TGD. Magnetic flux tubes were assumed to be accompanied by string world sheets. I did not consider the question whether string world sheets are always accompanied by flux tubes.

What could be the criterion for entanglement to be real? "Reality" of entanglement demands some space-time correlate. Could the presence of the flux tubes make the entanglement "real"? If statistical entanglement is accompanied by string connections without magnetic flux tubes, it would not be "real": only the presence of flux tubes would make it "real". Or is the presence of strings enough to make the statistical entanglement "real". In both cases the fermions associated with disjoint space-time surfaces or with disjoint CDs would not be indistinguishable. This looks rather sensible.

The space-time correlate for the reduction of entanglement would be the splitting of a flux tube and fermionic strings inside it. The fermionic strings associated with flux tubes carrying monopole flux are closed and the return flux comes back along parallel space-time sheet. Also fermionic string has similar structure. Reconnection of this flux tube with shape of very long flattened square splitting it to two pieces would be the correlate for the state function reduction reducing the entanglement with other fermions and would indeed decouple the fermion from the network.

See the article Comments about some unclear issues of TGD .

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

Articles and other material related to TGD.

Could the two photon emissions of dark particles reveal the value of heff?

Again a new anomaly! Photon pairs have been created by a new mechanism. Photons emerge at different points! See this.

Could this give support for the TGD based general model for elementary particle as a string like object (flux tube) with first end (wormhole contact) carrying the quantum numbers - in the case of gauge boson fermion and antifermion at opposite throats of the contact. Second end would carry neutrino-right-handed neutrino pair neutralizing the possible weak isospin. This would give only local decays. Also emissions of photons from charged particle would be local.

Could the bosonic particle be a mixture of two states. For the first state flux tube would have fermion and antifermion at the same end of the fluxtube: only local decays. For the second state fermion and antifermion would reside at the ends of the flux tubes residing at throats associated with different wormhole contacts. This state in state would give rise to non-local two-photon emissions. Mesons of hadron physics would correspond to this kind of states and in old-fashioned hadron physics one speaks about photon-vector meson mixing in the description of the photon-hadron interactions.

If the Planck constant h_eff/h=n of the emitting particle is large, the distance between photon emissions would be long. The non-local days could make the visible both exotic decay and allow to deduce the value of n! This would how require the transformation of emitted dark photon to ordinary (same would happen when dark photons transform to biophotons).

Can one say anything about the length of fux tube? Magnetic flux tube contains fermionic string. The length of this string is of order Compton length and of the order of p-adic length scale.

What about photon itself - could it have non-local fermion-antifermion decays based on the same mechanism? What the length of photonic string is is not clear. Photon is massless, no scales! One identification of length would be as wavelength defining also the p-adic length scale.

To sum up: the nonlocal decays and emissions could lend strong support for both flux tube identification of particles and for hierarchy of Planck constants. It might be possible to even measure the value of n associated with quantum critical state by detecting decays of this kind.

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

Articles and other material related to TGD.

Quantum computations without definite causal structure and TGD

I encountered a link to a interesting popular article "Causal Witness" Provides First Experimental Evidence Of Indefinite Causal Order (see this). The article tells about an article Experimental verification of an indefinite causal order by Rubio et al (see this). In the following are my first impressions. Probably I mest update impressions. The findings are however so fascinating that I cannot resists the temptation to
post them in blog.

1. Indefinite causal order

The abstract of the article Rubio et al might give some idea about what is involved.

Investigating the role of causal order in quantum mechanics has recently revealed that the causal relations of events may not be a priori well-defined in quantum theory. Although this has triggered a growing interest on the theoretical side, creating processes without a causal order is an experimental task. We report the first decisive demonstration of a process with an indefinite causal order. To do this, we quantify how incompatible our setup is with a definite causal order by measuring a "causal witness">. This mathematical object incorporates a series of measurements that are designed to yield a certain outcome only if the process under examination is not consistent with any well-defined causal order. In our experiment, we perform a measurement in a superposition of causal orders—without destroying the coherence—to acquire information both inside and outside of a causally non-ordered process. Using this information, we experimentally determine a causal witness, demonstrating by almost 7 SDs that the experimentally implemented process does not have a definite causal order.

Unfortunately, I do not have prerequisites to say anything interesting about the delicacies of the experiment itself. Since causal order is fixed by that associated with space-time in standard physics, the implications of the experiment could be world view changing.

The key quantum information theoretic notions are causal order, causal separability, quantum wittness, quantum process called SWITCH changing causal order, and superposition of causal orders.

1. The notion of causal order is discussed in the article "Quantum correlations with no causal order" by Oreshkov et al (see this). One has two events A and B. If they are causally separable, one can tell which causes which. In Minkowski space causally separable events would connected by a time-like curve. If not, one cannot speak about causal order. One can tell whether A precedes B or vice versa. For light-like distances, the situation is not so clear.
   

   Relaxing the standard assumption about fixed arrow of time one can at quantum level consider also a situation in which one has quantum superposition of different causal orders. One has causal non-separability.
   
   

2. The notion of causal wittness (see this) provides a method allowing to deduce experimentally whether the process is causally separable or not. The notion is similar to that of entanglement wittness (see this) allowing to deduce whether the two systems are entangled. Essentially one has observable whose expectation is negative for states with indefinite causal order and positive for those with definite causal order. Causal wittness is not universal but must be constructed for each causally indefinite state separately. The construction of causal wittness expectation value of operator is far from trivial and requires deeper understanding of operator theory. The abstract definition goes as follows:
   

   Causal wittness represents a set of quantum operations, such as unitaries, channels, state preparations, and measurements, whose expectation value is non-negative as long as all the operations are performed in a definite causal order, i.e., as long as only causally separable resources are used. The observation of a negative expectation value is thus sufficient to conclude that the operations were not performed in a definite order.
   

   Causal witness can be constructed efficiently and the construction is discussed here).
   
   

3. SWITCH is a further basic notion. One has two events A and B, which can be connected by a time-like curve. One can tell whether A precedes B or vice versa. SWITCH is a quantum operation switching the causal order. The obvious manner to do this would permute A and B and would require "time travel" not allowed in standard physics. Obviously, SWITCH cannot be realized as operation respecting fixed causal order.
   
   
4. If superpositions of causal orders are possible, one can have a situation in which causal order is indefinite. Also this is something which does not conform the ordinary view about physics in fixed space-time but is allowed by postulates for quantum computation and SWITCH represents an example of quantum computation impossible with a fixed causal order.
   
   

Needless to say, the notions of causal order and superposition of causal orders are revolutionary ideas and the article claims that they have been experimentally verified. Therefore there are excellent motivations to find whether they could be be understood in TGD framework.

In TGD Zero Energy Ontology (ZEO) replaces ordinary ontology and the arrow of time is not fixed, and it is interesting to see whether superposition of different causal orders related by time inversion T for causal diamond (CD) and SWITCH could be realized in ZEO. The twistor lift of TGD leads to the proposal that CD is accompanied by a Minkowskian generalization of self-dual Kähler form J(CD). Although the moduli space of CDs allows to avoid breaking of Poincare invariance, self-duality of J(CD) leads to violation of T implying that different causal orders correspond to disjoint sectors of "world of classical worlds". This makes possible also superposition of different causal orders and SWITCH would map these sectors to each other.

2. ZEO and discrete symmetries for twistor lift of TGD

Some background about TGD is necessary in order to to proceed.

1. Zero Energy Ontology (ZEO) is the cornerstone of TGD and TGD inspired theory of consciousness. Zero energy states appear as two variants and correspond to different WCW spinor fields (WCW for "world of classical worlds"). I have proposed that they correspond also to WCW spinor fields localized to different sectors of WCW but this might be un-necessarily strong assumption.
   

   Zero energy states for given basis have been subject to a state function reduction at either boundary of CD - passive boundary. Neither the passive boundary nor the members of state pairs at it appearing in the superposition of state pairs are affected in repeated state function reductions. One has what I have called generalized Zeno effect identified as conscious entity - self.
   

   At the opposite boundary the states evolve: every state function reduction at active boundary is preceded by a unitary time evolution ending to a localization of the active boundary of CD, which can be also seen as a state function reduction (see this). The temporal distance between the tips of CD increases in this process and gives rise to clock time and experienced flow of time.
   

   Eventually the first reduction to the opposite boundary occurs and the roles of active and passive boundary of CD are changed. One can say that time reversed zero energy state is obtained and begins to evolve. The first reduction to the opposite boundary would mean death of the conscious entity defined by the sequence of state function reductions at the same boundary and generation of time reversed re-incarnation of self.
   
   

2. Violation of T - to be discussed below - would also imply asymmetry between selves and their time reversals. For instance, the average duration for the state reduction sequences keeping boundary fixed could be different and second causal order could dominate giving rise to a dominating arrow of time. Since these reduction sequences are identified as correlates for conscious selves, the time reversed re-incarnations would live much shorter time. Biological systems might be an exception: in TGD inspired theory of consciousness sensory perception and motor action are time reversals of each other.
   
   
3. Fermionic oscillator operators associated with induced spinor fields allow to represent WCW gamma matrices as their linear combinations: Fermi statistics is geometrized. Fermionic oscillator operators define also quantum Boolean algebra in the sense that fermion numbers 1/0 correspond to the two Boolean values. One could say that quantum logic is square root of WCW Kähler geometry. This allows to interpretation the S-matrix for fermions as quantum Boolean map between quatum Boolean algebras at opposite boundaries. This is obviously important when one talks about quantum computation.
   

   In the approach to twistor amplitudes (see this, this, and this) fermions are localized at the boundaries of string world sheets defining light-like curves at the 3-D light-like orbits of partonic 2-surfaces, at which the signature of the induced metric changes from Minkowskian to Euclidian and has a vanishing determinant so that tangent space is effectively 3-D. The interpretation is in terms of strong form of holography (SH) stating that the data determining both space-time surface as preferred extremal and modes of the induced spinor field in the interior of space-time surface. SH predicts that both the bosonic and fermionic 4-D actions reduce to 2-D effective actions for string world sheets.
   

   The implication is that fermion states at the boundaries of CD are localized at discrete points of partonic 2-surfaces. One has of course amplitude over different locations of fermions at partonic 2-surfaces. The presence of fermionic string world sheets correlates the fermions at different partonic 2-surfaces and serves as correlate for entanglement in fermionic degrees of freedom.
   
   

4. The twistor lift of TGD (see this) has led to a rather detailed understanding of discrete symmetries CP, P,T. If M⁴ factor of imbedding space - or more precisely CD - is endowed with a generalized self-dual Kähler form J(M⁴) (analogs of magnetic and electric fields of same magnitude and direction), new violations of CP, P and T occurring in long scales and having no counterpart in standard model emerge. The reason is that CP, P and T do not respect the self-duality of M⁴ Kähler form. The violations of Poincare invariance are avoided if one assumes moduli space for CDs containing the Lorentz boosts and translations of CD.
   

   The first guess is that T leaving the center point of CD invariant applied to the CD maps the 3-surfaces at the boundaries of CD to each other. The violation of T however implies that the image of the pair need not allow preferred extremal (space-time surface) connecting its members.
   

   One can however define the temporal mirror image of pair by mapping only the 3-surface at the passive boundary to the opposite boundary: preferred extremal property would determine the 3-surface at the passive boundary. This could imply that the sub-WCWs formed by pairs and the time reversals are disjoint and form different sectors of WCW (see this). This realization of T allows also the possibility that the dynamics of preferred extremals is not strictly deterministic (true at least for p-adic space-time sheets).
   

   In absence of T violation T operation would also permute the values fermionic states partonic 2-surfaces at the boundaries of CD but if T is violated, can map only the state at the passive boundary to the opposite boundary and determine the state at original boundary from the hermitian conjugate of S-matrix in opposite time direction. The fermionic state at the opposite boundary would be superposition of states having only same total quantum numbers as the state at the passive boundary. The quantum numbers of individual fermions would not be sharp for non-trivial S-matrix if the zero energy states and their T images correspond to same sector of WCW.
   

   If T is not violated, zero energy states and their T-images would not correspond to the same sector of WCW. Obviously they would correspond to opposite causal orders, see below) This would force to give up the assumption that states at passive boundary are state function reduced. If T is violated globally, the zero energy states and their T-reversals would correspond to disjoint sectors of WCW, and the sectors would only correspond to different arrows of time. This option gives hopes about WCW localization the outcome of the measurement of causal order.
   

   The union of sub-WCWs with opposite arrow of time is a space with fixed causal order plus additional binary digit characterizing the causal order. The state function for this binary digit could fix the causal order and quantum computation generating superposition of causal orders should generate entanglement with this bit.
   
   

5. What about the situation for a union of CDs? Different CDs should be able to have their own arrow of time. For instance, there are reasons to think that in living matter one can have subsystems with non-standard arrow of time. Also phase conjugate laser ray could be also example of this. This requires that WCW spinor fields associated with a union of CDs form a tensor product. Causal order need not be same for all CDs but characterizes the sub-WCW associated with CD forming a Cartesian factor of WCW so that WCW spinors for the CDs form tensor product.
   
   

3. Two views about SWITCH and superposition of causal orders

One can imagine two approaches to the identification of SWITCH operation and superposition of causal orders.

3.1 Option I: Unitary SWITCH as time reversal

The first option corresponds to corresponds to unitary "time travel" option.

1. As already proposed, SWITCH as a unitary operation could correspond to T. Time reversal operation T applied to the 3-surfaces at the boundaries of CD would naturally change the causal order for the zero energy state. If T is violated the state and its T-image belong to separate sectors of WCW. One could also have a superposition of zero energy states related by T and having different causal orders and localizable to the two sectors of WCW.
   
   
2. Is it possible to perform SWITCH as a unitary (as a matter fact, antiunitary) operation? If T maps the disjoint sectors to each other and maps fermionic time evolutions to their time reversals, SWITCH maps the two sectors of WCW to each other.
   

   Can one realize SWITCH mathematically as a unitary operation between fermionic state spaces. This seems possible: the tensor product of S⊗ S^† on tensor product of fermionic Fock spaces would realize this map. Whether SWITCH can be realized physically is of course another question.
   
   

3.2 Option II: Non-unitary SWITCH as the first state function reduction to the opposite boundary of CD

Could non-unitary SWITCH be realized as the first state function reduction to the opposite boundary - death of self followed by a re-incarnation as time-reversed self? In this case SWITCH is neither unitary nor deterministic. This SWITCH corresponds to non-unitary "time travel" option in the sense that self identified as passive boundary makes a time travel to the opposite boundary of CD by re-incarnating in non-deterministic manner.

What about the superposition of self and time reversed self as a superposition of causal orders? Schrödinger cat would be more than a catchy metaphor: it would indeed be a superposition of cat and re-incarnated cat! Should one take this seriously?

If the CDs with different arrow of time correspond to different sectors of WCW, different causal orders correspond to states localized in these sectors. A superposition of causal orders would correspond to WCW spinor field having component in both these sectors. If the state function reduction to opposite boundary of CD takes place at the level of entire WCW - or more realistically, for a Cartesian factor of WCW, it must be accompanied by a localization to either sector of WCW in order to avoid paradoxes.

I have already earlier work with TGD inspired ideas related to quantum computation. For more than decade ago I developed rather speculative model topological quantum computation in TGD framework (see this). One speculation about super-effective quantum computations is inspired by the analogy of selves with quantum computations halting, when the self dies and re-incarnates as time reversed self with clock time running in opposite direction at opposite boundary of CD (see this). This would mean using the non-unitary SWITCH realized as state function reduction in quantum computation.

This allows to imagine a series of quantum computations at opposite boundaries proceeding as a sequence of re-incarnations so that the size of CD and thus clock time would grow in opposite directions during subsequent incarnations (see this). Although the re-incarnation as time reversed self could have long life-time, it would not be seen by an observer near the former re-incarnation and the re-re-incarnation would appear at clock-time, which need not be much later than the time of previous death. One could imagine that the time-reversed selves could have very long life-time or that the self could die and re-incarnate very many times without any-one noticing it! The large amount of time spent in time reversed mode could explain the miraculous cognitive feats of mathematicians like Ramajunan and also the magic computational abilities of idiot savants able to factorize large integers without any idea about the notion of prime.

4. Higher level quantum computations and ZEO

From the article of Rubio et al one ends up to an article Quantum computations without definite causal structure by Chiribella et al (see this). The article considers a rather far reaching generalization of quantum computation. Ordinary quantum computation is a time evolution of quantum states followed by a state function reduction. Since the outcome of state function reduction halting the quantum computer program is non-deterministic, the extraction of the result involves statistical averaging over a large enough number of quantum computations to get the outcome, say prime factorization or a period of periodic function.

The notion of classical computation is generalized by Church. The computation need not be a function but can assign function to a function. One can continue this abstraction hierarchy indefinitely and it is realized formally in terms of so called Λ calculus (see this). Could this hierarchy be extended to quantum computations? The quantum computation in question would be kind of super-computation assigning to quantum computation a quantum computation and entire hierarchy of quantum computations.

In TGD this kind of hierarchies emerge naturally. At space-time level there is hierarchy of space-time sheets: space-time sheet is (topologically) condensed to a larger space-time sheet and contains smaller space-time sheets condensed at it. The hierarchy of infinite primes corresponds to an infinite hierarchy of second quantizations and could relate to this hierarchy (see this). In each scale space-time sheets would be particles consisting of smaller particles consisting of ... Even galaxy could be seen as elementary particle in some scale characterizing the galactic space-time sheet.

The analog of quantum computational hierarchy emerges quite concretely in ZEO. The simplest zero energy states have positive and negative energy states with opposite total quantum numbers at the opposite boundaries of CD (intersection of future and past directed light-cones). One can have CDs within CDs. Furthermore, the positive/negative energy states assignable to the boundaries of CD could be also zero energy states associated with smaller CDs near the boundaries of CD. The simplest zero energy states correspond to quantum evolution representing ordinary quantum computation. Higher level zero energy stats would represent time evolution assigning to a quantum computation represented as zero energy state at the boundary of CD a second quantum computation at opposite boundary of CD.

The fermionic representation of quantum Boolean algebra makes this hierarchy quite concrete. At lowest level unitary evolution connects positive and negative energy fermionic states at opposite boundaries of CD and unitary S-matrix characterizes the computation. Higher level computations connect zero energy states assignable to sub-CDs near the boundaries of CD.

See the chapter Topological Quantum Computation in TGD Universe and the article Quantum computations without definite causal structure: TGD view

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

Articles and other material related to TGD.

Early galactic collision gives support for TGD based model of galactic dark matter

The discoveries related to galaxies and dark matter emerge with an accelerating pace, and from TGD point of view it seems that puzzle of galactic dark matter is now solved.

The newest finding is described in popylar article This Gigantic Ring of Galaxies Could Bring Einstein's Gravity Into Question. What has been found that in a local group of 54 galaxies having Milky Way and Andromeda near its center the other dwarf galaxies recede outwarts as a ring. The local group is in good approximation in plane and the situation is said to look like having a spinning umbrella from which the water droplets fly radially outwards.

The authors of the article Anisotropic Distribution of High Velocity Galaxies in the Local Group argue that the finding can be understood aif Milky Way and Andromeda had nearly head-on collision about 10 billion light-years ago. The Milky Way and Andromeda would have lost the radially moving dwarf galaxies in this collision during the rapid acceleration turning the direction of motion of both. Coulomb collision is good analog.

There are however problems. The velocities of the dwards are quite too high and the colliding Milky Way and Andromeda would have fused together by the friction caused by dark matter halo.

What says TGD? In TGD galactic dark matter (actually also energy) is at cosmic strings thickened to magnetic flux tubes like pearls along necklace. The finding could be perhaps explained if the galaxies in same plane make a near hit and generate in the collision the dwarf galaxies by the spinning umbrella mechanism.

In TGD Universe dark matter is at cosmic strings and this automatically predicts constant velocity distribution. The friction created by dark matter is absent and the scattering in the proposed manner could be possible. The scattering event could be basically a scattering of approximately parallel cosmic strings with Milky Way and Andromeda forming one pearl in their respective cosmic necklaces.

But were Milky Way and Andromeda already associated with cosmic strings at that time? The time would be about 10 billion years. One annot exclude this possibility. Note however that the binding to strings might have helped to avoid the fusion. The recent finding about effective absence of dark matter about 10 billion light years ago - velocity distributions decline at large distances - suggests that galaxies formed bound states with cosmic strings only later. This would be like formation of neutral atoms from ions as energies are not too high! How fast the things develope becomes clear from the fact that I posted TGD explanation to my blog yesterday and replaced with it with a corrected version this morning!.

See the chapter TGD and Astrophysics of "Physics in Many-Sheeted Space-time" or the article TGD interpretation for the new discovery about galactic dark matter.

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

Articles and other material related to TGD.

Velocity curves of galaxies flatten for large redshifts

Sabine Hossenfelder gave a link to a popular article "Declining Rotation Curves at High Redshift" (see this) telling about a new strange finding about galactic dark matter. The rotation curves are declining in the early Universe meaning distances about 10 billion light years (see this). In other words, the rotation velocity of distant stars decreases with radius rather than approaching constant - as if dark matter would be absent and galaxies were baryon dominated. This challenges the halo model of dark matter. For the illustrations of the rotation curves see the article. Of course, the conclusions of the article are uncertain.

Some time ago also a finding about correlation of baryonic mass density with density of dark matter emerged: the ScienceDaily article "In rotating galaxies, distribution of normal matter precisely determines gravitational acceleration" can be found here. The original article can be found in arXiv.org (see this). TGD explanation (see this) involves only the string tension of cosmic strings and predicts the behavior of baryonic matter on distance from the center of the galaxy.

In standard cosmology based on single-sheeted GRT space-time large redshifts mean very early cosmology at the counterpart of single space-time sheet, and the findings are very difficult to understand. What about the interpretation of the results in TGD framework? Let us first summarize the basic assumptions behind TGD inspired cosmology and view about galactic dark matter.

1. The basic difference between TGD based and standard cosmology is that many-sheeted space-time brings in fractality and length scale dependence. In zero energy ontology (ZEO) one must specify in what length scale the measurements are carried out. This means specifying causal diamond (CD) parameterized by moduli including the its size. The larger the size of CD, the longer the scale of the physics involved. This is of course not new for quantum field theorists. It is however a news for cosmologists. The twistorial lift of TGD allows to formulate the vision quantitatively.
   
   
2. TGD view resolves the paradox due to the huge value of cosmological constant in very small scales. Kähler action and volume energy cancel each other so that the effective cosmological constant decreases like inverse of the p-adic length scale squared because these terms compensate each other. The effective cosmological constant suffers huge reduction in cosmic scales and solves the greatest (the "most gigantic" would be a better attribute) quantitative discrepancy that physics has ever encountered. The smaller value of Hubble constant in long length scales finds also an explanation (see this). The acceleration of cosmic expansion due to the effective cosmological constant decreases in long scales.
   
   
3. In TGD Universe galaxies are located along cosmic strings like pearls in necklace, which have thickened to magnetic flux tubes. The string tension of cosmic strings is proportional to the effective cosmological constant. There is no dark matter hallo: dark matter and energy are at the magnetic flux tubes and automatically give rise to constant velocity spectrum for distant stars of galaxies determined solely by the string tension. The model allows also to understand the above mentioned finding about correlation of baryonic and dark matter densities (see this) .
   
   

What could be the explanation for the new findings about galactic dark matter?

1. The idea of the first day is that the string tension of cosmic strings depends on the scale of observation and this means that the asymptotic velocity of stars decreases in long length scales. The asymptotic velocity would be constant but smaller than for galaxies in smaller scales. The velocity graphs show that in the velocity range considered the velocity decreases. One cannot of course exclude the possibility that velocity is asymptotically constant.
   

   The grave objection is that the scale is galactic scale and same for all galaxies irrespective of distance. The scale characterizes the object rather than its distance for observer. Fractality suggests a hierarchy of string like structures such that string tension in long scales decreases and asymptotic velocity associated with them decreases with the scale.
   
   

2. The idea of the next day is that the galaxies at very early times have not yet formed bound states with cosmic strings so that the velocities of starts are determined solely by the baryonic matter and approach to zero at large distances. Only later the galaxies condense around cosmic strings - somewhat like water droplets around blade of grass. The formation of these gravitationally bound states would be analogous to the formation of bound states of ions and electrons below ionization temperature or formation of hadrons from quarks but taking place in much longer scale. The early galaxies are indeed baryon dominated and decline of the rotation velocities would be real.
   
   

See the chapter TGD and Astrophysics of "Physics in Many-Sheeted Space-time" or the article TGD interpretation for the new discovery about galactic dark matter.

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

Articles and other material related to TGD.

Getting quantitative about violations of CP, T, and P

The twistor lift of TGD led to the introduction of Kähler form also in M⁴ factor of imbedding space M⁴×CP₂. The moduli space of causal diamonds (CDs) introduced already early allow to save Poincare invariance at the level of WCW. One of the very nice things is that the self-duality of J(M⁴) leads to a new mechanism of breaking for P,CP, and T in long scales, where these breakings indeed take place. P corresponds to chirality selection in living matter, CP to matter antimatter asymmetry and T could correspond to preferred arrow of clock time. TGD allows both arrows but T breaking could make other arrow dominant. Also the hierarchy of Planck constant is expected to be important.

Can one say anything quantitative about these various breakings?

1. J(M⁴) is proportional to Newton's constant G in the natural scale of Minkowski coordinates defined by twistor sphere of T(M⁴). Therefore CP breaking is expected to be proportional to l_P²/R² or to its square root l_P/R. The estimate for l_P/R is X== l_P/R≈ 2^-12≈ 2.5× 10^-4.
   

   The determinant of CKM matrix is equal to phase factor by unitarity (UU^†=1) and its imaginary part characterizes CP breaking. The imaginary part of the determinant should be proportional to the Jarlskog invariant J= +/- Im(V_usV_cbV*_ub V*_cs) characterizing CP breaking of CKM matrix (see this).
   

   The recent experimental estimate is J≈ 3.0× 10^-5. J/X≈ 0 .1 so that there is and order of magnitude deviation. Earlier experimental estimate used in p-adic mass calculations was by almost order of magnitude larger consistent with the value of X. For B mesons CP breading is about 50 times larger than for kaons and it is clear that Jarlskog invariant does nto distinguish between different meson so that it is better to talk about orders of magnitude only.
   

   The parameter used to characterize matter antimatter asymmetry (see this) is the ratio R=[n(B-n(B*)]/n(γ)) ≈ 9× 10^-11 of the difference of baryon and antibaryon densities to photon density in cosmological scales. One has X³ ≈ 1.4 × 10^-11, which is order of magnitude smaller than R.
   
   

2. What is interesting that P is badly broken in long length scales as also CP. The same could be true for T. Could this relate to the thermodynamical arrow of time? In ZEO state function reductions to the opposite boundary change the direction of clock time. Most physicist believe that the arrow of thermodynamical time and thus also clock time is always the same. There is evidence that in living matter both arrows are possible. For instance, Fantappie has introduced the notion of syntropy as time reversed entropy. This suggests that thermodynamical arrow of time could correspond to the dominance of the second arrow of time and be due to self-duality of J(M⁴) leading to breaking of T. For instance, the clock time spend in time reversed phase could be considerably shorter than in the dominant phase. A quantitative estimate for the ratio of these times might be given some power of the the ratio X =l_P/R.
   

For background see chapter Some questions related to the twistor lift of TGD of "Towards M-matrix" or the article with the same title.

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

Articles and other material related to TGD.

Is there a duality between associative and co-associative space-time surfaces?

A more appropriate title of this posting would be "A new duality or an old duality seen from a number theoretic perspective?". The original proposal turned out to be partially wrong and I can only blame myself for breaking the rule "Wait for a week before posting!".

M⁸-H duality maps the preferred extremals in H to those M⁴× CP₂ and vice versa. The tangent spaces of an associative space-time surface in M⁸ would be quaternionic (Minkowski) spaces.

In M⁸ one can consider also co-associative space-time surfaces having associative normal space. Could the co-associative normal spaces of associative space-time surfaces in the case of preferred extremals form an integrable distribution therefore defining a space-time surface in M⁸ mappable to H by M⁸-H duality? This might be possible but the associative tangent space and the normal space correspond to the same CP₂ point so that associative space-time surface in M⁸ and its possibly existing co-associative companion would be mapped to the same surface of H.

This dead idea however inspires an idea about a duality mapping Minkowskian space-time regions to Euclidian ones. This duality would be analogous to inversion with respect to the surface of sphere, which is conformal symmetry. Maybe this inversion could be seen as the TGD counterpart of finite-D conformal inversion at the level of space-time surfaces. There is also an analogy with the method of images used in some 2-D electrostatic problems used to reflect the charge distribution outside conducting surface to its virtual image inside the surface. The 2-D conformal invariance would generalize to its 4-D quaterionic counterpart. Euclidian/Minkowskian regions would be kind of Leibniz monads, mirror images of each other.

1. If strong form of holography (SH) holds true, it would be enough to have this duality at the informational level relating only 2-D surfaces carrying the holographic information. For instance, Minkowskian string world sheets would have duals at the level of space-time surfaces in the sense that their 2-D normal spaces in X⁴ form an integrable distribution defining tangent spaces of a 2-D surface. This 2-D surface would have induced metric with Euclidian signature.
   

   The duality could relate either a) Minkowskian and Euclidian string world sheets or b) Minkowskian/Euclidian string world sheets and partonic 2-surfaces common to Minkowskian and Euclidian space-time regions. a) and b) is apparently the most powerful option information theoretically but is actually implied by b) due to the reflexivity of the equivalence relation. Minkowskian string world sheets are dual with partonic 2-surfaces which in turn are dual with Euclidian string world sheets.
   
   

   1. Option a): The dual of Minkowskian string world sheet would be Euclidian string world sheet in an Euclidian region of space-time surface, most naturally in the Euclidian "wall neighbour" of the Minkowskian region. At parton orbits defining the light-like boundaries between the Minkowskian and Euclidian regions the signature of 4-metric is (0,-1,-1,-1) and the induced 3-metric has signature (0,-1,-1) allowing light-like curves. Minkowskian and Euclidian string world sheets would naturally share these light-like curves aas common parts of boundary.
      
      
   2. Option b): Minkowskian/Euclidian string world sheets would have partonic 2-surfaces as duals. The normal space of the partonic 2-surface at the intersection of string world sheet and partonic 2-surface would be the tangent space of string world sheets so that this duality could make sense locally. The different topologies for string world sheets and partonic 2-surfaces force to challenge this option as global option but it might hold in some finite region near the partonic 2-surface. The weak form of electric-magnetic duality could closely relate to this duality.
      
   In the case of elementary particles regarded as pairs of wormhole contacts connected by flux tubes and associated strings this would give a rather concrete space-time view about stringy structure of elementary particle. One would have a pair of relatively long (Compton length) Minkowskian string sheets at parallel space-time sheets completed to a parallelepiped by adding Euclidian string world sheets connecting the two space-time sheets at two extremely short (CP₂ size scale) Euclidian wormhole contacts. These parallelepipeds would define lines of scattering diagrams analogous to the lines of Feynman diagrams.

This duality looks like new but as already noticed is actually just the old electric-magnetic duality seen from number-theoretic perspective.

For background see chapter Some questions related to the twistor lift of TGD of "Towards M-matrix" or the article with the same title.

About the generalization of dual conformal symmetry and Yangian in TGD

The discovery of dual of the conformal symmetry of gauge theories was crucial for the development of twistor Grassmannian approach. The D=4 conformal generators acting on twistors have a dual representation in which they act on momentum twistors: one has dual conformal symmetry, which becomes manifest in this representation. These two separate symmetries extend to Yangian symmetry providing a powerful constraint on the scattering amplitudes in twistor Grassmannian approach fo N=4 SUSY.

In TGD the conformal Yangian extends to super-symplectic Yangian - actually, all symmetry algebras have a Yangian generalization with multi-locality generalized to multi-locality with respect to partonic 2-surfaces. The generalization of the dual conformal symmetry has however remained obscure. In the following I describe what the generalization of the two conformal symmetries and Yangian symmetry would mean in TGD framework.

One also ends up with a proposal of an information theoretic duality between Euclidian and Minkowskian regions of the space-time surface inspired by number theory: one might say that the dynamics of Euclidian regions is mirror image of the dynamics of Minkowskian regions. A generalization of the conformal reflection on sphere and of the method of image charges in 2-D electrostatics to the level of space-time surfaces allowing a concrete construction reciple for both Euclidian and Minkowskian regions of preferred extremals is in question. One might say that Minkowskian and Euclidian regions are analogous to Leibnizian monads reflecting each other in their internal dynamics.

See the chapter Some Questions Related to the Twistor Lift of TGD of "Towards M-matrix" or the article with the same title.

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

Articles and other material related to TGD.

Could second generation of weak bosons explain the reduction of proton charge radius?

The discovery by Pohl et al (2010) was that the charge radius of proton deduced from the muonic version of hydrogen atom - is .842 fm and about 4 per cent smaller than .875 fm than the charge radius deduced from hydrogen atom is in complete conflict with the cherished belief that atomic physics belongs to the museum of science (for details see the Wikipedia article). The title of the article Quantum electrodynamics-a chink in the armour? of the article published in Nature expresses well the possible implications, which might actually go well extend beyond QED.

Quite recently (2016) new more precise data has emerged from Pohl et al (see this). Now the reduction of charge radius of muonic variant of deuterium is measured. The charge radius is reduced from 2.1424 fm to 2.1256 fm and the reduction is .012 fm, which is about .8 per cent (see this). The charge radius of proton deduced from it is reported to be consistent with the charge radius deduced from deuterium. The anomaly seems therefore to be real. Deuterium data provide a further challenge for various models.

The finding is a problem of QED or to the standard view about what proton is. Lamb shift is the effect distinguishing between the states hydrogen atom having otherwise the same energy but different angular momentum. The effect is due to the quantum fluctuations of the electromagnetic field. The energy shift factorizes to a product of two expressions. The first one describes the effect of these zero point fluctuations on the position of electron or muon and the second one characterizes the average of nuclear charge density as "seen" by electron or muon. The latter one should be same as in the case of ordinary hydrogen atom but it is not. Does this mean that the presence of muon reduces the charge radius of proton as determined from muon wave function? This of course looks implausible since the radius of proton is so small. Note that the compression of the muon's wave function has the same effect.

Before continuing it is good to recall that QED and quantum field theories in general have difficulties with the description of bound states: something which has not received too much attention. For instance, van der Waals force at molecular scales is a problem. A possible TGD based explanation and a possible solution of difficulties proposed for two decades ago is that for bound states the two charged particles (say nucleus and electron or two atoms) correspond to two 3-D surfaces glued by flux tubes rather than being idealized to points of Minkowski space. This would make the non-relativistic description based on Schrödinger amplitude natural and replace the description based on Bethe-Salpeter equation having horrible mathematical properties.

The basic idea of the original model of the anomaly (see this) is that muon has some probability to end up to the magnetic flux tubes assignable to proton. In this state it would not contribute to the ordinary Schrödinger amplitude. The effect of this would be reduction of |Ψ|² near origin and apparent reduction of the charge radius of proton. The weakness of the model is that it cannot make quantitative prediction for the size of the effect. Even the sign is questionable. Only S-wave binding energy is affected considerably but does the binding energy really increase by the interaction of muon with the quarks at magnetic flux tubes? Is the average of the charge density seen by muon in S wave state larger, in other words does it spend more time
near proton or do the quarks spend more time at the flux tubes?

In the following a new model for the anomaly will be discussed.

1. The model is inspired by data about breaking of universality of weak interactions in neutral B decays possibly manifesting itself also in the anomaly in the magnetic moment of muon. Also the different values of
   the charge radius deduced from hydrogen atom and muonium could reflect the breaking of universality. In the original model the breaking of universality is only effective.
   
   
2. TGD indeed predicts a dynamical U(3) gauge symmetry whose 8+1 gauge bosons correspond to pairs of fermion and anti-fermion at opposite throats of wormhole contact. Throats are characterized by genus g=0,1,2, so that bosons are superpositions of states labelled by (g₁,g₂). Fermions correspond to single wormhole throat carrying fermion number and behave as U(3) triplet labelled by g.
   

   The charged gauge bosons with different genera for wormhole throats are expected to be very massive. The 3 neutral gauge bosons with same genus at both throats are superpositions of states (g,g) are expected to be lighter. Their charge matrices are orthogonal and necessarily break the universality of electroweak interactions. For the lowest boson family - ordinary gauge bosons - the charge matrix is proportional to unit matrix. The exchange of
   second generation bosons Z⁰₁ and γ₁ would give rise to Yukawa potential increasing the binding energies of S-wave states. Therefore Lamb shift defined as difference between energies of S and P waves is increased and the charge radius deduced from Lamb shift becomes smaller.
   
   

3. The model thus predicts a correct sign for the effect but the size of the effect from naive estimate assuming only γ₂ and α₂=α₁== α for M=2.9 TeV is almost by an order of magnitude too small. The values of the gauge couplings α₂ and α_Z,2 are free parameters as also the mixing angles between states (g,g). The effect is also proportional to the ratio (m_μ/M(boson)². It turns out that the inclusion of Z⁰₁ contribution and assumption α₁ and α_Z,1 are near color coupling strength α_s gives a correct prediction.
   
   

Motivations for the breaking of electroweak universality

The anomaly of charge radius could be explained also as breaking of the universality of weak interactions. Also other anomalies challenging the universality exists. The decays of neutral B-meson to lepton pairs should be same apart from corrections coming from different lepton masses by universality but this does not seem to be the case (see this). There is also anomaly in muon's magnetic moment discussed briefly here. This leads to ask whether the breaking of universality could be due to the failure of universality of electroweak interactions.

The proposal for the explanation of the muon's anomalous magnetic moment and anomaly in the decays of B-meson is inspired by a recent very special di-electron event and involves higher generations of weak bosons predicted by TGD leading to a breaking of lepton universality. Both Tommaso Dorigo (see this) and Lubos Motl (see this) tell about a spectacular 2.9 TeV di-electron event not observed in previous LHC runs. Single event of this kind is of course most probably just a fluctuation but human mind is such that it tries to see something deeper in it - even if practically all trials of this kind are chasing of mirages.

Since the decay is leptonic, the typical question is whether the dreamed for state could be an exotic Z boson. This is also the reaction in TGD framework. The first question to ask is whether weak bosons assignable to Mersenne prime M₈₉ have scaled up copies assignable to Gaussian Mersenne M₇₉. The scaling factor for mass would be 2^(89-79)/2= 32. When applied to Z mass equal to about .09 TeV one obtains 2.88 TeV, not far from 2.9 TeV. Eureka!? Looks like a direct scaled up version of Z!? W should have similar variant around 2.6 TeV.

TGD indeed predicts exotic weak bosons and also gluons.

1. TGD based explanation of family replication phenomenon in terms of genus-generation correspondence forces to ask whether gauge bosons identifiable as pairs of fermion and antifermion at opposite throats of wormhole contact could have bosonic counterpart for family replication. Dynamical SU(3) assignable to three lowest fermion generations labelled by the genus of partonic 2-surface (wormhole throat) means that fermions are combinatorially SU(3) triplets. Could 2.9 TeV state - if it would exist - correspond to this kind of state in the tensor product of triplet and antitriplet? The mass of the state should depend besides p-adic mass scale also on the structure of SU(3) state so that the mass would be different. This difference should be very small.
   
   
2. Dynamical SU(3) could be broken so that wormhole contacts with different genera for the throats would be more massive than those with the same genera. This would give SU(3) singlet and two neutral states, which are analogs of η' and η and π⁰ in Gell-Mann's quark model. The masses of the analogs of η and π⁰ and the the analog of η', which I have identified as standard weak boson would have different masses. But how large is the mass difference?
   
   
3. These 3 states are expected top have identical mass for the same p-adic mass scale, if the mass comes mostly from the analog of hadronic string tension assignable to magnetic flux tube. connecting the two wormhole contacts associates with any elementary particle in TGD framework (this is forced by the condition that the flux tube carrying monopole flux is closed and makes a very flattened square shaped structure with the long sides of the square at different space-time sheets). p-Adic thermodynamics would give a very small contribution genus dependent contribution to mass if p-adic temperature is T=1/2 as one must assume for gauge bosons (T=1 for fermions). Hence 2.95 TeV state could indeed correspond to this kind of state.
   
   

The sign of the effect is predicted correctly and the order of magnitude come out correctly

Could the exchange of massive M_G,79 photon and Z⁰ give rise to additional electromagnetic interaction inducing the breaking of Universality? The first observation is that the binding energy of S-wave state increases but there is practically no change in the energy of P wave state. Hence the effective charge radius r_p as deduced from the parameterization of binding energy different terms of proton charge radius indeed decreases.

Also the order of magnitude for the effect must come out correctly.

1. The additional contribution in the effective Coulomb potential is Yukawa potential. In S-wave state this would give a contribution to the binding energy in a good approximation given by the expectation value of the Yukawa potential, which can be parameterized as
   

   V(r)= g² e^-Mr/r ,&g² = 4π kα .
   

   The expectation differs from zero significantly only in S-wave state characterized by principal quantum number n. Since the exponent function goes exponentially to zero in the p-adic length scale associated with 2.9 TeV mass, which is roughly by a factor 32 times shorter than intermediate boson mass scale, hydrogen atom wave function is constant in excellent approximation in the effective integration volume. This gives for the energy shift
   

   Δ E= g²| Ψ(0)|² × I ,
   

   Ψ(0) ² =[2²/n²]×(1/a₀³) ,
   

   a₀= 1/(mα) ,
   

   I= ∫ (e^-Mr/r) r²drdΩ =4π/3M².
   

   For the energy shift and its ratio to ground state energy
   

   E_n= α²/2n²× m
   

   one obtains the expression
   

   Δ E_n= 64π² α/n² α³ (m/M)² × m ,
   

   Δ E_n/E_n= (2⁷/3) π²α² k²(m/M)² .
   

   For k=1 and M=2.9 one has Δ E_n/E_n ≈ 3× 10^-11 for muon.
   

Consider next Lamb shift.

1. Lamb shift as difference of energies between S and P wave states (see this) is approximately given by
   

   Δ_n (Lamb)/E_n= 13α³/2n .
   

   For n=2 this gives Δ₂ (Lamb)/E₂= 4.9× 10^-7.
   
   

2. The parameterization for the Lamb shift reads as
   

   Δ E(r_p) =a - br_p² +cr_p³
   = 209.968(5) - 5.2248 × r²_p + 0.0347 × r³_p meV ,
   

   where the charge radius r_p=.8750 is expressed in femtometers and energy in meVs.
   
   

3. The reduction of r_p by 3.3 per cent allows to estimate the reduction of Lamb shift (attractive additional potential reduces it). The relative change of the Lamb shift is
   

   x=[Δ E(r_p))-Δ E(r_p(exp))]/Δ E(r_p)
   

   = [- 5.2248 × (r²_p- r²_p(exp)) + 0.0347 × ( r³_p-r³_p(exp))]/[209.968(5) - 5.2248 × r²_p + 0.0347 × r³_p(th)] .
   

   The estimate gives x= 1.2× 10^-3.
   
   

This value can be compared with the prediction. For n=2 ratio of Δ E_n/Δ E_n(Lamb) is

x=Δ E_n/Δ E_n (Lamb)= k² × [2⁹π²/3×13α] × (m/M)² .

For M=2.9 TeV the numerical estimate gives x≈ (1/3)×k² × 10^-4. The value of x deduced from experimental data is x≈ 1.2× 10^-3. There is discrepancy of one order of magnitude. For k≈ 5 a correct order of magnitude is obtained. There are thus good hopes that the model works.

The contribution of Z⁰₁ exchange is neglected in the above estimate. Is it present and can it explain the discrepancy?

1. In the case of deuterium the weak isospins of proton and deuterium are opposite so that their contributions to the Z⁰₁ vector potential cancel. If Z⁰₁ contribution for proton can be neglected, one has Δ r_p=Δ r_d.
   

   One however has Δ r_p≈ 2.75 Δ r_d. Hence Z⁰₁ contribution to Δ r_p should satisfy Δ r_p(Z⁰₁)≈ 1.75×Δ r_p(γ₁). This requires α_Z,1>α₁, which is true also for the ordinary gauge bosons. The weak isospins of electron and proton are opposite so that the atom is weak isospin singlet in Abelian sense, and one has I³_pI³_μ= -1/4 and attractive interaction. The condition relating r_p and r_Z suggests
   

   α_Z,1/α₁≈ 286=4+13 .
   

   In standard model one has α_Z/α= 1/[sin²(θ_W)cos²(θ_W)] =5.6 for sin²(θ_W)=.23 . One has upper bound α_Z,1/α₁ ≥ 4 saturated for sin²(θ_W,1) =1/2. Weinberg angle can be expressed as
   

   sin²(θ_W,1)= (1/2)[1 - (1-4( α₁/α_Z,1)^1/2] .
   

   α_Z,1/α₁≈ 28/6 gives sin²(θ_W,1) = (1/2)[1 -(1/7)^1/2] ≈ .31.
   

   The contribution to the axial part of the potential depending on spin need not cancel and could give a spin dependent contribution for both proton and deuteron.
   
   

2. If the scale of α₁ and α_Z,1 is that of α_s and if
   the factor 2.75 emerges in the proposed manner, one has k²≈ 2.75× 10= 27.5 rather near to the rough estimate k²≈ 27 from data for proton.
   

   Note however than there are mixing angles involved corresponding to the diagonal hermitian family charge matrix Q= (a,b,c) satisfying a²+b²+c²=1 and the condition a+b+c=0 expressing the orthogonality with the electromagnetic charge matrix (1,1,1)/3^1/2 expressing electroweak universality for ordinary electroweak bosons. For instance, one could have (a,b,c)= (0,1,-1)/2^1/2 for the second generation and (a,b,c)= (2,-1,-1)/6^1/2 for the third generation. In this case the above estimate would would be scaled down: α₁→ 2α₁/3≈ 1/20.5.
   
   

To sum up, the proposed model is successful at quantitative level allowing to understand the different changes for charge radius for proton and deuteron and estimate the values of electroweak couplings of the second generation of weak bosons apart from the uncertainty due to the family charge matrix. Muon's magnetic moment anomaly and decays of neutral B allow to test the model and perhaps fix the remaining two mixing angles.

See the article Could second generation of weak bosons explain the reduction of proton charge radius?

For background see the chapters New Physics Predicted by TGD: Part I and New Physics Predicted by TGD: Part II.

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

Articles and other material related to TGD.

What about actual realization of Lorentz invariant synchronization?

I wrote one day ago about synchronization of clocks and found that the clocks distributed at the hyperboloids of light-cone assignable to CD can in principle be synchronized in Lorentz invariant manner (see this). But what about actual Lorentz invariant synchronization of the clocks? Could TGD say something non-trivial about this problem? I received an interesting link relating to this (see this). The proposed theory deals with fundamental uncertainty of clock time due to quantum-gravitational effects. There are of course several uncertainties involved since quantum theory of gravity does not exist (officially) yet!

1. Operationalistic definition of time is adopted in the spirit with the empiristic tradition. Einstein was also empirist and talked about networks of synchronized clocks. Nowadays particle physicists do not talk much about them. Symmetry based thinking dominates and Special Relativity is taken as a postulate about symmetries.
   
   
2. In quantum gravity situation becomes even rather complex. If quantization attempt tries to realize quantum states as superpositions of 3-geometries one loses time totally. If GRT space-time is taken to be small deformation of Minkowski space one has path integral and classical solutions of Einstein's equation define the background.
   

   The difficult problem is the identification of Minkowski coordinates unless one regards GRT as QFT in Minkowski space. In astrophysical scales QFT picture one must consider solutions of Einstein's equations representing astrophysical objects. For the basic solutions of Einstein's equations the identification of Minkowski coordinates is obvious but in general case such as many-particle system this is not anymore so. This is a serious obstacle in the interpretation of the classical limit of GRT and its application to planetary systems.
   
   

What about the situation in TGD? Particle physicist inside me trusts symmetry based thinking and has been somewhat reluctant to fill space-time with clocks but I am ready to start the job if necessarily! Since I am lazy I of course hope that Nature might have done this already and the following argument suggests that this might be the case!

1. Quantum states can be regarded as superpositions of space-time surfaces inside causal diamond of imbedding space H= M⁴× CP₂ in quantum TGD. This raises the question how one can define universal time coordinate for them. Some kind of absolute time seems to be necessary.
   
   
2. In TGD the introduction of zero energy ontology (ZEO) and causal diamonds (CDs) as perceptive fields of conscious entities certainly brings in something new, which might help. CD is the intersection of future and past directed light-cones analogous to a big bang followed by big crunch. This is however only analogy since CD represents only perceptive field not the entire Universe.
   

   The imbeddability of space-time as to CD× CP₂ ⊂ H= M⁴× CP₂ allows the proper time coordinate a² =t²-r² near either CD bouneary as a universal time coordinate, "cosmic time". At a= constant hyperboloids Lorentz invariant synchronisation is possible. The coordinate a is kind of absolute time near a given boundary of CD representing the perceptive field of a particular conscious observer and serves as a common time for all space-time surfaces in the superposition. Newton would not have been so wrong after all.
   

   Also adelic vision involving number theoretic arguments selects a as a unique time coordinate. In p-adic sectors of adele number theoretic universality (NTU) forces discretization since the coordinates of hyperboloid consist of hyperbolic angle and ordinary angles. p-Adicallhy one cannot realize either angles nor their hyperbolic counterparts. This demands discretization in terms of roots of unity (phases) and roots of e (exponents of hyperbolic angles) inducing finite-D extension of p-adic number fields in accordance with finiteness of cognition. a as Lorentz
   invariant would be genuine p-adic coordinate which can in principle be continuous in p-adic sense. Measurement resolution however discretizes also a.
   

   This discretization leads to tesselations of a=constant hyperboloid having interpretation in terms of cognitive representation in the intersection of real and various p-adic variants of space-time surface with points having coordinates in the extension of rationals involved. There are two choices for a. The correct choice
   corresponds to the passive boundary of CD unaffected in state function reductions.
   
   

3. Clearly, the vision about space-time as 4-surface of H and NTU show their predictive power. Even more, adelic physics itself might solve the problem of Lorentz invariant synchronization in terms of a clock network assignable to the nodes of tesselation!
   

   Suppose that tesselation defines a clock network. What synchronization could mean? Certainly strong correlations between the nodes of the network Could the correlation be due to maximal quantum entanglement (maximal at least in p-adic sense) so that the network of clocks would behave like a single quantum clock? Bose-Einstein condensate of clocks as one might say? Could quantum entanglement in astrophysical scales predicted by TGD via h_gr= h_eff=n× h hypothesis help to establish synchronized clock networks even in astrophysical scales? Could Nature guarantee Lorentz invariant synchronization automatically?
   

   What would be needed would be not only 3-D lattice but also oscillatory behaviour in time. This is more or less time crystal (see this and this)! Time crystal like states have been observed but they require feed of energy in contrast to what Wilzek proposed. In TGD Universe this would be due to the need to generate large h_eff/h=n phases since the energy of states with n increases with n. In biological systems this requires metabolic energy feed. Can one imageine even cosmic 4-D lattice for which there would be the analog of metabolic energy feed?
   

   I have already a model for tensor networks and also here a appears naturally (see this). Tensor networks would correspond at imbedding space level to tesselations of hyperboloid t²-r²=a² analogous to 3-D lattices but with recession velocity taking the role of quantized position for the point of lattice. They would induce tesselations of space-time surface: space-time surface would go through the points of the tesselation (having also CP₂ counterpart). The number of these tesselations is huge. Clocks would be at the nodes of these lattice like structures. Maximal entanglement would be key feature of this network. This would make the clocks at the nodes one big cosmic clock.
   

   If astrophysical objects serving as clocks tend to be at the nodes of tesselation, quantization of cosmic redshifts is predicted! What is fascinating is that there is evidence for this: for TGD based model for this see (see this and this)! Maybe dark matter fraction of Universe might have taken care of the Lorentz invariant synchronization so that we need not worry about that!
   

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

Articles and other material related to TGD.

Is Lorentz invariant synchronization of clocks possible?

I participated an FB discussion with several anti-Einsteinians. As a referee I have expressed my opinion about numerous articles claiming that Einstein's special or general relativity contains a fatal error not noticed by any-one before. I have tried to tell that colleagues are extremely eager to find a mistake in the work of colleague (unless they can silence the colleague) so that logical errors can be safely excluded. If something goes wrong it is at the level of basic postulates. In vain.

Once I had a long email discussion with a professor of logic who claimed to have found logical mistake in the deduction of time dilation formula. It was easy to find that he thought in terms of Newtonian space-time and this was of course in conflict with relativistic view. The logical error was his, not Einstein's. I tried to tell this. In vain again.

At this time I was demanded to explain why the 2 page article of Stephen Crothers (see this). This article was a good example of own logical error projected to that of Einstein. The author assumed besides the basic formulas for Lorentz transformation also synchronization of clocks so that they show the same time everywhere (about how this is achieved see this).

Even more: Crothers assumes that Einstein assumed that this synchronization is Lorentz invariant. Lorentz invariant synchronization of clocks is not however possible for the linear time coordinate of Minkowski space as also Crothers demonstrates. Einstein was wrong! Or was he? No!: Einstein of course did not assume Lorentz invariant synchronization!

The assumption that the synchronization of clock network is invariant under Lorentz transformations is of course in conflict with SR. In Lorentz boosted system the clocks are not in synchrony. This expresses just Einstein's basic idea about the relativity of simultaneity. Basic message of Einstein is misunderstood! The Newtonian notion of absolute time again!

The basic predictions of SR - time dilation and Lorentz contraction - do not depend on the model of synchronization of clocks. Time dilation and Lorentz contraction follow from basic geometry of Minkowskian space-time extremely easily.

Draw system K and K' moving with constant velocity with respect to K. The t' and x' axis of K' have angle smaller than π/2 and are in first quadrant.

1. Assume first that K corresponds to the rest system of particle. You see that the projection of segment=(0,t') t'-axis to t-axis is shorter than the segment (0,t'): time dilation.
   
   
2. Take K to be the system of stationary observer. Project the segment L=(0,x') to segment on x axis. It is shorter than L: Lorentz contraction.
   
   

There is therefore no need to build synchronized networks of clocks to deduce time dilation and Lorentz contraction. They follow from Minkowskian geometry.

This however raises a question. Is it possible to find a system in which synchronization is possible in Lorentz invariant manner? The quantity a²=t²-x² defines proper time coordinate a along time like geodesics as Lorentz invariant time coordinate of light-one. a = constant hyper-surfaces are now hyperboloids. If you have a synchronized network of clocks, its Lorentz boost is also synchronized. General coordinate invariance of course allows this choice of time coordinate.

For Robertson-Walker cosmologies with sub-critical mass time coordinate a is Lorenz invariant so that one can have Lorentz invariant synchronization of clocks. General Coordinate Invariance allows infinitely many choices of time coordinate and the condition of Lorentz invariant synchronization fixes the time coordinate to cosmic time (or its function to be precise). To my opinion this is rather intesting fact.

What about TGD? In TGD space-time is 4-D surface in H=M⁴×CP₂. a²= t²-r² defines Lorentz invariant time coordinate a in future light-cone M⁴₊ ⊂ M⁴ which can be used as time-coordinate also for space-time surfaces.

Robertson-Walker cosmologies can be imbedded as 4-surfaces to H=M⁴×CP₂. The empty cosmology would be just the lightcone M⁴₊ imbedded in H by putting CP₂ coordinates constant. If CP₂ coordinates depend on M⁴₊ proper time a, one obtains more general expanding RW cosmologies. One can have also sub-critical and critical cosmologies for which Lorentz transformations are not isometries of a= constant section. Also in this case clocks are synchronized in Lorentz invariant manner. The duration of these cosmologies is finite: the mass density diverges after finite time.

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

Articles and other material related to TGD.

Are viruses fragments of topological quantum computer code?

I was listening a highly interesting talk about viruses in Helsinki by Dr. Matti Jalasvuori, a molecular biologist working in the University of Jyväskylä as a researcher (see this). He has published a book about viruses in finnish titled "Virus. Elämän synnyttäjä, kuoleman kylväjä, ajatusten tartuttaja" (see this).

I learned an extremely interesting new-to-me fact about viruses. They might be far from a mere nuisance, In TGD Universe they could be quantum memes, short pieces of a code of quantum computer code, wandering around and attaching to the existing quantum computer code represented by DNA! Replication of viruses would be replication of memes. If the infected organism survives the virus attack by taming the virus and making it part of its non-coding DNA, it will gain more strength! If my computer survives the updating of the operating system, it works better!

Some basic facts

Viruses are very small, few nanometers is the size scale. Virus contains short pieces of RNA or DNA coding for the virus, in particular the protein shell around it, which virus must have in the "non-living" state outside the host cell to which it can penerate. Inside its host this shell melts and virus attaches to DNA and is able to to replicate. The copies of virus leave the host cell to search for their own host cells.

Usually viruses are regarded as a nuisance. But a new more holistic vision is evolving about viruses and their actual role. Viruses have been present perhaps even before the cell was present in its recent form, they might have been crucial for the emergence of life as we know it and would be also now. The system would consist of various kinds of cells, not necesary those of single organism. The contain several kinds of DNA and RNA: cell nucleus and mitochondria contain their own genomes; there are circular plasmids, and also viruses.

There is a continual exchange of information between cells including viruses as form form of information exchange. In this framework virus represents a meme represented by its DNA ,which does not code for protein shell. This meme wants to replicate and must use the genetic machinery to achieve this. But does virus do this to only replicate and produce more nuisance?

The organism manages to survive the virus attack if it is able to transform the virus so that it cannot replicate. One manner to achieve this would be transformation of the DNA portion due to the attached virus DNA (possible reverse transcribed from the RNA of virus) to a non-coding DNA often referred to as "junk" DNA. Non-coding DNA includes both intragenic regions - introns - and intergenic regions containing for instance promoters and enhancers crucial for the control of gene expression as proteins (see this). Introns are portions of genes, whose contribution to mRNA is sliced away in translation to proteins. The decomposition to introns and translated regions is dynamical, which gives rise to a rich spectrum of different translations of the gene.

In fact, most of non-coding DNA might be due to viruses! The portion of non-coding DNA increases for speciei at higher evolutionary level. For our species it is estimated to be 98 percent! Most of our genome is "junk" as many biologists still would put it. But can this really be the case? On might think that immune system would have invented some mechanism to prevent the infection of DNA by junk DNA? The size of the trash bin cannot be a measure for evolutionary level! It is also known that virus infections force the organism to change and in some cases to become a better surviver. Viruses would drive evolution.

One can speculate that during the very early period in evolution there were only viruses and proto-cells. There is no need for them to be coded by genes. Self-organization can produce cell membrane like structures: soap films represent an example. The DNA fragments could survive inside these proto-cells but according to simulations done by the Jyäskylä group in which Matti Jalasvuori is working, eventually the evolution would lead to the emergence of parasitic DNA strands, which would soon begin to dominate and kill the protocell.

Viruses might solve the problem. Viruses would attract DNA fragments and replicate with themto build a protein wall around the fragment containg also a piece of DNA of proto-cell. Viruses would leave the proto cell before its death and find another protocell. Gradually genome would be formed as viruses would steal pieces of DNA fragments from protocells. One step in the later evolution could be the elimination of the part of virus coding for the protein shell and the use of the rest as protein coding DNA. For eukariotes the transformation to non-coding DNA including intronic and intergenic DNA becomes possible.

Viruses as pieces of quantum computer code?

Computational thinking would suggest that viruses might make possible the emergence of new biological program modules allowing to use existing program modules coding for proteins more effectively. The different slicings of mRNA dropping some pieces away would correspond to different manners to transform DNA sequences to proteins. But what about intragenic portions of DNA: are they just junk?

Could the non-coding DNA and viruses have a much deeper purpose of existence than mere replication? In TGD Universe this kind of purpose is easy to imagine if the system formed by DNA - say intragenic portions of DNA - and nuclear membrane (or cell membrane) system serves as a topological quantum computer. DNA codons would be connected to lipids of the lipid layer of cell nucleus by magnetic flux tubes carrying dark charged particles. These connections could be also to cell membrane and even to cell membranes of other cells.

The braiding of the flux tubes would define the space-time realization of a quantum computer program. This would represent a new expression of DNA and would explain why so small differences between our DNA and that of our cousins give rise to so huge differences. What is important that genetic code would not be terribly important: it is braiding that matters now. The realization as quantum computer programs would give rise to cultural evolution, the realization as proteins to biological evolution. There would be a transition from the level of genes to that of memes.

Viruses would correspond to pieces of quantum computer code - memes. They would be wandering between cells and infecting them to get fused to the DNA. If DNA is able to transform them to introns it gets the code. Otherwise it dies. Infection is the necessary price for achieving meme replication. Living cells could be seen quantum computer programs updating them continually. Sounds somehow familiar!

See the chapters DNA as topological quantum computer, Three new physics realizations of the genetic code and the role of dark matter in bio-systems, and More Precise TGD Based View about Quantum Biology and Prebiotic Evolution of "Genes and Memes".

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

Articles and other material related to TGD.