Friday, January 21, 2022

Bubble-like structures in TGD Universe

Membrane-like structures appear in all length scales from soap bubbles to large cosmic voids and it would be nice if they were fundamental objects in the TGD Universe. The Fermi bubble in the galactic center is an especially interesting membrane-like structure also from the TGD point of view as also the membrane-like structure presumably associated with the analog of horizon of the TGD counterpart of blackhole. Cell membrane is an example of a biological structure of this kind. I have however failed to identify candidates for the membrane-like structures.

In M8-H duality surface with M4 projections smaller than four appear as singularities of algebraic surfaces in M^8. The dimension of M4 projection varies and known extremals can be interpreted in terms of singularities.

An especially interesting singularity would be a static 3-D singularity M1× X2 with a geodesic circle S1 ⊂ CP2 as a local blow-up.

  1. The simplest guess is as product M1× S2× S1. The problem is that a soap bubble is not a minimal surface: a pressure difference between interior and exterior of the bubble is required so that the trace of the second fundamental form is constant. Quite generally, closed 2-D surfaces cannot be minimal surfaces in a flat 3-space since the vanishing curvature of the minimal surface forces the local saddle structure.
  2. A correlation between M4 and CP2 degrees of freedom is required. In order to obtain a minimal surface, one must achieve a situation in which the S2 part of the second fundamental form contains a contribution from a geodesic circle S1 ⊂ CP2 so that its trace vanishes. A simple example would correspond to a soap bubble-like minimal surface with M4 projection M1× X2, which has having geodesic circle S1 as a local CP2 projection, which depends on the point of M1× X2.
  3. The simplest candidate for the minimal surface M1× S2⊂ M4. One could assign a geodesic circle S1⊂ CP2 to each point of S2 in such a manner that the orientation of S1⊂ CP2 depends on the point of S2.

  4. A natural simplifying assumption is that one has S1⊂ S21⊂ CP2, where S21 is a geodesic sphere of CP2 which can be either homologically trivial or non-trivial. One would have a map S2→ S21 such that the image point of point of S2 defines the position of the North pole of S21 defining the corresponding geodesic circle as the equatorial circle.

    The maps S2→ S21 are characterized by a winding number. The map could also depend on the time coordinate for M1 so that the circle S1 associated with a given point of S1 would rotate in S21. North pole of S21 defining the corresponding geodesic circle as an equatorial circle. These maps are characterized by a winding number. The map could also depend on the time coordinate for M1 so that the circle S1 associated with a given point of S1 would rotate in S21.

    The minimal surface property might be realized for maximally symmetric maps. Isometric identification using map with winding number n=+/- 1 is certainly the simplest imaginable possibility.

See What could 2-D minimal surfaces teach about TGD? or the chapter Zero Energy Ontology.

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

Articles and other material related to TGD.

About the relationship of Kaehler approach to the standard picture

The replacement of the notion of unitary S-matrix with Kähler metric of fermionic state space generalizes the notion of unitarity. The rows of the matrix defined by the contravariant metric are orthogonal to the columns of the covariant metric in the inner product (T○ U)ABbar = TACbar ηCbarDUDBbar, where ηCbarD is flat contravariant Kähler metric of state space. Although the probabilities are complex, their real parts sum up to 1 and the sum of the imaginary parts vanishes.

The counterpart of the optical theorem in TGD framework

The optical theorem generalizes. In the standard form of the optical theorem i(T-T)mm=2Im(T) = TTmm states that the imaginary part of the forward scattering amplitude is proportional to the total scattering rate. Both quantities are physical observables.

In the TGD framework the corresponding statement

TABbarηBbar CABbarTBbar C +TABbarTBbar C=0 .

Note that here one has G= η+ T, where G and T are hermitian matrices. The correspondence with the standard situation would require the definition G= η +iU. The replacement T→ T= iU, where U is antihermitian matrix, gives

One has

i[UABbarηBbarC + ηABbarIBbarC] = UABbarUBbar C .

This statement does not reduce to single condition but gives two separate conditions.

  1. The first condition is analogous to Optical Theorem:

    Im[ηABbarUCBbar+UABbarηBbarC]=- Re[UABbarUBbar C] = Re[UABbarUCBbar] .

  2. Second condition is new and reflects the fact that the probabilities are complex. It is necessary to guarantee that the sum of the probabilities reduces to the sum of their real parts.

    Re[ηABbarUCBbar+UABbarηBbarC]= -Im[UABbarUCBbar] .

    The challenge would be to find a physical meaning for the imaginary parts of scattering probabilities. This is discussed in (see this). The idea is that the imaginary parts could make themselves visible in a Markov process involving a power of the complex probability matrix.

In the applications of the optical theorem, the analytic properties of the scattering matrix T make it possible to construct the amplitude as a function of mass shell momenta using its discontinuity at the real axis. Indeed, 2Im(T) for the forward scattering amplitude can be identified as the discontinuity Disc(T).

In the recent case, this identification would suggests the generalization

Disc[TABbarηBbarC]= TABbarηBbar CABbarTCBbar .

Therefore covariant and contravariant Kähler metric could be limits of the same analytic function from different sides of the real axis. One assigns the hermitian conjugate of S-matrix to the time reflection. Are covariant and contravariant forms of Kähler metric related by time reversal? Does this mean that T symmetry is violated for a non-flat Kähler metric.

The emergence of QFT type scattering amplitudes at long length scale limit

The basic objection against the proposal for the scattering amplitudes is that they are non-vanishing only at mass shells with m2=n. A detailed analysis of this objection improves the understanding about how the QFT limit of TGD emerges.

  1. The restriction to the mass shells replaces cuts of QFT approach with a discrete set of masses. The TGD counterpart of unitarity and optical theorem holds true at the discrete mass shells.
  2. The p-adic mass scale for the reaction region is determined by the largest ramified prime RP for the functional composite of polynomials characterizing the Galois singlets participating in the reaction. For large values of ramified prime RP for the reaction region, the p-adic mass scale increases and therefore the momentum resolution improves.
  3. For large enough RP below measurement resolution, one cannot distinguish the discrete sequence of poles from a continuum, and it is a good approximation to replace the discrete set of mass shells with a cut. The physical analogy for the discrete set of masses along the real axis is as a set of discrete charges forming a linear structure. When their density becomes high enough, the description as a line charge is appropriate and in 2-D electrostatistics this replaces the discrete set of poles with a cut.
This picture suggests that the QFT type description emerges at the limit when RP becomes very large. This kind of limit is discussed in the article considering the question whether a notion of a polynomial of infinite degree as an iterate of a polynomial makes sense (see this). It was found that the number of the roots is expected to become dense in some region of the real line so that effectively the QFT limit is approached.
  1. If the polynomial characterizing the scattering region corresponds to a composite of polynomials participating in the reaction, its degree increases with the number of external particles. At the limit of an infinite number of incoming particles, the polynomial approaches a polynomial of infinite degree. This limit also means an approach to a chaos as a functional iteration of the polynomial defining the space-time surface (see this). In the recent picture, the iteration would correspond to an addition of particles of a given type characterized by a fixed polynomial. Could the characteristic features for the approach of chaos by iteration, say period doubling, be visible in scattering in some situations. Could p-adic length scale hypothesis stating that p-adic primes near powers of two are favored, relate to this.
  2. For a large number of identical external particles, the functional composite defining RG involves iteration of polynomials associated with particles of a particular kind, if their number is very large. For instance, the radiation of IR photons and IR gravitons in the reaction increases the degree of RP by adding to P very high iterates of a photonic or gravitonic polynomial.

    Gravitons could have a large value of ramified prime as the approximately infinite range of gravitational interaction and the notion of gravitational Planck constant (see this) originally proposed by Nottale suggest. If this is the case, graviton corresponds to a polynomial of very high degree, which increases the p-adic length scale of the reaction region and improves the momentum resolution. If the number of gravitons is large, this large RP appears at very many steps of the SFR cascade.

A connection with dual resonance models

There is an intriguing connection with the dual resonances models discussed already in (see this).

  1. The basic idea behind the original Veneziano amplitudes (see this) was Veneziano duality. The 4-particle amplitude of Veneziano was generalized by Yoshiro Nambu, Holber-Beck Nielsen, and Leonard Susskind to N-particle amplitude (see this) based on string picture, and the resulting model was called dual resonance model. The model was forgotten as QCD emerged.
  2. Recall that Veneziano duality (1968) was deduced by assuming that scattering amplitude can be described as sum over s-channel resonances or t-channel Regge exchanges and Veneziano duality stated that hadronic scattering amplitudes have a representation as sums over s- or t-channel resonance poles identified as excitations of strings. The sum over exchanges defined by t-channel resonances indeed reduces at larger values of s to Regge form.
  3. The resonances have zero width and the imaginary part of the amplitude has a discontinuity only at the resonance poles, which is not consistent with unitarity so that one must force unitarity by hand by an iterative procedure. Further, there were no counterparts for the sum of s-, t-, and u-channel diagrams with continuous cuts in the kinematical regions encountered in QFT approach. What puts bells ringing is the u-channel diagrams would be non-planar and non-planarity is the problem of the twistor Grassmann approach.
It is interesting to compare this picture with the twistor Grassman approach and TGD picture.
  1. In the TGD framework, one just picks up the residue of what would be analogous to stringy scattering amplitude at mass shells. In the dual resonance models, one keeps the entire amplitude and encounters problems with the unitarity outside the poles. In the twistor Grassmann approach, one assumes that the amplitudes are completely determined by the singularities whereas in TGD they are the residues at singularities. At the limit of an infinite-fold iterate the amplitudes approach analogs of QFT amplitudes.
  2. In the dual resonance model, the sums over s- and t-channel resonances are the same. This guarantees crossing symmetry. An open question is whether this can be the case also in the TGD framework. If this is the case, the continuum limit of the scattering amplitudes should have a close resemblance with string model scattering amplitudes as the M4× CP2 picture having magnetic flux tubes in a crucial role indeed suggests.
  3. In dual resonance models, only the cyclic permutations of the external particles are allowed. As found, the same applies in TGD if the scattering event is a cognitive measurement (see this), only the cyclic permutations of the factors of a fixed functional composite are allowed. Non-cyclic permutations would produce the counterparts of non-planar diagrams and the cascade of cognitive state function reductions could not make sense for all polynomials in the superposition simultaneously. Remarkably, in the twistor Grassmann approach just the non-planar diagrams are the problem.
See the article About TGD counterparts of twistor amplitudes or the chapter with the same title.

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

Articles and other material related to TGD. 


Sunday, January 16, 2022

Bosonic strange metals and oscillatory behavior of magnetoresistance

Strange metals are difficult to understand in the standard paradigm. Linear dependence with respect to temperature instead of quadratic in fermionic liquids is one problem. In the TGD framework, dark particles as ordinary particles with effective Planck constant heff=nh0>h at magnetic flux tubes, explains this: see the blog posting or the article TGD and condensed matter.

In the article "Signatures of a strange metal in a bosonic system" by Yang et al published in Nature, bosonic strange metals are studied instead of fermionic ones. The system can also be superconducting and this seems to be essential.

The linear dependence on magnetoresistance in an external magnetic field B is the second interesting phenomenon.

  1. Below the onset of temperature Tc1>Tc, the low-field magneto-resistance varies with a periodic dictated by superconducting flux quantum suggesting that the density of charge carriers varies with this period.
  2. What comes to mind is the De Haas-Van Alphen effect in field B (see this).

    The magnetic susceptibility of the system varies periodically with the inverse of the magnetic flux Φ = e ∫ BdS defined by extremal orbit of electrons at the Fermi surface in field B. Φ is measured in units defined by elementary flux quantum h/2e.

  3. Could spin=0 Cooper pairs be formed from the electrons at the Fermi surface and lead to the De Haas-Van Alphen effect. They would go to the flux tubes of the external magnetic field B with a rate determined by the magnetic flux.

    The rate for this highest, when the extremal orbit at the Fermi surface corresponds to a quantized flux. Otherwise, energy is needed to kick the electrons from the Fermi surface to a larger orbit in order to satisfy the flux quantization condition.

Now one considers magnetoresistance rather than susceptibility. The linearity in magnetoresistance suggests that the resistance in the external field is mostly due to magnetoresistance.
  1. Could the analog of the De Haas-Van Alphen effect be present so that the density of Cooper pairs as current carriers at "endogenous" magnetic flux tubes has an oscillatory behavior as a function of the external magnetic field B? Could there be a competition for the Cooper pairs between the magnetic fields of flux tubes and external magnetic field B?
  2. When the flux Φ for the external B is near the multiple of the elementary flux quantum at extremal orbits at Fermi surface, the formation of spin=0 Cooper pair and transgder to the flux tubes of B would become probable by De-Haas-van Alphen effect. The number of Cooper pairs at "endogenous" flux tubes is therefore reduced and the current therefore reduced.
See the article TGD and condensed matter or the chapter with the same title.

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

Articles and other material related to TGD.

Friday, January 14, 2022

Why the resistance of strange metals is linear in temperature?

The so-called strange metals are a real troublemaker for condensed matter theorists. This posting was inspired by the most recent findings about strange metals (see this).

Could one understand the somewhat mysterious looking linear high T dependence of the resistivity of strange metals in TGD the framework?

In the TGD based model of high T superconductivity (see this) charge carriers are dark electrons, or rather Cooper pairs of them, at magnetic flux tubes which are effectively 1-D systems. Magnetic flux tubes are much more general aspect of TGD based model of condensed matter (see this).

Could magnetic flux tubes carrying dark matter with heff=nh0> h also explain the resistance of strange metals. I have actually asked this question earlier.

More precisely: Could the effective 1-dimensionality of flux tubes, darkness of charge carriers, and isolation from the rest of condensed matter together explain the finding?

Isolation would mean that only the collisions of dark electrons with each other cause resistance.

One can make a dimensional estimate.

  1. Assume that the resistance ρ can be written in the form

    ρ= (4π/ω2)/τ= (me/ne e2)/τ .

    Here ω is the plasma frequency

    ω2= 4πne e2/me.

    ne is 3-D electron density.

    What happens for 3-D ne in the case of 1-D flux tube? It would seem that ne must be replaced with linear density divided by the transversal area S of the flux tube: ne= (dne/dl)×(1/S).

  2. τ is the time spent by the charge carrier in free motion between collisions. Charge carrier is in thermal motion with thermal velocity vth= kT/m . The length Lf of the free path is determined non-thermally. Hence one has

    τ= Lf/vth= mLf/kT .

    This gives 1/τ= kT/mLf.

  3. For the resistivity ρ one obtains

    ρ= (me/nee2)kT/mLf,

    which indeed depends linearly on T as it does for strange metals.

    For m=me, one would have

    ρ= kT/(nee2Lf) .

For the TGD based view of condensed matter, see for instance this.

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

Articles and other material related to TGD.

Wednesday, January 12, 2022

Critical questions related to the number theoretical view about fundamental dynamics

One can pose several critical questions helping to further develop the proposed number theoretic picture.

Is mere recombinatorics enough as fundamental dynamics?

Fundamental dynamics as mere re-combination of free quarks to Galois singlets is attractive in its simplicity but might be an over-simplification. Can quarks really continue with the same momenta in each SSFR and even BSFR?

  1. For a given polynomial P, there are several Galois singlets with the same incoming integer-valued total momentum pi. Also quantum superpositions of different Galois singlets with the same incoming momenta pi but fixed quark and antiquark numbers are in principle possible. One must also remember Galois singlet property in spin degrees of freedom.
  2. WCW integration corresponds to a summation over polynomials P with a common ramified prime (RP) defining the p-adic prime. For each P of the Galois singlets have different decomposition to quark momenta. One can even consider the possibility that only the total quark number as the difference of quark and antiquark numbers is fixed so that polynomials P in the superposition could correspond to different numbers of quark-antiquark pairs.
  3. One can also consider a generalization of Galois confinement by replacing classical Galois singlet property with a Galois-singlet wave function in the product of quark momentum spaces allowing classical Galois non-singlets in the superposition.

    Hydrogen atom serves as an illustration: electron at origin would correspond to classical ground state and s-wave correspond to a state invariant under rotations such that the position of electron is not anymore invariant under rotations. The proposal for transition amplitudes remains as such otherwise.

Note however that the basic dynamics at the level of a single polynomial would be recombinatorics for all these options.

General number theoretic picture of scattering

Only the interaction region has been considered hitherto. One must however understand how the interaction region is determined by the 4-surfaces and polynomials associated with incoming Galois singlets. Also the details of the map of p-adic scatting amplitude to a real one must be understood.

The intuitive picture about scattering is as follows.

  1. The incoming particle "i" is characterized by p-adic prime pi, which is RP for the corresponding 4-surface in M8. Also the "adelic" option that all RPs characterize the particle, is considered below.
  2. The interaction region corresponds to a polynomial P. The integration over WCW corresponds to a sum over several P:s with at least one common RP allowing to map the superposition of amplitudes to real amplitude by canonical identification I: ∑ xnpn→ ∑ xnp-n.

    If one gives up the assumption about a shared RP, the real amplitude is obtained by applying I to the amplitudes in the superposition such that RP varies. Mathematically, this is an ugly option.

  3. If there are several shared RPs, in the superposition over polynomials P, one can consider an adelic picture. The amplitude would be mapped by I to a product of the real amplitudes associated with the shared RP:s. This brings in mind the adelic theorem stating that rational number is expressible as a product of the inverses of its p-adic norms. The map I indeed generalizes the p-adic norm as a map of p-adics to reals. Could one say that the real scattering amplitude is a product of canonical images of the p-adic amplitudes for the shared RP:s? Witten has proposed this kind of adelic representation of real string vacuum amplitude.

    Whether the adelization of the scattering amplitudes in this manner makes sense physically is far from clear. In p-adic thermodynamics one must choose a single p-adic prime p as RP. Sum over ramified primes for mass squared values would give CP2 mass scale if there are small p-adic primes present.

The incoming polynomials Pi should determine a unique polynomial P assignable to the interaction regions as CD to which particles arrive. How?
  1. The natural requirement would be that P possess the RPs associated with Pi:s. This can be realized if the condition Pi=0 is satisfied and P is a functional composite of polynomials Pi. All permutations π of 1,...,n are allowed: P= Pi1○ Pi2○ ....Pin with (i1,...in)=(π(1),...,π(n)). P possesses the roots of Pi.

    Different permutations π could correspond to different permutations of the incoming particles in the proposal for scattering amplitudes so that the formation of area momenta xi+1= ∑k=1ipk in various orders would corresponds to different orders of functional compositions.

  2. Number theoretically, interaction would mean composition of polynomials. I have proposed that so-called cognitive measurements as a model for analysis could be assigned with this kind of interaction (see this and this). The preferred extremal property realized as a simultaneous extremal property for both K\"ahler action and volume action suggests that the classical non-determinism due to singularities as analogs of frames for soap films serves as a classical correlate for quantum non-determinism (see this).
  3. If each incoming state "i" corresponds to a superposition of Pi:s with some common RPs, only the RP:s shared by all compositions P from these would appear in the adelic image. If all polynomials Pi are unique (no integration over WCW for incoming particles), the canonical image of the amplitude could be the product over images associated with common RPs.

    The simplest option is that a complete localization in WCW occurs for each external state, perhaps as a result of cognitive state preparation and reduction, so that P has the RP:s of Pis as RP:s and adelization could be maximal.

See the article About TGD counterparts of twistor amplitudes or the chapter with the same title.

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

Articles and other material related to TGD.

Tuesday, January 11, 2022

Fundamental quark dynamics as recombinatorics for Galois singlets

The proposal that unitary S-matrix should be replaced with the Kähler metric for the fermionic sector of the state space is extremely attractive. In the sequel the explicit expressions for the scattering probabilities are deduced and are of the same form as in the case of unitary S-matrix. The natural question is whether the notions of virtual and real particles can be geometrized.

Virtual states correspond to Galois non-singlets with momentum which is algebraic integer and real states to Galois singlets with momentum which is ordinary integer. The scattering amplitudes are shown to have basic properties as in QFT. Scattering amplitudes associated with mere re-combinations of quark states to different Galois singlets as in the initial states. Quarks move as free particles.This corresponds to OZI rule and conforms with the assumption that all particles are Galois composites of quarks

One can also ask whether the counterpart of S-matrix has on mass shell virtual states as singularities. This turns out to be the case. Also the analogs of non-planar amplitudes are allowed.

Explicit expressions for scattering probabilities

The proposed identification of scattering probabilities as P(A→B)= gABbargABbar in terms of components of the Kähler metric of the fermionic state space.

Contravariant component gABbar of the metric is obtained as a geometric series ∑n&ge 0 Tn from from the deviation TABbar= gABbarABbar of the covariant metric gABbar from δABbar.

g this is not a diagonal matrix. It is convenient to introduce the notation ZA, A=1,...,n, ZAbar=Zn+k, k=n+1,...,2n. So that the gBbarC corresponds to gk+n,l= δk,l+Tk,l.

and one has

gABbar to gk,l+n= δk,l+T1k,l.

The condition gABbargBbarC= δAC gives

gk,l+ngl+n,m= δkm .

giving

lk,l+T1k,l)(δl,m+Tl,m)

= δk,m + (T1+ T + T1T)km = δk,m ,

which resembles the corresponding condition guaranteeing unitarity. The condition gives

T1= -T/(1+T)>=- ∑n>1 ((-1)nTn .

The expression for PA→B reads as

P(A→B)=gABbargBbarA

=[1-T/(1+T)+T -(T/(1+T))ABT]AB .

It is instructive to compare the situation with unitary S-matrix S=1+T. Unitarity condition SS= 1 gives

T=-T/(1+T) ,

and

P(A→B)=δAB+ TAB+TAB+ TABTAB= [δAB-(T/(1+T))AB+TAB -(T/(1+T))ABTAB .

The formula is the same as in the case of Kähler metric.

Do the notions of virtual state, singularity and resonance have counterparts?

Is the proposal physically acceptable? Does the approach allow to formulate the notions of virtual state, singularity and resonance, which are central for the standard approach?

  1. The notion of virtual state plays a key role in the standard approach. On-mass-shell internal lines correspond to singularities of S-matrix and in a twistor approach for N=4 SUSY, they seem to be enough to generate the full scattering amplitudes.

    If only off-mass-shell scattering amplitudes between on- mass-shell states are allowed, one can argue that only the singularities are allowed, which is not enough.

  2. Resonance should correspond to the factorization of S-matrix at resonance, when the intermediate virtual state reduces to an on-mass-shell state. Can the approach based on Kähler metric allow this kind of factorization if the building brick of the scattering amplitudes as the deviation of the covariant Kähler metric from the unit matrix δABbar is the basic building bricks and defined between on mass shell states?

    Note that in the dual resonance model, the scattering amplitude is some over contribution of resonances and I have proposed that a proper generalization of this picture could make sense in the TGD framework.

The basic question concerns the number theoretical identification of on-mass-shell and off-mass-shell states.
  1. Galois singlets with integer valued momentum components are the natural identification for on-mass-shell states. Galois non-singlet would be off-mass-shell state naturally having complex quark masses and momentum components as algebraic integers.

    Virtual states could be arbitrary states without any restriction to the components of quark momentum except that they are in the extension of rationals and the condition coming from momentum conservation, which forces intermediate states to be Galois singlets or products of them.

    Therefore momentum conservation allows virtual states as on mass shell states, that is intermediate states, which are Galois singlets but consist of Galois non-singlets identified as off-mass-shell lines. The construction of bound states formed from Galois non-singlets would indeed take place in this way.

  2. The expansion of the contravariant part of the scattering matrix T1 = T/(1+T) appearing in the probability

    PA→B=gABbargABbar

    =[1-T/(1+T)]AB+TAB -[T/(1+T)]ABT]AB .

    would give a series of analogs of diagrams in which Galois singlets of intermediate states are deformed to non-singlets states.

  3. Singularities and resonances would correspond to the reduction of an intermediate state to a product of Galois singlets.
What about the planarity condition in TGD?

The simplest proposal inspired by the experience with the twistor amplitudes is that only planar polygon diagrams are possible since otherwise the area momenta are not well-defined. In the TGD framework, there is no obvious reason for not allowing diagrams involving permutations of external momenta with positive energies resp. negative energies since the area momenta xi+1= ∑k=1i pk are well-defined irrespective of the order. The only manner to uniquely order the Galois singlets as incoming states is with respect to their mass squared values given by integers.

Generalized OZI rule

In TGD, only quarks are fundamental particles and all elementary particles and actually all physical states in the fermionic sector are composites of them. This implies that quark and antiquark numbers are separately conserved in the scattering diagrams and the particle reaction only means the-arrangement of the quarks to a new set of Galois singlets.

At the level of quarks, the scattering would be completely trivial, which looks strange. One would obtain a product of quark propagators connecting the points at mass shells with opposite energies plus entanglement coefficients arranging them at positive and negative energy light-cones to groups which are Galois singlets.

This is completely analogous to the OZI role. In QCD it is of course violated by generation of gluons decaying to quark pairs. In TGD, gauge bosons are also quark pairs so that there is no problem of principle.

See the article About TGD counterparts of twistor amplitudes or the chapter with the same title.

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

Articles and other material related to TGD.

Friday, January 07, 2022

Horizontal Gene Transfer by Remote Replication?

This chapter was inspired by the discovery that a horizontal gene transfer (HGT) between eukaryotes is possible. The belief has been that HGT is possible only from prokaryotes to prokaryotes or eukaryotes. The basic obstacles are that the host DNA is within the cell nucleus and that DNA is tightly bound to chromosomes. The transfer should also occur to germ cells in order to have a lasting effect.

The case considered is HGT of antifreezing gene (AFG) from herring to smelt, which could have occurred during simultaneous spawning of herring and smelt in the same area. The AFT of herring associated with a transposon could have somehow attached to the sperm cell of the smelt and carried by it to the egg of the smelt. Vector carrying AFT to the sperm cell of smelt is needed and there are only guesses about what it might be.

That HGT however occurs, justifies a heretical question. Could it be only the genetic information, which is transferred and used to construct DNA in the host as a kind of remote replication analogous to quantum transportation? The findings of Gariaev and Montagnier indeed suggest remote replication and TGD provides a new physics model for it.

See the article Horizontal Gene Transfer by Remote Replication? or the chapter with the same title.

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

Articles and other material related to TGD. 


Sunday, January 02, 2022

More about the replacement of S-matrix with a generalized Kaehler metric in fermionic degrees of freedom

The following is almost as such a response to a comment of Stephen Crowley relating to unitarity.

I explain first the origins for the postulate about unitary S-matrix (introduced 1937 by Wheeler and 1940 independently by Heisenberg) as an encoder of the predictions of quantum theory. I consider the problems associated with this notion and then briefly summarize the TGD based solution of the problems discussed in here.

  1. Wave mechanics starts from a non-relativistic situation: there is a preferred time coordinate and Hamiltonian time evolution. In the relativistic situation this is not the case.

    In wave mechanics, time evolution is classically obtained by exponentiating a Hamiltonian: one has a flow. In wave-mechanics Hamiltonian time evolution is replaced with that generated by a Hamiltonian as an operator. There is of course normal ordering non-uniqueness but the real problem is that Hilbert space allows an endless variety of different unitary evolutions. Any hermitian operator generates such. This looks really ugly.

  2. Unitarity reduces to a belief already in quantum field theory based on path integral formalism and the situation is not made easier by the divergence problems. One can try to build unitarity by hand using unitarity conditions. Also here there is non-uniqueness since one is forced to take the discontinuity of the amplitude and use dispersion relations to deduce the entire amplitude.
  3. When space-time itself becomes topologically non-trivial and 3-space is dynamic changing its topology, the idea about Hamiltonian unitary evolution in a fixed spacetime becomes totally obsolete. This and the failure of the path integral approach, were the reasons why I ended up with the extensions of Einstein's program: geometrize entire quantum theory. Later the number theoretic dual of this program emerged and both are crucial in the twistorial construction of scattering amplitudes in the TGD framework.
  4. Also the twistor approach suffers from the unitarity problem: there is no proof for the unitarity. Second problem is that only planar amplitudes can be constructed. Could it be that these problems could have a common solution?
Something seems to go wrong with the entire QFT, and the natural guess is that the notion of unitarity is wrong. Also Nima Arkani-Hamed challenges the notion but cannot provide anything, which would replace it.

The failure of locality is the second cornerstone assumption, which seems to fail in the twistor approach: Yangian algebras have multi-local generators and multilocal Noether charges are definitely in conflict with locality of QFT. In TGD, the replacement of point-like with 3-surface means giving up locality from the very beginning.

In the unitarity problem, the geometrization of the quantum physics program came into rescue.

  1. Encode the physics, not by unitary S-matrix, but by the Kähler geometry of the state space. K\"ahler geometry has been already introduced for WCW but can can do it also for the state space: I proposed this here.

    It turned out that this is not yet quite a correct idea. A more precise statement would be following. Encode the transition amplitudes which define zero energy states in the fermionic degrees of freedom as the analog of Kähler geometry. Bosonic degrees of freedom would correspond to WCW. The resulting generalization of Kähler geometry would be somewhat analogous to what might be called super-WCW.

    The fermionic operator monomials consisting of creation (annihilation) operators creating positive (negative) energy parts of many fermion states would be multiplied by complex coordinate Zi (their conjugates) would define analogs of super fields. Monomials for theta parameters would be replaced by oscillator operator monomials. The monomials with odd fermion number need not be multiplied with anticommuting parameters since fermion number conservation is forced by vacuum expectation value.

    The generalization of exponent exp(-K) of Kähler action K obtained by adding to K this linear combination of these monomials would be formally analogous to QFT action expential containing also the fermionic part. What would matter, would be the vacuum expectation value of the expansion of the exponential giving rise to scattering amplitudes at the limit Zi=0. It is the Zi→ 0 limit that one considers in QFT for the action to deduce n-point functions. Zi &neq;0 would be something different and in QFT interaction terms would correspond to this kind of terms. Could Zi &neq;0 represent real physical situations?

  2. What is of extreme importance is that the situation is infinite-D. The experience with WCW geometry (already Freed noticed that loops spaces have unique Kähler geometry with maximal isometries from the mere existence of Riemann connection) strongly suggests that a non-trivial "super-Kaehler" geometry is unique if it exists at all.

    It must have maximal symmetries and is necessary a constant curvature geometry so that the generalization of Kähler metric, whose components are transition amplitudes, allow to code the entire Kähler geometry for arbitrary values of complex coordinates Zi.

  3. Unitarity conditions are replaced by almost identical conditions stating simply that the contravariant Kähler metric is the inverse of the covariant metric as a matrix. There is however an important difference: probabilities are in general complex. Probability conservation however holds true for the real parts of probabilities so that physics is OK.
  4. Allowance of complex probabilities creates however an interpretational problem and the solution became clear now the Kähler metric itself has an interpretation as Fisher information matrix. The usual probability interpretation is secondary but overall important for the testing of the theory. The information theoretic interpretation is more fundamental. It is accompanied by geometric interpretation so that infinite-D Kähler geometry both at the level of WCW and Hilbert space, number theory, information theory and probability theory meet and lead to a generalization of the notion of the fermionic Hilber space.
Zero energy ontology (ZEO) allows us to interpret the situation.
  1. In ZEO, the interpretation of quantum theory changes from "western" to "eastern". One gives up the western idea about a fixed reality. In ZEO only events are real both as zero energy states and as quantum jumps identified as moments of consciousness. To me this looks like Buddhism.

    Conserved probabilities for particle reactions still provide an empirical source of information about the state in thermo-dynamical sense. This picture of course conforms with the TGD based view about consciousness as a continual recreation of reality.

  2. The exponent of Kähler function with a fermionic part determined by a superposition of operator monomials creating the positive (negative) energy parts of zero energy states multiplied by complex coordinates Zi (their conjugates) becomes the analog of thermodynamic state. I have used to speak of a complex square root of thermodynamics. The complex coordinates Zi parametrize a given state as an analog for a phase of a quantum theory or of a coherent state.

    In the "Buddhist" view there are only events A → B but no fixed reality exists. The ratios of probabilities for the occurrence of events A → B are fundamental from the experimental point of view and deduced at the limit Zi they are enough in the constant curvature case. From these probabilities one can in principle deduce what one can know about the values of Zi.

    This conforms with ZEO, where time evolution associated with a transition A → B becomes a key element: behavior in biology and neuroscience, computer program in computer science. Could exact holography at classical level mean that the eastern and western views are nearly equivalent. In any case, the exact holography is broken: the minimal surfaces as preferred extremals are not completely deterministic but have singularities as analogs of frames and at frames determinism is violated.

  3. Cosmologists are probably not happy about this ontological relativism. In TGD cosmology indeed becomes a hierarchy of sub-cosmologies assignable to causal diamonds (CDs): there is no absolute reality lasting from time=-&infty; to time=+&infty;.

    I am however sure that the extreme mathematical beauty, elegance and uniqueness of this view leave no other option than to accept it. This view only repeats at the level of WCW and at the level of the fermionic state space what Einstein did for space-time.

See the article About TGD counterparts of twistor amplitudes or the chapter with the same title.

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

Articles and other material related to TGD.