Both arrows of time are possible microscopically

Sabine Hossenfelder comments in her Youtube talk (see this) about the recent theoretical claim that microscopically both arrows of time are possible.  In the article "Emergence of opposing arrows of time in open quantum systems" (see this), Thomas Guff, Chintalpati Umashankar Shastry, Andrea Rocco  argue that  the thermodynamic arrow of time can be understood as a property of the initial state without assuming that that it is assumed at the level of dynamics as usually believe. This means that at the microscopic level both arrows of time are allowed.

It seems that the time is becoming mature for TGD: TGD indeed predicts that both arrows of time are possible non only in microscopic, but also  macroscopic, scales.   Zero energy ontology (ZEO) is the quantum ontology of TGD,  implied by the new view of space-time as a 4-surface.  In  TGD  point-like particles are replaced with 3-surfaces rather than strings.

General coordinate invariance without a mathematically non-existing path integral requires holography and  classical physics becomes an exact part of quantum physics. Space-time surfaces become   analogs of Bohr orbits for 3-surfaces as particles  and classical physics represents an exact part of  quantum physics. 

Holography= holomorphy principle allows to reduce field equations to minimal surface  equations for any action as long as it is general coordinate invariance and constructible in terms of the induced geometry. Hence the classical theory is universal.  Remarkably, the holography is slightly non-deterministic. 

 Quantum states are now quantum superpositions of Bohr orbits rather than 3-surfaces and  this  solves the basic problem of quantum measurement theory.  Space-time surfaces are located inside causal diamonds CD=cd×CP₂, where cd is a causal diamond of M⁴.  The cds form a scale hierarchy and CD has interpretation as a counterpart for the perceptive field of a self, defined by the superposition of Bohr orbits changing in "small" state function reductions (SSFRs), which define the TGD counterpart for the sequence of measurements of the same observables in standard QM. In TGD, the Zeno effect means   that the 3-D states and 3-surfaces at the passive boundary (PA) of CD do not change in the sequence of SSFRs. They however change at  the active boundary (AB) of CD (since the superpositions of  Bohr orbits (4-D minimal surfaces) made possible by the slight classical non-determinism  change present already for 2-D minimal surfaces.  This gives rise to a conscious entity sel having CD as its perceptive field.

CD  increases in a statistical sense during the sequence of SSFRs and this gives a correlation between the subjective time  of self defined by the sequence of SSFRs and  the geometric time defined by the distance between the tips of the CD.

Either boundary of the CD can be   passive.  In "big" SFRS (BSFRs) as counterparts of ordinary SFRs, the roles of PB and AB change and the arrow of geometric time changes. This can occur   on all scales.  In the article motivating the Youtube talk this is claimed to happen in microscopic scales.

The number theoretic view of TGD   predicts an entire hierarchy of effective Planck constants and therefore quantum coherence becomes possible in all scales. Also the  arrow of time can change  in even macroscopic scales.  

In  TGD inspired theory of consciousness,  this allows us to interpret the sleep state as a state with a non-standard arrow of time. We do not remember anything from the period of deep sleep since the classical signals traveling in the wrong direction of time do not reach us the next day. Roughly, one half of the Universe is sleeping and wake-up and sleep periods characterize all systems even in cosmological scales. Biological death is also falling in sleep but on a longer time scale and with more dramatic consequences.

See  for instance  the articles   TGD as it is towards end of 2024: part I, TGD as it is towards end of 2024: part II, and Some comments related to Zero Energy Ontology (ZEO).

A more detailed view of topological qubit in the TGD framework

The Zoom discussion with Tuomas Sorakivi about Microsoft's claimed realization of a topological qubit was very inspiring and led to a generalization of the notion of Majorana qubit characterized by Z₂ group acting as reflection so that one an assign parity to Majora qubit. In TGD Z₂ is replaced by a generalization of the Galois group and this leads to a discrete group bringing in mind anyons with a larger number of internal states. This also involves the notion of Galois confinement discussed earlier. What would be achieved would be a dual interpretation as topological qubit or as number theoretic qubit. This conforms with the notion of geometric Langlands duality realized in the TGD framework as M⁸-H duality (see this and this).

1. Background

The basic idea is that the Majorana fermions of condensed matter are assumed to define a qubit. A Majorana fermion would be a superposition of an electron and a hole. The idea is not pretty because it violates the superselection rule for fermions and the conservation of the fermion number is also questionable. It has also been found that the existence of the Majorana fermion claimed by the Microsoft research group and the superconductivity it requires have not been demonstrated.

A hole must physically correspond to the electron being "somewhere else". In the case of an insulating band, it could be in the conduction band, or in the case of a conduction band, in another conduction band: this description would hold in wave vector space.

In TGD, the electron corresponding to a hole could be in another space-time plane. The equivalent of a Majorana fermion would be a superposition of states where the fermion would be on two space-time sheets. It would be a topological qubit because small deformations of the space-time surfaces do not cause contact between the surfaces. Of course, one can argue that the energies must be the same on different sheets. In the case of condensed matter, this would correspond to the branches of the Fermi surface touching each other.

This idea can be realized concretely: a transfer is an operation that, when repeated, produces the original state, i.e. acts like a unitary operator. The square of the Majorana fermion creation operator is correspondingly a unitary operator. This leads to a concrete model (see this) and the idea that OH-O^-+p qubits could realize topological qubits, at least in biology.

Yesterday's discussion led to a review of holography=holomorphic vision.

2. About Galois groups and their TGD counterparts

How to define a Galois group when we are in dimension 4 and not in the complex plane? Is it possible to define a generalization of the concept of ramified primes: these would give a generalization of p-adic primes that label elementary particles in TGD?

2.1 Space-time surfaces as solutions of the equations (f₁,f₂)=(0,0)

Holography= holomorphy vision leads to the following picture.

1. Space-time surfaces are roots (f₁,f₂)=(0,0) of two complex values functionsf_i defining an analytic map from H=M⁴× CP₂ to C². f_i, i=1,2 is an analytic function of 3 complex coordinates of H=M⁴× CP₂ and one hypercomplex coordinate of M⁴. The Taylor coefficients of f_i are in an extension E of rationals. A very important special case corresponds to a situation in which f_i are polynomials. There are good physical reasons to believe that f₂ is the same for a very large class of space-time surfaces and its roots actually define a slowly varying analog of cosmological constant.

   The roots (f₁,f₂)=(0,0) correspond to space-time sheets, which are algebraic surfaces. The space-time surface need not be connected. The Hamilton-Jacobi coordinates (see this) serve the coordinates of H: there is one hypercomplex coordinate u and its dual v and 3 complex coordinates w for M⁴ and ξ₁ and ξ₂ for CP₂. The coordinate curves for u and v of M⁴ have light-like tangent vectors.

2. Dimensional reduction occur because the hypercomplex coordinates are separated from the dynamics and take role of parameters appearing as coefficients of f_i interpreted as functions of w,ξ₁ξ₂ so that only three complex coordinates ξ₁,ξ₂ and w would effectively remain dynamical. For partonic orbits as the interfaces between Minkowskian regions and CP₂-like regions with Euclidean signature of the induced metric, u= constant would be a natural condition. At these 3-surfaces, the dimensional reduction would be complete: the roots would not depend on u. In the interior of CP₂ like region u would be also constant and Minkowskian contribution to the induced metric would vanish as for CP₂ type extremals.
3. If f_i is polynomial P_i with coefficients in the rational expansion E, analytic flows as analogs of homotopies that take roots as regions of the space-time surface to each other would correspond to a 4-D version of the Galois group. The definition of the Galois group operation would be as a flow rather than an automorphism of an algebraic extension leaving E unaffected as usual. Definition as flow is used in braid representations of groups.

   This is new mathematics for me and perhaps for mathematicians as well. It would be a generalization of the 2-D Galois group.

4. The 4-surfaces corresponding to different roots would have lower-dimensional surfaces interfaces. The hypercomplex sector effectively decouples this gives 2 conditions in 4-D space stating that the complex coordinates, say w, are identical at the boundary so that interfaces are string world sheets. This fixes w(u) at the interface.
   1. The roots as 4-surfaces could correspond to branches of a fold taking place along a string world sheet. This suggests a complexification of a cusp catastrophe. For cusp catastrophe, the catastrophe curve is a V-shaped curve along which two real roots of a polynomial of degree 3 depending on a real coordinate x and real parameters a,b co-incide. Now x is replaced with a complex coordinate w which at the string world sheet depends on the coordinate hypercomplex coordinate u. One can say that the 1-D boundary of V is replaced with string world sheets. What happens in the vertex of V is an interesting question. The boundaries of V having coinciding root pairs as analogs co-incide. Does this mean that two string world sheets fuse. Could this be regarded as a reaction in which strings fuse along their full length?
   2. Could the space-time regions defined by the roots genuinely intersect along a string world sheet? This kind of intersection would be analogous to a self-intersection of a 1-dimensional curve. The basic example is the curve x²-y²=0 splitting to the curves x-y=0 and x+y=0.

      If for instance, f₁=P₁ fails to be irreducible and decomposes to a product P₁=Q₁Q₂ of two polynomials Q_i, the roots Q₁=0 and Q₂=0 intersect at the common root Q₁=Q₂=0. These kinds of intersections are excluded if one allows only irreducible polynomials. The irreducibility can fail for some values of the coefficients of the polynomials.

      The space-time surface would decompose to a union of 2 surfaces represented as roots of Q₁ and Q₂ and do not interact unless they intersect along a string world sheet. The dimensional reduction due to the same Hamilton-Jacobi structure implies that 2 2-surfaces intersect in 6-dimensional space. This does not happen in the generic case. Hence this option does not seem possible.

Analytical flows take the points corresponding to the roots from one sheet to another through string world sheets: here cusp catastrophe helps to visualize. String world sheets correspond to the common values of ξ₁, ξ₂, w. For instance w can serve as coordinate and at the intersection w the value is fixed.

The ends of the strings correspond to complex numbers that depend on the time parameter u: the complex number, say w, would represent the intersection of the space-time sheets as a root. The complex roots depend on u through polynomial coefficients. If one has u=constant at the parton trajectories at which the signature of induced metric changes, the u-dependence disappears at the paths of the string ends at which fermions are attached in the physical picture about the situation. Under very mild assumption about the polynomials P_i(w,ξ₂,ξ₂,u=0), the roots can be algebraic numbers in an extension of E and would characterize the intersections of the roots of the equation (P₁,P₂)=(0,0).

These complex numbers are considered a generalization of complex roots and would be related to quantum criticality, i.e., the fact that the two roots are the same and the system is at the interface between space-time regions. The criticality would correspond to a fold of the cusp catastrophe.

If it is possible to attach a Galois group to the set of string world sheets transforming them to each other, it would transform different string world sheets into each other. Could this group serve as an algebraization for the generalized Galois group represented as a geometric flow?

What about the counterparts of p-adic primes? The product of the differences of the roots defines the discriminant D. Can it be decomposed into the product of powers of algebraic primes of the extension E? If so , this would generalize the concept of a p-adic prime. The intersections of the sheets of the space-time surface, or rather their intersections with partonic 2-surfaces, could be associated with p-adic primes. This has just been a physical picture.

2.2 The analogs of Galois group associated with dynamic symmetries

The descriptions g: C²→ C² define dynamic symmetries f=(f₁,f₂) → g(f) , which produce new space-time surfaces of higher complexity.

1. What happens in the operation (f: H→ C²)→ (g\circ f: H→ C²), f H→ C² and g: C²→ C²? The surface g(f) =0 would correspond to the surface (g₁(f₁,f₂), g₂(f₁,f₂))=(0,0).

   The intuitive picture is that complexity increases the in these dynamical symmetries. For example, in the case of C, iterations produce fractals. These descriptions would provide a geometric model for the abstraction and can be combined and iterated.

2. If g(0,0)=(0,0) then (f₁,f₂)=(0,0) remains a root and in the "G\"odelian" view of the classical dynamics of the space-time surfaces produces analogies to theorems (see this). Other roots represent more complex space-time surfaces: the non-trivial action of g brings in the meta-level and makes the composition with g provides statements about statements represented by(f₁,f₂)=(0,0). "Simple" spacetime sheets, which do not allow a decomposition to f=g(h) , would represent lowest level statements. The associated magnetic bodies could correspond to the surfaces (g₁(f₁,f₂), g₂(f₁,f₂))=(0,0). Entire hierarchies of meta-levels are possible.

   Magnetic bodies indeed represent a higher level in the number theoretic hierarchies and correspond to larger values of the effective Planck constant as dimension of extension associated with E. In the TGD inspired quantum biology, the magnetic body serves as a controller of the biological body.

Can the concept of Galois group be generalized in this case?

1. The regions of the surface (g₁(f₁,f₂), g₂(f₁,f₂)=(0,0) correspond to roots. 2+2 conditions fix the roots f₁= a and f₂=b are 6-surfaces, and their intersection is a 4-surface.

   If the consideration is restricted to the surface u=constant, assumed to correspond to a partonic orbit, then the roots do not depend on u and can be algebraic numbers and perhaps a generalization of the Galois group could be defined.

   The condition g₂(f₁,f₂)=0 gives f₁ =h(f₂), where h is an algebraic function. The condition g₁(f₁,h(f₁))=0 gives f₁=a and f₂=b, where a and b are algebraic numbers. They correspond to 6-surfaces: the space-time surface is the intersection of two algebraic 6-surfaces. If (a,b) and (c,d) are not identical, then the corresponding surfaces are disjoint.

2. Is it possible to define a Galois group using the algebraic extension of E defined by the roots? The Galois group would permute the surfaces (f₁=a,f₂=b), which would correspond to pairs of complex numbers and would be disjoint.

   Now the element of the Galois group would not correspond to a flow permuting the pairs (a,b). It should act as an automorphism of E× E. Is this possible? One cannot provide E× E with the structure of a number field. It is however enough to have algebra structure involving component-wise sum (a,b)+(c,d)= (a+c,b+d) and product (a,b)*(c,d)= (ac,bd).

   The algebraic extension of E× E defined by the roots of g(f) as pairs (r_i,1,r_i,2) would have an automorphism group identifiable as the Galois group. Also discriminant D=(D₁,D₂) could be defined using the component-wise product for the differences of the root pairs. It would have two components and one can ask whether D₁ and D₂ could be decomposed to products of algebra primes of E.

3. Is it possible to generalize the concept of ramified prime? They would define generalized p-adic primes. The discriminant can be defined as the product of the differences of the roots, which would factor into the product of algebraic primes in the extension E. The roots (a,b) would be in E× E so that the structure of the number field would be required. For quaternions the lack of commutativity implies that the product of the root differences depends on their order.

It was already noticed that there are good physical motivations for decomposing WCW to sub-WCWs for which f₂ is fixed. The counterpart of the ordinary Galois group is obtained in the sub-WCWs: g=(g₁,I) reduces to a map g₁: C→ C. The roots of g₁(f₁)=0 are surfaces (g₁(f₁),f₂)=(0,0). g₁ has n surfaces as roots. The transitions between these disjoint surfaces would generate the analog of the ordinary Galois group acting as a number-theoretic dynamical symmetry group. Also ramified primes as primes of algebraic extension of E are obtained.

1. Representations of the Galois group transfer fermions between space-time regions corresponding to different roots of g₁. The Galois group is generally non-Abelian and its elements could appear in topological quantum computation as basic operations for the topological qubits. The analogs of anyons would be irreducible representations of the Galois group.
2. If the degree n is prime, g is a prime polynomial. It cannot be represented as a composite of polynomials, whose degree is a product of smaller integers.

   Remark: If P is irreducible then it cannot be a product, in which case the degree would be the sum of their degrees. Therefore one has two kinds of primeness.

3. The surfaces corresponding to different roots of g₁ are disjoint. If the roots are the same then the surfaces are the same. If g(0,0)=0 then (f₁,f₂)=(0,0) is a root. As two roots approach each other. the two separate surfaces merge into one. What does this mean physically? Should one regard the identical copies of the surface as different surfaces, members of a double, and carrying different many-fermion states? In any case, the order of the Galois group is reduced in this case.

3. On the intersections of 4-surfaces

There are several options to consider.

1. The 2 4-surfaces X⁴ and Y⁴ correspond to different pairs (f₁,f₂). If the Hamilton-Jacobi structures are different so that the hypercomplex coordinates (u,v) are different, the intersection X⁴\cap Y⁴ is a discrete set of points. Field theory suggests itself as a natural description of fermions assigned with the interaction points.

   If the Hamilton-Jacobi structures are the same, the dimensional reduction occurs and one has effective intersection of 2 complex surfaces in 6-D complex space. In the generic case the intersection is empty.

2. One can also consider the analogs of self-intersections as interfaces for 2 4-D roots for the same pair (f₁,f₂). The intersection consists of string world sheets. As found, genuine self-intersection is exclude so that only the analogy of a complexified cusp catastrophe remains.

   String model is a natural description of the interactions of 4-surfaces and the self-interaction of 4-surfaces in the fermionic sector. Fermion propagators can be calculated because the induced spinor field is a restriction of the corresponding H.

The analogy of TGD based physics with formal systems discussed in \cite{btar/Gtgd} led to ask whether the interaction of space-time surfaces involves the fusion of the 3-surfaces with different Hamilton-Jacobi structures to a single connected 3-surface with a common Hamilton-Jacobi structure for the components. Physically the tusion could mean a generation of monopole flux tube contacts between the 3-surfaces.

In the G\"odelian framework, this interaction would have an interpretation as a morphism realized as an action of the composite space-time surfaces on each other. In the connected intermediate state, string model type description might apply in the fermionic degrees of freedom. Even stronger condition would be that fermions reside at the string ends at partonic orbits.

4. Galois group as as group of possible transfer operations for fermions and a generalization of the Majorana qubit

4.1 Roots for the condition (f₁,f₂)=(0,0) as space-time sheets

Generalization of the Galois group. Galois generalizes Z₂ for Majorana fermions. Classical equivalent of the transfer operation between space-time sheets. A particle is transported through a string word sheet corresponding to a common root pair to another sheet.

Topological/number-theoretic qubit. Transfer through a string world sheet. What is the physical interpretation. String 1-D object in 3-space. Could the Riemann surface for z^1/n serve as an analogy. Anyons and braid statistics. Since hypercomplex coordinates are passive, we get effective 2-dimensionality and braid statistics.

4.2 Roots in the special case g=(g₁,Id)

Ordinary roots of a polynomial represented as 4-surfaces. Disjoint or identical. However, the representation of the Galois group of g₁ is non-trivial. These would correspond to abstractions. Fermion transfer between disjoint surfaces Galois group operation represented using oscillator operators.

When does this work?

1. This happens only if f₁ allows the decomposition f₁= g₁(h₁). When could this be possible? In the case of polynomials, this means that the degree of f₁ for a given H complex coordinate ξ₁,ξ₂, or w polynomial is the product of the degrees of n₁× n₂× n₃ for the lower degree polynomials n₁,n₂,n₃.
2. If the degrees of the polynomial for different coordinates are primes, then the decomposition is not possible. These would be "prime polynomials". The 3 prime numbers p₁,p₂,p₃ characterize these. If it is a homogeneous polynomial, then one prime number p is enough. These polynomials would be in a special position physically. They would correspond to "elementary particles". The tetrahedra associated with them would be uniform.

4.3 Concrete realization of topological/number-theoretic qubit and generalization of qubits

The TGD based view leads to generalization of bit to n-ary digit or pinary digit, where n or p corresponds to a degree of a polynomial g₁ in g=(f₁,Id) defining a dynamical symmetry and associated Galois group whose elements would correspond to transfers of fermions between different branches of the space-time surface.

1. Roots as regions of an n-sheeted space-time surface correspond to roots (f₁,f₂)=(0,0) and would correspond to different values of an n-ary digit. They are glued together along string world sheets as analogs of folds.

   The functional composition f→ g(f) gives rise to hierarchies of Galois groups. The Galois group, represented as analytic flows, replaces the group Z₂ of the Majorana case. Analytic flows define braiding operations, which define the 4-D Galois group.

2. Also the dynamical symmetries g give rise to an analog of a Galois group. The non-vanishing roots of g are disjoint. The Galois group can be defined in the usual way for g= (g₁,I) but the definition generalizes g= (g₁,g₂).

   For OH-O^-+p qubits (see this and this) they could correspond to different pairs because h_eff would be of different magnitude.

4.4 Generalization of a bit to n-ary digit and pinary-digit

The replacement of bit with n-ary digit would take place when the degree d of the polynomial P₁ (or g₁ in g=(g₁,Id)) is d=n and bit → pinary digit when the d is a prime: d=p. Polynomials for which the degrees with respect to complex coordinates of H are primes are primes with respect to the functional composition and could physically correspond to fundamental objects appearing at the bottom of the hierarchy obtained by a functional composition with maps g.

These primes should not be confused with ramified primes. One can of course ask whether the p-adic primes appearing in p-adic mass calculations could actually correspond to these primes.

This allows us to consider a possible definition for a topological/number-theoretic qubit. For g(0)=0, the original surface is included in the set of g\circ f=0 surfaces. In the case of OH-O^-+p qubits, the magnet monopole flux tubes could correspond to the non-vanishing root f\neq 0 of g. In this case the Galois group of g would be Z₂ and correspond to the parity of Majorana fermions. In the general case more complex Galois groups are possible.

4.5 A more precise connection to the Majorana qubit of condensed matter

The definition of a Majorana qubit involves the observation that when two branches of the Fermi surface that correspond to an insulator and to a conduction band touch each other, the gap energy disappears. In superconductivity, this gap energy is very small but non-vanishing. If this energy vanishes, Majorana type excitation becomes possible and is interpreted as a quantum superposition of an electron and a hole.

What could this situation correspond to or how could it generalize in TGD?

1. M⁸-H duality (see this) strongly suggests that Fermi surfaces determined as an energy constant surface in momentum space have space-time counterparts.
2. The group Z₂ defining the parity of Majorana qubit would be generalized to Galois group and and one can consider two options corresponding 1) to the 4-D Galois group realized as analytic flows assignable to a connected 4-surface (f₁,f₂) and 2) to the Galois group assignable to g= (g₁,Id) or even g= (g₁,g₂) acting as a dynamical symmetry.

Consider option 1) first.

1. The Galois group would relate string world sheets to each other. The branches of the Fermi surface could at the space-time level correspond to 2-D string world sheets at which the roots associated with the different space-time surface sheets (f₁,f₂)=(0,0) coincide . One could move from one branch of the space-time sheet to another through the string world sheets. Each string world sheet would correspond to a discrete complex point (ξ₁,ξ₂,w).
2. The E³ projection of the string world sheet would be a string, which would have apparent ends at the "boundary" of the 3-surface. The 2-D "boundaries" of the 3-surfaces are surfaces, where the 3-surface has a fold, i.e. the normal M⁴ coordinate has a maximum value. One can say that the string effectively ends at these surfaces although it actually has a fold.

   String would sheet would also have an end at the partonic orbit, where the signature of the space-time metric changes. Since the coordinate u would be constant inside the CP₂ type extremals, the 2-D string world sheet reduces to a 1-D light-like curve inside it.

   In the case of topological qubits, the superconducting wire could correspond to the string identifiable as the superconducting wire whose ends correspond to the points of the Fermi surface at which the branches of the Fermi surface touch. The ends of the wire, assumed to carry Majorana fermions, would correspond to the real ends of the string at partonic orbits to which fermions are assigned or to an apparent end at the fold.

3. The situation would correspond to quantum criticality, since even a small perturbation will move the particle to one of the branches.

For option 2), the space-time surfaces related by the Galois group for g=(g₁,Id) and more general g=(g₁, g₂) would be disjoint. This does not conform with the assumption that Fermi surfaces touch at a point. This picture could however work for OH-H^- topological qubits for which the two surfaces related by Z₂ Galois group for g=(g₁,g₂) would have different "internal" Galois groups represented as flows leaving the space-time surface invariant.

See the article The realization of topological qubits in many-sheeted space-time or the chapter Quartz crystals as a life form and ordinary computers as an interface between quartz life and ordinary life?.

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

Some TGD inspired comments about biocatalysis

It seems that the Pollack effect could play a fundamental role in living matter. In the TGD framework, Pollack effect has several applications generalizations. OH-bonds, typically associated with acids, are fundamental and they could be dynamical so that Pollack effect and its reversal, that is the transformation OH↔ O^-+ p, where p is dark proton at the monopole flux tube, could be central in quantum biology (see this). Pollack effect would generate exclusion zones (EZs) with negative charge and also the electrons could be dark. What follows is an attempt to test this proposal.

There are good reasons to believe that this qubit is topological and TGD analog of condensed matter Majorana fermion requiring respecting fermion number superselection rule (see this). This topological qubit would make possible fully topological quantum computations based on braidings of monopole flux tubes (see this, this and this).

Catalyst action by a gel phase bounding water is necessary for the Pollack effect. It could be needed to kick the OH bond near to the criticality against the splitting to O^-+p induced by the Pollack photon. One should also understand catalyst action in the TGD framework. I have proposed that here magnetic monopole flux tubes and large value of h_eff behaving like dark matter could play a central role: the latest discussion can be found in (see this). Magnetic body could serve in the role of midwife or energy investor in bio-catalysis which together with the Pollack effect would make it possible to overcome the potential barrier making the reaction very slow.

Biocatalysis, Pollack effect, and catabolism and anabolism as aspects of metabolism are considered from the TGD point of view. In particular, the mysterious notions of high energy phosphate bond and the existence of two different phosphates, the organic and inorganic phosphate are discussed.

See the article Some TGD inspired comments about biocatalysis or the chapter The based view about dark matter at the level of molecular biology.

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

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

Levy flight and TGD

I encountered an interesting Science Daily article (see this) about the finding that in high energy heavy ion collisions in which quark-gluon plasma is created. What is studied is the distribution for the distances of hadrons travelled between two collisions during the rescattering phase during which hadrons collide, fuse and long-lived resonances decay. This distribution is studied for pions (see this). It is found that the distribution does not obey Gaussian distribution as in Brownian motion but has a long tail obeying a power law.

This distribution is associated with Levy flight (see this) for which the distribution parameter corresponds to a Levy walk for which the second moment of the distribution is finite. Levy walk is the analog of Brownian motion for the distances travelled between subsequent collisions characterized by the distribution for the time durations for the periods during which the motion occurs with a constant velocity. The tail obeys a power law, which suggests a scaling symmetry and fractality associated with criticality.

1. TGD view about the origin of random walks

Zero energy ontology (ZEO) replaces standard ontology of quantum theory and solves the basic problem of quantum measurement theory. ZEO is forced by the slight non-determinism of classical field equations of TGD which are exactly solvable by holography= holomorphy hypothesis. This non-determinism could explain Brownian motions and also Levy flight as a signature of quantum criticality.

1.1 Could random motion reflects the structure of space-time at the fundamental level?

The key point is that in TGD point-like particles are replaced by 3-surfaces and their orbits define space-time surfaces in H=M⁴× CP₂ as analogs of slightly non-deterministic Bohr orbits satisfying holography forced by the realization of general coordinate invariance without path integral (see this and this). Holography= holomorphy principle implies that the solution of field equations is a minimal surface irrespective of the action principle as long as it is general coordinate invariant and constructed in terms of the geometry induced from H. One can wonder whether Brownian motion and Levy flight could at the fundamental level relate to the slight failure of the classical determinism in the TGD framework.

Space-time surfaces as solutions of field equations (see this) are determined as a root f=(f₁,f₂)=(0,0) of two analytic functions of one hypercomplex coordinate and 3 complex coordinates of H defining what I call Hamilton-Jacobi coordinates (see this). Space-time as a minimal surface is expected to be slightly non-deterministic. The non-determinism would be located at 3-D surfaces analogous to the 1-D frames spanning 2-D soap film where the same occurs. At the frames the space-time surface can branch in several ways.

This could give rise to an analog of Brownian motion and Levy flight with point-like particles replaced with a 3-surface and the orbits with the preferred extremals as a minimal surface defining the analog of Bohr orbits.

1.2 How to describe classical non-determinism elegantly?

There are strong constraints to be satisfied at the loci of non-determinism since the field equations code for conservation laws of the isometry charges of H and these must be satisfied. This kind of conditions must be satisfied also for 2-D minimal surfaces at frames and they pose very strong conditions for what can happen at the frames. A natural guess is that the space-time surface decomposes to regions charracterized by different function pairs f=(f₁,f₂) having the loci of non-determinism as interfaces.

In TGD number theoretic vision is complementary to the geometric vision of physics. It involves p-adic physics for various values of p-adic prime characterizing the p-adic numbers field in question. Also the extensions of p-adic numbers, induced by extensions E of rationals, are allowed. In the holography= holomorphy vision, the extension of rationals would characterize the Taylor coefficients of analytic functions f₁,f₂ of the Hamilton-Jacobi coordinates of H. This is not enough to give p-adic primes without additional assumptions.

For ordinary polynomials of a complex argument the product for the difference of the roots defines what is known as discriminant. For rational coefficients it decomposes to a product of powers of so called ramified primes which define special p-adic primes and the proposal is that these primes define the p-adic primes characterizing the system. These p-adic primes near powers of 2 were found to characterize elementary particles in p-adic mass calculations that I performed around 1995 (see this).

1.3 How could the roots of polynomials of complex coordinates emerge?

One should somehow introduce naturally the roots of 2-D complex polynomials in order to get p-adic primes as ramified primes.

1. The basic observation is that the analytic maps g: C²→ C² allow to generate new solutions from the existing ones by the map f=(f₁,f₂)→ g(f)= (g₁(f₁,f₂),g₂(f₁,f₂)).
2. As a special case, one has g= (P,Id), P a polynomial, giving rise to map f=(f₁,f₂)→ (P(f₁),f₂). There are physical motivations for considering this restriction: the proposal is that f₂=0 represents a very long length scale constraint on the physics (see this), defining a slowly varying cosmological constant. The "world of classical worlds" (WCW) as the space of these 4-surfaces inside causal diamond (CD) would decompose to sub-WCWs inside with f₂ is fixed. One can calculate the roots for the polynomial P and assign to it ramified primes. The roots (g₁(f₁),f₂)=(0,0) give 4-surfaces representing roots of r of P as 4-surfaces f₁=r.
3. The maps g play a key role in the TGD based view about the physical analog of metamathematics and Gödel's incompleteness theorem (see this). Space-time surfaces as almost deterministic systems would be analogous to theorems with the assumptions of the theorem represented by holographic data at the so called passive boundary of CD remaining unaffected in the sequence of small state function reductions (SSFRs) defining the analog of Zeno effect.

   The axiom system would be represented by the variational principle, which also gives minimal surfaces as its solutions by holography= holomorphy correspondence therefore one can say that classical physics defines the analog of logic, which cannot depend on axiomatics. The map g would mean a transition to a meta level and the surfaces obtained in this way would represent theorems about theorems. One would obtain hierarchies of composite maps representing statements about statements about....

   If this picture is correct, the p-adic primes as ramified primes would be associated with a metalevel accompanying already elementary particles and represented by g. Hierarchies of these metalevels are predicted the iterations of g gives rise to the analogs of Mandelbrot fractals and Julia sets and for the approach to chaos as increase of complexity.

1.4 Does the p-adic non-determinism of differential equations correspond to the classical non-determinism of Bohr orbits?

p-Adic differential equations differ from their real counterparts in that the integration constants are pseudo constants having vanishing p-adic derivatives. They depend on a finite number of pinary digits. This generalizes also to partial differential equations. Suppose that the space-time surfaces can be characterized by a p-adic prime p, or its analog for algebraic extension of rationals, identified as a ramified prime of f=P.

1. Could the classical non-determinism of field equations correspond to the p-adic nondeterminism for some ramified prime p in the sense that the coefficients of (f₁,f₂) are p-adic pseudo constants for p? Could the real coefficients and their p-adic counterparts be related by a canonical identification x_p=∑ x_npⁿ→ x_R= ∑ x_np^-n or be identical? Canonical identification would guarantee a continuous correspondence: otherwise one would obtain huge fluctuations.
2. Could the Brownian motion and Levy flight reflect the underlying p-adic non-determinism at the fundamental level at which particles are replaced by 3-surfaces which in turn are replaced by the analogs of Bohr orbits which are slightly non-deterministic.
3. Could the parameters characterizing the TGD analog of Brownian motion and Levy flight be p-adic pseudo constants depending on finite number of pinary digits for the Hamilton-Jacobi coordinates of H. Could they change their values at points with Hamilton-Jacobi coordinates with a finite number of pinary digits: this would conform with the idea of discretization as a representation for a finite measurement resolution.

One can consider several ways of modelling ordinary random walks from this perspective. One can consider two basic options.

1. The above picture suggests that one considers random walks as a smooth dynamical evolution at the p-adic level and that the replacement of the initial values or other parameters characterizing the orbits with p-adic pseudo constants as functions of time gives rise to discontinuities of say velocity in the random walk. The p-adic scale should make itself visible in statistical sense as a natural scale associated with the motion.

   This seems to require that first the real configuration space is mapped to its p-adic counterpart by the inverse of canonical identification. After the smooth p-adic orbit, say free motion or motion in gravitational field, is mapped to its real counterpart by canonical identification. One can require the orbit x₊vt is continuous (x₀ is constant) but v is pseudo constant. The orbit would be a zigzag curve having also the characteristic p-adic fractality.

2. Probability distributions suggests an alternative, perhaps more elegant approach. Also the p-adic calculations based on p-adic thermodynamics define this kind of approach. The key idea would be that everything is smooth at the p-adic level and canonical identification brings in fractallity.

   The probability distributions of durations T_n= t_n-t_n-1 and velocities v is a way to statistically characterize the random walk. Brownian walk and Levy flight serve as basic examples.

   Restrict first the consideration to the durations T_R of motion with a constant velocity. The inverse I^-1 of the canonical identification would map the real durations T_R to p-adic durations T_p=I^-1T_R. The p-Adic distribution function P_p(T_p) would be a smooth function in accordance with the idea that the fractality at the real side derives from p-adic smoothness via canonical identification. On the real side, canonical identification would give the real distribution as P_R(T_R)= (P_p(T_p))_R by I. Normalization would be required to get probability interpretation.

   One would have T_R→ T_p → P_p(T_p)→ (P_p(T_p))_R= P_R(T_R) One would obtain a hierarchy of distributions labelled by the parameters of P_p and by p. A similar map v_R→ v_pP_p(v_p)→ (P_p(v_p))_R= P_R(v_R) would give a fractal velocity distribution.

2. The role of quantum criticality

Classical non-determinism alone need not produce the Levy flight or Levy walk. In the particle physics experiments involving high energy ion collisions the total path lengths for particles during the period involving final state interactions are considered. It is indeed found that this distribution has a long tail obeying a scaling law. Since scaling laws mean fractality and are associated with the criticality, the natural question is whether quantum criticality could be involved.

The TGD based view of the generation of quark-gluon plasma suggests an interpretation in terms of quantum criticality against transition from the ordinary hadron physics characterized by the Mersenne prime M₁₀₇ to M₈₉ hadron physics characterized by a mass scales which is 512 times higher. By quantum criticality this hadron physics would however correspond to a large value of h_eff. For h_eff= 512= L(107)/L(89) the Compton length of M₈₉ and M₁₀₇ mesons a would be equal (see this and this). The transition to ordinary hadron physics could take place by p-adic cooling gradually, which would mean that p-adic prime corresponding to M₈₉ is gradually reduced to M₁₀₇ and the p-adic hadron mass scale reduced and Compton length would be preserved. I have considered the possibility M₈₉ hadron physics and p-adic cooling as a possible explanation of the solar wind and energy production at the surface of the Sun (see this).

See the article Levy flight and TGD or the chapter TGD and condensed matter.

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

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

Why life would be experienced as a panorama near the moment of death?

My email friend DU asked a highly interesting question: Why would the brain go through a recall of important life events before death?

The question is an excellent challenge for the TGD based view of conscious experience. It splits into many sub-questions. What is conscious experience? What is life? What happens in death? What happens in memory recall? Why would life be seen as a panorama before the moment of death?

1. What memories and memory recall are in the TGD Universe?

1. Memories are not possible in standard QM based theory of consciousness. Since the state after the quantum jump carries no information about the state before it.
2. In TGD the small failure of classical determinism plus holography predict that quantum states in zero energy ontology (ZEO) are superpositions of 4-d Bohr orbits for 3-surfaces as analogs of particles. In standard ontology one would have superpositions of 3-D surfaces.

   In ZEO, quantum states are superpositions of these Bohr orbits. However, all Bohr orbits start from the same 3-surface and repeated small SFRs (SSFRs)do not change this surface at the passive boundary (PB) of causal diamond. At the active boundary (AB) state changes and this gives rise to self as a conscious entity experiencing the flow of time. This corresponds to the Zeno effect in which the quantum state does not change in repeated quantum jumps.

3. Conscious experience is about space-regions where the non-determinism is. The 3-D loci of non-determinism would serve as sources of conscious experience. They would serve as memory seats. Small state function reductions (SSFRs) for a given seat with time t_n would replace the Bohr orbit with a new one at times t>t_n. The superpositions of Bohr orbits would be like quantum stories of life. These stories would begin from the same 3-D quantum state at PB.

   Each SSFR would mean memory recall would mean a quantum measurement for some memory seat as a locus of non-determinism. For instance, a localization to one alternative state of the memory locus would occur: this would mean selection of one possible classical future.

2. How to achieve the memory recall as a quantum measurement?

Option 1: The first view is that in an active memory recall a negative energy signal to geometric past is sent and this signal is received by the memory state and induces a quantum jump changing the 3-D state of the memory state and therefore also the future. This induces a signal propagating to the future brain and induces memory as conscious experience.

Option 2: The second view is that the memory loci with t=t_n are quantum entangled as they would be if a superposition of Bohr orbits is in question. What would be measured, would be the state of memory locus at t_n. This would mean a selection of one future.

3. What would happen in death?

In the TGD Universe, death and birth would be completely universal phenomena occurring in all scales and biological death and birth would be only a special instance of it.

1. In ZEO death would mean "big" SFR (BSFR) in which the arrow of time changes. This measurement is induced by interaction with the external world at AP.

   Density matrix is a fundamental observable and is measured in BSSF. In the interaction of AB with the external world, the density matrix of AB ceases to commute with the density matrix and observables whose eigenstates the state associated with PB is.

2. The measurement of density of AB occurs and necessarily changes the state at PB. The roles of AB and PB are changed and the arrow of geometric time changes. Old self dies and the reincarnated self with an opposite geometric arrow of time is born.

4. Why would death make it possible to recall the entire life?

The total memory recall should occur before the death of BSFR. All memory seats of self, which define subselves, should be activated simultaneously and involve SSFR.

Option 1: If negative energy signals to the geometric past induce a memory recall then a very strong signal of negative energy covering the entire frequency spectrum could generate this. A kind of white noise. All memory seats would be activated in time order propagating from now to the geometric past and the last activated seat would be near or at the moment of birth. Life would be lived in a reverse order. The arrow of geometric time for memory seats as subselves would change: also they would die and reincarnate with a reversed arrow of time!

After that the even bigger BSFR, in which self itself dies, would/could occur and also the arrow of time of self would change.

Option 2: In this case the entanglement between the neighboring memory seats would be reduced by a sequence of SSFRs propagating to the geometric past step by step and stop at PB. The outcome would be life experienced in reverse time order.

In the blog discussion a person who had experienced the life panorama told that the life review was in some sense objective: as if the life would have been seen from a higher perspective. This could be possible if the NDE involves transition to a higher reflective level at which the person is a subself. The memories would be memories of the higher level self about the person. This transition might mean that the third person perspective, which could be present always, is not masked by the sensory, motor and cognitive input anymore. Also a phase transition in which the algebraic complexity of the magnetic body increases can be considered.

The holography= holomorphy vision leads to a concrete proposal what reflective levels of consciousness could mean (see this). They could reduce to a functional composition f→ g(f) of an analytic map g: C²→ C² with a map f: H=M⁴× CP₂ → C², where f is an analytic map with respect to hypercomplex coordinate and 3 complex coordinates of H. When g reduces to (g,Id) acting trivially in the second factor of C², this gives compositions of complex maps of C→ C and iteration of g produces Mandelbrot fractals and Julia sets.

See the article Does Consciousness Survive Bodily Death? or the chapter with the same title.

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

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

The realization of topological qubits in many-sheeted space-time

Microsoft has unveiled Majorana 1, claimed to be the world’s first quantum processor powered by topological qubits (see this)

1. How could one stabilize the computations and qubits?

The basic problem is how to realize computations in a stable way and how to make stable enough qubits? Concerning computation, topology comes to rescue.

1. Topological quantum computations (see this can be represented as braidings which are topologically stable under small deformations. Each braid strand represent a unitary evolution of a particle representing a unitary evolution if a qubit and the braiding operation would represent the computation. Braiding can be either time-like dynamical operation for point-like particles in plane or space-like for a braid connects two planes.
2. Since the 2-D plane containing particles as punctures, the homotopy group is non-abelian. This means that the rotation of a puncture around a second puncture of say bound state can transform the state such that transformation is not a mere phase factor but is a rotation which change the directions of the spins of the particles involved. Therefore the exchange of particles which can be seen as basic braiding operation changing the braid strands can induce an operation, which can be used as a basic building brick for a topological quantum computation.

How could one obtain stable qubits? Qubit represented as a spin is not thermodynamically stable and extremely low temperatures are required. This is the case also for the proposed topological quantum computation: the reason is now that superconductivity is required and this is possible only at temperatures of order milli Kelvins. In any case, the notion of qubit should be topologized. How to achieve this? Here Majorana bound bound states have been proposed as an answer (see this).

1. Non-Abelian braid statistics, which means that their exchange realized as a 2-D rotational flow generated by braiding induces, instead of change of a sign in Fermi statistics, a non-Abelian unitary transformation of the state. It could be used to change the directions of their spins and affect the anyons. 2π rotation would induce a non-Abelian rotation instead of a mere sign change or phase factor in brain statistics. This is only possible in dimension 2 where the homotopy group can be non-abelian if there are punctures in the plane that the braids would represent. Similarly, swapping two Majorana fermions in braid produces a SU(2) rotation and can flip the spins and thus the qubits. This swap would be an essential operation in quantum computing. In order to have non-trivial topological quantum computation, one must have non-Abelian braid statistics characterized by a Lie group. Rotation group SO(2) or its covering SU(2) are the minimal options
2. The bound state of two Majorana fermions associated with planar punctures, anyons, would thus obey non-Abelian braid stastistics. It is also possible to affect the second fermion of Majorana bound state by rotating a puncture containing a fermion around the second fermion. Braidings could therefore represent unitary transformations having an interpretation as topological quantum computations. Wikipedia article mentions several realizations of Majorana bound states in superconductors. Quantum vortices in super conductors can provide this kind of states. The ends of the super-conducting wire or of line defects can contain the Majorana fermions. Also fractional Hall effect can provide this kind of states. The realization studied by Microsoft has the fermions of the Majorana fermion at the ends of a superconducting wire.
3. As I understand it, a condensed matter Majorana fermion would correspond formally to a superposition of an electron and a hole. The statistics would no longer be normal but non-Abelian Fermi statistic but would be that of a non-abelian anion. The weird sounding property of this statistics is that the the creation operator is equal to annihilation operator. One obtains two creation operators corresponding to two spin statse and square the creation operator of is unit operator: for fermions it vanishes. This implies that Majorana fermion number is defined only modulo 2 and only the number of fermions modulo 2 matters. Also the anticommutator of two creation operators at different points is equal to unit operator so that the system is highly nonlocal.
4. How the braiding could be realized? One can consider two options. Dance metaphor allows to understand the situation. Imagine that particles are dancers at the parquette. The dance would give rise to a time like braiding. If the feet of the dancers are tied to a wall of the dancing house by threads, also a space-like braiding is induced since the threads get tangled.
5. In the TGD framework, dancers would correspond to particle-like 3-surfaces moving in the plane and the dance would define the dancing pattern as a time-like braiding. This classical view is actually exact in the TGD framework since classical physics is an exact part of quantum physics in TGD. If thee particles are connected to the wall by threads realized as monopole flux tubes, a space-like braiding is induced.
6. These threads bring in mind the wires connecting superconductor and another object and containing Majorana fermions at its ends. Now the second end would be fixed and second would correspond to a moving particle. Majorana bound states would correspond to the ends of the thread and the superconducting flow of the second end would correspond to the dynamical braiding.

2. Algebraic description of Majorana fermions

The dissertation of Aran Sivagure contains a nice description of Majorana fermions (see this). Majorana fermions would be quasiparticles possible in a many-fermion state. They would create from a fermion state with N fermions a superposition of states with fermion numbers N+1 and N-1. They would be created by hermitian operators γ_i^+/-= a^†_i+/- a_i formed from the fermionic oscillator operators satisfying the standard anticommutation relations {a^†_i,a_j}= δ_i,j. Note that one consider also more general Hermitian operators γ_i^+/-= exp(iφ)a^†_i+/- exp(-iφ)a_i.

The anticommutations would be {γ_i ^ε₁, γ_i^ε₂}= 2× Id, where one has ε₁=+1 and ε₂=-1 and Id denotes the unit operator. Therefore the statistics is not the ordinary Bose or Fermi statistics and non-Abelian statistics.

What is so remarkable is that that also the anticommutators {γ_i,γ_j} satisfy {γ_i^+/-,γ_j^+/-}= 2Id even when i and j label different points. Therefore these operators are highly non-local meaning long range quantum coherence.

3. Could many-sheeted spacetime allow a more fundamental description of Majorana like states?

The problematic aspect of the notion of Majorana fermion is that the manyfermion states in this kind of situation do not in general have a well-defined fermion number. Physically, fermion conservation is a superselection rule so that the superposition of fermion and hole must physically correspond to a superposition of fermion states, where the hole corresponds to a fermion which is outside the system. TGD suggests an elegant solution of the problem.

1. In condensed matter physics Majorana fermions could be assigned with the vortices of superconductors. In the TGD Universe, these vortices could correspond to monopole flux tubes as body parts of field body. The states created by γ_i would be superpositions of states in which the fermion is at the monopole flux tube or at the normal space-time sheet representing the part of the condensed matter system that we see. The Majorana description would be effective description.
2. The Majorana creation operators γ_i would be replaced with operators which shift the fermion from ordinary space-time sheet to the monopole flux tube and vice versa. From the geometric interpretation it is clear that this operation must be idempotent. This operation must be representatable in terms of annihilation and creation operators. The operators γ_i would be expressible products of creation and annihilation operators acting at the space-time sheets 1 and 2 and one would have

   γ_i^ε= a^†₁a₂+ε a^†₂a₁.

   One can consider either commutation or anticommutation relations for these operators. Since the operation does not change the total fermion number, the interpretations as a bosonic operator is natural and therefore commutations relations look more plausible.

   Neglecting for a moment the indices labelling positions and spins, a rather general expression for the operator γ^ε would be

   γ^ε= a₁^†a₂+ ε a₂^†a₁ .

   This operator is hermitian. If fermionic anticommutations are true, one has for anticommutator

   (γ^ε)²= X+Y ,

   X=a₁^†a₂a₁^†a₂+ ε² a₂^†a₁a₂^†a₁ ,

   Y=ε₁ε₂ (a₁^†a₂a₂^†a₁+ a₂^†a₁a₁^†a₂) .

   If one can assume anticommutativity for the oscillator operators associated with flux tube and ordinary space-time sheet one has X=0. For Y one obtains

   Y= ε₁ε₂ (N₁a₂a₂^†+N₂ a₁a₁^†)= ε₁ε₂(-N₁-N₂+2N₁N₂) .

   The eigen values of (-N₁-N₂+2N₁N₂) vanish for (N₁,N₂) ∈ {(1,1),(0,0)} and are equal to -1 for (N₁,N₂) ∈ {(1,0),(0,1)} so that the eigenvalues are equal to -ε₁ε₂.

   One can consider also the commutator, which is perhaps more natural on the basis of the physical interpretation. The commutator obviously vanishes for ε₁=ε₂. For (ε₁,ε₂)=(1,-1) one has

   [γ¹,γ^-1]= ε₁ε₂ (a₁^†a₂a₂^†a₁- a₂^†a₁a₁^†a₂) =-(N₁a₂a₂^†-N₂a₁a₁^†)= N₂-N₁ .

   For ε₁=-ε₂, the eigen values of ε₁ε₂(N₂-N₁) vanish for (N₁,N₂) ∈ {(1,1),(0,0)} and belong to {1 ,-1} for (N₁,N₂) ∈ {(1,0),(0,1)}.

   Both the anticommutator and commutator resemble that in the Majorana case but are not identical to a unit operator since the eignvalues belong to the set {0,1} for the anticommutator and to the set {0,-1,1} for the commutator.

4. OH-OH⁺+p as topological qubits?

While writing this, I noticed that the OH-OH⁺+p qubits, where p is a dark proton ag monopole flux tubes, that I proposed earlier to play fundamental role in biology and perhaps even make quantum counterparts of ordinary computes possible, are to some degree analogous to Majorana fermions. The extremely nice feature of these qubits would be that superconductivity, in particular biosuperconductivity. would be possible at room temperatures. This is would be possible by the new physics predicted by TGD both at the space-time level and at the level of quantum theory.

1. In TGD space-times are surfaces in H=M^{× CP₂ and many-sheetedness is the basic prediction. Another related prediction is the notion of field body (magnetic/electric) body. Number theoretica view of TGD predicts a hierarchy of effective Planck constants making possible quantum coherence in arbitrarily long length scales. Second new element is zero energy ontology modifying profoundly quantum measurement theory and solving its basic problem.}
2. OH-OH⁺+p qubit means that one considers protons but also electrons can be considered. Now the proton is either in the OH group associated with water molecule in the simplest situation in which Pollack effect occurs or the proton is a dark proton at a monopole flux tube. A proton in OH would be analog of non-hole state and the dark proton in the flux tube be the anaog of hole state.
3. What is new is that the proton being on/off the spacetime surface would represent  a bit.  For Majorana fermions,  the situation is rather similar: the hole  corresponds to the electron being "somewhere else", which could also correspond to being on a monopole flux tube as I have suggested. In standard quantum computation, a qubit would correspond to a spin. The analog of  the Majorana qubit would be a quasiparticle which is superposition of transitions OH ↔ O^-+p transitions and transitions in which nothing happens: OH goes to itself and O^-+p goes to itself. The Majorana property would correspond to the fact that the transfer between two space-time sheets repeated twice is trivial.
4. If the energies for OH and OH⁺+p are close to each other, the situation is quantum critical and the qubits can be flipped and a process similar to quantum computation becomes possible. Also superconductivity becomes possible at the magnetic flux tubes analogous to magnetic vortices appearing in superconductivity and in fractional Quantum Hall effect. These are truly topological qubits also because the topologies of the spacetime surface for different bit values ​​are different. However, the energy difference must be larger than the thermal energy, otherwise the qubits become unstable. With the help of electric fields, qubits can be sensitized to quantum criticality and their inversion becomes possible.
5. The above argument suggests that a non-abelian statistics could be understood for OH-OH⁺+p qubits. The anticommutation/commutation relations for the operators trasferring protons to the magnetic body would not be identical to those for Majorana oscillator operators the squares of these operators would be proportional to unit operator which is essentially the Majorana property. I have proposed a possible realization for this in a more general case. The exchange of dark protons/qubits would be induced by reconnection of monopole flux tubes:    it would therefore be a purely topological process. Nothing would be done to the dark electrons, but the flux tubes would be reconnected. Strands AB and CD would become strands AD and BC. At the same time, the unilluminated protons would become associated with different O^-. In this exchange, could the final result be represented as an SU(2) rotation for the entire space.

See the article The realization of topological qubits in many-sheeted space-time or the chapter Quartz crystals as a life form and ordinary computers as an interface between quartz life and ordinary life?.

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

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

Gödel, Lawvere and TGD

The tweets of Curt Jaimungal (see this) inspired an attempt to understand Gödel's incompleteness theorem and related constructions from the TGD point of view.

It has remained somewhat unclear to me how the notion of conscious self is defined in theories based on pure mathematics. I however understand that the conscious system is identified as an object in a category X and the view of self about itself would be a set of morphisms of f_x of X→ X as structure-preserving descriptions, morphisms, which would give information about x to the other selves y as objects of X. One can define X^Y as an object having as objects the morphisms Y→ X. X^Y would correspond to X as seen by object Y.

This associates to every object x \in X morphism f_x\in X^X of the category X into itself. One could say that X embedded in X^X and f_x corresponds to models of x for other selves of y \in X. Under conditions formulated by Lawvere, any morphism f in X^X has a fixed point y_f. In particular, for f_x one can find y_x such that f_x(y_x)=y_x is satisfied. In some cases this might be the case. Under the assumptions of Lawvere, one can have y_x=x and this might be the case always. These kinds of objects x are very special and one can wonder what its interpretation is.

In particular, Gödel's sentence is a fixed point for a sentence f_x, which associates to a sentence y a sentence f_x(y) stating that y is not provable in the formal system considered. It turns out that f(x)=x is true. Therefore x is not provable but is true. Could this mean that this kind of object is self-conscious and has a self model?

On the other hand, self-reflection, which means that one becomes aware of the content of one's own consciousness at least partially, can be claimed to create descriptions of itself and fixed point property suggests an infinite number of levels or possibly limit cycles: for Julia sets only non-trivial limit cycles are present. Infinite regression however means a paradox. On the other hand one can argue that self-representation is trivial for a fixed point.

What is the situation in TGD? In the following the idea about physics laws, identified in the TGD frameworks as the dynamics of space-time surfaces, is discussed in detail from the perspective of the metamathematics or metaphysics.

The laws of physics as analogs for the axioms of a formal system

The basic idea is that the laws of physics, as they are formulated in the TGD framework (see this and this), can be regarded as analogs for the axioms of a formal system.

1. Space-time surface, which by holography= holomorphy vision is analogous to a Bohr orbit of particles represented as a 3-surface is analogous to a theorem. The slight classical non-determinism forces zero energy ontology (ZEO)(see this): instead of 3 surfaces the analogs of Bohr orbits for a 3-surfaces at the the passive boundary (PB) of the causal diamond (CD) are fundamental objects. By the slight classical non-determinism, there are several Bohr orbits associated with the same 3-surface X³ at the PB remaining un-affected in the sequence of "small" state function reductions (SSFRs). This is the TGD counterpart of the Zeno effect. The sequences of SSFRs defines conscious entity, self.
2. The adelization of physics means that real space-time surfaces obtained using extension of E of rationals are extended to adelic space-time surfaces. The p-adic analogs of the space-time surface would be correlates for cognition and cognitive representations correspond to the intersections of the real space-time surface and its p-adic variants with points having Hamilton-Jacobi coordinates in E (see this).  
3. Concerning Gödel, the most important question is how self reference as a metamathematical notion is realized: how space-time surfaces can represent analogs of statements about space-time surfaces. In the TGD framework holography= holomorphy vision provides an exact solution of the classical field equations in terms of purely algebraic conditions. Space-time surfaces correspond to the roots function pairs (f₁,f₂), where f_i are analytic functions of the Hamilton Jacobi coordinates of H=M⁴\times CP₂ consisting of one hypercomplex and 3 complex coordinates.

   The meta level could correspond to the maps g= (g₁,g₂): C²→ C², where g_i are also analytic functions or Hamilton-Jacobi coordinates, mapping the function pairs f=(f₁,f₂): H→ C² and giving new, number theoretically more complex, solutions. The space-time surfaces obtained in this way correspond to the roots of the composites gºf = (g₁(f₁,f₂),g₂(f₁,f₂)).

   g should act trivially at the PB of CD in order to leave X³ invariant. One can construct hierarchies of composites of maps g having an interpretation as hierarchies of metalevels. Iteration using the same g repeatedly would be a special case and give rise to the generalization of Mandelbrot fractals and Julia sets.

4. Second realization would be in terms of the hierarchy of infinite primes (see this) analogous to a repeated second quantization of a supersymmetric arithmetic quantum field theory for an extension E of rationals and starting from a theory with single particle boson and fermions states labelled by ordinary primes. Here one can replace ordinary primes with the prime of an algebraic extension E of rationals. This gives a second hierarchy. Also the Fock basis of WCW spinor fields relates to WCW like the set of statements about statements to the set of statements.

How space-time surfaces could act on space-time surfaces as morphisms

Could one, by assuming holography= holomorphy principle, construct a representation for the action of space-time X⁴ surface on other space-time surfaces Y⁴ as morphisms in the sense that at least holomorphy is respected. In what sense this kind of action could leave a system associated with X⁴ fixed? Can the entire X⁴ remain fixed or does only the 3-D end X³ of X³ at the PB remain fixed? In ZEO this is indeed true in the sequence of SSFRs made possible by the slight failure of the classical determinism.

What the action of X⁴ on Y⁴ could be?

1. The action of X⁴ on Y⁴ would be a morphism respecting holomorphy if X⁴ on Y⁴ have a common Hamilton-Jacobi structure (see this). This requirement is extremely strong and cannot be satisfied for a generic pair of disjoint surfaces X⁴ and Y⁴. The interpretation would be that this morphism defines a kind of perception of Y⁴ about X⁴, a representation of X⁴ by Y⁴. Ψ

   A naive proposal for the action of X⁴ on Y⁴ assumes a fixed point action for Y⁴=X⁴. The self-perception of X⁴ would be trivial. Non-triviality of self-representation since is in conflict with the fixed point property: this can be seen as the basic weakness of the proposal that conscious experience could be described using a formal system involving only the symbolic description but no semantics level.

2. The classical non-determinism of TGD comes to rescue here. It makes possible conscious memory and memory recall (see this and this) and the slightly non-deterministic space-time surface X⁴ as an analog of Bohr orbit can represent geometrically the data making possible conscious memories about the sequence of SSFRs. The memory seats correspond to loci of non-determinism analogous to the frames spanning 2-D soap films. In the approach based on algebraic geometry, the non-determinism might be forced by the condition that space-time surfaces have non self-intersections. Second possibility is that space-time surfaces consist of regions, which correspond to different choices of (f₁,f₂) glued together along 3-D surfaces.
3. Purely classical self-representation would be replaced at the quantum level by a quantum superposition of the Bohr orbits for a given X³. A sequence of "small" state function reductions (SSFRs) in which the superposition of Bohr orbits having the same end at the PB is replaced with a new one. SSFRs leave the 3-surfaces X³ appearing as ends of the space-time surface at the PB invariant. The sequence of SSFRs giving rise to conscious entity self, would give rise to conscious self-representation.
4. The fixed point property for X⁴ making the self-representation trivial would be weakened to a fixed point property for X³, and more generally of 3-D holographic data.

How zero energy states identified as selves could act on each other as morphisms?

How the superposition Ψ(X³) of Bohr orbits associated with X³ can act as a morphism on Ψ(Y³)? The physical interpretation would be that Ψ(X³) and Ψ(Y³) interact: Ψ(X³) "perceives" Ψ(Y³) and vice versa and sensory representations are formed. This sensory representation is also analogous with the quantum counterpart of the learning process of language models producing associations and association sequences as analogs of sensory perceptions (see this).

1. These "sensory" representations must originate from a self-representation. This requires a geometric and topological interaction X⁴ and Y⁴ as a temporary fusion of X⁴ and Y⁴ to form a connected 4-surface Z⁴. This would serve as a universal model for sensory perception. In the TGD inspired quantum biology, a temporary connection by monopole flux tubes serves as a model for this interaction. If the flux tubes serve as prerequisites and correlates for entanglement, entanglement could also be generated.
2. The holomorphy for Z⁴ requires that X⁴ on Y⁴ have a common Hamilton-Jacobi structure during the fusion but not necessarily before and after the fusion. Therefore the defining analytic function pairs (f₁,f₂) (see this) can be different before and after the fusion and during the fusion and also for X⁴ and Y⁴ after and before the fusion. This might be an essential element of classical non-determinism. Continuity requirement poses very strong conditions on the function pairs involved. The representations produced in the interaction would be highly unique. As already mentioned, also the absence of self-intersections could force classical non-determinism.

   The outcome of the temporary fusion would give rise to a representation of the action of X⁴ on Y⁴ and vice versa. The representation would be a morphism in the sense that outcomes are holomorphic surfaces and the ends of X⁴ and Y⁴ at the PB of CD remain unaffected.

3. The fixed point property for Z⁴ making the self-representation trivial would be replaced with the fixed point property for Z³ and therefore also X³ and Y³.
4. The time reversed variant of sensory perception has an interpretation as motor action between them and would involve a pair of BSFRs induced by a subsystem of Z⁴. Now the end of Z⁴ at the PB of CD would be changed. X⁴ would affect Y⁴ in a non-deterministic way. The construction of the representation of X⁴ on Y⁴ would reduce to a construction of a self-representation for Z⁴.

This view is inspired by the TGD view in which self is identified as a sequence of non-deterministic SSFRs and is thus not "provable" and has also free will. The holographic data would be in the role of the assumptions of a theorem, which need not to be proved and reduce to axioms, and the Bohr orbits would correspond to theorems deducible from these assumptions. In the interaction of X³ and Y³ a larger self Z³ would be created and would involve quantum entanglement. In this view, the infinite self reflection hierarchy is replaced with a finite sequence of SSFRs providing new reflective levels and self is a dynamical object.

See the article Gödel, Lawvere and TGD or the chapter with the same title.

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

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