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TGD Inspired Theory of Consciousness

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Year 2006



What really distinguishes between future and past?

Our knowledge about geometric future is very uncertain as compared to that about geometric past. Hence we usually use words like plan/hunch/hope/... in the case of geometric future and speak about memories in the case of geometric past. We also regard geometric past as something absolutely stable. Why we cannot remember geometric future as reliably as the geometric past? Is it that geometric future is highly unstable as compared to the geometric past? Why this should be the case? This provides a possible TGD based articulation for the basic puzzles relating to time experience. The latest progress in the understanding of quantum TGD allows a more detailed consideration of these questions.

1. Is p-adic-to-real phase transition enough?

The basic idea is that the flow of subjective time corresponds to a phase transition front representing a transformation of intentions to actions and propagating towards the geometric future quantum jump by quantum jump. All quantum states have vanishing total quantum numbers in zero energy ontology which now forms the basis of quantum TGD and this ontology allows to imagine models for what could happen in this process.

This starting point is the interpretation of fermions as correlates for cognition bosons as correlates for intentions/actions (see this). Fermions correspond to pairs of real and p-adic space-time sheets with opposite quantum numbers with p-adic space-time sheet providing a cognitive representation of the real space-time sheet. Bosonic space-time sheets would be either p-adic or real and thus represent intentions or actions. Fermionic world and its cognitive representations would be common to future and geometric past and the asymmetry would relate only to the intention-action dichotomy.

Geometric future contains a lot of p-adic space-time sheets representing intentions which transform to real space-time sheets allowing interpretation as desires inducing eventually neuronal activities. Time mirror mechanism for intentional action assumes that the phase transition gives rise to negative energy space-time sheets representing propagation of signals to geometric past where they induce neuronal activities. From Libet's experiments relating to neuronal correlates of volition the time scale involved is a fraction of second but an infinite hierarchy of time scales is implied by fractality.

Conservation of quantum numbers poses strong conditions on p-adic-to-real phase transition. Noether charges are in the real context given by integrals over partonic 2-surfaces. The problem is that these integrals do not make sense p-adically. There are two options.

a) Give up the notion of p-adic Noether charge so that it would not make sense to speak about four-momentum and other conserved quantum numbers in case of p-adic space-time sheet. This implies zero energy ontology in the real sector. All real space-time sheets would have vanishing conserved quantum numbers and p-adic-to real transition generates real space-time sheet complex with vanishing total energy. Negative energy signal must be somehow compensated by a positive energy state.

b) It might be however possible to assign charges to p-adic space-time sheets. The equations characterizing p-adic space-time sheet representing intention and corresponding real space-time sheet representing action are assumed to be given in terms of same rational functions with coefficients which are algebraic numbers consistent with the extension of p-adic numbers used so that the points common to real and p-adic space-time sheets are in this extension. If real charges belong to the algebraic extension used, one could identify the p-adic charges as real charges. Zero energy ontology requires the presence of positive energy real space-time sheets whose charges compensate those of negative energy space-time sheets. One possibility is that real and corresponding p-adic space-time sheets appear in pairs with vanishing total quantum numbers just as fermionic space-time sheets are assumed to occur (see this. In the case of fermions p-adic-to-real phase transition is impossible by Exclusion Principle so that a stable cognitive representation results.

The minimal option would be that p-adic space-time sheets possess negative energy and are transformed to negative energy signals inducing neuronal activities. The flow of subjective time would involve a transformation of the universe to zero energy universe in the sense that total conserved quantum numbers vanish in the real sense in bosonic sector but in fermionic sector real and p-adic charges compensate each other.

This picture is probably too simple. Robertson-Walker cosmology has vanishing density of inertial energy. Hence it would seem that real bosons and fermions should appear in both positive and negative energy states and the arrow of time defined by the direction of the propagation of the intention-to-action wave front would be local.

The transition of the geometric past back to intentional phase would involve transformation of real bosons to p-adic ones and is in principle possible for this option. For the first option the transition could occur only for real states with vanishing total quantum numbers which would make this transition highly improbable and thus imply irreversibility.

The basic criticism is that since intentions in the proposed sense do not involve any selection, one could argue that this picture is not enough to explain the instability of the geometric future unless the instability is due to the instability of p-adic space-time sheets in quantum jumps.

2. Does intentional action transform quantum critical phase to non-quantum critical phase?

It is far from clear whether the proposed model is not able to explain the uncertainty of the geometric future and relative stability of the geometric past related very intimately to the possibility to select between different options. TGD based view about dark matter as a hierarchy of phases characterized by M4 and CP2 Planck constants quantized in integer multiples of minimum value hbar0 of hbar (see this) suggests a more refined view about what happens in the quantum jump transforming intention to action.

  1. The geometric future of the living system corresponds to a quantum critical state which is a superposition of (at least) two phases. Quantum criticality means that future is very uncertain and universe can be in dramatically different macroscopic quantum states.

  2. Experienced flow of time corresponds to a phase transition front proceeding towards the geometric future quantum jump by quantum jump. In this transition intentional action represented by negative energy bosonic signals transforms the quantum critical phase to either of the two phases present. This selection between different phases would be the basic element of actions involving choice. The geometric past is stabilized so that geometric memories about geometric past are relatively stable. This picture applies always in some time scale and there is an entire hierarchy of time and spatial scales corresponding to the hierarchies of p-adic length scales and of Planck constants. Note that Compton length and time are proportional to hbar as is also the span of long term memories and time scale of planned actions.

The (at least) two phases present at quantum criticality would have different values of Planck constants. In the simplest case the values of M4 and CP2 Planck constants for the second phase would correspond to the minimal value hbar0 of Planck constants. For instance, cell could be in quantum superposition of ordinary and high Tc super-conducting phase, with high Tc superconductor characterized by a large M4 Planck constant.

Intentional action would induce a transition to either of these two phases. Sub-system would chose either the lower or higher level in the hierarchy of consciousness with level characterized by the values of Planck constants. This unavoidably brings in mind a moral choice. Intentional actions involve often a choice between good and bad and this choice could reduce to a choice between values of Planck constant. Good deed would lead to higher value of Planck constant and bad deed to a lower one. This interpretation conforms with the earlier view about quantum ethics stating that good deeds are those which support evolution. The earlier proposal was however based on the assumption that evolution means a gradual increase of a typical p-adic length scale and seems to be too restricted in the recent framework.

For instance, in cell length scale the cells of the geometric future could be in quantum critical phase such that large hbar phase corresponds to high Tc super-conductivity and low hbar phase to its absence. In quantum jump cell would transform to either of these phases. The natural interpretation for the transition to low hbar phase is as cell death since the communications of the cell to and quantum control by the magnetic body are lost. Ageing could be seen as a process in which the transitions to small hbar phase begin to dominate or even the quantum criticality is lost. A model for the quantum criticality based on zeros of Riemann zeta developed here,here, here, here, and here allows a more quantitative view about what could happen in the phase transition.

For more details see the chapter Time, Space-Time and Consciousness.



What really distinguishes between future and past?

Our knowledge about geometric future is very uncertain as compared to that about geometric past. Hence we usually use words like plan/hunch/hope/... in the case of geometric future and speak about memories in the case of geometric past. We also regard geometric past as something absolutely stable. Why we cannot remember geometric future as reliably as the geometric past? Is it that geometric future is highly unstable as compared to the geometric past? Why this should be the case? This provides a possible TGD based articulation for the basic puzzles relating to time experience. These questions have been already discussed in this chapter but I want to close the chapter with considerations inspired by the latest progress in the understanding of quantum TGD.

1. Is p-adic-to-real phase transition enough?

The basic idea is that the flow of subjective time corresponds to a phase transition front representing a transformation of intentions to actions and propagating towards the geometric future quantum jump by quantum jump. All quantum states have vanishing total quantum numbers in zero energy ontology which now forms the basis of quantum TGD and this ontology allows to imagine models for what could happen in this process.

This starting point is the interpretation of fermions as correlates for cognition bosons as correlates for intentions/actions (see this). Fermions correspond to pairs of real and p-adic space-time sheets with opposite quantum numbers with p-adic space-time sheet providing a cognitive representation of the real space-time sheet. Bosonic space-time sheets would be either p-adic or real and thus represent intentions or actions. Fermionic world and its cognitive representations would be common to future and geometric past and the asymmetry would relate only to the intention-action dichotomy.

Geometric future contains a lot of p-adic space-time sheets representing intentions which transform to real space-time sheets allowing interpretation as desires inducing eventually neuronal activities. Time mirror mechanism for intentional action assumes that the phase transition gives rise to negative energy space-time sheets representing propagation of signals to geometric past where they induce neuronal activities. From Libet's experiments relating to neuronal correlates of volition the time scale involved is a fraction of second but an infinite hierarchy of time scales is implied by fractality.

Conservation of quantum numbers poses strong conditions on p-adic-to-real phase transition. Noether charges are in the real context given by integrals over partonic 2-surfaces. The problem is that these integrals do not make sense p-adically. There are two options.

a) Give up the notion of p-adic Noether charge so that it would not make sense to speak about four-momentum and other conserved quantum numbers in case of p-adic space-time sheet. This implies zero energy ontology in the real sector. All real space-time sheets would have vanishing conserved quantum numbers and p-adic-to real transition generates real space-time sheet complex with vanishing total energy. Negative energy signal must be somehow compensated by a positive energy state.

b) It might be however possible to assign charges to p-adic space-time sheets. The equations characterizing p-adic space-time sheet representing intention and corresponding real space-time sheet representing action are assumed to be given in terms of same rational functions with coefficients which are algebraic numbers consistent with the extension of p-adic numbers used so that the points common to real and p-adic space-time sheets are in this extension. If real charges belong to the algebraic extension used, one could identify the p-adic charges as real charges. Zero energy ontology requires the presence of positive energy real space-time sheets whose charges compensate those of negative energy space-time sheets. One possibility is that real and corresponding p-adic space-time sheets appear in pairs with vanishing total quantum numbers just as fermionic space-time sheets are assumed to occur (see this. In the case of fermions p-adic-to-real phase transition is impossible by Exclusion Principle so that a stable cognitive representation results.

The minimal option would be that p-adic space-time sheets possess negative energy and are transformed to negative energy signals inducing neuronal activities. The flow of subjective time would involve a transformation of the universe to zero energy universe in the sense that total conserved quantum numbers vanish in the real sense in bosonic sector but in fermionic sector real and p-adic charges compensate each other.

This picture is probably too simple. Robertson-Walker cosmology has vanishing density of inertial energy. Hence it would seem that real bosons and fermions should appear in both positive and negative energy states and the arrow of time defined by the direction of the propagation of the intention-to-action wave front would be local.

The transition of the geometric past back to intentional phase would involve transformation of real bosons to p-adic ones and is in principle possible for this option. For the first option the transition could occur only for real states with vanishing total quantum numbers which would make this transition highly improbable and thus imply irreversibility.

The basic criticism is that since intentions in the proposed sense do not involve any selection, one could argue that this picture is not enough to explain the instability of the geometric future unless the instability is due to the instability of p-adic space-time sheets in quantum jumps.

2. Does intentional action transform quantum critical phase to non-quantum critical phase?

It is far from clear whether the proposed model is not able to explain the uncertainty of the geometric future and relative stability of the geometric past related very intimately to the possibility to select between different options. TGD based view about dark matter as a hierarchy of phases characterized by M4 and CP2 Planck constants quantized in integer multiples of minimum value hbar0 of hbar (see this) suggests a more refined view about what happens in the quantum jump transforming intention to action.

  1. The geometric future of the living system corresponds to a quantum critical state which is a superposition of (at least) two phases. Quantum criticality means that future is very uncertain and universe can be in dramatically different macroscopic quantum states.

  2. Experienced flow of time corresponds to a phase transition front proceeding towards the geometric future quantum jump by quantum jump. In this transition intentional action represented by negative energy bosonic signals transforms the quantum critical phase to either of the two phases present. This selection between different phases would be the basic element of actions involving choice. The geometric past is stabilized so that geometric memories about geometric past are relatively stable. This picture applies always in some time scale and there is an entire hierarchy of time and spatial scales corresponding to the hierarchies of p-adic length scales and of Planck constants. Note that Compton length and time are proportional to hbar as is also the span of long term memories and time scale of planned actions.

The (at least) two phases present at quantum criticality would have different values of Planck constants. In the simplest case the values of M4 and CP2 Planck constants for the second phase would correspond to the minimal value hbar0 of Planck constants. For instance, cell could be in quantum superposition of ordinary and high Tc super-conducting phase, with high Tc superconductor characterized by a large M4 Planck constant.

Intentional action would induce a transition to either of these two phases. Sub-system would chose either the lower or higher level in the hierarchy of consciousness with level characterized by the values of Planck constants. This unavoidably brings in mind a moral choice. Intentional actions involve often a choice between good and bad and this choice could reduce to a choice between values of Planck constant. Good deed would lead to higher value of Planck constant and bad deed to a lower one. This interpretation conforms with the earlier view about quantum ethics stating that good deeds are those which support evolution. The earlier proposal was however based on the assumption that evolution means a gradual increase of a typical p-adic length scale and seems to be too restricted in the recent framework.

For instance, in cell length scale the cells of the geometric future could be in quantum critical phase such that large hbar phase corresponds to high Tc super-conductivity and low hbar phase to its absence. In quantum jump cell would transform to either of these phases. The natural interpretation for the transition to low hbar phase is as cell death since the communications of the cell to and quantum control by the magnetic body are lost. Ageing could be seen as a process in which the transitions to small hbar phase begin to dominate or even the quantum criticality is lost. A model for the quantum criticality based on zeros of Riemann zeta developed here,here, here, here, and here allows a more quantitative view about what could happen in the phase transition.

For more details see either the chapter p-Adic Phycics as Physics of Cognition and Intentionality or the chapter Quantum Model for Memory.



Zero energy ontology, cognition, and intentionality

In TGD inspired theory of consciousness the space-time correlates for intentions are provided by p-adic space-time sheets whereas actions correspond to real space-time sheets. The transformation of intention to action corresponds to a quantum jump in which p-adic space-time sheet transforms to a real one: these two space-time sheets have exactly the same analytic representation. The larger the number of rational (algebraic points) in the intersection of the space-time sheets, the more probable the transition is expected to be.

One could however argue that conservation laws forbid p-adic-real phase transitions in practice so that cognitions (intentions) realized as real-to-padic (p-adic-to-real) transitions is not be possible. The situation changes if one accepts what might be called zero energy ontology (see this and this).

1. Zero energy ontology classically

In TGD inspired cosmology the imbeddings of Robertson-Walker cosmologies are vacuum extremals. Same applies to the imbeddings of Reissner-Nordström solution and in practice to all solutions of Einstein's equations imbeddable as extremals of Kähler action. Since four-momentum currents define a collection of vector fields rather than a tensor in TGD, both positive and negative signs for energy corresponding to two possible assignments of the arrow of the geometric time to a given space-time surface are possible. This leads to the view that all physical states have vanishing net energy classically and that physically acceptable universes are creatable from vacuum.

The result is highly desirable since one can avoid unpleasant questions such as "What are the net values of conserved quantities like rest mass, baryon number, lepton number, and electric charge for the entire universe?", "What were the initial conditions in the big bang?", "If only single solution of field equations is selected, isn't the notion of physical theory meaningless since in principle it is not possible to compare solutions of the theory?". This picture fits also nicely with the view that entire universe understood as quantum counterpart 4-D space-time is recreated in each quantum jump and allows to understand evolution as a process of continual re-creation.

2. Zero energy ontology at quantum level

Also the construction of S-matrix leads to the conclusion that all physical states possess vanishing conserved quantum numbers. Furthermore, the entanglement coefficients between positive and negative energy components of the state define a unitary S-matrix. S-matrix thus becomes a property of the zero energy state and physical states code by their structure what is usually identified as quantum dynamics.

Also the transitions between zero energy states are possible but general arguments lead to the conclusion that the corresponding S-matrix is almost trivial. This finding, which actually forced the new view about S-matrix, is highly desirable since it explains why positive energy ontology works so well if one forgets effects related to intentional action.

At space-time level this would mean that positive energy component and negative energy component are at a temporal distance characterized by an appropriate p-adic time scale and the integer characterizing the value of Planck constant for the state in question. The scale in question would also characterize the geometric duration of quantum jump and the size scale of space-time region contributing to the contents of conscious experience. The interpretation in terms of a mini bang followed by a mini crunch suggests itself also.

3. Hyper-finite factors of type II1 and new view about S-matrix

The representation of S-matrix as unitary entanglement coefficients would not make sense in ordinary quantum theory but in TGD the von Neumann algebra in question is not a type I factor as for quantum mechanics or a type III factor as for quantum field theories, but what is called hyper-finite factor of type II1. This algebra is an infinite-dimensional algebra with the almost defining, and at the first look very strange, property that the infinite-dimensional unit matrix has unit trace. The infinite dimensional Clifford algebra spanned by the configuration space gamma matrices (configuration space understood as the space of 3-surfaces, the "world of classical worlds") is indeed very naturally algebra of this kind since infinite-dimensional Clifford algebras provide a canonical representations for hyper-finite factors of type II1.

4. The new view about quantum measurement theory

This mathematical framework leads to a new kind of quantum measurement theory. The basic assumption is that only a finite number of degrees of freedom can be quantum measured in a given measurement and the rest remain untouched. What is known as Jones inclusions N in M} of von Neumann algebras allow to realize mathematically this idea (see this). N characterizes measurement resolution and quantum measurement reduces the entanglement in the non-commutative quantum space M/N. The outcome of the quantum measurement is still represented by a unitary S-matrix but in the space characterized by N. It is not possible to end up with a pure state with a finite sequence of quantum measurements.

The obvious objection is that the replacement of a universal S-matrix coding entire physics with a state dependent unitary entanglement matrix is too heavy a price to be paid for the resolution of the above mentioned paradoxes. Situation could be saved if the S-matrices have fractal structure. The quantum criticality of TGD Universe indeed implies fractality. The possibility of an infinite sequence of Jones inclusions for hyperfinite type II1 factors isomorphic as von Neumann algebras expresses this fractal character algebraically. Thus one can hope that the S-matrix appearing as entanglement coefficients is more or less universal in the same manner as Mandelbrot fractal looks more or less the same in all length scales and for all resolutions. Whether this kind of universality must be posed as an additional condition on entanglement coefficients or is an automatic consequence of unitarity in type II1 sense is an open question.

5. The S-matrix for p-adic-real transitions makes sense

In zero energy ontology conservation laws do not forbid p-adic-real transitions and one can develop a relatively concrete vision about what happens in these kind of transitions. The starting point is the generalization of the number concept obtained by gluing p-adic number fields and real numbers along common rationals (expressing it very roughly). At the level of the imbedding space this means that p-adic and real space-time sheets intersect only along common rational points of the imbedding space and transcendental p-adic space-time points are infinite as real numbers so that they can be said to be infinite distant points so that intentionality and cognition become cosmic phenomena.

In this framework the long range correlations characterizing p-adic fractality can be interpreted as being due to a large number of common rational points of imbedding space for real space-time sheet and p-adic space-time sheet from which it resulted in the realization of intention in quantum jump. Thus real physics would carry direct signatures about the presence of intentionality. Intentional behavior is indeed characterized by short range randomness and long range correlations.

One can even develop a general vision about how to construct the S-matrix elements characterizing the process (see this). The basic guideline is the vision that real and various p-adic physics as well as their hybrids are continuable from the rational physics. This means that these S-matrix elements must be characterizable using data at rational points of the imbedding space shared by p-adic and real space-time sheets so that more or less same formulas describe all these S-matrix elements. Note that also p1→ p2 p-adic transitions are possible.

The interpretation of infinite primes leads to a detailed vision about space-time correlates of quantum states and cognition and intentionality. Intentions correspond to p-adic space-time sheets and actions to their real counterparts being related by a mere algebraic continuations of the exlicit analytic representsions as surfaces. Cognitions correspond to pairs of real space-time sheet and correspond p-adic space-time sheet obtained in the same manner providing also a representation for a state generated by appropriate generator of super algebra.

The discreteness of the intersection of the real space-time sheet and its p-adic variant obtained by algebraic continuation would be a completely universal phenomenon associated with all fermionic states. This suggests that also real-to-real S-matrix elements involve instead of an integral a sum with the arguments of an n-point function running over all possible combinations of the points in the intersection. S-matrix elements would have a universal form which does not depend on the number field at all and the algebraic continuation of the real S-matrix to its p-adic counterpart would trivialize. Note that also fermionic statistics favors strongly discretization unless one allows Dirac delta functions.

The chapter p-Adic Physics as Physics of Cognition and Intention contains a more detailed text about this topic.



Infinite primes, cognition, and intentionality

Somehow it is obvious that infinite primes (see this) must have some very deep role to play in quantum TGD and TGD inspired theory of consciousness. What this role precisely is has remained an enigma although I have considered several detailed interpretations (see the link above).

In the following an interpretation allowing to unify the views about fermionic Fock states as a representation of Boolean cognition and p-adic space-time sheets as correlates of cognition is discussed. Very briefly, real and p-adic partonic 3-surfaces serve as space-time correlates for the bosonic super algebra generators, and pairs of real partonic 3-surfaces and their algebraically continued p-adic variants as space-time correlates for the fermionic super generators. Intentions/actions are represented by p-adic/real bosonic partons and cognitions by pairs of real partons and their p-adic variants and the geometric form of Fermi statistics guarantees the stability of cognitions against intentional action.

1. Infinite primes very briefly

Infinite primes have a decomposition to infinite and finite parts allowing an interpretation as a many-particle state of a super-symmetric arithmetic quantum field theory for which fermions and bosons are labelled by primes. There is actually an infinite hierarchy for which infinite primes of a given level define the building blocks of the infinite primes of the next level. One can map infinite primes to polynomials and these polynomials in turn could define space-time surfaces or at least light-like partonic 3-surfaces appearing as solutions of Chern-Simons action so that the classical dynamics would not pose too strong constraints.

The simplest infinite primes at the lowest level are of form mBX/sF + nBsF, X=∏i pi (product of all finite primes). mB, nB, and sF are defined as mB= ∏ipimi, nB= ∏iqini, and sF= ∏iqi, mB and nB have no common prime factors. The simplest interpretation is that X represents Dirac sea with all states filled and X/sF + sF represents a state obtained by creating holes in the Dirac sea. The integers mB and nB characterize the occupation numbers of bosons in modes labelled by pi and qi and sF= ∏iqi characterizes the non-vanishing occupation numbers of fermions.

The simplest infinite primes at all levels of the hierarchy have this form. The notion of infinite prime generalizes to hyper-quaternionic and even hyper-octonionic context and one can consider the possibility that the quaternionic components represent some quantum numbers at least in the sense that one can map these quantum numbers to the quaternionic primes.

The obvious question is whether configuration space degrees of freedom and configuration space spinor (Fock state) of the quantum state could somehow correspond to the bosonic and fermionic parts of the hyper-quaternionic generalization of the infinite prime as proposed here. That hyper-quaternionic (or possibly hyper-octonionic) primes would define as such the quantum numbers of fermionic super generators does not make sense. It is however possible to have a map from the quantum numbers labelling super-generators to the finite primes. One must also remember that the infinite primes considered are only the simplest ones at the given level of the hierarchy and that the number of levels is infinite.

2. Precise space-time correlates of cognition and intention

The best manner to end up with the proposal about how p-adic cognitive representations relate bosonic representations of intentions and actions and to fermionic cognitive representations is through the following arguments.

  1. In TGD inspired theory of consciousness Boolean cognition is assigned with fermionic states. Cognition is also assigned with p-adic space-time sheets. Hence quantum classical correspondence suggets that the decomposition of the space-time into p-adic and real space-time sheets should relate to the decomposition of the infinite prime to bosonic and fermionic parts in turn relating to the above mention decomposition of physical states to bosonic and fermionic parts.

    If infinite prime defines an association of real and p-adic space-time sheets this association could serve as a space-time correlate for the Fock state defined by configuration space spinor for given 3-surface. Also spinor field as a map from real partonic 3-surface would have as a space-time correlate a cognitive representation mapping real partonic 3-surfaces to p-adic 3-surfaces obtained by algebraic continuation.

  2. Consider first the concrete interpretation of integers mB and nB. The most natural guess is that the primes dividing mB=∏ipmi characterize the effective p-adicities possible for the real 3-surface. mi could define the numbers of disjoint partonic 3-surfaces with effective pi-adic topology and associated with with the same real space-time sheet. These boundary conditions would force the corresponding real 4-surface to have all these effective p-adicities implying multi-p-adic fractality so that particle and wave pictures about multi-p-adic fractality would be mutually consistent. It seems natural to assume that also the integer ni appearing in mB=∏iqini code for the number of real partonic 3-surfaces with effective qi-adic topology.

  3. Fermionic statistics allows only single genuinely qi-adic 3-surface possibly forming a pair with its real counterpart from which it is obtained by algebraic continuation. Pairing would conform with the fact that nF appears both in the finite and infinite parts of the infinite prime (something absolutely essential concerning the consistency of interpretation!).

    The interpretation could be as follows.

    1. Cognitive representations must be stable against intentional action and fermionic statistics guarantees this. At space-time level this means that fermionic generators correspond to pairs of real effectively qi-adic 3-surface and its algebraically continued qi-adic counterpart. The quantum jump in which qi-adic 3-surface is transformed to a real 3-surface is impossible since one would obtain two identical real 3-surfaces lying on top of each other, something very singular and not allowed by geometric exclusion principle for surfaces. The pairs of boson and fermion surfaces would thus form cognitive representations stable against intentional action.

    2. Physical states are created by products of super algebra generators Bosonic generators can have both real or p-adic partonic 3-surfaces as space-time correlates depending on whether they correspond to intention or action. More precisely, mB and nB code for collections of real and p-adic partonic 3-surfaces. What remains to be interpreted is why mB and nB cannot have common prime factors (this is possible if one allows also infinite integers obtained as products of finite integer and infinite primes).

    3. Fermionic generators to the pairs of a real partonic 3-surface and its p-adic counterpart obtained by algebraic continuation and the pictorial interpretation is as a pair of fermion and hole.

    4. This picture makes sense if the partonic 3-surfaces containing a state created by a product of super algebra generators are unstable against decay to this kind of 3-surfaces so that one could regard partonic 3-surfaces as a space-time representations for a configuration space spinor field.

  4. Are alternative interpretations possible? For instance, could q=mB/mB code for the effective q-adic topology assignable to the space-time sheet as suggested here. That q-adic numbers form a ring but not a number field casts however doubts on this interpretation as does also the general physical picture.

3. Number theoretical universality of S-matrix

The discreteness of the intersection of the real space-time sheet and its p-adic variant obtained by algebraic continuation would be a completely universal phenomenon associated with all fermionic states. This suggests that also real-to-real S-matrix elements involve instead of an integral a sum with the arguments of an n-point function running over all possible combinations of the points in the intersection. S-matrix elements would have a universal form which does not depend on the number field at all and the algebraic continuation of the real S-matrix to its p-adic counterpart would trivialize. Note that also fermionic statistics favors strongly discretization unless one allows Dirac delta functions.

For a more detailed view about intentionality and cognition as a basic element of TGD based physics even at elementary particle level and also about zero energy ontology as the prerequisite of the whole approach infinite see the chapter p-Adic Physics as Physics of Cognition and Intention.





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