The relation between U- and M-matrices

U- and M-matrices are key objects in zero energy ontology (ZEO). M-matrix for large causal diamonds (CDs) is the counterpart of thermal S-matrix and proportional to scale dependent S-matrix: the dependence on the scale of CD characterized by integer is S(n)=Sⁿ in accordance with the idea that S corresponds to the counterpart of ordinary unitary time evolution operator. U-matrix characterizes the time evolution as dispersion in the moduli space of CDs tending to increase the size of CD and giving rise to the experience arrow of geometric time and also to the notion of self in TGD insprired theory of consciousness.

The original view about the relationship between U- and M-matrices was a purely formal guess: M-matrices would define the orthonormal rows of U-matrix. This guess is not correct physically and one must consider in detail what U-matrix really means.

1. First about the geometry of CD. The boundaries of CD will be called passive and active: passive boundary correspond to the boundary at which repeated state function reductions take place and give rise to a sequence of unitary time evolutions U followed by localization in the moduli of CD each. Active boundary corresponds to the boundary for which U induces delocalization and modifies the states at it.
   

   The moduli space for the CDs consists of a discrete subgroup of scalings for the size of CD characterized by the proper time distance between the tips and the sub-group of Lorentz boosts leaving passive boundary and its tip invariant and acting on the active boundary only. This group is assumed to be represented unitarily by matrices Λ forming the same group for all values of n.
   

   The proper time distance between the tips of CDs is quantized as integer multiples of the minimal distance defined by CP₂ time: T= nT₀. Also in quantum jump in which the size scale n of CD increases the increase corresponds to integer multiple of T₀. Using the logarithm of proper time, one can interpret this in terms of a scaling parametrized by an integer. The possibility to interpret proper time translation as a scaling is essential for having a manifest Lorentz invariance: the ordinary definition of S-matrix introduces preferred rest system.
   
   

2. The physical interpretation would be roughly as follows. M-matrix for a given CD codes for the physics as we usually understand it. M-matrix is product of square root of density matrix and S-matrix depending on the size scale of CD and is the analog of thermal S-matrix. State function at the opposite boundary of CD corresponds to what happens in the state function reduction in particle physics experiments. The repeated state function reductions at same boundary of CD correspond to TGD version of Zeno effect crucial for understanding consciousness. Unitary U-matrix describes the time evolution zero energy states due to the increase of the size scale of CD (at least in statistical sense). This process is dispersion in the moduli space of CDs: all possible scalings are allowed and localization in the space of moduli of CD localizes the active boundary of CD after each unitary evolution.
   
   

In the following I will proceed by making questions. One ends up to formulas allowing to understand the architecture of U-matrix and to reduce its construction to that for S-matrix having interpretation as exponential of the generator L_-1 of the Virasoro algebra associated with the super-symplectic algebra.

What one can say about M-matrices?

1. The first thing to be kept in mind is that M-matrices act in the space of zero energy states rather than in the space of positive or negative energy states. For a given CD M-matrices are products of hermitian square roots of hermitian density matrices acting in the space of zero energy states and universal unitary S-matrix S(CD) acting on states at the active end of CD (this is also very important to notice) depending on the scale of CD:
   

   Mⁱ=HⁱS(CD) .
   

   Hⁱ is hermitian square root of density matrix and the matrices Hⁱ must be orthogonal for given CD from the orthonormality of zero energy states associated with the same CD. The zero energy states associated with different CDs are not orthogonal and this makes the unitary time evolution operator U non-trivial.
   
   

2. Could quantum measurement be seen as a measurement of the observables defined by the Hermitian generators Hⁱ? This is not quite clear since their action is on zero energy states. One might actually argue that the action of this kind of observables on zero energy states does not affect their vanishing net quantum numbers. This suggests that Hⁱ carry no net quantum numbers and belong to the Cartan algebra. The action of S is restricted at the active boundary of CD and therefore it does not commute with Hⁱ unless the action is in a separate tensor factor. Therefore the idea that S would be an exponential of generators Hⁱ and thus commute with them so that Hⁱ would correspond to sub-spaces remaining invariant under S acting unitarily inside them does not make sense.
   
   
3. In TGD framework symplectic algebra actings as isometries of WCW is analogous to a Kac-Moody algebra with finite-dimensional Lie-algebra replaced with the infinite-dimensional symplectic algebra with elements characterized by conformal weights. There is a temptation to think that the Hⁱ could be seen as a representation for this algebra or its sub-algebra. This algebra allows an infinite fractal hierarchy of sub-algebras of the super-symplectic algebra isomorphic to the full algebra and with conformal weights coming as n-ples of those for the full algebra. In the proposed realization of quantum criticality the elements of the sub-algebra characterized by n act as a gauge algebra. An interesting question is whether this sub-algebra is involved with the realization of M-matrices for CD with size scale n. The natural expectation is that n defines a cutoff for conformal weights relating to finite measurement resolution.
   
   

How does the size scale of CD affect M-matrices?

1. In standard quantum field theory S-matrix represents time translation. The obvious generalization is that now scaling characterized by integer n is represented by a unitary S-matrix that is as n:th power of some unitary matrix S assignable to a CD with minimal size: S(CD)= Sⁿ. S(CD) is a discrete analog of the ordinary unitary time evolution operator with n replacing the continuous time parameter.
   
   
2. One can see M-matrices also as a generalization of Kac-Moody type algebra. Also this suggests S(CD)= Sⁿ, where S is the S-matrix associated with the minimal CD. S becomes representative of phase exp(iφ). The inner product between CDs of different size scales can n₁ and n₂ can be defined as
   
   
   

   ⟨ Mⁱ(m), M^j(n)⟩ =Tr(S^-m• HⁱH^j• Sⁿ) × θ(n-m) ,

   θ(n)=1 for n≥ 0 , θ (n)=0 for n<0 .
   

   Here I have denoted the action of S-matrix at the active end of CD by "•" in order to distinguish it from the action of matrices on zero energy states which could be seen as belonging to the tensor product of states at active and passive boundary.
   

   It turns out that unitarity conditions for U-matrix are invariant under the translations of n if one assumes that the transitions obey strict arrow of time expressed by n_j-n_i≥ 0. This simplifies dramatically unitarity conditions. This gives orthonormality for M-matrices associated with identical CDs. This inner product could be used to identify U-matrix.
   
   

3. How do the discrete Lorentz boosts affecting the moduli for CD with a fixed passive boundary affect the M-matrices? The natural assumption is that the discrete Lorentz group is represented by unitary matrices λ: the matrices Mⁱ are transformed to Mⁱ•λ for a given Lorentz boost acting on states at active boundary only.
   

   One cannot completely exclude the possibility that S acts unitarily on zero energy states. In this case the scaling would be interpreted as acting on zero energy states rather than those at active boundary only. The zero energy state basis defined by M_i would depend on the size scale of CD in more complex manner. This would not affect the above formulas except by dropping away the "•".
   
   

Unitary U must characterize the transitions in which the moduli of the active boundary of causal diamond (CD) change and also states at the active boundary (paired with unchanging states at the passive boundary) change. The arrow of the experienced flow of time emerges during the period as state function reductions take place to the fixed ("passive") boundary of CD and do not affect the states at it. Note that these states form correlated pairs with the changing states at the active boundary. The physically motivated question is whether the arrow of time emerges statistically from the fact that the size of CD tends to increase in average sense in repeated state function reductions or whether the arrow of geometric time is strict. It turns out that unitarity conditions simplify dramatically if the arrow of time is strict.

What can one say about U-matrix?

1. Just from the basic definitions the elements of a unitary matrix, the elements of U are between zero energy states (M-matrices) between two CDs with possibly different moduli of the active boundary. Given matrix element of U should be proportional to an inner product of two M-matrices associated with these CDs. The obvious guess is as the inner product between M-matrices
   

   U^ij_m,n= ⟨Mⁱ(m,λ₁), M^j(n,λ₂)⟩

   =Tr(λ₁^† S^-m• HⁱH^j• Sⁿ λ₂)

   =Tr(S^-m• HⁱH^j • Sⁿ λ₂λ₁^-1)θ(n-m) .
   

   Here the usual properties of the trace are assumed. The justification is that the operators acting at the active boundary of CD are special case of operators acting non-trivially at both boundaries.
   
   

2. Unitarity conditions must be satisfied. These conditions relate S and the hermitian generators Hⁱ serving as square roots of density matrices. Unitarity conditions
   UU^†=U^†U=1 is defined in the space of zero energy states and read as
   

   ∑j1n1 Uij1mn1(U†)j1jn1n = δi,jδm,nδλ1,λ2
   

   To simplify the situation let us make the plausible hypothesis contribution of Lorentz boosts in unitary conditions is trivial by the unitarity of the representation of discrete boosts and the independence on n.
   
   

3. In the remaining degrees of freedom one would have
   

   ∑_{j1,k≥ Max(0,n-m)} Tr(S^k• HⁱH^j₁) Tr(H^j₁H^j• S^n-m-k)= δ^i,jδ_m,n .
   

   The condition k≥ Max(0,n-m) reflects the assumption about a strict arrow of time and implies that unitarity conditions are invariant under the proper time translation (n,m)→ (n+r,m+r). Without this condition n back-wards translations (or rather scalings) to the direction of geometric past would be possible for CDs of size scale n and this would break the translational invariance and it would be very difficult to see how unitarity could be achieved. Stating it in a general manner: time translations act as semigroup rather than group.
   
   

4. Irreversibility reduces dramatically the number of the conditions. Despite this their number is infinite and correlates the Hermitian basis and the unitary matrix S. There is an obvious analogy with a Kac-Moody algebra at circle with S replacing the phase factor exp(inφ) and Hⁱ replacing the finite-dimensional Lie-algebra. The conditions could be seen as analogs for the orthogonality conditions for the inner product. The unitarity condition for the analog situation would involve phases exp(ikφ₁)↔ S^k and exp(i(n-m-k)φ₂)↔ S^n-m-k and trace would correspond to integration ∫ dφ₁ over φ₁ in accordance with the basic idea of non-commutative geometry that trace corresponds to integral. The integration of φ_i would give δ_k,0 and δ_m,n. Hence there are hopes that the conditions might be satisfied. There is however a clear distinction to the Kac-Moody case since Sⁿ does not in general act in the orthogonal complement of the space spanned by Hⁱ.
   
   
5. The idea about reduction of the action of S to a phase multiplication is highly attractive and one could consider the possibility that the basis of Hⁱ can be chosen in such a manner that Hⁱ are eigenstates of of S. This would reduce the unitarity constraint to a form in which the summation over k can be separated from the summation over j₁.
   

   ∑_{k≥ Max(0,n-m)} exp(iks_i-(n-m-k)s_j)∑_j1Tr(HⁱH^j₁) Tr(H^j₁H^j)= δ^i,jδ_m,n .
   

   The summation over k should gives a factor proportional to δ_si,sj. If the correspondence between Hⁱ and eigenvalues s_i is one-to-one, one obtains something proportional to δ (i,j) apart from a normalization factor. Using the orthonormality Tr(HⁱH^j)=δ^i,j one obtains for the left hand side of the unitarity condition
   

   exp(is_i(n-m)) ∑_j1Tr(HⁱH^j₁) Tr(H^j₁H^j)= exp(is_i(n-m)) δ_i,j .
   

   Clearly, the phase factor exp(is_i(n-m)) is the problem. One should have Kronecker delta δ_m,n instead. One should obtain behavior resembling Kac-Moody generators. Hⁱ should be analogs of Kac-Moody generators and include the analog of a phase factor coming visible by the action of S.
   

It seems that the simple picture is not quite correct yet. One should obtain somehow an integration over angle in order to obtain Kronecker delta.

1. A generalization based on replacement of real numbers with function field on circle suggests itself. The idea is to the identify eigenvalues of generalized Hermitian/unitary operators as Hermitian/unitary operators with a spectrum of eigenvalues, which can be continuous. In the recent case S would have as eigenvalues functions λ_i(φ) = exp(is_iφ). For a discretized version φ would have has discrete spectrum φ(n)= 2π k/n. The spectrum of λ_i would have n as cutoff. Trace operation would include integration over φ and one would have analogs of Kac-Moody generators on circle.
   
   
2. One possible interpretation for φ is as an angle parameter associated with a fermionic string connecting partonic 2-surface. For the super-symplectic generators suitable normalized radial light-like coordinate r_M of the light-cone boundary (containing boundary of CD) would be the counterpart of angle variable if periodic boundary conditions are assumed.
   

   The eigenvalues could have interpretation as analogs of conformal weights. Usually conformal weights are real and integer valued and in this case it is necessary to have generalization of the notion of eigenvalues since otherwise the exponentials exp(is_i) would be trivial. In the case of super-symplectic algebra I have proposed that the generating elements of the algebra have conformal weights given by the zeros of Riemann zeta. The spectrum of conformal weights for the generators would consist of linear combinations of the zeros of zeta with integer coefficients. The imaginary parts of the conformal weights could appear as eigenvalues of S.
   
   

3. It is best to return to the definition of the U-matrix element to check whether the trace operation appearing in it can already contain the angle integration. If one includes to the trace operation appearing the integration over φ it gives δ_m,n factor and U-matrix has elements only between states assignable to the same causal diamond. Hence one must interpret U-matrix elements as functions of φ realized factors exp(i(s_n-s_m)φ). This brings strongly in mind operators defined as distributions of operators on line encountered in the theory of representations of non-compact groups such as Lorentz group. In fact, the unitary representations of discrete Lorentz groups are involved now.
   
   
4. The unitarity condition contains besides the trace also the integrations over the two angle parameters φ_i associated with the two U-matrix elements involved. The left hand side of the unitarity condition reads as
   

   ∑_{k≥ Max(0,n-m)} =I(ks_i)I((n-m-k)s_j) × ∑ _j1Tr(HⁱH^j₁) Tr(H^j₁H^j) = δ^i,jδ_m,n ,
   

   I(s)=(1/2π)× ∫ dφ exp(isφ) =δ_s,0 .
   

   Integrations give the factor δ_k,0 eliminating the infinite sum obtained otherwise plus the factor δ_n,m. Traces give Kronecker deltas since the projectors are orthonormal. The left hand side equals to the right hand side and one achieves unitarity. It seems that the proposed ansatz works and the U-matrix can be reduced by a general ansatz to S-matrix.
   
   

What about the identification of S?

1. S should be exponential of time the scaling operator whose action reduces to a time translation operator along the time axis connecting the tips of CD and realized as scaling. In other words, the shift t/T₀=m→ m+n corresponds to a scaling t/T₀=m→ km giving m+n=km in turn giving k= 1+ n/m. At the limit of large shifts one obtains k≈ n/m→ ∞, which corresponds to QFT limit. nS corresponds to (nT₀)× (S/T₀)= TH and one can ask whether QFT Hamiltonian could corresponds to H=S/T₀.
   
   
2. It is natural to assume that the operators Hⁱ are eigenstates of radial scaling generator L₀=ir_Md/dr_M at both boundaries of CD and have thus well-defined conformal weights. As noticed the spectrum for super-symplectic algebra could also be given in terms of zeros of Riemann zeta.
   
   
3. The boundaries of CD are given by the equations r_M=m⁰ and r_M= T-m⁰, m⁰ is Minkowski time coordinate along the line between the tips of CD and T is the distance between the tips. From the relationship between r_M and m⁰ the action of the infinitesimal translation H== i∂/∂m⁰ can be expressed as conformal generator L_-1= i∂/∂r_M = r_M^-1 L₀ . Hence the action is non-diagonal in the eigenbasis of L₀ and multiplies with the conformal weights and reduces the conformal weight by one unit. Hence the action of U can change the projection operator. For large values of conformal weight the action is classically near to that of L₀: multiplication by L₀ plus small relative change of conformal weight.
   
   
4. Could the spectrum of H be identified as energy spectrum expressible in terms of zeros of zeta defining a good candidate for the super-symplectic radial conformal weights. This certainly means maximal complexity since the number of generators of the conformal algebra would be infinite. This identification might make sense in chaotic or critical systems. The functions (r_M/r₀)^1/2+iy and (r_M/r₀)^-2n, n>0, are eigenmodes of r_M/dr_M with eigenvalues (1/2+iy) and -2n corresponding to non-trivial and trivial zeros of zeta.
   

   There are two options to consider. Either L₀ or iL₀ could be realized as a hermitian operator. These options would correspond to the identification of mass squared operator as L₀ and approximation identification of Hamiltonian as iL₁ as iL₀ making sense for large conformal weights.
   
   

   1. Suppose that L₀= r_Md/dr_M realized as a hermitian operator would give harmonic oscillator spectrum for conformal confinement. In p-adic mass calculations the string model mass formula implies that L₀ acts essentially as mass squared operator with integer spectrum. I have proposed conformal confinent for the physical states net conformal weight is real and integer valued and corresponds to the sum over negative integer valued conformal weights corresponding to the trivial zeros and sum over real parts of non-trivial zeros with conformal weight equal to 1/2. Imaginary parts of zeta would sum up to zero.
      
      
   2. The counterpart of Hamiltonian as a time translation is represented by H=iL₀= ir_M d/dr_M. Conformal confinement is now realized as the vanishing of the sum for the real parts of the zeros of zeta: this can be achieved. As a matter fact the integration measure dr_M/r_M brings implies that the net conformal weight must be 1/2. This is achieved if the number of non-trivial zeros is odd with a judicious choice of trivial zeros. The eigenvalues of Hamiltonian acting as time translation operator could correspond to the linear combination of imaginary part of zeros of zeta with integer coefficients. This is an attractive hypothesis in critical systems and TGD Universe is indeed quantum critical.
      

What about quantum classical correspondence and zero modes?

The one-one correspondence between the basis of quantum states and zero modes realizes quantum classical correspondence.

1. M-matrices would act in the tensor product of quantum fluctuating degrees of freedom and zero modes. The assumption that zero energy states form an orthogonal basis implies that the hermitian square roots of the density matrices form an orthonormal basis. This condition generalizes the usual orthonormality condition.
   
   
2. The dependence on zero modes at given boundary of CD would be trivial and induced by 1-1 correspondence
   |m⟩ → z(m) between states and zero modes assignable to the state basis |m_+/-⟩ at the boundaries of CD, and would mean the presence of factors δ_z+,f(m+) × δ_z-,f(n-) multiplying M-matrix Mⁱ_m,n.
   
   

To sum up, it seems that the architecture of the U-matrix and its relationship to the S-matrix is now understood and in accordance with the intuitive expectations the construction of U-matrix reduces to that for S-matrix and one can see S-matrix as discretized counterpart of ordinary unitary time evolution operator with time translation represented as scaling: this allows to circumvent problems with loss of manifest Poincare symmetry encountered in quantum field theories and allows Lorentz invariance although CD has finite size. What came as surprise was the connection with stringy picture: strings are necessary in order to satisfy the unitary conditions for U-matrix. Second outcome was that the connection with super-symplectic algebra suggests itself strongly. The identification of hermitian square roots of density matrices with Hermitian symmetry algebra is very elegant aspect discovered already earlier. A further unexpected result was that U-matrix is unitary only for strict arrow of time (which changes in the state function reduction to opposite boundary of CD).

See the article The relation between U-matrix and M-matrices.

For a summary of earlier postings see Links to the latest progress in TGD.

p-Adic physics as physics of cognition and imagination

The vision about p-adic physics as physics of cognition and imagination has gradually established itself as one of the key idea of TGD inspired theory of consciousness. There are several motivations for this idea.

The vision has developed fromthe vision about living matter as something residing in the intersection of real and p-adic worlds. One of the earliest motivations was p-adic non-determinism identified tentatively as a space-time correlate for the non-determinism of imagination. p-Adic non-determinism follows from the fact that functions with vanishing derivatives are piecewise constant functions in the p-adic context.

More precisely, p-adic pseudo constants depend on the pinary cutoff of their arguments and replace integration constants in p-adic differential equations. In the case of field equations this means roughly that the initial data are replaced with initial data given for a discrete set of time values chosen in such a manner that unique solution of field equations results. Solution can be fixed also in a discrete subset of rational points of the imbedding space. Presumably the uniqueness requirement implies some unique pinary cutoff. Thus the space-time surfaces representing solutions of p-adic field equations are analogous to space-time surfaces consisting of pieces of solutions of the real field equations. p-Adic reality is much like the dream reality consisting of rational fragments glued together in illogical manner or pieces of child's drawing of body containing body parts in more or less chaotic order.

The obvious looking interpretation for the solutions of the p-adic field equations would be as a geometric correlate of imagination. Plans, intentions, expectations, dreams, and cognition in general could have p-adic space-time sheets as their geometric correlates. A deep principle could be involved: incompleteness is characteristic feature of p-adic physics but the flexibility made possible by this incompleteness is absolutely essential for imagination and cognitive consciousness in general.

The original idea was that p-adic space-time regions can suffer topological phase transitions to real topology and vice versa in quantum jumps replacing space-time surface with a new one is given up as mathematically awkward: quantum jumps between different number fields do not make sense. The new adelic view states that both real and p-adic space-time sheets are obtained by continuation of string world sheets and partonic 2-surfaces to various number fields by strong form of holography.

The idea about p-adic pseudo constants as correlates of imagination is however too nice to be thrown away without trying to find an alternative interpretation consistent with strong form of holography. Could the following argument allow to save p-adic view about imagination in a mathematically respectable manner?

1. Construction of preferred extremals from data at 2-surfaces is like boundary value problem. Integration constants are replaced with pseudo-constants depending on finite number pinary digits of variables depending on coordinates normal to string world sheets and partonic 2-surfaces.
   
   
2. Preferred extremal property in real context implies strong correlations between string world sheets and partonic 2-surfaces by boundary conditions a them. One cannot choose these 2- surfaces completely independently. Pseudo-constant could allow a large number of p-adic configurations involving string world sheets and partonic 2-surfaces not allowed in real context and realizing imagination.
   
   
3. Could imagination be realized as a larger size of the p-adic sectors of WCW? Could the realizable intentional actions belong to the intersection of real and p-adic WCWs? Could the modes of WCW spinor fields for which 2-surfaces are extandable to space-time surfaces only in some p-adic sectors make sense? The real space-time surface for them be somehow degenerate, for instance, consisting of string world sheets only.
   

   Could imagination be search for those collections of string world sheets and partonic 2-surfaces, which allow extension to (realization as) real preferred extremals? p-Adic physics would be there as an independent aspect of existence and this is just the original idea. Imagination could be realized in state function reduction, which always selects only those 2-surfaces which allow continuation to real space-time surfaces. The distinction between only imaginable and also realizable would be the extendability by using strong form of holography.
   
   

I have the feeling that this view allows respectable mathematical realization of imagination in terms of adelic quantum physics. It is remarkable that strong form of holography derivable from - you can guess, strong form of General Coordinate Invariance (the Big E again!), plays an absolutely central role in it.

See the article How Imagination Could Be Realized p-Adically?.

For a summary of earlier postings see Links to the latest progress in TGD.

How time reversed mental images differ from mental images?

The basic predictions of ZEO based quantum measurement theory is that self corresponds to a sequence of state function reductions to a fixed boundary of CD (passive boundary) and that the first reduction to the opposite boundary means death of self and re-incarnation at the opposite boundary. The re-incarnated self has reversed arrow of geometric time. This applies also to sub-selves of self giving rise to mental images. One can raise several questions.

Do we indeed have both mental images and time-reversed mental images? How the time-reversed mental image differs from the original one? Does the time flow in opposite direction for it? The roles of boundaries of CD have changed. The passive boundary of CD define the static back-ground the observed whereas the non-static boundary defines kind of dynamic figure. Does the change of the arrow of time change the roles of figure and background?

I have also proposed that motor action and sensory perception are time reversals of each other. Could one interpret this by saying that sensory perception is motor action affecting the body of self (say emotional expression) and motor action sensory perception of the environment about self.

In the sequel reverse speech and figure-background illusion is represented as examples of what time reversal for mental images could mean.

Time reversed cognition

Time reflection yields time reversed and spatially reflected sensory-cognitive representations. When mental image dies it is replaced with its time-reversal at opposite boundary of its CD. The observation of these representations could serve as a test of the theory.

There is indeed some evidence for this rather weird looking time and spatially reversed cognition.

1. I have a personal experience supporting the idea about time reversed cognition. During the last psychotic episodes of my "great experience" I was fighting to establish the normal direction of the experienced time flow. Could this mean that for some sub-CDs the standard arrow of time had reversed as some very high level mental images representing bodily me died and was re-incarnated?
   
   
2. The passive boundary of CD corresponds to static observing self - kind of background - and active boundary the dynamical - kind of figure. Figure-background division of mental image in this sense would change as sub-self dies and re-incarnates since figure and background change their roles. Figure-background illusion could be understood in this manner.
   
   
3. The occurrence of mirror writing is well known phemonenon (my younger daughter was reverse writer when she was young). Spatial reflections of MEs are also possible and might be involved with mirror writing. The time reversal would change the direction of writing from right to left.
   
   
4. Reverse speech would be also a possible form of reversed cognition. Time reversed speech has the same power spectrum as ordinary speech and the fact that it sounds usually gibberish means that phase information is crucial for storing the meaning of speech. Therefore the hypothesis is testable.
   
   

Reverse speech

Interestingly, the Australian David Oates claims that so called reverse speech is a real phenomenon, and he has developed entire technology and therapy (and business) around this phenomenon. What is frustrating that it seems impossible to find comments of professional linguistics or neuro-scientits about the claims of Oates. I managed only to find comments by a person calling himself a skeptic believer but it became clear that the comments of this highly rhetoric and highly arrogant commentator did not contain any information. This skeptic even taught poor Mr. Oates in an aggressive tone that serious scientists are not so naive that they would even consider the possibility of taking seriously what some Mr. Oates is saying. The development of science can often depend on ridiculously small things: in this case one should find a shielded place (no ridiculing skeptics around) to wind tape recorder backwards and spend few weeks or months to learn to recognize reverse speech if it really is there! Also computerized pattern recognition could be used to make speech recognition attempts objective since it is a well-known fact that brain does feature recognition by completing the data into something which is familiar.

The basic claims of Oates are following.

1. Reverse speech contains temporal mirror images of ordinary words and even metaphorical statements, that these words can be also identified from Fourier spectrum, that brain responds in unconscious manner to these words and that this response can be detected in EEG. Oates classifies these worlds to several categories. These claims could be tested and pity that no professional linguist nor neuroscientist (as suggested by web search) has not seen the trouble of finding whether the basic claims of Oates are correct or not.
   
   
2. Reverse speech is complementary communication mode to ordinary speech and gives rise to a unconscious (to us) communication mechanism making lying very difficult. If person consciously lies, the honest alter ego can tell the truth to a sub-self understanding the reverse speech. Reverse speech relies on metaphors and Oates claims that there is general vocabulary. Could this taken to suggest that reverse speech is communication of right brain whereas left brain uses ordinary speech? The notion of semitrance used to model bicameral mind suggests that reverse speech could be communication of higher levels of self hierarchy dispersed inside the ordinary speech. There are also other claims relating the therapy using reverse speech, which sound rather far-fetched but one should not confuse these claims to those which are directly testable.
   
   

Physically reverse speech could correspond to phase conjugate sound waves which together with their electromagnetic counterparts can be produced in laboratory . Phase conjugate waves have rather weird properties due the fact that second law applies in a reversed direction of geometric time. For this reason phase conjugate waves are applied in error correction. ZEO predicts this phenomenon.

Negative energy topological light rays are in a fundamental role in the TGD based model for living matter and brain. The basic mechanism of intentional action would rely on time mirror mechanism utilizing the TGD counterparts of phase conjugate waves producing also the nerve pulse patterns generating ordinary speech. If the language regions of brain contain regions in which the the arrow of psychological time is not always the standard one, they would induce phase conjugates of the sound wave patterns associated with the ordinary speech and thus reverse speech.

ZEO based quantum measurement theory, which is behind the recent form of TGD inspired theory of consciousness, provides a rigorous basis for this picture. Negative energy signals can be assigned with sub-CDs representing selves with non-standard direction of geometric time and every time when mental image dies, a mental images with opposite arrow of time is generated. It would be not surprising if the reverse speech would be associated with these time reversed mental images.

Figure-background rivalry and time reversed mental images

The classical demonstration of figure-background rivalry is is a pattern experienced either as a vase or two opposite faces. This phenomenon is not the same thing as bi-ocular rivalry in which the percepts associated with left and right eyes produced by different sensory inputs are rivalling. There is also an illusion in which one perceices the dancer to make a pirouette in either counter-clockwise or clockwise direction althought the figure is static. The direction of pirouette can change. In this case time-reversal would naturally change the direction of rotation.

Figure-background rivalry gives a direct support for the TGD based of self relying on ZEO if the following argument is accepted.

1. In ZEO the state function reduction to the opposite boundary of CD means the death of the sensory mental image and birth of new one, possibly the rivalling mental image. During the sequence of state function reductions to the passive boundary of CD defining the mental image a boundary quantum superposition of rivalling mental images associated with the active boundary of CD is generated.
   

   In the state function reduction to the opposite boundary the previous mental image dies and is replaced with new one. In the case of bin-ocular rivalry this might be the either of the sensory mental images generated by the sensory inputs to eyes. This might happen also now but also different interpretation is possible.
   
   

2. The basic questions concern the time reversed mental image. Does the subject person as a higher level self experience also the time reversed sensory mental image as sensory mental image as one might expect. If so, how the time reversed mental image differs from the mental image? Passive boundary of CD defines quite generally the background - the static observer - and active boundary the figure so that their roles should change in the reduction to the opposite boundary.In sensory rivalry situation this happens at least in the example considered (vase and two faces).
   

   I have also identified motor action as time reversal of sensory percept. What this identification could mean in the case of sensory percepts? Could sensory and motor be interpreted as an exchange of experiencer (or sub-self) and environment as figure and background?

If this interpretation is correct, figure-background rivalry would tell something very important about consciousness and would also support ZEO. Time reversal would permute figure and background. This might happen at very abstract level. Even subjective-objective duality and first - and third person aspects of conscious experience might relate to the time reversal of mental images. In near death experiences person sees himself as an outsider: could this be interpreted as the change of the roles of figure and background indentified as first and third person perspectives? Could the first moments of the next life be seeing the world from the third person perspective?

An interesting question is whether right- and left hemispheres tend to have opposite directions of geometric time. This would make possible metabolic energy transfer between them making possible kind of flip-flop mechanism. The time-reversed hemisphere would receive negative energy serving as metabolic energy resource for it and the hemisphere sending negative energy would get in this manner positive metabolic energy. Deeper interpretation would be in terms of periodic transfer of negentropic entanglement. This would also mean that hemispheres would provide two views about the world in which figure and background would be permuted.

See the article Time Reversed Self.

For a summary of earlier postings see Links to the latest progress in TGD.

Voevoedski's univalent foundations of mathematics

I found a very nice article about the work of mathematician Voevoedski: he talks about univalent foundations of mathematics. To put in nutshell: the world deals with mathematical proofs and the deep idea is that proofs are like paths in some abstract space. One deals with paths also in homotopy theory. What is remarkable that Voevoedski's work leads to computer programs allowing to check that proof does not contain an error. Something very badly needed when the proofs are getting increasingly complex. I dare to guess that the article is understandable by laymen too.

Russell's problem

The article tells about type theory originated already by Russell, who was led to his paradox with the set consisting of sets which do not contain itself as an element. The fatal question was "Does this set contain itself?".

Russell proposed a solution of the problems by introducing a hierarchy of types. Sets are at the bottom and defined so that they do not contain as set collection of sets. In usual applications of set theory - say in manifold theory - this is always true. Type hierarchy guarantees that you do not put apples and oranges in the same basket.

Voevoedski's idea

Voevoedski's work relates to proof theory and to formalising what mathematical proof does mean.

Consider demonstration that two sets A and B are equivalent. This means simple thing: construct a one-to-one map between them. Usually one is only interested in the existence of this map but one can also get interested on all manners to perform this map. All manners to make this map define in rather abstract sense a collection of paths between A and B regarded as objects. Single path consists of a collection of the arrows connecting element in A with element in B.

More generally, in mathematical proof theory all proofs of theorem define this kind of collection. In topology all paths connecting two points defined this kind of collection. In this framework Goedel's theorem becomes obvious: given axioms define rules for constructing paths and cannot give the paths connecting arbitrarily chosen two truths.

One can again abstract this process. Just as one can make statements about statements about..., one can build paths between paths, and paths between paths between paths.... Voevoedsky studied this problem in his attempts to formalise mathematics and with a practical goal to develop eventually tools for checking by computer that mathematical proofs are correct.

A more rigorous manner to define path is in terms of groupoid. It is more general notion that that of group since the inverse of groupoid element need not exist. For intense open paths between two points form a group but only closed paths can be given a structure of group.

Voevoedski introduced the notion of infinite groupoid containing paths, paths between paths, .... ad infinitum. Voevoedski talks about univalent foundations. The great idea is that homotopy theory becomes a foundation of mathematics: proofs are paths in some abstract structure. This suggests in my non-professional mind that one can talk about continuous deformations of proofs and that one can classify them to homotopy types with proofs of same homotopy type deformable to each other.

How this relates to quantum TGD

What made me fascinated was how closely the basic hierarchies of quantum TGD relate to the objects studies in type theory and Voevoedski's approach.

Let us start with set theory and type theory. TGD provides a non-trivial example about types, which by the way distinguishes TGD from super string models. Imbedding space is set, space-time surface are its subsets. "World of classical worlds" (WCW) is the key abstraction distinguishing TGD from super string models where one still tries to deal by working at space-time level.

What has surprised me against and again that super string modellers have spend decades in the landscape instead of making super string models a real theory by introducing loop space as a key notion although it has very nice mathematics: just the existence of Kähler geometry fixes it uniquely: this observation actually led to the realisation that quantum TGD might be unique just from its mathematical existence.

The points of WCW are 3-surfaces and its sets are collections of 3-surfaces. They are of higher type than the sets of imbedding space. There would be no sense in putting points of WCW and of imbedding space in the same basket. But in the set theory before Russell you could in principle do this. We have got as as birth gift the ability to not put cows and tooth brushes into same set. But the ability to take seriously the existence of more abstract types does not seem to be a birth gift.

Voevoedski and others deal with statements about statements about statements…. What is amusing that this vision has direct counterparts in TGD based quantum physics where various hierarchies have taken key role. Some deep ideas seem to burst out simultaneously in totally different contexts! Voevoedski noticed the same thing in his work but with within the realm of mathematics.

Just a list of examples should be enough. Consider first type theory.

1. The hierarchy of infinite primes (integers, rationals) was the first mathematical discovery inspired by TGD inspired theory of consciousness. Infinite primes are constructed by a process analogous to a repeated second quantisation of arithmetic quantum field theory having interpretation as making statements about statements about..... up to arbitrary high order. Hierarchy of space-time sheets of many-sheeted space-time is the classical counterpart. Physics prediction is that higher level of quantisation are part of generalised quantum physics and allow quantum description of macroscopic and even astrophysical objects. The map of the sheets of many-sheeted space-time to single region of Minkowski space defines the contraction of TGD to GRT and is approximate operation: it maps a hierarchy of types to single type and is a violent procedure meaning a loss of information.
   

   Infinite integers could provide a generalisation of Goedel number numbering in a quantum mathematics based on the replacement of axiomatics with anti-axiomatics: specify what you cannot do instead of what you can do! I wrote about this in earlier posting.
   

   Infinite rationals of unit real norm lead also to a generalisation of the real number. Each real point becomes infinite dimensional space consisting of all infinite rationals of unit norm but well-defined number theoretic anatomy.
   
   

2. There are several very closely related infinite hierarchies. Fractal hierarchy of quantum criticalities ( ball at the top of a hill at the top...) and isomorphic super-symplectic sub-algebras with conformal structure. There is infinite fractal hierarchy of conformal gauge symmetry breakings. This defines infinite hierarchy of dark matters with Planck constant h_eff=n× h. The algebraic extensions of rationals giving rise to evolutionary hierarchy for physics and perhaps explaining biological evolution define a hierarchy. The inclusions of hyper-finite factors realizing finite measurement resolution define a hierarchy. Hierarchy of infinite integers and rationals relates also closely to these hierarchies.
   
   
3. In TGD inspired theory of conscioussness hierarchy of selves having sub-selves (experienced as mental images) having.... This hierarchy relates also very closely to the above hierarchies.
   
   

The notion of mathematical operation sequences as path is second key idea in Voevoedski's work. The idea about paths representing mathematical computations, proofs, etc.. is realised quite concretely in TGD quantum physics. Scattering amplitudes are identified as representations for sequences of algebraic operations of Yangian leading from an initial collection of elements of super-symplectic Yangian (physical states) to a final one. The duality symmetry of old fashioned string models generalises to a statement that any two sequences connecting same collections are equivalent and correspond to same amplitudes. This means extremely powerful predictions and it seems that in twistor programs analogous results are obtained too: very many twistor Grassmann diagrams represent the same scattering amplitude.

More about physical interpretation of algebraic extensions of rationals

The number theoretic vision has begun to show its power. The basic hierarchies of quantum TGD would reduce to a hierarchy of algebraic extensions of rationals and the parameters - such as the degrees of the irreducible polynomials characterizing the extension and the set of ramified primes - would characterize quantum criticality and the physics of dark matter as large h_eff phases. The identification of preferred p-adic primes as ramified primes of the extension and generalization of p-adic length scale hypothesis as prediction of NMP are basic victories of this vision (see this and this).

By strong form of holography the parameters characterizing string world sheets and partonic 2-surfaces serve as WCW coordinates. By various conformal invariances, one expects that the parameters correspond to conformal moduli, which means a huge simplification of quantum TGD since the mathematical apparatus of superstring theories becomes available and number theoretical vision can be realized. Scattering amplitudes can be constructed for a given algebraic extension and continued to various number fields by continuing the parameters which are conformal moduli and group invariants characterizing incoming particles.

There are many un-answered and even un-asked questions.

1. How the new degrees of freedom assigned to the n-fold covering defined by the space-time surface pop up in the number theoretic picture? How the connection with preferred primes emerges?
   
   
2. What are the precise physical correlates of the parameters characterizing the algebraic extension of rationals? Note that the most important extension parameters are the degree of the defining polynomial and ramified primes.
   
   

1. Some basic notions

Some basic facts about extensions are in order. I emphasize that I am not a specialist.

1.1. Basic facts

The algebraic extensions of rationals are determined by roots of polynomials. Polynomials be decomposed to products of irreducible polynomials, which by definition do not contain factors which are polynomials with rational coefficients. These polynomials are characterized by their degree n, which is the most important parameter characterizing the algebraic extension.

One can assign to the extension primes and integers - or more precisely, prime and integer ideals. Integer ideals correspond to roots of monic polynomials P_n(x)=xⁿ+..a₀ in the extension with integer coefficients. Clearly, for n=0 (trivial extension) one obtains ordinary integers. Primes as such are not a useful concept since roots of unity are possible and primes which differ by a multiplication by a root of unity are equivalent. It is better to speak about prime ideals rather than primes.

Rational prime p can be decomposed to product of powers of primes of extension and if some power is higher than one, the prime is said to be ramified and the exponent is called ramification index. Eisenstein's criterion states that any polynomial P_n(x)= a_nxⁿ+a_n-1x^n-1+...a₁x+ a₀ for which the coefficients a_i, i<n are divisible by p and a₀ is not divisible by p² allows p as a maximally ramified prime. mThe corresponding prime ideal is n:th power of the prime ideal of the extensions (roughly n:th root of p). This allows to construct endless variety of algebraic extensions having given primes as ramified primes.

Ramification is analogous to criticality. When the gradient potential function V(x) depending on parameters has multiple roots, the potential function becomes proportional a higher power of x-x₀. The appearance of power is analogous to appearance of higher power of prime of extension in ramification. This gives rise to cusp catastrophe. In fact, ramification is expected to be number theoretical correlate for the quantum criticality in TGD framework. What this precisely means at the level of space-time surfaces, is the question.

1.2 Galois group as symmetry group of algebraic physics

I have proposed long time ago that Galois group acts as fundamental symmetry group of quantum TGD and even made clumsy attempt to make this idea more precise in terms of the notion of number theoretic braid. It seems that this notion is too primitive: the action of Galois group must be realized at more abstract level and WCW provides this level.

First some facts (I am not a number theory professional, as the professional reader might have already noticed!).

1. Galois group acting as automorphisms of the field extension (mapping products to products and sums to sums and preserves norm) characterizes the extension and its elements have maximal order equal to n by algebraic n-dimensionality. For instance, for complex numbers Galois group acs as complex conjugation. Galois group has natural action on prime ideals of extension mapping them to each other and preserving the norm determined by the determinant of the linear map defined by the multiplication with the prime of extension. For instance, for the quadratic extension Q(5^1/2) the norm is N(x+5^1/2y)=x²-5y²: not that number theory leads to Minkowkian metric signatures naturally. Prime ideals combine to form orbits of Galois group.
   
   
2. Since Galois group leaves the rational prime p invariant, the action must permute the primes of extension in the product representation of p. For ramified primes the points of the orbit of ideal degenerate to single ideal. This means that primes and quite generally, the numbers of extension, define orbits of the Galois group.

Galois group acts in the space of integers or prime ideals of the algebraic extension of rationals and it is also physically attractive to consider the orbits defined by ideals as preferred geometric structures. If the numbers of the extension serve as parameters characterizing string world sheets and partonic 2-surfaces, then the ideals would naturally define subsets of the parameter space in which Galois group would act.

The action of Galois group would leave the space-time surface invariant if the sheets co-incide at ends but permute the sheets. Of course, the space-time sheets permuted by Galois group need not co-incide at ends. In this case the action need not be gauge action and one could have non-trivial representations of the Galois group. In Langlands correspondence these representation relate to the representations of Lie group and something similar might take place in TGD as I have indeed proposed.

Remark: Strong form of holography supports also the vision about quaternionic generalization of conformal invariance implying that the adelic space-time surface can be constructed from the data associated with functions of two complex variables, which in turn reduce to functions of single variable.

If this picture is correct, it is possible to talk about quantum amplitudes in the space defined by the numbers of extension and restrict the consideration to prime ideals or more general integer ideals.

1. These number theoretical wave functions are physical if the parameters characterizing the 2-surface belong to this space. One could have purely number theoretical quantal degrees of freedom assignable to the hierarchy of algebraic extensions and these discrete degrees of freedom could be fundamental for living matter and understanding of consciousness.
   
   
2. The simplest assumption that Galois group acts as a gauge group when the ends of sheets co-incide at boundaries of CD seems however to destroy hopes about non-trivial number theoretical physics but this need not be the case. Physical intuition suggests that ramification somehow saves the situation and that the non-trivial number theoretic physics could be associated with ramified primes assumed to define preferred p-adic primes.
   
   

2. How new degrees of freedom emerge for ramified primes?

How the new discrete degrees of freedom appear for ramified primes?

1. The space-time surfaces defining singular coverings are n-sheeted in the interior. At the ends of the space-time surface at boundaries of CD however the ends co-incide. This looks very much like a critical phenomenon.
   

   Hence the idea would be that the end collapse can occur only for the ramified prime ideals of the parameter space - ramification is also a critical phenomenon - and means that some of the sheets or all of them co-incide. Thus the sheets would co-incide at ends only for the preferred p-adic primes and give rise to the singular covering and large h_eff. End-collapse would be the essence of criticality! This would occur, when the parameters defining the 2-surfaces are in a ramified prime ideal.
   
   

2. Even for the ramified primes there would be n distinct space-time sheets, which are regarded as physically distinct. This would support the view that besides the space-like 3-surfaces at the ends the full 3-surface must include also the light-like portions connecting them so that one obtains a closed 3-surface. The conformal gauge equivalence classes of the light-like portions would give rise to additional degrees of freedom. In space-time interior and for string world sheets they would become visible.
   

   For ramified primes n distint 3-surfaces would collapse to single one but the n discrete degrees of freedom would be present and particle would obtain them. I have indeed proposed number theoretical second quantization assigning fermionic Clifford algebra to the sheets with n oscillator operators. Note that this option does not require Galois group to act as gauge group in the general case. This number theoretical second quantization might relate to the realization of Boolean algebra suggested by weak form of NMP (see this).
   
   

3. About the physical interpretation of the parameters characterizing algebraic extension of rationals in TGD framework

It seems that Galois group is naturally associated with the hierarchy h_eff/h=n of effective Planck constants defined by the hierarchy of quantum criticalities. n would naturally define the maximal order for the element of Galois group. The analog of singular covering with that of z^1/n would suggest that Galois group is very closely related to the conformal symmetries and its action induces permutations of the sheets of the covering of space-time surface.

Without any additional assumptions the values of n and ramified primes are completely independent so that the conjecture that the magnetic flux tube connecting the wormhole contacts associated with elementary particles would not correspond to very large n having the p-adic prime p characterizing particle as factor (p=M₁₂₇=2¹²⁷-1 for electron). This would not induce any catastrophic changes.

TGD based physics could however change the situation and reduce number theoretical degrees of freedom: the intuitive hypothesis that p divides n might hold true after all.

1. The strong form of GCI implies strong form of holography. One implication is that the WCW Kähler metric can be expressed either in terms of Kähler function or as anti-commutators of super-symplectic Noether super-charges defining WCW gamma matrices. This realizes what can be seen as an analog of Ads/CFT correspondence. This duality is much more general. The following argument supports this view.
   
   
   1. Since fermions are localized at string world sheets having ends at partonic 2-surfaces, one expects that also Kähler action can be expressed as an effective stringy action. It is natural to assume that string area action is replaced with the area defined by the effective metric of string world sheet expressible as anti-commutators of Kähler-Dirac gamma matrices defined by contractions of canonical momentum currents with imbedding space gamma matrices. It string tension is proportional to h_eff², string length scales as h_eff.
      
      
   2. AdS/CFT analogy inspires the view that strings connecting partonic 2-surfaces serve as correlates for the formation of - at least gravitational - bound states. The distances between string ends would be of the order of Planck length in string models and one can argue that gravitational bound states are not possible in string models and this is the basic reason why one has ended to landscape and multiverse non-sense.
      
      
2. In order to obtain reasonable sizes for astrophysical objects (that is sizes larger than Schwartschild radius r_s=2GM) For h_eff=h_gr=GMm/v₀ one obtains reasonable sizes for astrophysical objects. Gravitation would mean quantum coherence in astrophysical length scales.
   
   
3. In elementary particle length scales the value of h_eff must be such that the geometric size of elementary particle identified as the Minkowski distance between the wormhole contacts defining the length of the magnetic flux tube is of order Compton length - that is p-adic length scale proportional to p^1/2. Note that dark physics would be an essential element already at elementary particle level if one accepts this picture also in elementary particle mass scales. This requires more precise specification of what darkness in TGD sense really means.
   

   One must however distinguish between two options.
   
   

   1. If one assumes n≈ p^1/2, one obtains a large contribution to classical string energy as Δ ∼ m_CP2²L_p/hbar²_eff ∼ m_CP2/p^1/2, which is of order particle mass. Dark mass of this size looks un-feasible since p-adic mass calculations assign the mass with the ends wormhole contacts. One must be however very cautious since the interpretations can change.
      
      
   2. Second option allows to understand why the minimal size scale associated with CD characterizing particle correspond to secondary p-adic length scale. The idea is that the string can be thought of as being obtained by a random walk so that the distance between its ends is proportional to the square root of the actual length of the string in the induced metric. This would give that the actual length of string is proportional to p and n is also proportional to p and defines minimal size scale of the CD associated with the particle. The dark contribution to the particle mass would be Δ m ∼ m_CP2²L_p/hbar²_eff∼ m_CP2/p, and completely negligible suggesting that it is not easy to make the dark side of elementary visible.
      
      
4. If the latter interpretation is correct, elementary particles would have huge number of hidden degrees of freedom assignable to their CDs. For instance, electron would have p=n=2¹²⁷-1 ≈ 10³⁸ hidden discrete degrees of freedom and would be rather intelligent system - 127 bits is the estimate- and thus far from a point-like idiot of standard physics. Is it a mere accident that the secondary p-adic time scale of electron is .1 seconds - the fundamental biorhythm - and the size scale of the minimal CD is slightly large than the circumference of Earth?
   

   Note however, that the conservation option assuming that the magnetic flux tubes connecting the wormhole contacts representing elementary particle are in h_eff/h=1 phase can be considered as conservative option.
   

   See the article More about physical interpretation of algebraic extensions of rationals.

   For a summary of earlier postings see Links to the latest progress in TGD.
   
   
   
   

Quantum Mathematics in TGD Universe

Some comments about quantum mathematics, quantum Boolean thinking and computation as they might happen at fundamental level.

1. One should understand how Boolean statements A→B are represented. Or more generally: How a computation like procedure leading from a collection A of math objects collection B of math objects takes place? Recall that in computations the objects coming in and out are bit sequences. Now one have computation like process. → is expected to correspond to the arrow of time.

   If fermionic oscillator operators generate Boolean basis, zero energy ontology is necessary to realize rules as rules connecting statements realized as bit sequences. Positive energy ontology would allow only statements A,B but not statements A→B about them. ZEO allows also to avoid restrictions due to fermion number conservation and its well-definedness.
   

   Collection A is at the passive boundary of CD and not changed in state function reduction sequence defining self and B is at the active one. As a matter fact, it is not single statement but a quantum superpositions of statements B, which resides there! In the quantum jump selecting single B at the active boundary, A is replaced with a superposition of A:s: self dies and re-incarnates as more negentropic entity. Q-computation halts.
   

   That both a and b cannot be known precisely is a quantal limitation to what can be known: philosopher would talk about epistemology here. The different pairs (a,b) in superposition over b:s are analogous to different implications of a. Thinker is doomed to always live in a quantum cognitive dust and never be quite sure of.
   
   

2. What is the computer like structure now? Turing computer is discretized 1-D time-like line. This quantum computer is superposition of 4-D space-time surfaces with the basic computational operations located along it as partonic 2-surfaces defining the algebraic operations and connected by fermion lines representing signals. Also string world sheets are involved. In some key aspects this is very similar to ordinary computer. By strong form of holography computations use only data at string world sheets and partonic 2-surfaces.
   
   
3. What is the computation? It is sequence of repeated state function reduction leaving the passive boundary of CD
   intact but affecting the position (moduli) of upper boundary of CD and also the parts of zero energy states there.
   It is a sequence of unitary processes delocalizing the active boundary of CD followed by localization but no reduction. This the counterpart for a sequence of reductions leaving quantum state invariant in ordinary measurement theory (Zeno etc). Commutation halts as the first reduction to the opposite boundary occurs. Self dies and re-incarnates at the opposite boundary. Negentropy gain results in general and can be see as the information gained in the computation. One might hope that the new self (maybe something at higher level of dark matter hierarchy) is a little bit wiser - at least statistically speaking this seems to be true by weak form of NMP!
   
   
   
4. One should understand the quantum counterparts for the basic rules of manipulation. ×,/,+, and - are the most familiar example.
   
   
   1. The basic rules correspond physically to generalized Feynman/twistor diagrams representing sequences of algebraic manipulations in the Yangian of super-symplectic algebra. Sequences correspond now to collections of partonic 2-surfaces defining vertices of generalized twistor diagrams.
      
      
   2. 3- vertices correspond to product and co-product for quantal stringy Noether charges. Geometrically the vertex - analog of algebraic operation - is a partonic 2-surface at with incoming and outgoing light-like 3-surfaces meet - like vertex of Feynman diagram. Co-product vertex is not encountered in simple algebraic systems, and is time reversed variant of vertex. Fusion instead of annihilation.
      
      
   3. This diagrammatics has a huge symmetry just like ordinary computations have. All computation sequences (note that the corresponding space-time surfaces are different!) connecting same collections A and B of objects produce the same scattering amplitude. This generalises the duality symmetry of hadronic string models. This is really gigantic simplification and the results of twistor Grassmann approach suggest that something similar is obtained there. This implication was so gigantic that I gave up the idea for years.
      
      
5. One should understand the analogs for the mathematical axioms. What are the fundamental rules of manipulation?
   
   
   1. The classical computation/deduction would obey deterministic rules at vertices. The quantal formulation cannot be deterministic for the simple reason that one has quantum non-determinism (weak form of NMP allowing also good and evil) . The quantum rules obey the format that God used when communicating with Adam and Eve: do anything else but do not the break the conservation laws. Classical rules would list all the allowed possibilities and this leads to difficulties as Goedel demonstrated. I think that chess players follow the "anti-axiomatics".
      
      
   2. I have the feeling that anti-axiomatics - not any well-established idea, it occurred to me as I wrote this - could provide a more natural approach to quantum computation and even allow a new manner to approach to the problematics of axiomatisations. It is also interesting to notice a second TGD inspired notion - the infinite hierarchy of mostly infinite integers (generated from infinite primes obtained by a repeated second quantization of an arithmetic QFT) - could make possible a generalisation of Gödel numbering for statements/computations. This view has at least one virtue: it makes clear how extremely primitive conscious entities we are in a bigger picture!
      
      
6. The laws of physics take care that the anti-axioms are obeyed. Quite concretely:
   
   
   1. Preferred extremal property of Kähler action and Käler-Dirac action plus conservation laws for charges associated with super-symplectic and other generalised conformal symmetries would define the rules not broken in vertices.
      
      
   2. At the fermion lines connecting the vertices the propagator would be determined by the boundary part of Kahler-Dirac action. K-D equation for spinors and consistency consistency conditions from Kahler action (strong form of holography) would dictate what happens to fermionic oscillator operators defining the analog of quantum Boolean algebra as super-symplectic algebra.
      
      

Breakthroughs in the number theoretic vision about TGD

Number theoretic universality states that besides reals and complex numbers also p-adic number fields are involved (they would provide the physical correlates of cognition). Furthermore, scattering amplitudes should be well-defined in all number fields be obtained by a kind of algebraic continuation. I have introduced the notion of intersection of realities and p-adicities which corresponds to some algebraic extension of rationals inducing an extension of p-adic numbers for any prime p. Adelic physics is a strong candidate for the realization of fusion of real and p-adic physics and would mean the replacement of real numbers with adeles. Field equations would hold true for all numer fields and the space-time surfaces would relate very closely to each other: one could say that p-adic space-time surfaces are cognitive representations of the real ones.

I have had also a stronger vision which is now dead. This sad event however led to a discovery of several important results.

1. The idea has been that p-adic space-time sheets would be not only "thought bubbles" representing real ones but also correlates for intentions and the transformation of intention to action would would correspond to a quantum jump in which p-adic space-time sheet is transformed to a real one. Alternatively, there would be a kind of leakage between p-adic and real sectors. Cognitive act would be the reversal of this process. It did not require much critical thought to realize that taking this idea seriously leads to horrible mathematical challenges. The leakage takes sense only in the intersection, which is number theoretically universal so that there is no point in talking about leakage. The safest assumption is that the scattering amplitudes are defined separately for each sector of the adelic space-time. This means enormous relief, since there exists mathematics for defining adelic space-time.
   
   
2. This realization allows to clarify thoughts about what the intersection must be. Intersection corresponds by strong form of holography to string world sheets and partonic 2-surfaces at which spinor modes are localized for several reasons: the most important reasons are that em charge must be well-defined for the modes and octonionic and real spinor structures can be equivalent at them to make possible twistorialization both at the level of imbedding space and its tangent space.

   The parameters characterizing the objects of WCW are discretized - that is belong to an appropriate algebraic extension of rationals so that surfaces are continuous and make sense in real number field and p-adic number fields. By conformal invariance they might be just conformal moduli. Teichmueller parameters, positions of punctures for partonic 2-surfaces, and corners and angles at them for string world sheets. These can be continued to real and p-adic sectors and
   
   

3. Fermions are correlates for Boolean cognition and anti-commutation relations for them are number theoretically universal, even their quantum variants when algebraic extension allows quantum phase. Fermions and Boolean cognition would reside in the number theoretically universal intersection. Of course they must do so since Boolean thought and cognition in general is behind all mathematics!
   
   
4. I have proposed this in p-adic mass calculations for two decades ago. This would be wonderful simplification of the theory: by conformal invariance WCW would reduce to finite-dimensional moduli space as far as calculations of scattering amplitudes are considered. The testing of the theory requires classical theory and 4-D space-time. This holography would not mean that one gives up space-time: it is necessary. Only cognitive and as it seems also fundamental sensory representations are 2-dimensional. All that one can mathematically say about reality is by using data at these 2-surfaces. The rest is needed but it require mathematical thinking and transcendence! This view is totally different from the sloppy and primitive philosophical idea that space-time could somehow emerge from discrete space-time.
   
   

This has led also to modify the ideas about the relation of real and p-adic physics.

1. The notion of p-adic manifolds was hoped to provide a possible realization of the correspondence between real and p-adic numbers at space-time level. It relies on the notion canonical identification mapping p-adic numbers to real in continuous manner and realizes finite measurement resolution at space-time level. p-Adic length scale hypothesis emerges from the application of p-adic thermodynamics to the calculation of particle masses but generalizes to all scales.
   
   
2. The problem with p-adic manifolds is that the canonical identification map is not general coordinate invariant notion. The hope was that one could overcome the problem by finding preferred coordinates for imbedding space. Linear Minkowski coordinates or Robertson-Walker coordinates could be the choice for M⁴. For CP₂ coordinates transforming linearly under U(2) suggest themselves. The non-uniqueness however persists but one could argue that there is no problem if the breaking of symmetries is below measurement resolution. The discretization is however also non-unique and makes the approach to look ugly to me although the idea about p-adic manifold as cognitive chargt looks still nice.
   
   
3. The solution of problems came with the discovery of an entirely different approach. First of all, realized discretization at the level of WCW, which is more abstract: the parameters characterizing the objects of WCW are discretized - that is assumed to belong to an appropriate algebraic extension of rationals so that surfaces are continuous and make sense in real number field and p-adic number fields.

   Secondly, one can use strong form of holography stating that string world sheets and partonic 2-surfaces define the "genes of space-time". The only thing needed is to algebraically extend by algebraic continuation these 2-surfaces to 4-surfaces defining preferred extremals of Kähler action - real or p-adic. Space-time surface have vanishing Noether charges for a sub-algebra of super-symplectic algebra with conformal weights coming as n-ples of those for the full algebra- hierarchy of quantum criticalities and Planck constants and dark matters!

   One does not try to map real space-time surfaces to p-adic ones to get cognitive charts but 2-surfaces defining the space-time genes to both real and p-adic sectors to get adelic space-time! The problem with general coordinate invariance at space-time level disappears totally since one can assume that these 2-surfaces have rational parameters. One has discretization in WCW, rather than at space-time level. As a matter fact this discretization selects punctures of partonic surfaces (corners of string world sheets) to be algebraic points in some coordinatization but in general coordinate invariant manner
   
   

4. The vision about evolutionary hierarchy as a hierarchy of algebraic extensions of rationals inducing those of p-adic number fields become clear. The algebraic extension associated with the 2-surfaces in the intersection is in question. The algebraic extension associated with them become more and more complex in evolution. Of course, NMP, negentropic entanglement (NE) and hierarchy of Planck constants are involved in an essential manner too. Also the measurement resolution characterized by the number of space-time sheets connecting average partonic 2-surface to others is a measure for "social" evolution since it defines measurement resolution.
   
   

There are two questions, which I have tried to answer during these two decades.

1. What makes some p-adic primes preferred so that one can say that they characterizes elementary particles and presumably any system?
   
   
2. What is behind p-adic length scale hypothesis emerging from p-adic mass calculations and stating that primes near but sligthly below two are favored physically, Mersenne primes in particular. There is support for a generalization of this hypothesis: also primes near powers of 3 or powers of 3 might be favored as length sand time scales which suggests that powers of prime quite generally are favored.
   
   

The adelic view led to answers to these questions. The answer to the first question has been staring directly to my eyes for more than decade.

1. The algebraic extension of rationals allow so called ramified primes. Rational primes decompose to product of primes of extension but it can happen that some primes of extension appear as higher than first power. In this case one talks about ramification. The product of ramified primes for rationals defines an integer characterizing the ramification. Also for extension allows similar characteristic. Ramified primes are an extremely natural candidate for preferred primes of an extension (I know that I should talk about prime ideals, sorry for a sloppy language): that preferred primes could follow from number theory itself I had not though earlier and tried to deduce them from physics. One can assign the characterizing integers to the string world sheets to characterize their evolutionary level. Note that the earlier heuristic idea that space-time surface represents a decomposition of integer is indeed realized in terms of holography!
   
   
2. Also infinite primes seem to find finally the place in the big picture. Infinite primes are constructed as an infinite hierarchy of second quantization of an arithmetic quantum field theory. The infinite primes of the previous level label the single fermion - and boson states of the new level but also bound states appear. Bound states can be mapped to irreducible polynomials of n-variables at n:th level of infinite obeying some restrictions. It seems that they are polynomials of a new variable with coefficients which are infinite integers at the previous level.
   

   At the first level bound state infinite primes correspond to irreducible polynomials: these define irreducible extensions of rationals and as a special case one obtains those satisfying so called Eistenstein criterion: in this case the ramified primes can be read directly from the form of the polynomial. Therefore the hierarchy of infinite primes seems to define algebraic extension of rationals, that of polynomials of one variables, etc.. What this means from the point of physics is a fascinating question. Maybe physicist must eventually start to iterate second quantization to describe systems in many-sheeted space-time! The marvellous thing would be the reduction of the construction of bound states - the really problematic part of quantum field theories - to number theory!
   
   

The answer to the second question requires what I call weak form of NMP.

1. Strong form of NMP states that negentropy gain in quantum jump is maximal: density matrix decompose into sum of terms proportional to projection operators: choose the sub-space for which number theoretic negentropy is maximal. The projection operator containing the largest power of prime is selected. The problem is that this does not allow free will in the sense as we tend to use: to make wrong choices!
   
   
2. Weak NMP allows to chose any projection operator and sub-space which is any sub-space of the sub-space defined by the projection operator. Even 1-dimensional in which case standard state function reduction occurs and the system is isolated from the environment as a prize for sin! Weak form of NMP is not at all so weak as one might think. Suppose that the maximal projector operator has dimension n_max which is product of large number of different but rather small primes. The negentropy gain is small. If it is possible to choose n=n_max-k, which is power of prime, negentropy gain is much larger!
   

   It is largest for powers of prime defining n-ary p-adic length scales. Even more, large primes correspond to more refined p-adic topology: p=1 (one could call it prime) defines discrete topology, p=2 defines the roughest p-adic topology, the limit p→ ∞ is identified by many mathematicians in terms of reals. Hence large primes p<n_max are favored. In particular primes near but below powers of prime are favored: this is nothing but a generalization of p-adic length scale hypothesis from p=2 to any prime p.
   
   

See the article What Could Be the Origin of Preferred p-Adic Primes and p-Adic Length Scale Hypothesis?.

For a summary of earlier postings see Links to the latest progress in TGD.

Updated Negentropy Maximization Principle

Quantum TGD involves "holy trinity" of time developments. There is the geometric time development dictated by the preferred extremal of Kähler action crucial for the realization of General Coordinate Invariance and analogous to Bohr orbit. There is what I originally called unitary "time development" U: Ψ_i→ UΨ_i→ Ψ_f, associated with each quantum jump. This would be the counterpart of the Schrödinger time evolution U(-t,t→ ∞). Quantum jump sequence itself defines what might be called subjective time development.

Concerning U, there is certainly no actual Schrödinger equation involved: situation is in practice same also in quantum field theories. It is now clear that in Zero Energy Ontology (ZEO) U can be actually identified as a sequence of basic steps such that single step involves a unitary evolution inducing delocalization in the moduli space of causal diamonds CDs) followed by a localization in this moduli space selecting from a superposition of CDs single CD. This sequence replaces a sequence of repeated state function reductions leaving state invariant in ordinary QM. Now it leaves in variant second boundary of CD (to be called passive boundary) and also the parts of zero energy states at this boundary. There is now a very attractive vision about the construction of transition amplitudes for a given CD, and it remains to be see whether it allows an extension so that also transitions involving change of the CD moduli characterizing the non-fixed boundary of CD.

A dynamical principle governing subjective time evolution should exist and explain state function reduction with the characteristic one-one correlation between macroscopic measurement variables and quantum degrees of freedom and state preparation process. Negentropy Maximization Principle is the candidate for this principle. In its recent form it brings in only a single little but overall important modification: state function reductions occurs also now to an eigen-space of projector but the projector can now have dimension which is larger than one. Self has free will to choose beides the maximal possible dimension for this sub-space also lower dimension so that one can speak of weak form of NMP so that negentropy gain can be also below the maximal possible: we do not live in the best possible world. Second important ingredient is the notion of negentropic entanglement relying on p-adic norm.

The evolution of ideas related to NMP has been slow and tortuous process characterized by misinterpretations, over-generalizations, and unnecessarily strong assumptions, and has been basically evolution of ideas related to the anatomy of quantum jump and of quantum TGD itself.

Quantum measurement theory is generalized to theory of consciousness in TGD framework by replacing the notion of observer as outsider of the physical world with the notion of self. Hence it is not surprising that several new key notions are involved.

1. ZEO is in central role and brings in a completely new element: the arrow of time changes in the counterpart of standard quantum jump involving the change of the passive boundary of CD to active and vice versa. In living matter the changes of the of time are inn central role: for instance, motor action as volitional action involves it at some level of self hierarchy.
   
   
2. The fusion of real physics and various p-adic physics identified as physics of cognition to single adelic physics is second key element. The notion of intersection of real and p-adic worlds (intersection of sensory and cognitive worlds) is central and corresponds in recent view about TGD to string world sheets and partonic 2-surfaces whose parameters are in an algebraic extension of rationals. By strong form of of holography it is possible to continue the string world sheets and partonic 2-surfaces to various real and p-adic surfaces so that what can be said about quantum physics is coded by them. The physics in algebraic extension can be continued to real and various p-adic sectors by algebraic continuation meaning continuation of various parameters appearing in the amplitudes to reals and various p-adics.
   

   An entire hierarchy of physics labeled by the extensions of rationals inducing also those of p-adic numbers is predicted and evolution corresponds to the increase of the complexity of these extensions. Fermions defining correlates of Boolean cognition can be said so reside at these 2-dimensional surfaces emerging from strong form of holography implied by strong form of general coordinate invariance (GCI).
   

   An important outcome of adelic physics is the notion of number theoretic entanglement entropy: in the defining formula for Shannon entropy logarithm of probability is replaced with that of p-adic norm of probability and one assumes that the p-adic prime is that which produces minimum entropy. What is new that the minimum entropy is negative and one can speak of negentropic entanglement (NE). Consistency with standard measurement theory allows only NE for which density matrix is n-dimensional projector.
   
   

3. Strong form of NMP states that state function reduction corresponds to maximal negentropy gain. NE is stable under strong NMP and it even favors its generation. Strong form of NMP would mean that we live in the best possible world, which does not seem to be the case. The weak form of NMP allows self to choose whether it performs state function reduction yielding the maximum possible negentropy gain. If n-dimensional projector corresponds to the maximal negentropy gain, also reductions to sub-spaces with n-k-dimensional projectors down to 1-dimensional projector are possible. Weak form has powerful implications: for instance, one can understand how primes near powers of prime are selected in evolution identified at basic level as increase of the complexity of algebraic extension of rationals defining the intersection of realities and p-adicities.
   
   
4. NMP gives rise to evolution. NE defines information resources, which I have called Akashic records - kind of Universal library. The simplest possibility is that under the repeated sequence of state function reductions at fixed boundary of CD NE at that boundary becomes conscious and gives rise to experiences with positive emotional coloring: experience of love, compassion, understanding, etc... One cannot exclude the possibility that NE generates a conscious experience only via the analog of interaction free measurement but this option looks un-necessary in the recent formulation.
   
   
5. Dark matter hierarchy labelled by the values of Planck constant h_eff=n× h is also in central role and interpreted as a hierarchy of criticalities in which sub-algebra of super-symplectic algebra having structure of conformal algebra allows sub-algebra acting as gauge conformal algebra and having conformal weights coming as n-ples of those for the entire algebra. The phase transition increasing h_eff reduces criticality and takes place spontaneously. This implies a spontaneous generation of macroscopic quantum phases interpreted in terms of dark matter. The hierarchies of conformal symmetry breakings with n(i) dividing n(i+1) define sequences of inclusions of HFFs and the conformal sub-algebra acting as gauge algebra could be interpreted in terms of measurement resolution.
   

   n-dimensional NE is assigned with h_eff=n× h and is interpreted in terms of the n-fold degeneracy of the conformal gauge equivalence classes of space-time surfaces connecting two fixed 3-surfaces at the opposite boundaries of CD: this reflects the non-determinism accompanying quantum criticality. NE would be between two dark matter system with same h_eff and could be assigned to the pairs formed by the n sheets. This identification is important but not well enough understood yet. The assumption that p-adic primes p divide n gives deep connections between the notion of preferred p-adic prime, negentropic entanglement, hierarchy of Planck constants, and hyper-finite factors of type II₁.
   
   

6. Quantum classical correspondence (QCC) is an important constraint in ordinary measurement theory. In TGD QCC is coded by the strong form of holography assigning to the quantum states assigned to the string world sheets and partonic 2-surfaces represented in terms of super-symplectic Yangian algebra space-time surfaces as preferred extremals of Kähler action, which by quantum criticality have vanishing super-symplectic Noether charges in the sub-algebra characterized by integer n. Zero modes, which by definition do not contribute to the metric of "world of classical worlds" (WCW) code for non-fluctuacting classical degrees of freedom correlating with the quantal ones. One can speak about entanglement between quantum and classical degrees of freedom since the quantum numbers of fermions make themselves visible in the boundary conditions for string world sheets and their also in the structure of space-time surfaces.
   
   

NMP has a wide range of important implications.

1. In particular, one must give up the standard view about second law and replace it with NMP taking into account the hierarchy of CDs assigned with ZEO and dark matter hierarchy labelled by the values of Planck constants, as well as the effects due to NE. The breaking of second law in standard sense is expected to take place and be crucial for the understanding of evolution.
   
   
2. Self hierarchy having the hierarchy of CDs as imbedding space correlate leads naturally to a description of the contents of consciousness analogous to thermodynamics except that the entropy is replaced with negentropy.
   
   
3. In the case of living matter NMP allows to understand the origin of metabolism. NMP demands that self generates somehow negentropy: otherwise a state function reduction to tjhe opposite boundary of CD takes place and means death and re-incarnation of self. Metabolism as gathering of nutrients, which by definition carry NE is the manner to avoid this fate. This leads to a vision about the role of NE in the generation of sensory qualia and a connection with metabolism. Metabolites would carry NE and each metabolite would correspond to a particular qualia (not only energy but also other quantum numbers would correspond to metabolites). That primary qualia would be associated with nutrient flow is not actually surprising!
   
   
4. NE leads to a vision about cognition. Negentropically entangled state consisting of a superposition of pairs can be interpreted as a conscious abstraction or rule: negentropically entangled Schrödinger cat knows that it is better to keep the bottle closed.
   
   
5. NMP implies continual generation of NE. One might refer to this ever expanding universal library as "Akaschic records". NE could be experienced directly during the repeated state function reductions to the passive boundary of CD - that is during the life cycle of sub-self defining the mental image. Another, less feasible option is that interaction free measurement is required to assign to NE conscious experience. As mentioned, qualia characterizing the metabolite carrying the NE could characterize this conscious experience.
   
   
6. A connection with fuzzy qubits and quantum groups with NE is highly suggestive. The implications are highly non-trivial also for quantum computation allowed by weak form of NMP since NE is by definition stable and lasts the lifetime of self in question.
   
   

For details see the chapter Negentropy Maximization Principleof "TGD Inspired Theory of Consciousness".

For a summary of the earlier postings see Links to the latest progress in TGD.