Von Neumann inclusions, quantum groups, and quantum model for beliefs

Configuration space spinor fields live in "the world of classical worlds", whose points correspond to 3-surfaces in H=M⁴×CP₂. These fields represent the quantum states of the universe. Configuration space spinors (to be distinguished from spinor fields) have a natural interpretation in terms of a quantum version of Boolean algebra obtained by applying fermionic operators to the vacuum state. Both fermion number and various spinlike quantum numbers can be interpreted as representations of bits. Once you have true and false you have also beliefs and the question is whether it is possible to construct a quantum model for beliefs.

1. Some background about number theoretic Clifford algebras

Configuration space spinors are associated with an infinite-dimensional Clifford algebra spanned by configuration space gamma matrices: spinors are created from vacuum state by complexified gamma matrices acting like fermionic oscillator operators carrying quark and lepton numbers. In a rough sense this algebra could be regarded as an infinite tensor power of M₂(F), where F would correspond to complex numbers. In fact, also F=H (quaternions) and even F=O (octonions) can and must(!) be considered although the definitions involve some delicacies in this case. In particular, the non-associativy of octonions poses an interpretational problem whose solution actually dictates the physics of TGD Universe.

These Clifford algebras can be extended local algebras representable as powers series of hyper-F coordinate (hyper-F is obtained by multiplying imaginary part of F number with a commuting additional imaginary unit) so that a generalization of conformal field concept results with powers of complex coordinate replaced with those of hyper-complex numerg, hyper-quaternion or octonion. TGD could be seen as a generalization of superstring models by adding H and O layers besides C so that space-time and imbedding space emerge without ad hoc tricks of spontaneous compactification and adding of branes non-perturbatively.

The inclusion sequence C in H in O induces generalization of Jones inclusion sequence for the local versions of the number theoretic Clifford algebras allowing to reduce quantum TGD to a generalized number theory. That is, classical and quantum TGD emerge from the natural number theoretic Jones inclusion sequence. Even more, an explicit master formula for S-matrix emerges consistent with the earlier general ideas. It seems safe to say that one chapter in the evolution of TGD is now closed and everything is ready for the technical staff to start their work.

2. Brahman=Atman property of hyper-finite type II₁ factors makes them ideal for realizing symbolic and cognitive representations

Infinite-dimensional Clifford algebras provide a canonical example of von Neumann algebras known as hyper-finite factors of type II₁ having rather marvellous properties. In particular, they possess Brahman= Atman property making it possible to imbed this kind of algebra within itself unitarily as a genuine sub-algebra. One obtains what infinite Jones inclusion sequences yielding as a by-product structures like quantum groups.

Jones inclusions are ideal for cognitive and symbolic representations since they map the fermionic state space of one system to a sub-space of the fermionic statespace of another system. Hence there are good reasons to believe that TGD universe is busily mimicking itself using Jones inclusions and one can identify the space-time correlates (braids connecting two subsystems consisting of magnetic flux tubes). p-Adic and real spinors do not differ in any manner and real-to-p-adic inclusions would give cognitive representations, real-to-real inclusions symbolic representations.

3. Jones inclusions and cognitive and symbolic representations

As already noticed, configuration space spinors provide a natural quantum model for the Boolean logic. When you have logic you have the notions of truth and false, and you have soon also the notion of belief. Beliefs of various kinds (knowledge, misbelief, delusion,...) are the basic element of cognition and obviously involve a representation of the external world or part of it as states of the system defining the believer. Jones inclusions for the mediating unitary mappings between the spaces of configuration spaces spinors of two systems are excellent candidates for these maps, and it is interesting to find what one kind of model for beliefs this picture leads to.

The resulting quantum model for beliefs provides a cognitive interpretation for quantum groups and predicts a universal spectrum for the probabilities that a given belief is true following solely from the commutation relations for the coordinates of complex quantum plane interpreted now as complex spinor components. This spectrum of probabilities depends only on the integer n characterizing the quantum phase q=exp(iπ/n) characterizing the Jones inclusion. For n < ∞ the logic is inherently fuzzy so that absolute knowledge is impossible. q=1 gives ordinary quantum logic with qbits having precise truth values after state function reduction.

One can make two conclusions.

1. Quantum logics might have most interesting applications in the realm of consciousness theory and quantum spinors rather than quantum space-times seem to be more natural for the inclusions of factors of type II₁.

2. For n< ∞ inclusions quantum physical constraints pose fundamental restriction on how precisely it is possible to know and are reflected by the quantum dimension d<2 of quantum spinors telling the effective number of truth values smaller than one by correlations between non-commuting spinor components representing truth values. One could speak about Uncertainty Principle of Cognition for these inclusions.

The reader interested in details is recommended to look at the chapter Was von Neumann Right After All? of the book "Mathematical Aspects of Consciousness".

Matti Pitkänen

Does TGD reduce to inclusion sequence of number theoretic von Neumann algebras?

The idea that the notion of space-time somehow from quantum theory is rather attractive. In TGD framework this would basically mean that the identification of space-time as a surface of 8-D imbedding space H=M⁴× CP₂ emerges from some deeper mathematical structure. It seems that the series of inclusions for infinite-dimensional Clifford algebras associated with classical number fields F=R,C,H,O defining von Neumann algebras known as hyper-finite factors of type II₁, could be this deeper mathematical structure.

1. Quaternions, octonions, and TGD

The dimensions of quaternions and octonions are 4 and 8 and same as the dimensions of space-time surface and imbedding space in TGD. It is difficult to avoid the feeling that TGD physics could somehow reduce to the structures assignable to the classical number fields. This vision is already now rather detailed. For instance, a proposal for a general solution of classical field equations is one outcome of this vision.

TGD suggests also what I call HO-H duality. Space-time can be regarded either as surface in H or as hyper-quaternionic sub-manifold of the space HO of hyper-octonions obtained by multiplying imaginary parts of octonions with a commuting additional imaginary unit.

The 2-dimensional partonic surfaces X² are of central importance in TGD and it seems that the inclusion sequence C in H in O (complex numbers, quaternions, octonions) somehow corresponds to the inclusion sequence X² in X⁴ in H. This inspires the that that whole TGD emerges from a generalized number theory and I have already proposed arguments for how this might happen.

2. Number theoretic Clifford algebras

Hyper-finite factors of type II₁ defined by infinite-dimensional Clifford algebras is one thread in the multiple strand of number-theoretic ideas involving p-adic numbers fields and their fusion with reals along common rationals to form a generalized number system, classical number fields, hierarchy of infinite primes and integers, and von Neumann algebras and quantum groups. The new ideas allow to fuse von Neumans strand with the classical number field strand.

1. The mere assumption that physical states are represented by spinor fields in the infinite-dimensional "world of classical worlds" implies the notion of infinite-dimensional Clifford algebra identifiable as generated by gamma matrices of infinite-dimensional separable Hilbert space. This algebra provides a standard representation for hyperfinite factors of type II₁.

2. Von Neumann algebras known as hyperfinite factors of type II₁ are rather miraculous objects. The almost defining property is that the trace of unit operator is unity instead of infinity. This justifies the attribute hyperfinite and gives excellent hopes that the resulting quantum theory is free of infinities. These algebras are strange fractal like creatures in the sense that they can be imbedded unitarily within itself endlessly and one obtains infinite hierarchies of Jones inclusions. This means what might be called Brahman=Atman property: subsystem can represent in its state the state of the entire universe and this indeed leads to the idea that symbolic and cognitive representations are realized as Jones inclusions and that Universe is busily mimicking itself in this manner.

3. Classical number fields F=R,C,H,O define four Clifford algebras using infinite tensor power of 2x2 Clifford algebra M₂(F) associated with 2-spinors. The tensor powers associated with R and C are straightforward to define. The non-commutativity of H with C requires Connes tensor product which by definition guarantees that left and right multiplications of tensor product M₂(H)×M₂(H) by complex numbers are equivalent. For F=O the matrix algebra is not anymore associative but this implies only interpretational problems and means a slight generalization of von Neumann algebras which as far as I know are usually assumed to be associative. Denote by Cl(F) the infinite-dimensional Clifford algebras obtained in this manner. Perhaps I should not have said "only interpretational" since the solution of these problems dictates the classical and quantum dynamics.

3. TGD does not quite emerge from Jones inclusions for number theoretic Clifford algebras

Physics as a generalized number theory vision suggests that TGD physics is contained by the Jones inclusion sequence Cl(C) in Cl(H) in Cl(O) induced by C in H in O. This sequence could alone explain partonic, space-time, and imbedding space dimensions as dimensions of classical number fields. The dream is that also imbedding space H=M⁴× CP₂ would emerge as a unique choice allowed by mathematical existence.

1. CP₂ indeed emerges naturally: it labels the possible H-planes of O and this observation stimulated the emergence idea for few years ago.

2. Also Minkowski space M⁴ is wanted. In particular, future lightcones are needed since the super-canonical algebra defining the second super-conformal invariance of TGD is associated with the canonical algebra of δM⁴× CP₂. The generalized conformal and symplectic structures of 4-D(!) lightcone boundary are crucial element here. Ordinary Super Kac-Moody algebra assignable with lightlike 3-D causal determinants is associated with the inclusion of partonic 2-surface X² to X⁴ corresponding to C in H. Imbedding space cannot be dynamical anymore since no 16-D number field exists.

3. The representation of space-times as surfaces of H should emerge as well as the space of configuration space spinor fields (not only spinors) defined in the space of 3-surfaces (or equivalently 4-surfaces which are generalizations of Bohr orbits).

4. These surfaces should also have interpretation as hyper-quaternionic sub-manifold of hyper-octonionic 8-space HO (this would dictate the classical dynamics).

This has been the picture before the lacking string of ideas emerged.

4. Number-theoretic localization of infinite-dimensional number theoretic Clifford algebras as a lacking piece of puzzle The lacking piece of the big argument is below.

1. Sequences of inclusions C in H in F allow to interpret infinite-D spinors in Cl(O) as a module having quaternionic spinors Cl(H) as coefficients multiplying quantum spinors with finite quantum dimension not larger than 16: this conforms with the fact that OH spinors indeed are complex 8+8 spinors (quarks, leptons). Configuration space spinors can be seen as quantized imbedding space spinors. Infinite-dimensional Cl(H) spinors in turn can be seen as 4-D quantum spinors having CL(C) spinors as coefficients. Quantum groups emerge naturally and relate to inclusions as does also Kac-Moody algebra.

2. The key idea is to extend infinite-dimensional Clifford algebras to local algebras by allowing power series in hyper-F numbers with coefficients in Cl(F). Using algebraic terminology this means a direct integral of the factors. The resulting objects are generalizations of conformal fields (or quantum fields) defined in the space of hyper-complex numbers (string orbits), hyper-quaternions (space-time surface), hyper-octonions (HO). Their argument is hyper-F number instead of z. Very natural number theoretic generalization of gamma matrix fields (generators of local Clifford algebra!) of super string model is thus in question.

3. Associativity at the space-time level becomes the fundamental physical law. This requires that physical Clifford algebra is associative. For Cl(O) this means that a quaternionic plane in O parametrized by a point of CP₂ is selected at each point hyper-quaternionic point. For the local version of Cl(O) this means that powers of hyper-octonions in powers series are restricted to be hyperquaternions assignable to some hyper-quaternionic sub-manifold of HO (classical dynamics!). But since ordinary inclusion assigns CP₂ point to given point of M₄ represented by a hyper-quaternion one can regard space-time surface also as a surface of H! This means HO-H duality. Parton level emerges from the requirement of commutativity implying that partonic 2-surface correspond to commutative sub-manifolds of HO and thus also of H.

4. Also the super-canonical invariance comes out naturally. The point is that light like hyper-quaternions do not possess inverse so that Laurent series for local Cl(F) elements does not exist at the boundaries lightcones of M⁴ which are thus causal determinants (note the analogy with pole of analytic function). Super-canonical algebra emerges at their boundaries and the intersections of space-time surfaces with the boundaries define a natural gauge fixing for the general coordinate invariance. Configuration space spinor fields are obtained by allowing quantum superpositions of these 3-surfaces (equivalently corresponding 4-surfaces).

Here is the entire quantum TGD believe it or not! I cannot tell whom I admire more: von Neumann or Chopin!

5. Explicit general formula for S-matrix emerges also

This picture leads also to an explicit master formula for S-matrix.

1. The resulting S-matrix is consistent with the generalized duality symmetry implying that S-matrix element can be always expressed using a single diagram having single vertex from which lines identified as space-time surfaces emanate. There is analogy with effective action formalism in the sense that one proceeds in a direction reverse to that in the ordinary perturbative construction of S-matrix: from the vertex to the points defining tips of the boundaries of lightcones assignable to the incoming and outgoing particles appearing in n-point function along the "lines". It remains to be shown that the generalized duality indeed holds true: now its basic implication is used to write the master formula for S-matrix.

2. Configuration space integral over the 3-surfaces appearing as vertex is involved and corresponds to bosonic degrees of freedom in super string models. It is free of divergences since the exponent of Kähler function is a nonlocal functional of 3-surface, since ill-defined metric determinant is cancelled by ill-defined Gauss determinant, and since Ricci tensor for the configuration space vanishes implying the vanishing of further divergences coming from the metric determinant. Hyper-finiteness of type II₁ factors (infinite-dimensional unit matrix has unit trace) is expected to imply the cancellation of the infinities in fermionic sector.

3. Diagrams obtained by gluing of space-time sheets along their ends at the vertex rather than stringy diagrams turn indeed be the Feynman diagrams in TGD framework as previously concluded on basis of physical and algebraic arguments. These singular four-manifolds are not real solutions of field equation but only a construct emerging naturally in the definition of S-matrix based on general coordinate invariance implying that configuration space spinor fields have same value for all Diff⁴ related 3-surfaces along the space-time surface. S-matrix is automatically non-trivial.

The reader interested in details is recommended to look at the chapter Was von Neumann Right After All? of the book "TGD: an Overview".

Matti Pitkänen

Shamanic travels and p-adic physics as physics of cognition and intentionality

Below an email sent this morning. I realized that I could add it to my blog as such (apart from correcting some typos, adding some clarifications, etc.) rather than wasting time to rewriting it and loosing some of the spontaneity of the response.

Dear X and Y,

I read the chapter of the forthcoming book by you and Z. This kind of book has social order. I am myself frustrated of not having seen any clear analysis written using language understandable by a layman and demonstrating the problems of materialistic view and showing that spiritual world view is by no means in conflict with basic tenets of science. The text happened also to resonate with what I have been working just now. I add some subtitles to the comments below to make clear the red thread.

1. World view induced depression

The observation that scientists are people suffering world view induced depression is to the point. As I told Luis, when I was younger my attempt to believe in the world view that I was taught made me literally sick. I have followed discussions in physics blogs and have found that the tone is pathologically negative: crackpot, idiot, imbecille, moron,...: these words are thrown again and again against the opponent. The language used is language of power and violence. I see also this a side effect of this world view induced depression and an attempt to overcome it by aggression. Kind of monoculture of consciousness, stucking to a theory/worldview without ability to detach from it, is in question.

2. Perennial philosophy and the new number concept

I liked very much about the representation of the basic ideas of perennial philosophy. I think that the basic challenge for theories of consciousness is to understand mathematically the division of reality to the sensory world which we can study by doing experiments and the spiritual world which we can approach by various spiritual practices.

Cognition and intentionality (I use "cognitive" somewhat loosely but I do not know any better word!) should have physical and space-time correlates if the notion of physics is properly extended. Even more: we should be able to show that the physics of spiritual world is visible in the physics of the material world. Just like directly invisible quarks are visible via the physics of hadrons.

Here I find strong resonance since TGD in its recent form involves generalization of number concept involving fusion of real numbers with various p-adic number fields, one for each prime p=2,3,5,7,... This fusion is along common rational numbers (very roughly): genuinely p-adic numbers are infinite as real numbers and are analogous to transcendental real numbers representing different manner to complete rational numbers to a continuum.

The point is that one can extend also the notion of spacetime, 8-dimensional space containing space-times as 4-surfaces and speak about p-adic space-time sheets as correlates for intentionality and, there are strong indications, also for cognition.

First point: What is remarkable that non-rational points of these space-time sheets are literally at infinity and only rational points belong to the physical universe. The interpretation is that our thoughts and intentions are literally cosmic or even a super-cosmic phenomenon: cognitive body somehow looks the material universe from outside. This fits very nicely with the idea of cosmic consciousness as consciousness in which sensory input is minimal and cosmic cognitive and intentional component dominates.

Second point: That cognitive space-time sheets have discrete rational projection to real imbedding space and intersect real space-time sheets at discrete set of rational points, conforms with the fact that all physical representations of thoughts are necessary discrete and based on rational numbers. Consider only numerical computation which is bound to satisfy this constraint although cognizing mathematician can perform exact calculations.

p-Adically infinitesimal means infinite in the real sense: that is very short p-adically means very long in real sense. Therefore the continuity and smoothness of local p-adic physics at infinity means that real space-time sheets having discrete set of rational intersection points with p-adic space-time sheets obey p-adic fractality meaning very special kind of long range correlations. Local randomness with long range spatial and temporal correlations can be seen as a direct physical correlate for the existence of cognition and intentionality.

Intentional behavior is indeed characterized by temporal long range correlations. Hence we can measure the immediate implications of something which as such is not measurable! Spiritual and non-local intuitive mind would reflect itself in the properties and behavior of the material world. Without it, the material world would indeed be just a random soup of particles as materialist try hardly to believe.

Even better: these predictions are very spesific. In particular, they lead to successful elementary particle mass calculations and to quantitative understanding of basic spatial and temporal scales in nuclear, atomic and molecular physics, biology, cosmology. This is something completely new.

A possible model for the realization of intentional action is as a quantum jump transforming p-adic space-time sheet to a real one. This is possible if real space-time sheet has vanishing conserved charges such as energy, momentum, electromagnetic charge,... In TGD framework this is possible since conserved inertial energy can be both negative and positive. In principle it would seem that we could really create our physical universe by this kind of intentional action so that Eastern view about reality as a purely mental construct would be correct. I have even proposed S-matrix describing this process and in principle predicting probabilities for different intentional actions.

A question, which just occurred to me, is how reversible the transition from intention to action is: it might be that transitions from action to intention (from matter to thought) is very rare since initial system must have vanishing net quantum numbers, in particular energy and this is extremely difficult to arrange. This could mean that our geometric future is mostly p-adic and past rather stably real: dreams would be the stuff that the reality is made of! If so the flow of experienced time would correspond to the front of the p-adic-to-real phase transition propagating towards geometric future as I have proposed. Of course, infinite number of these kind of wave fronts would be there and the direction of geometric future could be also non-standard.

3. Brahman=Atman and infinite primes

This idea is second element of Perennial Philosophy. Infinite primes, integers, and rationals represent a further extension of number concept besides the fusion of p-adic and real number fields. What is fascinating from the point of physics is that the construction of infinite primes is structurally equivalent with a repeated second quantization of super-symmetric arithmetic quantum field theory. Furthermore, just as 0-dimensional points represents number, 4-D space-time surfaces represent infinite primes, integers,...

This generalization leads also to a generalization of finite numbers: one can construct infinite number of ratios of infinite rationals which are equal to 1 as real numbers but p-adically finite for any prime p. Hence the number 1 and obviously all other numbers and also space-time points have infinitely rich number-theoretical anatomy not detectable by any physical measurement. Single point of space-time can represent in its structure the quantum state of entire material Universe! Brahman=Atman in the most literal and maximal sense that one can imagine!

4. Limits of quantum theory

My view concerning the capacity of standard quantum theory to solve the riddle of consciousness should be already clear. I think that wave mechanics is far too simplistic to allow to understand consciousness. Quantum measurement theory where quantum jump, a good candidate for moment of consciousness as elementary act of creation/re-creation is taken as a fact, is a set of mere phenomenological rules, and in conflict with Schrödinger equation.

My own proposal is simple basically: quantum states are actually entire time evolutions of Schrödinger equation and quantum jumps occur between these (or their generalizations) and thus outside the realm of space-time and given quantum state. Quantum jump means a re-creation of entire time evolution of the cosmology meaning in particular that both geometric past and future are re-created but in accordance with field equations. The experienced time identified as sequence of quantum jumps is something different from geometric time and these two coincide only in certain states of consciousness. Western mode of consciousness is this kind of mode but also in this case long term memories are actually communications with the geometric past: classically as in the case of declarative memories and by quantum entanglement making possible sharing of mental images as in the case of sensory memories.

Second point. Planck constant hbar is the symbol of quantum mechanics and taken usually to be absolute constant which can be put to hbar=1 by a suitable choice of units. Quantum classical correspondence in TGD however predicts that space-time sheets which can be arbitrarily large define quantum coherence regions. This is in conflict with standard quantum mechanics predicting that macroscopic quantum coherence regions should not exist. The resolution of problem is that Planck constant is actually dynamical and quantized: the larger the value of hbar the larger the Compton length so that for instance electron can be zoomed up to an arbitrarily large size and these zoomed up electrons can overlap and form Cooper pairs and superconductor.

The implications are rather dramatic: there is entire hierarchy of values of Planck constant and these correspond to dark matter phases which are macroscopically and even astrophysically quantum coherent. TGD can "predict" the value spectrum of Planck constant and this has led to a surprisingly precise model for living matter including band and resonance structure of EEG.

This gives justification also to the notion of magnetic body (actually onion like hierarchy of them) having astrophysical size in the case of brain. These magnetic bodies carry dark matter and act as intentional agents having biological bodies as sensory receptors and motor instruments. For instance, the time delays of consciousness found by Libet can be understood in this framework.

5. Microtubuli and what shamans do during their travels?

I believe that microtubuli are involved with the realization of long term memories and neural communications: for instance, it is very difficult to understand how high frequency sounds (higher than kHz) could be communicated by nerve pulse patterns since characteristic time scale is about ms. Microtubular conformational and em field patterns are ideal for this purpose. I however think that microtubuli represent only one important level in the hierarchy and that magnetic bodies carrying the dark matter are the star players in the real sector.

At the top of hierarchy wuold be p-adic space-time sheets, p-adic/spiritual bodies representing us as eternal cosmic beings in the real sense. The travels of shamans could result by the ability of shaman's p-adic/spiritual body to partially detach from biological body and direct attention to other parts of the infinite universe.

The direction of attention could mean that shaman as a master of intential action transforms part of his infinite p-adic body at some distant corner of the universe to a real zero energy space-time sheet which can then sensorily perceive the environment. Remote mental interactions would quite generally be based on this mechanism.

Best Regards,

Matti