Planar algebras and generalized Feynman diagrams

There has been an interesting discussion in Kea's blog about planar algebras and related concepts and I decided to add here the posting that I sent also there. You can find information about issues related to planar algebras in Kea's blog. I found also an article about planar algebras in Wikipedia.

What occurred to me is that planar algebras might have interpretation in terms of planar projections of generalized Feynman diagrams (these structures are metrically 2-D by presence of one light-like direction so that 2-D representation is especially natural).

1. Planar algebra very briefly

First a brief definition of planar algebra.

1. One starts from planar k-tangles obtained by putting disks inside a big disk. Inner disks are empty. Big disk contains 2k braid strands starting from its boundary and returning back or ending to the boundaries of small empty disks in the interior containing also even number of incoming lines. It is possible to have also loops. Disk boundaries and braid strands connecting them are different objects. A black-white coloring of the disjoint regions of k-tangle is assumed and there are two possible options. Equivalence of planar tangles under diffeomorphisms is assumed.

2. One can define a product of k-tangles by identifying k-tangle along its outer boundary with some inner disk of another k-tangle. Obviously the product is not unique when the number of inner disks is larger than 1. In the product one deletes the inner disk boundary but if one interprets this disk as a vertex-parton, it would be better to keep the boundary.

3. One assigns to the planar k-tangle a vector space V_k and a linear map from the tensor product of spaces V_ki associated with the inner disks to V_k such that this map is consistent with the decomposition k-tangles. Under certain additional conditions the resulting algebra gives rise to an algebra characterizing multi-step inclusion of HFFs of type II₁.

4. It is possible to bring in additional structure and in TGD framework it seems necessary to assign to each line of tangle an arrow telling whether it corresponds to a strand of a braid associated with positive or negative energy parton. One can also wonder whether disks could be replaced with closed 2-D surfaces characterized by genus if braids are defined on partonic surfaces of genus g. In this case there is no topological distinction between big disk and small disks. One can also ask why not allow the strands to get linked (as suggested by the interpretation as planar projecitons of generalized Feynman diagrams) in which case one would not have a planar tangle anymore.

2. General arguments favoring the assignment of a planar algebra to a generalized Feynman diagram

There are some general arguments in favor of the assignment of planar algebra to generalized Feynman diagrams.

1. Planar diagrams describe sequences of inclusions of HFF:s and assign to them a multi-parameter algebra corresponding indices of inclusions. They describe also Connes tensor powers in the simplest situation corresponding to Jones inclusion sequence. Suppose that also general Connes tensor product has a description in terms of planar diagrams. This might be trivial.

2. Generalized vertices identified geometrically as partonic 2-surfaces indeed contain Connes tensor products. The smallest sub-factor N would play the role of complex numbers meaning that due to a finite measurement resolution one can speak only about N-rays of state space and the situation becomes effectively finite-dimensional but non-commutative.

3. The product of planar diagrams could be seen as a projection of 3-D Feynman diagram to plane or to one of the partonic vertices. It would contain a set of 2-D partonic 2-surfaces. Some of them would correspond vertices and the rest to partonic 2-surfaces at future and past directed light-cones corresponding to the incoming and outgoing particles.

4. The question is how to distinguish between vertex-partons and incoming and outgoing partons. If one does not delete the disk boundary of inner disk in the product, the fact that lines arrive at it from both sides could distinguish it as a vertex-parton whereas outgoing partons would correspond to empty disks. The direction of the arrows associated with the lines of planar diagram would allow to distinguish between positive and negative energy partons (note however line returning back).

5. One could worry about preferred role of the big disk identifiable as incoming or outgoing parton but this role is only apparent since by compactifying to say S² the big disk exterior becomes an interior of a small disk.

3. A more detailed view

The basic fact about planar algebras is that in the product of planar diagrams one glues two disks with identical boundary data together. One should understand the counterpart of this in more detail.

1. The boundaries of disks would correspond to 1-D closed space-like stringy curves at partonic 2-surfaces along which fermionic anti-commutators vanish.

2. The lines connecting the boundaries of disks to each other would correspond to the strands of number theoretic braids and thus to braidy time evolutions. The intersection points of lines with disk boundaries would correspond to the intersection points of strands of number theoretic braids meeting at the generalized vertex.

   [Number theoretic braid belongs to an algebraic intersection of a real parton 3-surface and its p-adic counterpart obeying same algebraic equations: of course, in time direction algebraicity allows only a sequence of snapshots about braid evolution].

3. Planar diagrams contain lines, which begin and return to the same disk boundary. Also "vacuum bubbles" are possible. Braid strands would disappear or appear in pairwise manner since they correspond to zeros of a polynomial and can transform from complex to real and vice versa under rather stringent algebraic conditions.

4. Planar diagrams contain also lines connecting any pair of disk boundaries. Stringy decay of partonic 2-surfaces with some strands of braid taken by the first and some strands by the second parton might bring in the lines connecting boundaries of any given pair of disks (if really possible!).

5. There is also something to worry about. The number of lines associated with disks is even in the case of k-tangles. In TGD framework incoming and outgoing tangles could have odd number of strands whereas partonic vertices would contain even number of k-tangles from fermion number conservation. One can wonder whether the replacement of boson lines with fermion lines could imply naturally the notion of half-k-tangle or whether one could assign half-k-tangles to the spinors of the configuration space ("world of classical worlds") whereas corresponding Clifford algebra defining HFF of type II₁ would correspond to k-tangles.

For the recent TGD view about generalized Feynman graphics see the chapter Hyperfinite Factors and Construction of S-matrix of "Towards S-matrix".

TGD Universe from the condition that all possible statistics are possible

By simple physical arguments H=M⁴×CP₂ is the unique choice for the imbedding space in TGD Universe. This choice follows also from the number theoretic vision. M⁴ has interpretation as hyper-quaternions and CP₂ as space of space of quaternionic planes at a point of (hyper)-octonion space. Space-time surfaces can be seen as hyper-quaternionic sub-manifolds of hyper-octonionic space M⁸ and correspond to sub-manifolds of H. The basic conjecture is that hyper-quaternionicity corresponds to the property of being a certain preferred extremal of so called Kähler action with corresponding Kähler action defining Kähler function for the "world of classical worlds". The corresponding isometry group SO(3,1)× SU(3) and vielbein group SL(2,C)× U(2)_ew have therefore a distinguished position both in physics and quantum TGD.

A new observation induced by a little discussion with Kea is that also the general pattern for inclusions of hyperfinite factors of type II₁ selects these groups as something very special: the condition that all possible statistics are realized is guaranteed by the choice M⁴× CP₂.

1. Inclusions are partially characterized by quantum phase q=exp(i2π/n). n>2 for the quantum counterparts of the fundamental representation of SU(2) means that braid statistics for Jones inclusions cannot give the usual fermionic statistics. That Fermi statistics cannot "emerge" conforms with the role of infinite-D Clifford algebra as a canonical representation of HFF of type II₁. SO(3,1) as isometries of H gives Z₂ statistics via the action on spinors of M⁴ and U(2) holonomies for CP₂ realize Z₂ statistics in CP₂ degrees of freedom.

2. n>3 for more general inclusions [I learned this from the thesis "Hecke algebras of type A_n and subfactors" of Hans Wenzl (Inv. Math. 92, 349-383 (1988)] in turn excludes Z₃ statistics as braid statistics in the general case. SU(3) as isometries induces a non-trivial Z₃ action on quark spinors but trivial action at the imbedding space level so that Z₃ statistics becomes possible in quark sector quite generally.

For more details about general inclusions see the appendix of the chapter Was von Neumann Right After All? of "Towards S-matrix".

Inverting reductionism upside down

In the recent New Scientists there is an article about string nets. For a more technical description the article Photons and electrons as emergent phenomena of Michael Levin and Xiao-Gang Wen is recommended. I found myself resonating with several ideas appearing in this work but also enjoy of disagreeing.

1. Inverting reductionism upside down

1. What makes me happy is that the work challenges the basic dogmas of reductionism and locality which have led to the recent blind alley in the official theoretical physics. One might say that the standard reductionistic view is turned upside down: local structures such as gauge bosons and fermions emerge from non-local ones. Condensed matter physics is taken as starting point in attempt to build a model for fundamental physics. It is really enjoyable to see that genuine thinking is still taking occasionally place in theoretical physics.

2. The basic inspiration comes from topological quantum computation. The crucial element of quantum computation is quantum entanglement which in the case of topological quantum computation is robust against perturbations. This implies now non-locality at fundamental level. Universal (topological) quantum computers are able to mimic any dynamics at the level of discrete approximation. The point of view taken by the authors is that that there is no fundamental level involving elementary gauge bosons and fermions. Particle spectrum and dynamics of gauge theories could result as this kind of mimicry. Hence the breakdown of reductionism in the sense that everything emerges from the dance of quarks and leptons or some even smaller structures.

2. String nets as fundamental structures

Consider now a more concrete view about the model.

1. One might think that braid like structures would have been introduced as fundamental objects. This was not the case, and probably because they are not enough for the desired inverse reduction. Instead, strings are taken as the basic objects. Strings can form string nets by fusing together along their ends and string network becomes the fundamental notion giving rise to elementary particles as its excitations. There are different type of strings and only those fusion vertices are favored for which the interaction energy is small. Besides this Hamiltonian contains kinetic energy and string tension term. If string tension is very high one obtains large number of small string nets. If it is small, large string nets emerge.

2. These models predict excitations having interpretation in terms of gauge bosons and fermions. From string model point of view this is perhaps not surprising. For instance, transversal excitations of string would naturally give rise to bosonic excitations and ends of string would behave like fermions.

3. Some criticism

I do not try to pretend of being objective and I confess that my criticism reflects strongly my TGD based belief system also profoundly influenced by the beautiful ideas of topological quantum computation.

1. In order to pass as a unified theory the model should be able to provide fundamental dynamics based on some simple principle. The string net Hamiltonian can be however tailored to reproduce any gauge theory. This smells first like a reincarnation of string landscape. At the second thought there is nothing wrong with this kind of flexibility but I would take it as a reflection of the fact that universal topological quantum computer is able to mimic any system when discretization is allowed. To me the correct question would be "What is the fundamental dynamics allowing topological quantum computation as a fundamental process?".

   Perhaps it is un-necessary to tell the reader that the belief that standard model based physics really allows universal topological quantum computation is only a belief. Personally I regard this belief as wrong. The point is that much more than a mere construction of a discrete Hamiltonian characterizing the quantum computation is needed. Fundamental physics must allow the entire process culminating to a conscious experience about the result of the computation. The basic challenge is the identification of the principles of the fundamental dynamics allowing any topological computation. One cannot circumvent the fact before there can be a topological computer there must also be intention to build it and theory must describe also this.

2. Could string nets (or something more general) then be fundamental non-local structures? My answer based on my personal belief system is a qubit somewhere between yes and no.

   Most of the qubit consists of "No". I do not believe that strings of any kind are really the fundamental objects. In TGD framework strings are replaced by 3-dimensional light-like 3-surfaces as basic objects and they define generalized Feynman diagrams. The interpretation is as orbits of 2-dimensional partons (in honor of Feynman's deep intuition) which can have arbitrarily large size. Classical space-time dynamics emerges in a well-defined sense as a classical correlate for the quantum physics of these light-like 3-surfaces. Quantum physics is not anymore local at space-time level since 3-surface behaves as a single coherent whole (very important for the proper understanding of living systems). Interestingly, quantum physics however remains local and formally completely classical (apart from quantum jump) at the level of the "world of classical worlds".

   Qubit contains also some "Yes".

   • The dynamics of TGD involves definitely stringy aspects, in particular generalized super-conformal symmetries and the stringy character of fermionic anti-commutation relations.

   • Also TGD Universe predicts the existence of string like objects and of fractal string networks consisting of magnetic flux quanta. This network plays a key role in cosmology: for instance, dark energy corresponds to magnetic energy. In TGD inspired nuclear physics nuclei are identified as highly tangled nuclear strings.

3. I would have been happy to see braids, which served as starting point, as fundamental objects because they represent really deep mathematics.

4. Braid statistics is an essentially 2-dimensional phenomenon. Hence I would have expected that 2-dimensional surfaces would have been introduced explicitly as fundamental objects.

5. In the model of Levin and Wen one could see bosons (strings) and fermions (string ends) as dual manners to describe dynamics and this kind of eliminative reductionism makes me very skeptic. In TGD framework the counterpart of string is interior dynamics of space-time and corresponds to classical physics as exact part of quantum theory necessary for quantum measurement theory whereas bosons and fermions as counterparts of string ends correspond to quantum states of light-like partonic 3-surfaces.

6. Effective 2-dimensionality is absolutely essential for the braid statistics and is of course only an assumption subject to criticism. Standard model skeptic might argue that standard model quantum physics does not allow to achieve effective 2-dimensionality in such a good an approximation as to guarantee the effective topologization of the dynamics. Standard model skeptic might be right. It is of course experimental fact that anyons are there but their existence might only demonstrate that the belief system of standard model skeptic is in need of updating.

7. The idea of giving up fermions as fundamental dynamical objects looks to me questionable. For instance, for Jones inclusions for which braid statistics emerges naturally only q=exp(iπ/n), n≥3 is allowed as quantum phase and one does allow fermionic braid statistics. The second reason for my skepticism is more personal. In TGD framework the Clifford algebra of the world of classical worlds has interpretation in terms of fermionic oscillator operators so that a beautiful geometrization of fermion number results. This algebra is identifiable as fundamental hyper-finite factor of type II₁ responsible for the beauty of TGD and also lurking behind the magic of braids. I am ready to describe bosons as fermion-antifermion bound states but refuse to continue further.

4. What if one requires that Universe is topological quantum computer?

Topological quantum computation has strongly inspired also the development of quantum TGD. This inspires me to pose the key question differently. Suppose that Universe is topological quantum computer. What can one conclude about the fundamental dynamics? You can of course guess the outcome and for me this is still one exercise to deduce TGD Universe from some simple basic assumptions in the hope (I am really incurable optimist!) that the message could finally permeate through the magnificently effective cognitive immune (or its it insulation-?) system of main stream theoretical physicist.

1. The first requirement is that, as far as fermionic quantum dynamics is considered, the fundamental objects are effectively 2-dimensional and carry braids as fundamental objects. This makes braid statistics a genuine statistics and allow topological quantum computation at the fundamental level. In particular, topological braid dynamics would be genuinely topological and not only approximately so.

   Conformal symmetries are natural in 2-dimensional context and lead naturally to braid statistics. Requiring the generalization of super-conformal invariance as a fundamental symmetry leads to the identification of fundamental dynamical objects as light-like 3-surfaces identifiable as orbits of partonic 2-surfaces containing braids. Chern-Simons type action emerges as the only possible action principle and gives rise to almost (light-likeness!) topological QFT.

   A generalization of topological quantum computation emerges since parton replication accompanied by braid replication is involved and has interpretation in terms of copying of information. Particle exchanges in generalized Feynman diagrams have interpretation in terms of communication whereas incoming and outgoing lines could be interpreted as involving topological quantum computations.

2. The condition that braids emerge as fundamental structures is very strong and it is very difficult to imagine how they could emerge naturally in the standard mathematical framework of physics. In the physics based on the fusion of real physics and various p-adic physics interpreted in terms of cognition and intentionality, the fundamental braids emerge naturally as subsets of the rational (more generally algebraic) intersection of real parton and its p-adic counterpart obeying same algebraic equations. It seems that we have got a lot of TGD already. Bringing in the condition that standard model quantum numbers appear at fundamental level or accepting the vision about physics as a generalized number theory gives the rest of TGD (or all of it).

3. The quantum dynamics of TQC Universe need not rely on standard quantum theory. Indeed, new quantum physics based on hyper-finite factors of type II₁ with brand new quantum measurement theory with measurement resolution as a basic concept implying automatically non-commutative physics is involved. This physics emerges from both the Clifford algebra of the world of classical worlds and from the braid models of topological quantum computation. TGD suggests also strongly the quantization of Planck constant so that quantum entanglement in arbitrarily long length scales becomes possible and one can understand dark matter as macroscopic quantum phases responsible also for the very special properties of living matter.

Still about Higgs candidate

I have already discussed the slight indications for Higgs at 160 GeV in two postings (see this and this). In the original posting I mentioned briefly the possibility that the exotic pion of the 512-fold scaled up fractal copy of ordinary hadron physics might be responsible for the excess but by a stupid calculational error the mass estimate of the exotic pion came almost twice too large whereas correct estimate is 152 GeV not too far from 160 GeV. I attach below a short piece of the text from the first posting.

M₈₉ hadron physics might be required in TGD framework by the requirement of perturbative unitarity. Thus the mesons of M₈₉ hadron physics might be involved. By a very naive scaling by factor 2^(107-89)/2=2⁹ the mass of the pion of M₈₉ physics would be about 70 GeV. This estimate is not reliable since color spin-spin splittings distinguishing between pion and ρ mass do not scale naively. For M₈₉ mesons this splitting should be very small since color magnetic moments are very small (for calculation of color-magnetic splittings see this). The mass of ordinary pion in absence of splitting would be around 297 MeV and 512-fold scaling gives M(π₈₉)= 152 GeV which is not too far from 160 GeV. Could the decays of this exotic pion give rise to the excess of fermion pairs? (Note that the mass was erratically estimated to be 250 GeV in the original posting). This interpretation might also allow to understand why b-pair and t-pair excesses are not consistent. Monochromatic photon pairs with photon energy around 76 GeV would be the probably easily testable experimental signature of this option.

If this interpretation is correct, a whole new hadron physics might have been already seen in laboratory! Few of us can irritate their colleagues to the verge of madness by boasting with a discovery of a brand new physics and remaining still unemployed;-)! Be as it may, LHC will tell the truth.

For more details see the chapter p-Adic Particle Massivation: Elementary Particle Masses of "p-Adic length Scale Hypothesis and Dark Matter Hierarchy".

About the construction of vertices

The understanding about the construction of S-matrix has increased considerably and I can now attack seriously the challenge of writing down the generalized Feynman rules.

1. Generalized Feynman diagrams

Let us first summarize the general picture.

1. Feynman diagrams are replaced with their higher dimensional variants with lines replaced with lightlike 3-surfaces identifiable as partonic orbits and with vertices replaced with partonic 2-surfaces along which lines meet. Lightlike 3-surfaces corresponding of maxima of Kähler function define generalized Feynman diagrams. There is no summation over the diagrams and each reaction corresponds to single minimal diagram. Quantum dynamics is 2-dimensional in the sense that vertices are defined by partonic 2-surfaces and 3-dimensional in the sense that different maxima of Kähler function defining points of spin glass energy landscape give rise to additional degeneracy essentially due to the presence of light-like direction.

2. S-matrix reduces to a unitary S-matrix depending parametrically on points of M⁴ defining arguments of N-point function in QFT approach. The momentum representation of S-matrix is obtained by taking a Fourier transform of this and is also unitary.

3. S-matrix is a generalization of braiding S-matrix in the sense that one assigns to the incoming/outgoing and internal lines a unitary braiding matrix. To the vertices, where braids replicate, one assigns a unitary isomorphism between tensor product of hyper-finite II₁ factors (HFFs) associated with incoming resp. outgoing lines. A crucial element in the construction is that these tensor products are themselves HFFs of type II₁.

4. Since also bosons are fermion-antifermion states located at partonic 2-surfaces, the construction of vertices reduces basically to that in the fermionic Fock space associated with the vertex and the space of small deformations of the generalized Feynman diagram around the maximum of Kähler function. The discrete set of points defining number theoretic strand define the basic unitary S-matrix and these points carry various quantum numbers. The natural assumption is that one can use at the vertex same fermionic basis for all incoming and outgoing lines and that unitary braiding S-matrix associated with lines induces a unitary transformation of basis. Its presence in internal lines gives rise to propagators as one integrates over the positions for tips of future and past lightcones containing at their light-like boundaries incoming and outgoing partons.

One can proceed by making simple guesses about the unitary isomorphism associated with the vertex.

1. The simplest guess would be that vertices involve only simple Fock space inner product. This would be like old fashioned quark model in which the quarks of incoming hadrons are re-arranged to from outgoing hadrons without pair creation or gluon emission. This trial does not work since it would not allow bosons which can be regarded as fermion-antifermion pair with either of them having non-physical helicity. This observation however serves as a valuable guideline.

2. An alternative guess is based on the observation that partonic 2-surface with punctures defined by number theoretical braids is analogous to closed bosonic string emitting particles. This would suggest that unitary S-matrix could be assigned with some conformal field theory or possibly string model. At least for non-specialist in conformal field theories this approach looks too abstract.

2. Vertices from free field theory defined by the modified Dirac operator

Something more concrete is required and to proceed one can try to apply the mathematical constraints from the basic definition of TGD.

1. The vertices should come out naturally from the modified Dirac action which contains the classical coupling of the gauge potentials (induced spinor connection) to fermions. Hence the modified Dirac action defining the analog of free field theory should appear as a basic building block in the definition of the inner product. Perturbation theory with respect to the induced gauge potential would conform with standard QFT but does not make sense. There is simply no decomposition of the modified Dirac operator D to "free" part and interaction term.

2. The vacuum expectation for the exponent of the modified Dirac action gives vacuum functional identified as exponent of Kähler function. When one sandwiches the exponent of Dirac action between many-fermion states, one obtains an inner product analogous to that in free field theory Feynman rules. How however the states are are not annihilated by D but are its generalized eigenstates with eigenvalues λ depending on p-adic prime by an overall scaling factor log(p) responsible for the coupling constant evolution. The generalized eigenvalue equation reads as DΨ= λ t^kΓ_kΨ, where t^k is lightlike vector t^k defining the tangent vector of partonic 3-surface or its M⁴ dual fixed once rest system and quantization axis of angular momentum has been fixed (it is not yet quite clear which option is correct). The notion of generalized eigenmode allows also to define Dirac determinant without giving up the separate conservation of H-chiralities (B and L). The generalized eigenstates are analogs of solutions of massless wave equation in the sense that the square of D annihilates them. Between states created by a monomial of fermionic oscillator operators the inner product reduces to a product of propagators.

3. A strict correspondence with free field theory would require that the incoming and outgoing states correspond to zero modes with λ=0 whereas internal lines as off mass shell states would correspond to non-vanishing eigenvalues λ . This assumption is however un-necessary since the four-momentum dependence comes only through the Fourier transform and one can regard all generalized eigenmodes as counterparts of massless modes. The restriction might be also inconsistent with unitarity.

4. For generalized eigenstates of D the modified Dirac propagator 1/D reduces to o^kΓ_k/λ. o^k is the light-like M⁴ dual of the lightlike vector t^k and λ is the generalized eigenvalue of D proportional to log(p). The propagator can be non-vanishing between vacuum and a boson consisting of fermion with physical helicity and antifermion with non-physical helicity so that non-trivial boson emission vertices are possible. At first it would seem that the inverse of the generalized eigenvalue λ contributes to the p-adic coupling constant evolution an overall 1/log(p) proportionality factor. However, since the inner product of un-normalized "bare" boson states (just fermion pair) is proportional to 1/log(p), the normalization of bosonic states cancels this factor so that algebraic number results. Thus fermionic contributions to the vertices are extremely simple since only the matrix o^kΓ_k remains. The conclusion made already earlier is that the p-adic coupling constant evolution must be due to the time evolution along parton lines dictated by the modified Dirac operator.

5. The fermionic contribution to the vertex says nothings about gauge couplings. All gauge coupling strengths must be proportional to the RG invariant Kähler coupling strength α_K, which can emerge only from the functional integral over small fluctuations around maximum of Kähler function K when the operator inverse of the covariant configuration space Kähler metric defining propagator is contracted between bosonic vector fields generating Kac-Moody and super-canonical symmetries in terms of which the deformation of the partonic 3-surface can be expressed. Obviously the configuration space spinor fields representing bosonic states must vanish at the maximum of K: otherwise coupling strength is of order unity. Geometrically this means that the maxima of Kähler function correspond to fixed points of these isometries.

3. Number theoretical constraints

The condition that S-matrix elements are algebraic numbers is an additional powerful guideline.

1. The most straightforward manner to guarantee that S-matrix elements are algebraic numbers is that vertex factors and propagators are separately algebraic numbers. log(p)-factors are obviously problematic number theoretically but normalization of the Fock space inner products cancels these factors. Thus coupling constant evolution can come only from the unitary time evolution with respect to the light-like coordinate of propagator lines dictated by the modified Dirac operator. Fermionic oscillator operators suffer a non-trivial unitary transformation depending on the p-adic prime p since (expressing it schematically) e^iHt is replaced by p^iHt.

2. The fundamental number theoretic conjecture is that the numbers p^s_n, where s_n=1/2+iy_n correspond to non-trivial zeros of Riemann zeta (or of more general zetas possibly involved), are algebraic numbers. If this is the case, then also the products and sums with rational coefficients involving finite number of nontrivial zeros of zeta are algebraic numbers and define a commutative algebra. The effect of the unitary time evolution operator should be expressible as an element of this algebra. Also larger algebraic extensions can be considered.

3. A simplified picture is provided by the dynamics of free number theoretic Hamiltonian for which eigenstates are labelled by primes and energy eigenvalues are given by E_p= log(p). Time evolution gives rise to phase factors exp(iE_pt)=p^it which are algebraic numbers in given extension of rationals for some quantized values of light-like coordinate t. If the conjectures about zeros of zeta hold true this is achieved if t is a linear combination of imaginary parts of zeros of zeta with integer coefficients: t= ∑_n k(n) y_n.

For more details see the chapter Hyper-Finite Factors and Construction of S-Matrix of the book "Towards S-matrix".

P.S. The following saying of the week from Tommaso Dorigo's blog somehow resonates with my inner feelings.

I make a living as a lawyer, and I spend a lot of time in the world of physics. I have encountered a lot of sleazy scumbags in the world of lawyers, but none of them are as bad as the bad guys in physics, and, although the bad guys in law do a lot of damage, I really think that the bad guys in physics do more damage to human civilization.

Non-commutative geometry: warmly recommended

I love to visit the good old This Week's Finds . Down-to-earth explanations of mathematical concepts and ideas, good humor, and no putting down mentality. What more one could hope for.

This Week's Finds is not anymore the only one of its kind. A new highly interesting blog has appeared in the blogsphere at the boundary layer between mathematics and physics: Noncommutative Geometry. I warmly recommend it to myself and theoretical physicists who might still dare to dream about the day when M theory is not anymore the only possible theory of even more than everything and that the proponents of competing theories are not anymore regarded as mindless crackpots and human waste.

Number theory as generalized physics

Amusingly, and rather satisfyingly, the basic theme during last days has been number theory as generalized quantum physics whereas my own basic theme (at much lower level of sophistication) has been physics as a generalized number theory.

One of very interesting articles briefly discussed in the blog is Quantum Statistical Mechanics and Class Field Theory.

1. A physical model allowing to construct the generators of the maximal Abelian extension of rationals and identify the action of corresponding Galois group (call it Gal) on these generators, is proposed. Someone might build an association with Hilbert's twelth problem.
2. The idea is to assign a C^* algebra to so called Hecke algebra appearing in Langlands program. The quantal time evolution and thermodynamics of this simple number theoretical quantum system is essentially unique since it corresponds to a state in a hyperfinite factor of type III.
3. What is amazing is that the numerical values of states of this system at the infinite temperature limit code for the algebraic numbers generating the maximal Abelian extension of rationals. This kind of condition is fundamental in TGD program of physics as generalized number theory having universality with respect to number field. Furthermore, Gal acts as symmetries of the system.
4. Partition function is nothing but Riemann Zeta as a function of temperature. Rieman Zeta and more general zetas have become an essential element of quantum TGD although I am very far from rigorous formulations yet.
5. There are two phases: low temperature phase corresponding to 1/T> 1 and high temperature phase corresponding to critical strip 1/T≤ 1. At T=1 partition function diverges: a signature of phase transition. At low temperature phase the action of Gal is non-trivial and thermal expectations of observables transforming non-trivially under Gal are non-vanishing. At high temperature phase the action of Gal is trivial meaning that thermal expectations of observables transforming non-trivially under Gal vanish. There is complete analogy with non-trivial action of gauge group on Higgs vacuum expectation value below critical temperature.

6. In TGD framework somewhat different breaking of gauge symmetry occurs for the Galois group of closure or rationals identified as infinite symmetric group whose group algebra is..., you guessed correctly: hyper-finite factor of type II₁! The finite Galois groups to which symmetry breaking occurs correspond are analogous to unbroken subgroups of gauge groups and directly related to Jones inclusions.

The article inspires interesting questions which only an innocent novice blessed with deep ignorance about basics can articulate.

1. At the critical line at which partition function is real. Could one consider quantum field theory at a finite temperature 1/T= Re(s)=1/2 and finite time interval t=Im[s]?

2. The partition function vanishes for the values of t corresponding to non-trivial zeros of zeta. What could this mean physically? Could it be that along critical line symmetry is broken in step wise manner. At first zero the thermal expectations for some non-singlets under some subgroup H of Gal become non-vanishing. At the next zero H is extended. In accordance with some earlier TGD inspired speculations this process would gradually lead to the entire Abelian Galois group and emergence of all phases generating maximal abelian extension of rationals. This suggests also a possible connection with Jones inclusions coded by phases exp(iπ/n) and cyclic group Z_n.

3. The reckless number theoretic speculations inspired by TGD inspire further questions. Could it be that for zeros the numbers p^iy appearing in the product decomposition of zeta are algebraic numbers in maximal Abelian extension? Could also zeros be algebraic numbers as the most stringent speculations suggest?

Physics as generalized number theory

I find this article highly interesting for many reasons.

1. In TGD framework finite subgroups of Galois group of Galois group of algebraic closure of rationals play a fundamental role as group of symmetries of number theoretic braids (see earlier postings).

2. Number theoretic universality requires that everything reduces to algebraic numbers belonging to the algebraic extension of rationals or p-adics corresponding to the level of predicted number theoretic hierarchy. For instance, the maxima of Kähler function coding for the geometry of the world of classical worlds would be algebraic number (also Neper number e and its roots could be allowed in p-adic context if extensions of p-adics are required to be finite-dimensional). S-matrix elements would be algebraic numbers and so on. The model discussed realizes this dream for thermodynamical states at infinite temperature limit.

3. The notion of number theoretic braid defined as a subset of the intersection of real parton and its p-adic counterpart would consist of algebraic points so that the construction of S-matrix would reduce to discrete number theory.

4. It seems that number theoretical QFT at finite temperature based on hyperfinite factors of type III could be associated with the strands of number theoretic braids representing incoming partons propagating in generalized Feynman diagrams generalizing the braid diagrams. This would bring in p-adic thermodynamics at the fundamental level. Coupling constant evolution would be at the level of free states and no summation over loops would be needed.

Some ideas relating to physics, number theory, and biology

Before closing I want to repeat myself by recalling some associated ideas suggesting deep connections between number theory, quantum computation, and biology.

1. Vertices are direct generalizations of those for Feynman diagrams with point-like vertex being replaced with 2-dimensional partonic surface. They are totally different from string diagrams.

2. At the vertices number theoretic braids replicate and also DNA replication might involve this process as a deeper level process. Isomorphisms between tensor products of HFFs of type II₁ associated with incoming resp. outgoing lines would define vertices. Everything would be unitary and definitely non-trivial.

3. I ended up with braids from TGD inspired model for topological quantum computation using braids. Universe could be topological quantum computer at all levels of fractal hierarhcy. Copying of information represented by number theoretic braids would occur at vertices, its communication would takes place at propagator lines, and quantum computation would be performed at incoming lines.

To conclude, I think that Noncommutative Geometry might provide an excellent opportunity for a theoretical physicists to enjoy simple explanations of ideas which are difficult to extract from formal mathematical papers.

More about Higgs candidate

As I told in previous positng, there have been cautious claims (see New Scientist article, the postings in the blog of Tommaso Dorigo, and the postings of John Conway in Cosmic Variance) about the possible detection of first Higgs events.

According to simple argument of John Conway based on branching ratios of Z⁰ and standard model Higgs to τ-τbar and b-bbar, Z⁰→ τ-τbar excess predicts that the ratio of Higgs events to Z⁰ events for Z⁰→ b-bbar is related by a scaling factor

[B(H→ b-bbar)/B(H→ τ-τbar)]:[B(Z⁰→ b-bbar)/B(Z⁰→ τ-τbar)] ≈ 10/5.6=1.8

to that in Z⁰→ τ-τbar case. The prediction seems to be too high which raises doubts against the identification of the excecss in terms of Higgs.

In a shamelessly optimistic mood and forgetting that mere statistical fluctuations might be in question, one might ask whether the inconsistency of τ-τbar and b-bbar excesses could be understood in TGD framework.

1. The couplings of Higgs to fermions need not scale as mass in TGD framework. Rather, the simplest guess is that the Yukawa couplings scale like p-adic mass scale m(k)=1/L(k), where L(k) is the p-adic length scale of fermion. Fermionic masses can be written as m(F)= x(F)/L(k), where the numerical factor x(F)>1 depends on electro-weak quantum numbers and is different for quarks and leptons. If the leading contribution to the fermion mass comes from p-adic thermodynamics, Yukawa couplings in TGD framework can be written as h(F)= ε(F) m(F)/x(F), ε<< 1. The parameter ε should be same for all quarks resp. leptons but need not be same for leptons and quarks so that that one can write ε (quark)= ε_Q and ε (lepton)= ε_L. This is obviously an important feature distinguishing between Higgs decays in TGD and standard model.

2. The dominating contribution to the mass highest generation fermion which in absence of topological mixing correspond to genus g=2 partonic 2-surface comes from the modular degrees of freedom and is same for quarks and leptons and does not depend on electro-weak quantum numbers at all (p-adic length scale is what matters only). Topological mixing inducing CKM mixing affects x(F) and tends to reduce x(τ), x(b), and x(t).

3. In TGD framework the details of the dynamics leading to the final states involving Z⁰ bosons and Higgs bosons are different since one expects that it fermion-Higgs vertices suppressed to the degree that weak-boson-Higgs vertices could dominate in the production of Higgs. Since these details should not be relevant for the experimental determination of Z⁰→ τ-τbar and Z⁰→ b-bbar distributions, then the above argument can be modified in a straightforward manner by looking how the branching ratio R(b-bbar)/R(τ-τbar) is affected by the modification of Yukawa couplings for b and τ. What happens is following:

   B(H→ b-bbar)/B(H→ τ-τbar)= m_b²/m_τ² → B(H→ b-bbar)/B(H→ τ-τbar)×X ,

   X=(ε²(q)/ε² (L))× (x_τ²/x_b²).

   Generalizing the simple argument of Conway one therefore has

   (H/Z)⁰(b-bbar)= 1.8 × (ε²_Q/ε²_L )×(x_τ²/x_b²)× (H/Z)⁰(τ-τbar).

   Since the topological mixing of both charged leptons and quarks of genus 2 with lower genera is predicted to be very small (see this) , x_τ/x_b≈ 1 is expected to hold true. Hence the situation is not improved unless one has ε_Q/ε_L<1 meaning that the coupling of Higgs to the p-adic mass scale would be weaker for quarks than for leptons.

Can one then guess then value of r and perhaps even Yukawa coupling from general arguments?

1. The actual value of r should relate to electro-weak physics at very fundamental level. The ratio r=1/3 of Kähler couplings of quarks and leptons is certainly this kind of number. This would reduce the prediction for (H/Z⁰)(b-bbar) by a factor of 1/9. To my best understanding, this improves the situation considerably (see for yourself).

2. Kähler charge Q_K equals electro-weak U(1) charge Q_U(1). Furthermore, Kähler coupling strength which is RG invariant equals to U(1) coupling strength at the p-adic length scale of electron but not generally (see this). This observation encourages the guess that, apart from a numerical factor of order unity, ε² itself is given by either α_KQ_K² and thus RG invariant or by α_U(1)Q_U(1)². The contribution of Higgs vacuum expectation to fermionic mass would be roughly a fraction 10^-2-10^-3 about fermion mass in consistency with p-adic mass calculations.

Of course, it might turn out that fake Higgs is in question. What is however important is that the deviation of the Yukawa coupling allowed by TGD for Higgs from those predicted by standard model could manifest itself in the ratio of Z_0→ b-bbar and Z⁰→ τ-τbar excesses.

For more details see the chapter p-Adic Particle Massivation: Elementary Particle Masses of "p-Adic length Scale Hypothesis and Dark Matter Hierarchy".

Indications for Higgs with mass of 160 GeV

There have been cautious claims (see New Scientist article, the postings in the blog of Tommaso Dorigo, and the postings of John Conway in Cosmic Variance) about the possible detection of first Higgs events.

This inspires more precise considerations of the experimental signatures of TGD counterpart of Higgs. This kind of theorizing is of course speculative and remains on general qualitative level only since no calculational formalism exists and one must assume that gauge field theory provides an approximate description of the situation.

Has Higgs been detected?

The indications for Higgs comes from two sources. In both cases Higgs would have been produced as gluons decay to two b-bbar pairs and virtual b-bbar pair fuses to Higgs, which then decays either to tau-lepton pair or b-quark pair.

John Conway, the leader of CDF team analyzing data from Tevatron, has reported about a slight indication for Higgs with mass m_H=160 GeV as a small excess of events in the large bump produced by the decays of Z⁰ bosons with mass of m_Z≈ 94 GeV to tau-taubar pairs in the blog Cosmic Variance. These events have 2σ significance level meaning that the probability that they are statistical fluctuations is about 2 per cent.

The interpretation suggested by Conway is as Higgs of minimal super-symmetric extension of standard model (MSSM). In MSSM there are two complex Higgs doublets and this predicts three neutral Higgs particles denoted by h, H, and A. If A is light then the rate for the production of Higgs bosons is proportional to the parameter tan(β) define as the ratio of vacuum expectation values of the two doublets. The rate for Higgs production is by a factor tan(β)² higher than in standard model and this has been taken as a justification for the identification as MSSM Higgs (the proposed value is tan(β)≈ 50). If the identification is correct, about recorded 100 Higgs candidates should already exist so that this interpretation can be checked.

Also Tommaso Dorigo, the blogging member of second team analyzing CDF results, has reported at his blog site a slight evidence for an excess of b-bbar pairs in Z⁰→ b-bbar decays at the same mass m_H=160 GeV. The confidence level is around 2 sigma. The excess could result from the decays of Higgs to b-bbar pair associated with b-bbar production.

What forces to take these reports with some seriousness is that the value of m_H is same in both cases. John Conway has however noticed that if both signals correspond to Higgs then it is possible to deduce estimate for the number of excess events in Z⁰→ b-bbar peak from the excess in tau-taubar peak. The predicted excess is considerably larger than the real excess. Therefore a statistical fluke could be in question, or staying in an optimistic mood, there is some new particle there but it is not Higgs.

m_H=160 GeV is not consistent with the standard model estimate by D0 collaboration for the mass of standard model Higgs boson mass based on high precision measurement of electro-weak parameters sin(θ_W), α, α_s , m_t and m_Z depending on log(m_H) via the radiative corrections. The best fit is in the range 96-117 GeV. The upper bound from the same analysis for Higgs mass is 251 GeV with 95 per cent confidence level. The estimate m_t=178.0+/- 4.3 GeV for the mass of top quark is used. The range for the best estimate is not consistent with the lower bound of 114 GeV on m_H coming from the consistency conditions on the renormalization group evolution of the effective potential V(H) for Higgs (see the illustration here). Here one must of course remember that the estimates vary considerably.

TGD picture about Higgs briefly

Since TGD cannot yet be coded to precise Feynman rules, the comparison of TGD to standard model is not possible without some additional assumptions. It is assumed that p-adic coupling constant evolution reduces in a reasonable approximation to the coupling constant evolution predicted by a gauge theory so that one can apply at qualitative level the basic wisdom about the effects of various couplings of Higgs to the coupling constant evolution of the self coupling λ of Higgs giving upper and lower bounds for the Higgs mass. This makes also possible to judge the determinations of Higgs mass from high precision measurements of electro-weak parameters in TGD framework.

In TGD framework the Yukawa coupling of Higgs to fermions can be much weaker than in standard model. This has several implications.

1. The rate for the production of Higgs via channels involving fermions is much lower. This could explain why Higgs has not been observed even if it had mass around 100 GeV.

2. The radiative corrections to electro-weak parameters coming from fermion-Higgs vertices are much smaller than in standard model and cannot be used to deduce Higgs mass from the high precision measurements of electro-weak parameters. Hence one cannot anymore localize Higgs mass to the range 96-117 GeV.

3. In standard model the large Yukawa coupling of Higgs to top, call it h, tends to reduce the quartic self coupling constant λ for Higgs in ultraviolet. The condition that the minimum for Higgs potential is not transformed to a maximum gives a lower bound on the initial value of λ and thus to the value of m_H. In TGD framework the weakness of fermionic couplings implies that there is no lower bound to Higgs mass.

4. The weakness of Yukawa couplings means that self coupling of Higgs tends to increase λ faster than in standard model. Note also that when Yukawa coupling h_t to top is small (h_t²< λ, see arXiv:hep-ph/9409458), its contribution tends to increase the value of β_λ. Thus the upper bound from perturbative unitarity to the scalar coupling λ (and m_H) is reduced. This would force the value of Higgs mass to be even lower than in standard model.

   In TGD framework new physics can however emerge in the length scales corresponding to Mersenne primes M_n=2ⁿ-1. Ordinary QCD corresponds to M₁₀₇ and one cannot exclude even M₈₉ copy of QCD. M₆₁ would define the next candidate. The quarks of M₈₉ QCD would give to the beta function β_λ a negative contribution tending to reduce the value λ so that unitary bound would not be violated. If this new physics is accepted m_H=160 GeV can be considered.

Can one then identify the Higgs candidate with m_H=160 with the TGD variant of standard model Higgs? This is far from clear.

1. Even in standard model the rate for the production of Higgs is low. In TGD the rate for the production of the counterpart of standard model Higgs is reduced since the coupling of quarks to Higgs is expected to be much smaller than in standard model. This might exclude the interpretation as Higgs.

2. The slow rate for the production of Higgs could also allow the presence of Higgs at much lower mass and explain why Higgs has not been detected in the mass range m_H<114>

3. In TGD framework one can consider also other interpretations of the excess events at 160 GeV (taking the findings of both Dorigo's and Conway's group seriously and the fact that they do not seem to be consistent). p-Adically scaled up variants of ordinary quarks which might have something to do with the bumpy nature of top quark mass distribution.

   M₈₉ hadron physics might be required in TGD framework by the requirement of perturbative unitarity. Thus the mesons of M₈₉ hadron physics might be involved. By a very naive scaling by factor 2^(107-89)/2=2⁹ the mass of the pion of M₈₉ physics would be about 70 GeV. This estimate is not reliable since color spin-spin splittings distinguishing between pion and ρ mass do not scale naively. For M₈₉ mesons this splitting should be very small since color magnetic moments are very small. The mass of pion in absence of splitting would be around 297 MeV and 512-fold scaling gives M(π₈₉)≈ 152 GeV which is not too far from 160 GeV. Could the decays of this exotic pion give rise to the excess of fermion pairs? Note that he mass was given erratically in the original posting. This interpretation might also allow to understand why b-pair and t-pair excesses are not consistent.

For more details see the chapter p-Adic Particle Massivation: Elementary Particle Masses of "p-Adic length Scale Hypothesis and Dark Matter Hierarchy".

Hyper-finite factors and construction of S-matrix

During years I have spent a lot of time and effort to attempts to imagine various options for the construction of S-matrix. Contrary to my original belief, the real problem has not been the lack of my analytic skills but the failure of ordinary QFT based thinking in TGD framework.

Super-conformal symmetries generalized from string model context to TGD framework are symmetries of S-matrix. This is very powerful constraint to S-matrix but useless unless one has precisely defined ontology translated to a rigorous mathematical framework. The zero energy ontology of TGD is now rather well understood but differs dramatically from that of standard quantum field theories. Second deep difference is that path integral formalism is given up and the goal is to construct S-matrix as a generalization of braiding S-matrices with reaction vertices replaced with the replication of number theoretic braids associated with partonic 2-surfaces taking the role of vertices. Also number theoretic universality requiring fusion of real physics and various p-adic physics to single coherent whole is a completely new element.

The most recent vision about S-matrix combines ideas scattered in various chapters of various books and often drowned with details. A very brief summary would be as follows.

1. In TGD framework functional integral formalism is given up. S-matrix should be constructible as a generalization of braiding S-matrix in such a manner that the number theoretic braids assignable to light-like partonic 3-surfaces glued along their ends at 2-dimensional partonic 2-surfaces representing reaction vertices replicate in the vertex. This means a replacement of the free dynamics of point particles of quantum field theories with braiding dynamics associated with partonic 2-surfaces carrying braids and the replacement of particle creation with the creation of partons and replication of braids.

2. The construction of braiding S-matrices assignable to the incoming and outgoing partonic 2-surfaces is not a problem. The problem is to express mathematically what happens in the vertex. Here the observation that the tensor product of hyper-finite factors (HFFs) of type II is HFF of type II is the key observation. Many-parton vertex can be identified as a unitary isomorphism between the tensor product of incoming resp. outgoing HFFs. A reduction to HFF of type II₁ occurs because only a finite-dimensional projection of S-matrix in bosonic degrees of freedom defines a normalizable state. Most importantly, unitarity and non-triviality of S-matrix follows trivially.

3. HFFs of type III could also appear at the level of field operators used to create states but that at the level of quantum states everything reduces to HFFs of type II₁ and their tensor products giving these factors back. If braiding automorphisms reduce to the purely intrinsic unitary automorphisms of HFFs of type III then for certain values of the time parameter of automorphism having interpretation as a scaling parameter these automorphisms are trivial. These time scales could correspond to p-adic time scales so that p-adic length scale hypothesis would emerge at the fundamental level. In this kind of situation the braiding S-matrices associated with the incoming and outgoing partons could be trivial so that everything would reduce to this unitary isomorphism: a counterpart for the elimination of external legs from Feynman diagram in QFT. p-Adic thermodynamics and particle massivation could be also obtained when the time parameter of the automorphism is allowed to be complex as a generalization of thermal QFT.

4. One might hope that all complications related to what happens for space-like 3-surfaces could be eliminated by quantum classical correspondence stating that space-time view about particle reaction is only a space-time correlate for what happens in quantum fluctuating degrees of freedom associated with partonic 2-surfaces. This turns out to be the case only in non-perturbative phase. The reason is that the arguments of n-point function appear as continuous moduli of Kähler function. In non-perturbative phases the dependence of the maximum of Kähler function on the arguments of n-point function cannot be regarded as negligible and Kähler function becomes the key to the understanding of these effects including formation of bound states and color confinement.

5. In this picture light-like 3-surface would take the dual role as a correlate for both state and time evolution of state and this dual role allows to understand why the restriction of time like entanglement to that described by S-matrix must be made. For fixed values of moduli each reaction would correspond to a minimal braid diagram involving exchanges of partons being in one-one correspondence with a maximum of Kähler function. By quantum criticality and the requirement of ideal quantum-classical correspondence only one such diagram would contribute for given values of moduli. Coupling constant evolution would not be however lost: it would be realized as p-adic coupling constant at the level of free states via the log(p) scaling of eigen modes of the modified Dirac operator.

6. A completely unexpected prediction deserving a special emphasis is that number theoretic braids replicate in vertices. This is of course the braid counterpart for the introduction of annihilation and creation of particles in the transition from free QFT to an interacting one. This means classical replication of the number theoretic information carried by them. This allows to interpret one of the TGD inspired models of genetic code in terms of number theoretic braids representing at deeper level the information carried by DNA. This picture provides also further support for the proposal that DNA acts as topological quantum computer utilizing braids associated with partonic light-like 3-surfaces (which can have arbitrary size). In the reverse direction one must conclude that even elementary particles could be information processing and communicating entities in TGD Universe.

To sum up, my personal feeling is that the constraints identified hitherto might lead to a more or less unique final result and I can only hope that some young analytically blessed brain would bother to transform this picture to concrete calculational recipes.

For more details see the new chapter Hyper-Finite Factors and Construction of S-matrix of "Towards S-matrix".

About dogmas and world view as a disease

The PEAR Lab (Princeton Engineering Anomalies Research Lab) will be closing at the end of February of 2007. It is regrettable that the experimental research challenging the cherished dogmas of our scientific world view is not allowed to continue.

About PEAR

The research group was directed by Robert G. Jahn and studied both machine mind interactions and remote perception. Reader can find a brief description of these experiments in Wikipedia but just to make clear for myself what is involved I see the trouble of reproducing the description of machine mind interaction experiments here.

REG (Random Event Generator) experiment serves as a prototype for machine-mind interaction experiment.

1. Random noise was sampled with given frequency which varied from experiment to experiment and the outcome was coded into a bit.

2. The experiment involved three different intentions. Intention to produce bits 1 ("high"), bits 0 ("low"), and observing the data generation without any effort to affect the outcome.

3. The operator reported in the beginning of each trial her intention. The result of a particular trial was r= 200N(bit=1)/N(bits). In the absence of any effect the result should have been r= 100. The result of the run involving 8×10^5 trials per intention with 200 bits per trial was r(high)=100.026 and r(low)=99.984. The difference corresponds to 3.8 or 3.8 standard deviations. The deviation is 3.8 times large than the expected margin of error in the measurement and can be regarded as statistically significant.

This experiment served as a template for several other experiments such as remote experiment in which the device was influenced from distance; pseudo experiments in which operator was replaced with random analog noise source; a random mechanical cascade in which experimenter tried to affect the trajectories of macroscopic polystyrene balls falling through an array of pegs; a pendulum experiment in which operator tried to affect to motion of pendulum.

The conclusions were following. Human mind can affect random physical processes to a small but statistically significant degree. The effect seems to disappear when genuinely random sources are replaced with deterministic one (pseudo random sources). Different individuals produce different results. The effect shows long term fluctuations, which can be partly but not completely explained by changes in the operator pool.

Could quantum critical systems be more interesting than machines?

From the point of view of TGD inspired theory of consciousness, the attempt to affect random noise is certainly not the optimal experimental situation if one wants to detect strong effects. The optimal choice of system to be affected by intentional action would be a macroscopic quantum critical system. In TGD Universe high temperature super-conductors would be one example of such systems. Another system of this kind would be capacitor very near to the voltage at which di-electric breakdown occurs. Cell membrane provide one example of this kind of system.TGD also predicts that dark matter corresponds to a hierarchy of macroscopic quantum phases with increasing value of Planck constant responsible also for the very special properties of living matter.

The power of dogma

In TGD Universe, my own biological body would be a quantum critical system and I could argue that I experience the effects of intentions on living matter every day! I think that many readers would agree with me. Why should then people be burned on stake for suggesting that our intentions might have small effects on the material world outside our biological body? One can understand this irrational behavior only by realizing how enormous is the power of dogma.

So what happens when I raise my hand according to scientific explanation? "I decide to raise my hand and it raises!" would be the spontaneous answer of an innocent layman. Wrong! According to the belief system of an orthodox materialistic scientist wishing to keep his job, there is no intentional action involved. The scientist admits that there is some small quantum mechanical non-determinism in atomic length scales and below but in human scales all these effects give just random background noise. The initial conditions at the moment of big bang just happened to be such that my hand raises and I experience the illusion of having an intention to raise my hand. The reason why I have this illusion of free will and intention is not completely clear to the materialist. My complex initial value sensitive system and for some funny reason initial value sensitive systems have a tendency to create this kind of illusions. But not intentionally of course!

This example should demonstrate how difficult the challenge of proving that our intentions can affect the world outside us really is. I would actually talk abot mission impossible. Materialistic scientist can always fabulate a story explaining the outcome of any intentional action, whether it affects his own body or external world, as resulting from a deterministic laws of physics. They can always claim that statistical methods used have some flaw and it is always possible to say there is fraud involved.

The dogmas are what really matter for the average scientists as any average person willing to survive socially. Materialistic and reductionistic dogmas declare that in length scales above atomic length scale quantum effects are negligible and the world is in practice deterministic. Quantum world in macro scales is a random soup of matter with no long range quantal correlations otherwise made possible by quantum entanglement. Who argues something different is a crackpot (although I have heard this word so many times in physics blogs it still makes me almost puke!).

Dogmas and mathematics

The basic dogmas materialize themselves also mathematically. In standard quantum mechanics based on von Neumann algebras known as factors of type I. These algebras apply to the quantum theory of simple systems with finite number of degrees of freedom. Hydrogen atom is the classical example.

This is however not the only mathematical possibility. Von Neumann algebras known as hyper-finite factors of type II₁ about which I have been talking a lot during last two years are the mathematics for a quantum universe which behaves as a single cosmic organism. Quantum entanglement over arbitrarily long spatial (and temporal!) scale is always present and can be reduced only partially. That Planck constant can have arbitrarily large values realizes this as a new element of quantum physics. Everything is connected with everything. One of the basic implications at the level of consciousness theory are sharing and fusion of mental effects and collective pool of mental images. All this disgusting new age stuff is realized mathematically and even worse: the person responsible for all this scandalous mathematics is the father of classical computer architecture: you cannot trust anyone nowadays!

Theory is regarded as successful if it explains empirical facts. Eastern meditation practices are the empirical study of consciousness and this picture indeed confirms to surprising degree with the views about consciousness provided by these practices. Of course, this picture explains the facts about brain produced by western science: in particular a successful model for EEG emerges.

Dogmas materialize themselves even in the notion of number used. For physicists basically only the magnitude of the number matters, not its number theoretic anatomy. Numbers tell what some things weighs, nothing else. In fact that number theory has become basic element of TGD inspired theory of consciousness and number theoretical complexity becomes a quantitative measure for cognitive level.

Beliefs in crisis

All is after all about beliefs: which beliefs we raise to dogmas as we try to make sense of the world around us. Beliefs have however a finite life span. The explanatory power is what matters in the long run. Entire societies fall down when everyone knows that everyone knows that dogma is wrong. I strongly feel that materialistic and reductionistic dogma has now reached the end of its life span.

During last three decades the materialistic dogmatics combined with American pragmatism (kind of analog of quartal economy in science) has led to the deepest crisis that theoretical physics has ever experienced. The fate of super string theorists was to devote their lifetime for a theory which was not a beginning of something fantastic but an unavoidable culmination for the world view based on dogmas which do not work anymore. Super-string gurus cannot but continue to declare that tiny little strings of size Planck length (reductionism!) are the ultimate building blocks of matter. It does not matter that in its recent form this theory cannot even predict the dimension of space-time to say nothing about what we observe in laboratory! The reaction of the most fanatic gurus is that since this theory is the only possible one (by some misty arguments involving usually big names) we must give up the idea that physics can predict something. The reader has probably detected the deep irony here: after all, the materialistic dogma was based on the idea of complete predictability!

During these 28 years as out-of-law in science I have pondered many times how this kind of incredibly irrational behavior is possible. The people behind these prejudices do not look like lunatics, fanatics, or series killers. The only explanation I can imagine is that we are not able to tolerate social pressures. That ordinary decent people can by forced by social pressures to believe and even do almost everything has been demonstrated in many experiments. As an example consider the following social experiment. There is a group of subject persons. As a matter fact, some of them are actors and take the role of an influental social leader. These people are asked to tell what 1+1 is but in such manner that everyone knowns the answers of others. Actors tell first their opinion which is 1+1 =3. Surprisingly many of participants cannot but agree after a painful internal battle!

Spiritual IQ and world view as a disease

Some of us are more able to resist social pressures. How do they differ from the rest of us? I suggest a simple three-letter answer: SIQ, Spiritual Intelligence Quotient. At the level of individual the growth of SIQ often occurs through a turning point experiences in which previous world view is dramatically transformed and one realizes that old certainties are not much more than a result of a need to gain social acceptance. After the great change one central theme often rules the life of these people: the puzzle of consciousness. Just the mere attempt to understand this mystery can give a full meaning to the life and the realization that there are things larger than biological life gradually gives the courage to insist also the magic power of social pressures. Almost as a rule these people gain the respect of the people who know them personally.

Just to see how SIQ manifests itself in social situations go to a typical physics blog (if you want almost physical violence choose the blog from US: you can start from Not-Even-Wrong or Reference Frame and follow the links to other blogs), and follow the discussion or postings. You will find that a considerable portion of debate consists of exchanges of direct personal insults. You soon realize that many of the participants cannot be very happy: there is too much frustration, aggression, and arrogance. Of course, this kind of direct violence is not possible in everyday academic life but from personal experience I can tell that academic people are masters of the refined forms of implicit violence.

My explanation is that wrong world view is the disease that these people are suffering. The universe of materialistic and reductionistic science lacks both purpose and meaning. In this kind of world human life span is not more than a random spark between two darknesses. No wonder that a person believing that he really lives in this kind of world becomes sick.

To test this hypothesis you can make a comparison with a blog where SIQ should be higher at least if the interest in the mystery of consciousness is accepted as a rough criterion. For instance, you can go to the blog Conscious Entities. You find nice articles pondering various ideas about consciousness; discussions are civilized; there are no sudden bursts of violence; there are no ad hominem attacks. It seems that on the average these people are happy; they have a passion in their life but they are not fanatics; and they do not have any need to tell to the rest of the world that they belong to some kind of super-species and all those who think differently are crackpots (or "social waste", one of the newest verbal fruits in physics blog discussions!).

Jones inclusions and construction of S-matrix and U matrix

TGD leads naturally to zero energy ontology which reduces to the positive energy ontology of the standard model only as a limiting case. In this framework one must distinguish between the U-matrix characterizing the unitary process associated with the quantum jump (and followed by state function reduction and state preparation) and the S-matrix defining time-like entanglement between positive and negative energy parts of the zero energy state and coding the rates for particle reactions which in TGD framework correspond to quantum measurements reducing time-like entanglement.

1. S-matrix

In zero energy ontology S-matrix characterizes time like entanglement of zero energy states (this is possible only for HFFs for which Tr(SS⁺)=Tr(Id)=1 holds true). S-matrix would code for transition rates measured in particle physics experiments with particle reactions interpreted as quantum measurements reducing time like entanglement. In TGD inspired quantum measurement theory measurement resolution is characterized by Jones inclusion (the group G defines the measured quantum numbers), N subset M takes the role of complex numbers, and state function reduction leads to N ray in the space M/N regarded as N module and thus from a factor to a sub-factor.

The finite number theoretic braid having Galois group G as its symmetries is the space-time correlate for both the finite measurement resolution and the effective reduction of HFF to that associated with a finite-dimensional quantum Clifford algebra M/N. SU(2) inclusions would allow angular momentum and color quantum numbers in bosonic degrees of freedom and spin and electro-weak quantum numbers in spinorial degrees of freedom. McKay correspondence would allow to assign to G also compact ADE type Lie group so that also Lie group type quantum numbers could be included in the repertoire.

Galois group G would characterize sub-spaces of the configuration space ("world of classical worlds") number theoretically in a manner analogous to the rough characterization of physical states by using topological quantum numbers. Each braid associated with a given partonic 2-surface would correspond to a particular G that the state would be characterized by a collection of groups G. G would act as symmetries of zero energy states and thus of S-matrix. S-matrix would reduce to a direct integral of S-matrices associated with various collections of Galois groups characterizing the number theoretical properties of partonic 2-surfaces. It is not difficult to criticize this picture.

1. Why time like entanglement should be always characterized by a unitary S-matrix? Why not some more general matrix? If one allows more general time like entanglement, the description of particle reaction rates in terms of a unitary S-matrix must be replaced with something more general and would require a profound revision of the vision about the relationship between experiment and theory. Also the consistency of the zero energy ontology with positive energy ontology in time scales shorter than the time scale determined by the geometric time interval between positive and negative energy parts of the zero energy state would be lost. Hence the easy way to proceed is to postulate that the universe is self-referential in the sense that quantum states represent the laws of physics by coding S-matrix as entanglement coefficients.

2. Second objection is that there might a huge number of unitary S-matrices so that it would not be possible to speak about quantum laws of physics anymore. This need not be the case since super-conformal symmetries and number theoretic universality pose extremely powerful constraints on S-matrix. A highly attractive additional assumption is that S-matrix is universal in the sense that it is invariant under the inclusion sequences defined by Galois groups G associated with partonic 2-surfaces. Various constraints on S-matrix might actually imply the inclusion invariance.

3. One can of course ask why S-matrix should be invariant under inclusion. One might argue that zero energy states for which time-like entanglement is characterized by S-matrix invariant in the inclusion correspond to asymptotic self-organization patterns for which U-process and state function reduction do not affect the S-matrix in the relabelled basis. The analogy with a fractal asymptotic self-organization pattern is obvious.

2. U-matrix

In a well-defined sense U process seems to be the reversal of state function reduction. Hence the natural guess is that U-matrix means a quantum transition in which a factor becomes a sub-factor whereas state function reduction would lead from a factor to a sub-factor.

Various arguments suggest that U matrix could be almost trivial and has as a basic building block the so called factorizing S-matrices for integrable quantum field theories in 2-dimensional Minkowski space. For these S-matrices particle scattering would mean only a permutation of momenta in momentum space. If S-matrix is invariant under inclusion then U matrix should be in a well-defined sense almost trivial apart from a dispersion in zero modes leading to a superpositions of states characterized by different collections of Galois groups.

3. Relation to TGD inspired theory of consciousness

U-matrix could be almost trivial with respect to the transitions which are diagonal with respect to the number field. What would however make U highly interesting is that it would predict the rates for the transitions representing a transformation of intention to action identified as a p-adic-to-real transition. In this context almost triviality would translate to a precise correlation between intention and action.

The general vision about the dynamics of quantum jumps suggests that the extension of a sub-factor to a factor is followed by a reduction to a sub-factor which is not necessarily the same. Breathing would be an excellent metaphor for the process. Breathing is also a metaphor for consciousness and life. Perhaps the essence of living systems distinguishing them from sub-systems with a fixed state space could be cyclic breathing like process N→ M supset N → N₁ subset M→ .. extending and reducing the state space of the sub-system by entanglement followed by de-entanglement.

One could even ask whether the unique role of breathing exercise in meditation practices relates directly to this basic dynamics of living systems and whether the effect of these practices is to increase the value of M:N and thus the order of Galois group G describing the algebraic complexity of "partonic" 2-surfaces involved (they can have arbitrarily large sizes). The basic hypothesis of TGD inspired theory of cognition indeed is that cognitive evolution corresponds to the growth of the dimension of the algebraic extension of p-adic numbers involved.

If one is willing to consider generalizations of the existing picture about quantum jump, one can imagine that unitary process can occur arbitrary number of times before it is followed by state function reduction. Unitary process and state function reduction could compete in this kind of situation.

4. Fractality of S-matrix and translational invariance in the lattice defined by sub-factors

Fractality realized as the invariance of the S-matrix in Jones inclusion means that the S-matrices of N and M relate by the projection P: M→N as S_N=PS_MP. S_N should be equivalent with S_M with a trivial re-labelling of strands of infinite braid.

Inclusion invariance would mean translational invariance of the S-matrix with respect to the index n labelling strands of braid defined by the projectors e_i. Translations would act only as a semigroup and S-matrix elements would depend on the difference m-n only. Transitions can occur only for m-n≥ 0, that is to the direction of increasing label of strand. The group G leaving N element-wise invariant would define the analog of a unit cell in lattice like condensed matter systems so that translational invariance would be obtained only for translations m→ m+ nk, where one has n≥ 0 and k is the number of M(2,C) factors defining the unit cell. As a matter fact, this picture might apply also to ordinary condensed matter systems.

For more details see the chapters Construction of Quantum Theory: S-matrix and Was von Neumann Right After All? of "Towards S-matrix".

Langlands Program and TGD

Number theoretic Langlands program can be seen as an attempt to unify number theory on one hand and theory of representations of reductive Lie groups on the other hand. So called automorphic functions to which various zeta functions are closely related define the common denominator. Geometric Langlands program tries to achieve a similar conceptual unification in the case of function fields. This program has caught the interest of physicists during last years.

TGD can be seen as an attempt to reduce physics to infinite-dimensional Kähler geometry and spinor structure of the "world of classical worlds" (WCW). Since TGD ce be regarded also as a generalized number theory, it is difficult to escape the idea that the interaction of Langlands program with TGD could be fruitful.

More concretely, TGD leads to a generalization of number concept based on the fusion of reals and various p-adic number fields and their extensions implying also generalization of manifold concept, which inspires the notion of number theoretic braid crucial for the formulation of quantum TGD. TGD leads also naturally to the notion of infinite primes and rationals. The identification of Clifford algebra of WCW as a hyper-finite factors of type II₁ in turn inspires further generalization of the notion of imbedding space and the idea that quantum TGD as a whole emerges from number theory. The ensuing generalization of the notion of imbedding space predicts a hierarchy of macroscopic quantum phases characterized by finite subgroups of SU(2) and by quantized Planck constant. All these new elements serve as potential sources of fresh insights.

1. The Galois group for the algebraic closure of rationals as infinite symmetric group?

The naive identification of the Galois groups for the algebraic closure of rationals would be as infinite symmetric group S_∞ consisting of finite permutations of the roots of a polynomial of infinite degree having infinite number of roots. What puts bells ringing is that the corresponding group algebra is nothing but the hyper-finite factor of type II₁ (HFF). One of the many avatars of this algebra is infinite-dimensional Clifford algebra playing key role in Quantum TGD. The projective representations of this algebra can be interpreted as representations of braid algebra B_∞ meaning a connection with the notion of number theoretical braid.

2. Representations of finite subgroups of S_∞ as outer automorphisms of HFFs

Finite-dimensional representations of Gal(\overline{Q}/Q) are crucial for Langlands program. Apart from one-dimensional representations complex finite-dimensional representations are not possible if S_∞ identification is accepted (there might exist finite-dimensional l-adic representations). This suggests that the finite-dimensional representations correspond to those for finite Galois groups and result through some kind of spontaneous breaking of S_∞ symmetry.

1. Sub-factors determined by finite groups G can be interpreted as representations of Galois groups or, rather infinite diagonal imbeddings of Galois groups to an infinite Cartesian power of S_n acting as outer automorphisms in HFF. These transformations are counterparts of global gauge transformations and determine the measured quantum numbers of gauge multiplets and thus measurement resolution. All the finite approximations of the representations are inner automorphisms but the limit does not belong to S_∞ and is therefore outer. An analogous picture applies in the case of infinite-dimensional Clifford algebra.

2. The physical interpretation is as a spontaneous breaking of S_∞ to a finite Galois group. One decomposes infinite braid to a series of n-braids such that finite Galois group acts in each n-braid in identical manner. Finite value of n corresponds to IR cutoff in physics in the sense that longer wave length quantum fluctuations are cut off. Finite measurement resolution is crucial. Now it applies to braid and corresponds in the language of new quantum measurement theory to a sub-factor N subset M determined by the finite Galois group G implying non-commutative physics with complex rays replaced by N rays. Braids give a connection to topological quantum field theories, conformal field theories (TGD is almost topological quantum field theory at parton level), knots, etc...

3. TGD based space-time correlate for the action of finite Galois groups on braids and for the cutoff is in terms of the number theoretic braids obtained as the intersection of real partonic 2-surface and its p-adic counterpart. The value of the p-adic prime p associated with the parton is fixed by the scaling of the eigenvalue spectrum of the modified Dirac operator (note that renormalization group evolution of coupling constants is characterized at the level free theory since p-adic prime characterizes the p-adic length scale). The roots of the polynomial would determine the positions of braid strands so that Galois group emerges naturally. As a matter fact, partonic 2-surface decomposes into regions, one for each braid transforming independently under its own Galois group. Entire quantum state is modular invariant, which brings in additional constraints.

   Braiding brings in homotopy group aspect crucial for geometric Langlands program. Another global aspect is related to the modular degrees of freedom of the partonic 2-surface, or more precisely to the regions of partonic 2-surface associated with braids. Sp(2g,R) (g is handle number) can act as transformations in modular degrees of freedom whereas its Langlands dual would act in spinorial degrees of freedom. The outcome would be a coupling between purely local and and global aspects which is necessary since otherwise all information about partonic 2-surfaces as basic objects would be lost. Interesting ramifications of the basic picture about why only three lowest genera correspond to the observed fermion families emerge.

3. Correspondence between finite groups and Lie groups

The correspondence between finite and Lie group is a basic aspect of Langlands.

1. Any amenable group gives rise to a unique sub-factor (in particular, compact Lie groups are amenable). These groups act as genuine outer automorphisms of the group algebra of S_∞ rather than being induced from S_∞ outer automorphism. If one gives up uniqueness, it seems that practically any group G can define a sub-factor: G would define measurement resolution by fixing the quantum numbers which are measured. Finite Galois group G and Lie group containing it and related to it by Langlands correspondence would act in the same representation space: the group algebra of S_∞, or equivalently configuration space spinors. The concrete realization for the correspondence might transform a large number of speculations to theorems.

2. There is a natural connection with McKay correspondence which also relates finite and Lie groups. The simplest variant of McKay correspondence relates discrete groups Gsubset SU(2) to ADE type groups. Similar correspondence is found for Jones inclusions with index M:N≤ 4. The challenge is to understand this correspondence.
   1. The basic observation is that ADE type compact Lie algebras with n-dimensional Cartan algebra can be seen as deformations for a direct sum of n SU(2) Lie algebras since SU(2) Lie algebras appear as a minimal set of generators for general ADE type Lie algebra. The algebra results by a modification of Cartan matrix. It is also natural to extend the representations of finite groups Gsubset SU(2) to those of SU(2).

   2. The idea would that is that n-fold Connes tensor power transforms the direct sum of n SU(2) Lie algebras by a kind of deformation to a ADE type Lie algebra with n-dimensional Cartan Lie algebra. The deformation would be induced by non-commutativity. Same would occur also for the Kac-Moody variants of these algebras for which the set of generators contains only scaling operator L₀ as an additional generator. Quantum deformation would result from the replacement of complex rays with N rays, where N is the sub-factor.

   3. The concrete interpretation for the Connes tensor power would be in terms of the fiber bundle structure H=M⁴_+/-× CP₂→ H/G_a× G_b, G_a× G_b subset SU(2)× SU(2)subset SL(2,C)× SU(3), which provides the proper formulation for the hierarchy of macroscopic quantum phases with a quantized value of Planck constant. Each sheet of the singular covering would represent single factor in Connes tensor power and single direct SU(2) summand. This picture has an analogy with brane constructions of M-theory.

4. Could there exist a universal rational function giving rise to the algebraic closure of rationals?

One could wonder whether there exists a universal generalized rational function having all units of the algebraic closure of rationals as roots so that S_∞ would permute these roots. Most naturally it would be a ratio of infinite-degree polynomials.

With motivations coming from physics I have proposed that zeros of zeta and also the factors of zeta in product expansion of zeta are algebraic numbers. Complete story might be that non-trivial zeros of Zeta define the closure of rationals. A good candidate for this function is given by (ξ(s)/ξ(1-s))× (s-1)/s), where ξ(s)= ξ(1-s) is the symmetrized variant of zeta function having same zeros. It has zeros of zeta as its zeros and poles and product expansion in terms of ratios (s-s_n)/(1-s+s_n) converges everywhere. Of course, this might be too simplistic and might give only the algebraic extension involving the roots of unity given by exp(iπ/n). Also products of these functions with shifts in real argument might be considered and one could consider some limiting procedure containing very many factors in the product of shifted zeta functions yielding the universal rational function giving the closure.

5. What does one mean with S_∞?

There is also the question about the meaning of S_∞. The hierarchy of infinite primes suggests that there is entire infinity of infinities in number theoretical sense. Any group can be formally regarded as a permutation group. A possible interpretation would be in terms of algebraic closure of rationals and algebraic closures for an infinite hierarchy of polynomials to which infinite primes can be mapped. The question concerns the interpretation of these higher Galois groups and HFF:s. Could one regard these as local variants of S_∞ and does this hierarchy give all algebraic groups, in particular algebraic subgroups of Lie groups, as Galois groups so that almost all of group theory would reduce to number theory even at this level?

Be it as it may, the expressive power of HFFs seem to be absolutely marvellous. Together with the notion of infinite rational and generalization of number concept they might unify both mathematics and physics!

For more details see the new chapter TGD and Langlands Program of "TGD as a Generalized Number Theory".