What's new in

Topological Geometrodynamics: an Overview

Note: Newest contributions are at the top!

Year 2008

The relationship between super-canonical and Super Kac-Moody algebras, Equivalence Principle, and justification of p-adic thermodynamics

The relationship between super-canonical algebra (SC) acting at light-cone boundary and Super Kac-Moody algebra (SKM) acting on light-like 3-surfaces has remained somewhat enigmatic due to the lack of physical insights. This is not the only problem. The question to precisely what extent Equivalence Principle (EP) remains true in TGD framework and what might be the precise mathematical realization of EP is waiting for an answer. Also the justification of p-adic thermodynamics for the scaling generator L0 of Virasoro algebra -in obvious conflict with the basic wisdom that this generator should annihilate physical states- is lacking. It seems that these three problems could have a common solution.

Before going to describe the proposed solution, some background is necessary. The latest proposal for SC-SKM relationship relies on non-standard and therefore somewhat questionable assumptions.

  1. SKM Virasoro algebra (SKMV) and SC Virasoro algebra (SCV) (anti)commute for physical states.

  2. SC algebra generates states of negative conformal weight annihilated by SCV generators Ln, n < 0, and serving as ground states from which SKM generators create states with non-negative conformal weight.

This picture could make sense for elementary particles. On other hand, the recent model for hadrons [see this] assumes that SC degrees of freedom contribute about 70 per cent to the mass of hadron but at space-time sheet different from those assignable to quarks. The contribution of SC degrees of freedom to the thermal average of the conformal weight would be positive. A contradiction results unless one assumes that there exists also SCV ground states with positive conformal weight annihilated by SCV elements Ln, n < 0, but also this seems implausible.

1. New vision about the relationship between SCV and SKMV

Consider now the new vision about the relationship between SCV and SKMV.

  1. The isometries of H assignable with SKM are also symplectic transformations [see this] (note that I have used the term canonical instead of symplectic previously). Hence might consider the possibility that SKM could be identified as a subalgebra of SC. If this makes sense, a generalization of the coset construction obtained by replacing finite-dimensional Lie group with infinite-dimensional symplectic group suggests itself. The differences of SCV and SKMV elements would annihilate physical states and (anti)commute with SKMV. Also the generators On, n > 0, for both algebras would annihilate the physical states so that the differences of the elements would annihilate automatically physical states for n > 0.

  2. The super-generator G0 contains the Dirac operator D of H. If the action of SCV and SKMV Dirac operators on physical states are identical then cm of degrees of freedom disappear from the differences G0(SCV)-G0(SKMV) and L0(SCV)-L0(SKMV). One could interpret the identical action of the Dirac operators as the long sought-for precise realization of Equivalence Principle (EP) in TGD framework. EP would state that the total inertial four-momentum and color quantum numbers assignable to SC (imbedding space level) are equal to the gravitational four-momentum and color quantum numbers assignable to SKM (space-time level). Note that since super-canonical transformations correspond to the isometries of the "world of classical worlds" the assignment of the attribute "inertial" to them is natural.

  3. The analog of coset construction applies also to SKM and SC algebras which means that physical states can be thought of as being created by an operator of SKM carrying the conformal weight and by a genuine SC operator with vanishing conformal weight. Therefore the situation does not reduce to that encountered in super-string models.

This picture provides also a justification for p-adic thermodynamics.

  1. In physical states the p-adic thermal expectation value of the SKM and SC conformal weights would be non-vanishing and identical and mass squared could be identified to the expectation value of SKM scaling generator L0. There would be no need to give up Super Virasoro conditions for SCV-SKMV.

  2. There is consistency with p-adic mass calculations for hadrons [see this] since the non-perturbative SC contributions and perturbative SKM contributions to the mass correspond to space-time sheets labeled by different p-adic primes. The earlier statement that SC is responsible for the dominating non-perturbative contributions to the hadron mass transforms to a statement reflecting SC-SKM duality. The perturbative quark contributions to hadron masses can be calculated most conveniently by using p-adic thermodynamics for SKM whereas non-perturbative contributions to hadron masses can be calculated most conveniently by using p-adic thermodynamics for SC. Also the proposal that the exotic analogs of baryons resulting when baryon looses its valence quarks [see this] remains intact in this framework.

  3. The results of p-adic mass calculations depend crucially on the number N of tensor factors contributing to the Super-Virasoro algebra. The required number is N=5 and during years I have proposed several explanations for this number. It seems that holonomic contributions that is electro-weak and spin contributions must be regarded as contributions separate from those coming from isometries. SKM algebras in electro-weak degrees and spin degrees of of freedom, would give 2+1=3 tensor factors corresponding to U(2)ew×SU(2). SU(3) and SO(3) (or SO(2) SO(3) leaving the intersection of light-like ray with S2 invariant) would give 2 additional tensor factors. Altogether one would indeed have 5 tensor factors.

2. Can SKM be lifted to a sub-algebra of SC?

A picture introducing only a generalization of coset construction as a new element, realizing mathematically Equivalence Principle, and justifying p-adic thermodynamics is highly attractive but there is a problem. SKM is defined at light-like 3-surfaces X3 whereas SC acts at light-cone boundary dH=dM4×CP2. One should be able to lift SKM to imbedding space level somehow. Also SC should be lifted to entire H. This problem was the reason why I gave up the idea about coset construction and SC-SKM duality as it appeared for the first time.

A possible solution of the lifting problem comes from the observation making possible a more rigorous formulation of HO-H duality stating that one can regard space-time surfaces either as surfaces in hyper-octonionic space HO=M8 or in H=M4×CP2 [see this]. Consider first the formulation of HO-H duality.

  1. Associativity also in the number theoretical sense becomes the fundamental dynamical principle if HO-H duality holds true [see this]. For a space-time surface X4 HO=M8 associativity is satisfied at space-time level if the tangent space at each point of X4 is some hyper-quaternionic sub-space HQ=M4 M8. Also partonic 2-surfaces at the boundaries of causal diamonds formed by pairs of future and past directed light-cones defining the basic imbedding space correlate of zero energy state in zero energy ontology and light-like 3-surfaces are assumed to belong to HQ=M4 HO.

  2. HO-H duality requires something more. If the tangent spaces contain the same preferred commutative and thus hyper-complex plane HC=M2, the tangent spaces of X4 are parameterized by the points s of CP2 and X4 HO can be mapped to X4 M4×CP2 by assigning to a point of X4 regarded as point (m,e) of M40×E4=M8 the point (m,s). Note that one must also fix a preferred global hyper-quaternionic subspace M40 M8 containing M2 to be not confused with the local tangent planes M4.

  3. The preferred plane M2 can be interpreted as the plane of non-physical polarizations so that the interpretation as a number theoretic analog of gauge conditions posed in both quantum field theories and string models is possible.

  4. An open question is whether the resulting surface in H is a preferred extremal of Kähler action. This is possible since the tangent spaces at light-like partonic 3-surfaces are fixed to contain M2 so that the boundary values of the normal derivatives of H coordinates are fixed and field equations fix in the ideal case X4 uniquely and one obtains space-time surface as the analog of Bohr orbit.

  5. The light-like "Higgs term" proportional to O=gktk appearing in the generalized eigenvalue equation for the modified Dirac operator [see this] is an essential element of TGD based description of Higgs mechanism. This term can cause complications unless t is a covariantly constant light-like vector. Covariant constancy is achieved if t is constant light-like vector in M2. The interpretation as a space-time correlate for the light-like 4-momentum assignable to the parton might be considered.

  6. Associativity requires that the hyper-octonionic arguments of N-point functions in HO description are restricted to a hyperquaternionic plane HQ=M4 HO required also by the HO-H correspondence. The intersection M4int(X4) consists of a discrete set of points in the generic case. Partonic 3-surfaces are assumed to be associative and belong to M4. The set of commutative points at the partonic 2-surface X2 is discrete in the generic case whereas the intersection X3M2 consists of 1-D curves so that the notion of number theoretical braid crucial for the p-adicization of the theory as almost topological QFT is uniquely defined.

  7. The preferred plane M2 M4 HO can be assigned also to the definition of N-point functions in HO picture. It is not clear whether it must be same as the preferred planes assigned to the partonic 2-surfaces. If not, the interpretation would be that it corresponds to a plane containing the over all cm four-momentum whereas partonic planes M2i would contain the partonic four-momenta. M2 is expected to change at wormhole contacts having Euclidian signature of the induced metric representing horizons and connecting space-time sheets with Minkowskian signature of the induced metric.

The presence of globally defined plane M2 and the flexibility provided by the hyper-complex conformal invariance raise the hopes of achieving the lifting of SC and SKM to H. At the light-cone boundary the light-like radial coordinate can be lifted to a hyper-complex coordinate defining coordinate for M2. At X3 one can fix the light-like coordinate varying along the braid strands can be lifted to some hyper-complex coordinate of M2 defined in the interior of X4. The total four-momenta and color quantum numbers assignable to the SC and SKM degrees of freedom are naturally identical since they can be identified as the four-momentum of the partonic 2-surface X2 X3dM4×CP2. Equivalence Principle would emerge as an identity.

3. Questions

There are still several open questions.

  1. Is it possible to define hyper-quaternionic variants of the superconformal algebras in both H and HO or perhaps only in HO. A positive answer to this question would conform with the conjecture that the geometry of "world of classical worlds" allows Hyper-Kähler property in either or both pictures [see this].

  2. How this picture relates to what is known about the extremals of field equations [see this] characterized by generalized Hamilton-Jacobi structure bringing in mind the selection of preferred M2?

  3. Is this picture consistent with the views about Equivalence Principle and its possible breaking based on the identification of gravitational four-momentum in terms of Einstein tensor is interesting [see this] ?

For more details see the chapter Massless particles and particle massivation.

What goes wrong with string theories?

Something certainly goes wrong with super-string models and M-theory. But what this something is? One could of course make the usual list involving spontaneous compactification, landscape, non-predictivity, and all that. The point I however want to make relates to the relationship of string models to quantum field theories.

The basic wisdom has been that when Feynman graphs of quantum field theories are replaced by their stringy variants everything is nice and finite. The problem is that stringy diagrams do not describe what elementary particles are doing and quantum field theory limit is required at low energies. This non-renormalizable QFT limit is obtained by the ad hoc procedure called spontaneous compactification, and leads to all this misery that spoils the quality of our life nowadays.

My intention is not to ridicule or accuse string theorists. I feel also myself very very stupid since I realized only now what the relationship between super-conformal and super-symplectic QFTs and generalized Feynman diagrams is in TGD framework: I described this already in the previous posting but did not want to make noise of my stupidity. This discovery (discovery only at level of my own subjective experience) was just becoming aware about something which should have been absolutely obvious for anyone with IQ above 20;-).

A very brief summary goes as follows.

  1. M-matrix elements characterize the time like entanglement between positive and negative energy parts of zero energy states. The positive/negative energy part can be localized to the boundary of past/future directed light-cone and these light-cones form a causal diamond.

  2. M-matrix elements can be expressed in terms of generalized Feynman diagrams with the lines of Feynman diagrams replaced with light-like 3-surfaces glued together along their 2-D ends representing vertices. For a given Feynman diagram of this kind one assigns an n-point function with additional intermediate points coming from the generalized vertices. In hyper-octonionic conformal field theory approach these vertices are fixed uniquely.

  3. The amplitude associated with a given generalized Feynman diagram is calculated by a recursive procedure using the fusion rules of a combination of conformal and symplectic QFT:s as described in the previous posting.

What this means that a fusion of generalizations of stringy conformal QFT to conformal-symplectic QFT and of ordinary QFT gives M-matrix elements. Feynman diagrams are not given up! Only the manner how they are computed is completely new: instead of the iterative approach one uses recursive approach based on fusion rules and involving automatically the cutoff which has interpretation in terms of finite measurement resolution.

For more details see the blog posting, the chapter Overall View about Quantum TGD, and the article Topological Geometrodynamics: What Might Be the First Principles?.

Could a symplectic analog of conformal field theory be relevant for quantum TGD?

Symplectic (or canonical as I have called them) symmetries of dM4+×CP2 (light-cone boundary briefly) act as isometries of the "world of classical worlds". One can see these symmetries as analogs of Kac-Moody type symmetries with symplectic transformations of S2×CP2, where S2 is rM=constant sphere of lightcone boundary, made local with respect to the light-like radial coordinate rM taking the role of complex coordinate. Thus finite-dimensional Lie group G is replaced with infinite-dimensional group of symplectic transformations. This inspires the question whether a symplectic analog of conformal field theory at dM4+×CP2 could be relevant for the construction of n-point functions in quantum TGD and what general properties these n-point functions would have.

1 Symplectic QFT at sphere

Actually the notion of symplectic QFT emerged as I tried to understand the properties of cosmic microwave background which comes from the sphere of last scattering which corresponds roughly to the age of 5×105 years. In this situation vacuum extremals of Kähler action around almost unique critical Robertson-Walker cosmology imbeddable in M4×S2, where there is homologically trivial geodesic sphere of CP2. Vacuum extremal property is satisfied for any space-time surface which is surface in M4×Y2, Y2 a Lagrangian sub-manifold of CP2 with vanishing induced Kähler form. Symplectic transformations of CP2 and general coordinate transformations of M4 are dynamical symmetries of the vacuum extremals so that the idea of symplectic QFT emerges natural. Therefore I shall consider first symplectic QFT at the sphere S2 of last scattering with temperature fluctuation DT/T proportional to the fluctuation of the metric component gaa in Robertson-Walker coordinates.

  1. In quantum TGD the symplectic transformation of the light-cone boundary would induce action in the "world of classical worlds" (light-like 3-surfaces). In the recent situation it is convenient to regard perturbations of CP2 coordinates as fields at the sphere of last scattering (call it S2) so that symplectic transformations of CP2 would act in the field space whereas those of S2 would act in the coordinate space just like conformal transformations. The deformation of the metric would be a symplectic field in S2. The symplectic dimension would be induced by the tensor properties of R-W metric in R-W coordinates: every S2 coordinate index would correspond to one unit of symplectic dimension. The symplectic invariance in CP2 degrees of freedom is guaranteed if the integration measure over the vacuum deformations is symplectic invariant. This symmetry does not play any role in the sequel.

  2. For a symplectic scalar field n 3-point functions with a vanishing anomalous dimension would be functions of the symplectic invariants defined by the areas of geodesic polygons defined by subsets of the arguments as points of S2. Since n-polygon can be constructed from 3-polygons these invariants can be expressed as sums of the areas of 3-polygons expressible in terms of symplectic form. n-point functions would be constant if arguments are along geodesic circle since the areas of all sub-polygons would vanish in this case. The decomposition of n-polygon to 3-polygons brings in mind the decomposition of the n-point function of conformal field theory to products of 2-point functions by using the fusion algebra of conformal fields (very symbolically FkFl = cklmFm). This intuition seems to be correct.

  3. Fusion rules stating the associativity of the products of fields at different points should generalize. In the recent case it is natural to assume a non-local form of fusion rules given in the case of symplectic scalars by the equation

    Fk(s1)Fl(s2) =

    Here the coefficients cklm are constants and A(s1,s2,s3) is the area of the geodesic triangle of S2 defined by the sympletic measure and integration is over S2 with symplectically invariant measure dms defined by symplectic form of S2. Fusion rules pose powerful conditions on n-point functions and one can hope that the coefficients are fixed completely.

  4. The application of fusion rules gives at the last step an expectation value of 1-point function of the product of the fields involves unit operator term cklf(A(s1,s2,s))Id dms so that one has

    Fk(s1)Fl(s2) =

    Hence 2-point function is average of a 3-point function over the third argument. The absence of non-trivial symplectic invariants for 1-point function means that n=1- an are constant, most naturally vanishing, unless some kind of spontaneous symmetry breaking occurs. Since the function f(A(s1,s2,s3)) is arbitrary, 2-point correlation function can have both signs. 2-point correlation function is invariant under rotations and reflections.

2 Symplectic QFT with spontaneous breaking of rotational and reflection symmetries

CMB data suggest breaking of rotational and reflection symmetries of S2. A possible mechanism of spontaneous symmetry breaking is based on the observation that in TGD framework the hierarchy of Planck constants assigns to each sector of the generalized imbedding space a preferred quantization axes. The selection of the quantization axis is coded also to the geometry of "world of classical worlds", and to the quantum fluctuations of the metric in particular. Clearly, symplectic QFT with spontaneous symmetry breaking would provide the sought-for really deep reason for the quantization of Planck constant in the proposed manner.

  1. The coding of angular momentum quantization axis to the generalized imbedding space geometry allows to select South and North poles as preferred points of S2. To the three arguments s1,s2,s3 of the 3-point function one can assign two squares with the added point being either North or South pole. The difference

    DA(s1,s2,s3) A(s1,s2,s3,N)-A(s1,s2,s3,S)

    of the corresponding areas defines a simple symplectic invariant breaking the reflection symmetry with respect to the equatorial plane. Note that DA vanishes if arguments lie along a geodesic line or if any two arguments co-incide. Quite generally, symplectic QFT differs from conformal QFT in that correlation functions do not possess singularities.

  2. The reduction to 2-point correlation function gives a consistency conditions on the 3-point functions


    = cklr

    f(DA(s1,s2,s)) f(DA(s,s3,t))dmsdmt.

    Associativity requires that this expression equals to Fk(s1)(Fl(s2)Fm(s3)) and this gives additional conditions. Associativity conditions apply to f(DA) and could fix it highly uniquely.

  3. 2-point correlation function would be given by

    Fk(s1)Fl(s2) = ckl
    f(DA(s1,s2,s)) dms

  4. There is a clear difference between n > 3 and n=3 cases: for n > 3 also non-convex polygons are possible: this means that the interior angle associated with some vertices of the polygon is larger than p. n=4 theory is certainly well-defined, but one can argue that so are also n > 4 theories and skeptic would argue that this leads to an inflation of theories. TGD however allows only finite number of preferred points and fusion rules could eliminate the hierarchy of theories.
  5. To sum up, the general predictions are following. Quite generally, for f(0)=0 n-point correlation functions vanish if any two arguments co-incide which conforms with the spectrum of temperature fluctuations. It also implies that symplectic QFT is free of the usual singularities. For symmetry breaking scenario 3-point functions and thus also 2-point functions vanish also if s1 and s2 are at equator. All these are testable predictions using ensemble of CMB spectra.

3 Generalization to quantum TGD

Since number theoretic braids are the basic objects of quantum TGD, one can hope that the n-point functions assignable to them could code the properties of ground states and that one could separate from n-point functions the parts which correspond to the symplectic degrees of freedom acting as symmetries of vacuum extremals and isometries of the 'world of classical worlds'.

  1. This approach indeed seems to generalize also to quantum TGD proper and the n-point functions associated with partonic 2-surfaces can be decomposed in such a manner that one obtains coefficients which are symplectic invariants associated with both S2 and CP2 Kähler form.

  2. Fusion rules imply that the gauge fluxes of respective Kähler forms over geodesic triangles associated with the S2 and CP2 projections of the arguments of 3-point function serve basic building blocks of the correlation functions. The North and South poles of S2 and three poles of CP2 can be used to construct symmetry breaking n-point functions as symplectic invariants. Non-trivial 1-point functions vanish also now.

  3. The important implication is that n-point functions vanish when some of the arguments co-incide. This might play a crucial role in taming of the singularities: the basic general prediction of TGD is that standard infinities of local field theories should be absent and this mechanism might realize this expectation.

Next some more technical but elementary first guesses about what might be involved.

  1. It is natural to introduce the moduli space for n-tuples of points of the symplectic manifold as the space of symplectic equivalence classes of n-tuples.

    i) In the case of sphere S2 convex n-polygon allows n+1 3-sub-polygons and the areas of these provide symplectically invariant coordinates for the moduli space of symplectic equivalence classes of n-polygons (2n-D space of polygons is reduced to n+1-D space). For non-convex polygons the number of 3-sub-polygons is reduced so that they seem to correspond to lower-dimensional sub-space.

    ii) In the case of CP2 n-polygon allows besides the areas of 3-polygons also 4-volumes of 5-polygons as fundamental symplectic invariants. The number of independent 5-polygons for n-polygon can be obtained by using induction: once the numbers N(k,n) of independent k n-simplices are known for n-simplex, the numbers of k n+1-simplices for n+1-polygon are obtained by adding one vertex so that by little visual gymnastics the numbers N(k,n+1) are given by N(k,n+1) = N(k-1,n)+N(k,n). In the case of CP2 the allowance of 3 analogs {N,S,T} of North and South poles of S2 means that besides the areas of polygons (s1,s2,s3), (s1,s2,s3,X), (s1,s2,s3,X,Y), and (s1,s2,s3,N,S,T) also the 4-volumes of 5-polygons (s1,s2,s3,X,Y), and of 6-polygon (s1,s2,s3,N,S,T), X,Y {N,S,T} can appear as additional arguments in the definition of 3-point function.

  2. What one really means with symplectic tensor is not clear since the naive first guess for the n-point function of tensor fields is not manifestly general coordinate invariant. For instance, in the model of CMB, the components of the metric deformation involving S2 indices would be symplectic tensors. Tensorial n-point functions could be reduced to those for scalars obtained as inner products of tensors with Killing vector fields of SO(3) at S2. Again a preferred choice of quantization axis would be introduced and special points would correspond to the singularities of the Killing vector fields.

    The decomposition of Hamiltonians of the "world of classical worlds" expressible in terms of Hamiltonians of S2×CP2 to irreps of SO(3) and SU(3) could define the notion of symplectic tensor as the analog of spherical harmonic at the level of configuration space. Spin and gluon color would have natural interpretation as symplectic spin and color. The infinitesimal action of various Hamiltonians on n-point functions defined by Hamiltonians and their super counterparts is well-defined and group theoretical arguments allow to deduce general form of n-point functions in terms of symplectic invariants.

  3. The need to unify p-adic and real physics by requiring them to be completions of rational physics, and the notion of finite measurement resolution suggest that discretization of also fusion algebra is necessary. The set of points appearing as arguments of n-point functions could be finite in a given resolution so that the p-adically troublesome integrals in the formulas for the fusion rules would be replaced with sums. Perhaps rational/algebraic variants of S2×CP2=SO(3)/SO(2)×SU(3)/U(2) obtained by replacing these groups with their rational/algebraic variants are involved. Tedrahedra, octahedra, and dodecahedra suggest themselves as simplest candidates for these discretized spaces.

    Also the symplectic moduli space would be discretized to contain only n-tuples for which the symplectic invariants are numbers in the allowed algebraic extension of rationals. This would provide an abstract looking but actually very concrete operational approach to the discretization involving only areas of n-tuples as internal coordinates of symplectic equivalence classes of n-tuples. The best that one could achieve would be a formulation involving nothing below measurement resolution.

  4. This picture based on elementary geometry might make sense also in the case of conformal symmetries. The angles associated with the vertices of the S2 projection of n-polygon could define conformal invariants appearing in n-point functions and the algebraization of the corresponding phases would be an operational manner to introduce the space-time correlates for the roots of unity introduced at quantum level. In CP2 degrees of freedom the projections of n-tuples to the homologically trivial geodesic sphere S2 associated with the particular sector of CH would allow to define similar conformal invariants. This framework gives dimensionless areas (unit sphere is considered). p-Adic length scale hypothesis and hierarchy of Planck constants would bring in the fundamental units of length and time in terms of CP2 length.

The recent view about M-matrix described in is something almost unique determined by Connes tensor product providing a formal realization for the statement that complex rays of state space are replaced with N rays where N defines the hyper-finite sub-factor of type II1 defining the measurement resolution. M-matrix defines time-like entanglement coefficients between positive and negative energy parts of the zero energy state and need not be unitary. It is identified as square root of density matrix with real expressible as product of of real and positive square root and unitary S-matrix. This S-matrix is what is measured in laboratory. There is also a general vision about how vertices are realized: they correspond to light-like partonic 3-surfaces obtained by gluing incoming and outgoing partonic 3-surfaces along their ends together just like lines of Feynman diagrams. Note that in string models string world sheets are non-singular as 2-manifolds whereas 1-dimensional vertices are singular as 1-manifolds. These ingredients we should be able to fuse together. So we try once again!

  1. Iteration starting from vertices and propagators is the basic approach in the construction of n-point function in standard QFT. This approach does not work in quantum TGD. Symplectic and conformal field theories suggest that recursion replaces iteration in the construction of given generalized Feynman diagram. One starts from an n-point function and reduces it step by step to a vacuum expectation value of a 2-point function using fusion rules. Associativity becomes the fundamental dynamical principle in this process. Associativity in the sense of classical number fields has already shown its power and led to a hyper-octoninic formulation of quantum TGD promising a unification of various visions about quantum TGD .

  2. Let us start from the representation of a zero energy state in terms of a causal diamond defined by future and past directed light-cones. Zero energy state corresponds to a quantum superposition of light-like partonic 3-surfaces each of them representing possible particle reaction. These 3-surfaces are very much like generalized Feynman diagrams with lines replaced by light-like 3-surfaces coming from the upper and lower light-cone boundaries and glued together along their ends at smooth 2-dimensional surfaces defining the generalized vertices. One indeed has sum over all possible generalized Feynman diagrams.

  3. It must be emphasized that the generalization of ordinary Feynman diagrammatics arises and conformal and symplectic QFTs appear only in the calculation of single generalized Feynman diagram. Therefore one could still worry about loop corrections. The fact that no integration over loop momenta is involved and there is always finite cutoff due to discretization together with recursive instead of iterative approach gives however good hopes that everything works.

  4. One can actually simplify things by identifying generalized Feynman diagrams as maxima of Kähler function with functional integration carried over perturbations around it. Thus one would have conformal field theory in both fermionic and configuration space degrees of freedom. The light-like time coordinate along light-like 3-surface is analogous to the complex coordinate of conformal field theories restricted to some curve. If it is possible continue the light-like time coordinate to a hyper-complex coordinate in the interior of 4-D space-time sheet, the correspondence with conformal field theories becomes rather concrete. Same applies to the light-like radial coordinates associated with the light-cone boundaries. At light-cone boundaries one can apply fusion rules of a symplectic QFT to the remaining coordinates. Conformal fusion rules are applied only to point pairs which are at different ends of the partonic surface and there are no conformal singularities since arguments of n-point functions do not co-incide. By applying the conformal and symplectic fusion rules one can eventually reduce the n-point function defined by the various fermionic and bosonic operators appearing at the ends of the generalized Feynman diagram to something calculable.

  5. Finite measurement resolution defining the Connes tensor product is realized by the discretization applied to the choice of the arguments of n-point functions so that discretion is not only a space-time correlate of finite resolution but actually defines it. No explicit realization of the measurement resolution algebra N seems to be needed. Everything should boil down to the fusion rules and integration measure over different 3-surfaces defined by exponent of Kähler function and by imaginary exponent of Chern-Simons action. The continuation of the configuration space Clifford algebra for 3-surfaces with cm degrees of freedom fixed to a hyper-octonionic variant of gamma matrix field of super-string models defined in M8 (hyper-octonionic space) and M8 M4×CP2 duality leads to a unique choice of the points, which can contribute to n-point functions as intersection of M4 subspace of M8 with the counterparts of partonic 2-surfaces at the boundaries of light-cones of M8. Therefore there are hopes that the resulting theory is highly unique. Symplectic fusion algebra reduces to a finite algebra for each space-time surface if this picture is correct.

  6. Consider next some of the details of how the light-like 3-surface codes for the fusion rules associated with it. The intermediate partonic 2- surfaces must be involved since otherwise the construction would carry no information about the properties of the light-like 3-surface, and one would not obtain perturbation series in terms of the relevant coupling constants. The natural assumption is that partonic 2-surfaces belong to future/past directed light-cone boundary depending on whether they are on lower/upper half of the causal diamond. Hyper-octonionic conformal field approach fixes the nint points at intermediate partonic two-sphere for a given light-like 3-surface representing generalized Feynman diagram, and this means that the contribution is just N-point function with N=nout+nint+nin calculable by the basic fusion rules. Coupling constant strengths would emerge through the fusion coefficients, and at least in the case of gauge interactions they must be proportional to Kähler coupling strength since n-point functions are obtained by averaging over small deformations with vacuum functional given by the exponent of Kähler function. The first guess is that one can identify the spheres S2 dM4 associated with initial, final and, and intermediate states so that symplectic n-point functions could be calculated using single sphere.

These findings raise the hope that quantum TGD is indeed a solvable theory. Even if one is not willing to swallow any bit of TGD, the classification of the symplectic QFTs remains a fascinating mathematical challenge in itself. A further challenge is the fusion of conformal QFT and symplectic QFT in the construction of n-point functions. One might hope that conformal and symplectic fusion rules can be treated separately.

For details see the chapter Overall View about Quantum TGD and the article Topological Geometrodynamics: What Might Be the First Principles?.

Infinite primes and algebraic Brahman Atman identity

The hierarchy of infinite primes (and of integers and rationals) was the first mathematical notion stimulated by TGD inspired theory of consciousness. The construction recipe is equivalent with a repeated second quantization of super-symmetric arithmetic quantum field theory with bosons and fermions labeled by primes such that the many particle states of previous level become the elementary particles of new level. The hierarchy of space-time sheets with many particle states of space-time sheet becoming elementary particles at the next level of hierarchy and also the hierarchy of n:th order logics are also possible correlates for this hierarchy. For instance, the description of proton as an elementary fermion would be in a well defined sense exact in TGD Universe.

This construction leads also to a number theoretic generalization of space-time point since given real number has infinitely rich number theoretical structure not visible at the level of the real norm of the number a due to the existence of real units expressible in terms of ratios of infinite integers. This number theoretical anatomy suggest kind of number theoretical Brahman=Atman principle stating that the set consisting of number theoretic variants of single point of the imbedding space (equivalent in real sense) is able to represent the points of the world of classical worlds or even quantum states of the Universe . Also a formulation in terms of number theoretic holography is possible.

Just for fun and to test these ideas one can consider a model for the representation of the configuration space spinor fields in terms of algebraic holography. I have considered guesses for this kind of map earlier and it is interesting to find whether additional constraints coming from zero energy ontology and finite measurement resolution might give. The identification of quantum corrections as insertion of zero energy states in time scale below measurement resolution to positive or negative energy part of zero energy state and the identification of number theoretic braid as a space-time correlate for the finite measurement resolution give considerable additional constraints.

  1. The fundamental representation space consists of wave functions in the Cartesian power U8 of space U of real units associated with any point of H. That there are 8 real units rather than one is somewhat disturbing: this point will be discussed below. Real units are ratios of infinite integers having interpretation as positive and negative energy states of a super-symmetric arithmetic QFT at some level of hierarchy of second quantizations. Real units have vanishing net quantum numbers so that only zero energy states defining the basis for configuration space spinor fields should be mapped to them. In the general case quantum superpositions of these basis states should be mapped to the quantum superpositions of real units. The first guess is that real units represent a basis for configuration space spinor fields constructed by applying bosonic and fermionic generators of appropriate super Kac-Moody type algebra to the vacuum state.

  2. What can one say about this map bringing in mind Gödel numbering? Each pair of bosonic and corresponding fermionic generator at the lowest level must be mapped to its own finite prime. If this map is specified, the map is fixed at the higher levels of the hierarchy. There exists an infinite number of this kind of correspondences. To achieve some uniqueness, one should have some natural ordering which one might hope to reflect real physics. The irreps of the (non-simple) Lie group involved can be ordered almost uniquely. For simple group this ordering would be with respect to the sum N=NF+NF,c of the numbers NF resp. NF,c of the fundamental representation resp. its conjugate appearing in the minimal tensor product giving the irrep. The generalization to non-simple case should use the sum of the integers Ni for different factors for factor groups. Groups themselves could be ordered by some criterion, say dimension. The states of a given representation could be mapped to subsequent finite primes in an order respecting some natural ordering of the states by the values of quantum numbers from negative to positive (say spin for SU(2) and color isospin and hypercharge for SU(3)). This would require the ordering of the Cartesian factors of non-simple group, ordering of quantum numbers for each simple group, and ordering of values of each quantum number from positive to negative.

  3. The presence of conformal weights brings in an additional complication. One cannot use conformal as a primary orderer since the number of SO(3)×SU(3) irreps in the super-canonical sector is infinite. The requirement that the probabilities predicted by p-adic thermodynamics are rational numbers or equivalently that there is a length scale cutoff, implies a cutoff in conformal weight. The vision about M-matrix forces to conclude that different values of the total conformal weight n for the quantum state correspond to summands in a direct sum of HFFs. If so, the introduction of the conformal weight would mean for a given summand only the assignment n conformal weights to a given Lie-algebra generator. For each representation of the Lie group one would have n copies ordered with respect to the value of n and mapped to primes in this order.

  4. Cognitive representations associated with the points in a subset, call it P, of the discrete intersection of p-adic and real space-time sheets, defining number theoretic braids, would be in question. Large number of partonic surfaces can be involved and only few of them need to contribute to P in the measurement resolution used. The fixing of P means measurement of N positions of H and each point carries fermion or anti-fermion numbers. A more general situation corresponds to plane wave type state obtained as superposition of these states. The condition of rationality or at least algebraicity means that discrete variants of plane waves are in question.

  5. By the finiteness of the measurement resolution configuration space spinor field decomposes into a product of two parts or in more general case, to their superposition. The part Y+, which is above measurement resolution, is representable using the information contained by P, coded by the product of second quantized induced spinor field at points of P, and provided by physical experiments. Configuration space örbital" degrees of freedom should not contribute since these points are fixed in H.

  6. The second part of the configuration space spinor field, call it Y-, corresponds to the information below the measurement resolution and assignable with the complement of P and mappable to the structure of real units associated with the points of P. This part has vanishing net quantum numbers and is a superposition over the elements of the basis of CH spinor fields and mapped to a quantum superposition of real units. The representation of Y- as a Schrödinger amplitude in the space of real units could be highly unique. Algebraic holography principle would state that the information below measurement resolution is mapped to a Schrödinger amplitude in space of real units associated with the points of P.

  7. This would be also a representation for perceiver-external world duality. The correlation function in which P appears would code for the information appearing in M-matrix representing the laws of physics as seen by conscious entity about external world as an outsider. The quantum superposition of real units would represent the purely subjective information about the part of universe below measurement resolution.

  8. The condition that Y represents a state with vanishing quantum numbers gives additional constraints. The interpretation inspired by finite measurement resolution is that the coordinate h associated with Y corresponds to a zero energy insertion to a positive or negative energy state localizable to a causal diamond inside the upper or lower half of the causal diamond of observer. Below measurement resolution for imbedding space coordinates Y(h) would correspond to a nonlocal operator creating a zero energy state. This would mean that Brahman=Atman would apply to the mini-worlds below the measurement resolution rather than to entire Universe but by algebraic fractality of HFFs this would would not be a dramatic loss.
There is an objection against this picture. One obtains an 8-plet of arithmetic zero energy states rather than one state only. What this strange 8-fold way could mean?

  1. The crucial observation is that hyper-finite factor of type II1 (HFF) creates states for which center of mass degrees of freedom of 3-surface in H are fixed. One should somehow generalize the operators creating local HFF states to fields in H, and an octonionic generalization of conformal field suggests itself. I have indeed proposed a quantum octonionic generalization of HFF extending to an HFF valued field Y in 8-D quantum octonionic space with the property that maximal quantum commutative sub-space corresponds to hyper-octonions . This construction raises X4 M8 and by number theoretic compactification also X4 H in a unique position since non-associativity of hyper-octonions does not allow to identify the algebra of HFF valued fields in M8 with HFF itself.

  2. The value of Y in the space of quantum octonions restricted to a maximal commutative subspace can be expressed in terms of 8 HFF valued coefficients of hyper-octonion units. By the hyper-octonionic generalization of conformal invariance all these 8 coefficients must represent zero energy HFF states. The restriction of Y to a given point of P would give a state, which has 8 HFF valued components and Brahman=Atman identity would map these components to U8 associated with P. One might perhaps say that 8 zero energy states are needed in order to code the information about the H positions of points P.

For background see the chapter Was von Neumann right after all?. See also the article "Topological Geometrodynamics: an Overall View".

Configuration space gamma matrices as hyper-octonionic conformal fields having values in HFF?

The fantastic properties of HFFs of type II1 inspire the idea that a localized version of Clifford algebra of configuration space might allow to see space-time, embedding space, and configuration space as emergent structures.

Configuration space gamma matrices act only in vibrational degrees of freedom of 3-surface. One must also include center of mass degrees of freedom which appear as zero modes. The natural idea is that the resulting local gamma matrices define a local version of HFF of type II1 as a generalization of conformal field of gamma matrices appearing super string models obtained by replacing complex numbers with hyper-octonions identified as a subspace of complexified octonions. As a matter fact, one can generalize octonions to quantum octonions for which quantum commutativity means restriction to a hyper-octonionic subspace of quantum octonions . Non-associativity is essential for obtaining something non-trivial: otherwise this algebra reduces to HFF of type II1 since matrix algebra as a tensor factor would give an algebra isomorphic with the original one. The octonionic variant of conformal invariance fixes the dependence of local gamma matrix field on the coordinate of HO. The coefficients of Laurent expansion of this field must commute with octonions.

The world of classical worlds has been identified as a union of configuration spaces associated with M4 labeled by points of H or equivalently HO. The choice of quantization axes certainly fixes a point of H (HO) as a point remaining fixed under SO(1,3)×U(2) (SO(1,3)×SO(4)). The condition that hyper-quaternionic inverses of M4 HO points exist suggest a restriction of arguments of the n-point function to the interior of M4.

Associativity condition for the n-point functions forces to restrict the arguments to a hyper-quaternionic plane HQ=M4 of HO. One can also consider the commutativity condition by requiring that arguments belong to a preferred commutative sub-space HC of HO. Fixing preferred real and imaginary units means a choice of M2=HC interpreted as a partial choice of quantization axes. This has quite strong implications.

  1. The hyper-quaternionic planes with a fixed choice of M2 are labeled by points of CP2. If the condition M2 T4 characterizes the tangent planes of all points of X4 HO it is possible to map X4 HO to X4 H so that HO-H duality ("number theoretic compactification") emerges. X4 H should correspond to a preferred extremal of Kähler action. The physical interpretation would be as a global fixing of the plane of non-physical polarizations in M8: it is not quite clear whether this choice of polarization need not have direct counterpart for X4 H. Standard model symmetries emerge naturally. The resulting surface in X4 H would be analogous to a warped plane in E3. This new result suggests rather direct connection with super string models. In super string models one can choose the polarization plane freely and one expects also now that the generalized choice M2 M4 M8 of polarization plane can be made freely without losing Poincare invariance with reasonable assumption about zero energy states.

  2. One would like to fix local tangent planes T4 of X4 at 3-D light-like surfaces X3l fixing the preferred extremal of Kähler action defining the Bohr orbit. An additional direction t should be added to the tangent plane T3 of X3l to give T4. This might be achieved if t belongs to M2 and perhaps corresponds to a light-like vector in M2.

  3. Assume that partonic 2-surfaces X belong to dM4 HO defining ends of the causal diamond. This is obviously an additional boundary condition. Hence the points of partonic 2-surfaces are associative and can appear as arguments of n-point functions. One thus finds an explanation for the special role of partonic 2-surfaces and a reason why for the role of light-cone boundary. Note that only the ends of lightlike 3-surfaces need intersect M4 HO. A stronger condition is that the pre-images of light-like 3-surfaces in H belong to M4 HO.

  4. Commutativity condition is satisfied if the arguments of the n-point function belong to an intersection X2M2 HQ and this gives a discrete set of points as intersection of light-like radial geodesic and X2 perhaps identifiable in terms of points in the intersection of number theoretic braids with dH. One should show that this set of points consists of rational or at most algebraic points. Here the possibility to choose X2 to some degree could be essential. As a matter fact, any radial light ray from the tip of light-cone allows commutativity and one can consider the possibility of integrating over n-point functions with arguments at light ray to obtain maximal information. For the pre-images of light-like 3-surfaces commutativity would allow one-dimensional curves having interpretation as braid strands. These curves would be contained in plane M2 and it is not clear whether a unique interpretation as braid strands is possible (how to tell whether the strand crossing another one is infinitesimally above or below it?). The alternative assumption consistent with virtual parton interpretation is that light-like geodesics of X3 are in question.

To sum up, this picture implies HO-H duality with a choice of a preferred imaginary unit fixing the plane of non-physical polarizations globally, standard model symmetries, and number theoretic braids. The introduction of hyper-octonions could be however criticized: could octonions and quaternions be enough after all? Could HO-H duality be replaced with O-H duality and be interpreted as the analog of Wick rotation? This would mean that quaternionic 4-surfaces in E8 containing global polarization plane E2 in their tangent spaces would be mapped by essentially by the same map to their counterparts in M4×CP2,and the time coordinate in E8 would be identified as the real coordinate. Also light-cones in E8 would make sense as the inverse images of M4.

For background see the chapter Was von Neumann right after all? . See also the article "Topological Geometrodynamics: an Overall View".

How quantum classical correspondence is realized at parton level?

Quantum classical correspondence must assign to a given quantum state the most probable space-time sheet depending on its quantum numbers. The space-time sheet X4(X3) defined by the Kähler function depends however only on the partonic 3-surface X3, and one must be able to assign to a given quantum state the most probable X3 - call it X3max - depending on its quantum numbers.

X4(X3max) should carry the gauge fields created by classical gauge charges associated with the Cartan algebra of the gauge group (color isospin and hypercharge and electromagnetic and Z0 charge) as well as classical gravitational fields created by the partons. This picture is very similar to that of quantum field theories relying on path integral except that the path integral is restricted to 3-surfaces X3 with exponent of Kähler function bringing in genuine convergence and that 4-D dynamics is deterministic apart from the delicacies due to the 4-D spin glass type vacuum degeneracy of Kähler action.

Stationary phase approximation selects X3max if the quantum state contains a phase factor depending not only on X3 but also on the quantum numbers of the state. A good guess is that the needed phase factor corresponds to either Chern-Simons type action or a boundary term of YM action associated with a particle carrying gauge charges of the quantum state. This action would be defined for the induced gauge fields. YM action seems to be excluded since it is singular for light-like 3-surfaces associated with the light-like wormhole throats (not only (det(g3)1/2 but also det(g4)1/2 vanishes).

The challenge is to show that this is enough to guarantee that X4(X3max) carries correct gauge charges. Kind of electric-magnetic duality should relate the normal components Fni of the gauge fields in X4(X3max) to the gauge fields Fij induced at X3. An alternative interpretation is in terms of quantum gravitational holography. The difference between Chern-Simons action characterizing quantum state and the fundamental Chern-Simons type factor associated with the Kähler form would be that the latter emerges as the phase of the Dirac determinant.

One is forced to introduce gauge couplings and also electro-weak symmetry breaking via the phase factor. This is in apparent conflict with the idea that all couplings are predictable. The essential uniqueness of M-matrix in the case of HFFs of type II1 (at least) however means that their values as a function of measurement resolution time scale are fixed by internal consistency. Also quantum criticality leads to the same conclusion. Obviously a kind of bootstrap approach suggests itself.

For background see the chapter Overall View about Quantum TGD.

How p-adic coupling constant evolution and p-adic length scale hypothesis emerge from quantum TGD proper?

What p-adic coupling constant evolution really means has remained for a long time more or less open. The progress made in the understanding of the S-matrix of theory has however changed the situation dramatically.

1. M-matrix and coupling constant evolution

The final breakthrough in the understanding of p-adic coupling constant evolution came through the understanding of S-matrix, or actually M-matrix defining entanglement coefficients between positive and negative energy parts of zero energy states in zero energy ontology (see this). M-matrix has interpretation as a "complex square root" of density matrix and thus provides a unification of thermodynamics and quantum theory. S-matrix is analogous to the phase of Schrödinger amplitude multiplying positive and real square root of density matrix analogous to modulus of Schrödinger amplitude.

The notion of finite measurement resolution realized in terms of inclusions of von Neumann algebras allows to demonstrate that the irreducible components of M-matrix are unique and possesses huge symmetries in the sense that the hermitian elements of included factor N subset M defining the measurement resolution act as symmetries of M-matrix, which suggests a connection with integrable quantum field theories.

It is also possible to understand coupling constant evolution as a discretized evolution associated with time scales Tn, which come as octaves of a fundamental time scale: Tn=2nT0. Number theoretic universality requires that renormalized coupling constants are rational or at most algebraic numbers and this is achieved by this discretization since the logarithms of discretized mass scale appearing in the expressions of renormalized coupling constants reduce to the form log(2n)=nlog(2) and with a proper choice of the coefficient of logarithm log(2) dependence disappears so that rational number results.

2. p-Adic coupling constant evolution

One can wonder how this picture relates to the earlier hypothesis that p-adic length coupling constant evolution is coded to the hypothesized log(p) normalization of the eigenvalues of the modified Dirac operator D. There are objections against this normalization. log(p) factors are not number theoretically favored and one could consider also other dependencies on p. Since the eigenvalue spectrum of D corresponds to the values of Higgs expectation at points of partonic 2-surface defining number theoretic braids, Higgs expectation would have log(p) multiplicative dependence on p-adic length scale, which does not look attractive.

Is there really any need to assume this kind of normalization? Could the coupling constant evolution in powers of 2 implying time scale hierarchy Tn= 2nT0 induce p-adic coupling constant evolution and explain why p-adic length scales correspond to Lp propto p1/2R, p≈ 2k, R CP2 length scale? This looks attractive but there is a problem. p-Adic length scales come as powers of 21/2 rather than 2 and the strongly favored values of k are primes and thus odd so that n=k/2 would be half odd integer. This problem can be solved.

  1. The observation that the distance traveled by a Brownian particle during time t satisfies r2= Dt suggests a solution to the problem. p-Adic thermodynamics applies because the partonic 3-surfaces X2 are as 2-D dynamical systems random apart from light-likeness of their orbit. For CP2 type vacuum extremals the situation reduces to that for a one-dimensional random light-like curve in M4. The orbits of Brownian particle would now correspond to light-like geodesics γ3 at X3. The projection of γ3 to a time=constant section X2 subset X3 would define the 2-D path γ2 of the Brownian particle. The M4 distance r between the end points of γ2 would be given r2=Dt. The favored values of t would correspond to Tn=2nT0 (the full light-like geodesic). p-Adic length scales would result as L2(k)= D T(k)= D2kT0 for D=R2/T0. Since only CP2 scale is available as a fundamental scale, one would have T0= R and D=R and L2(k)= T(k)R.

  2. p-Adic primes near powers of 2 would be in preferred position. p-Adic time scale would not relate to the p-adic length scale via Tp= Lp/c as assumed implicitly earlier but via Tp= Lp2/R0= p1/2Lp, which corresponds to secondary p-adic length scale. For instance, in the case of electron with p=M127 one would have T127=.1 second which defines a fundamental biological rhythm. Neutrinos with mass around .1 eV would correspond to L(169)≈ 5 μm (size of a small cell) and T(169)≈ 104 years. A deep connection between elementary particle physics and biology becomes highly suggestive.

  3. In the proposed picture the p-adic prime p≈ 2k would characterize the thermodynamics of the random motion of light-like geodesics of X3 so that p-adic prime p would indeed be an inherent property of X3.

  4. The fundamental role of 2-adicity suggests that the fundamental coupling constant evolution and p-adic mass calculations could be formulated also in terms of 2-adic thermodynamics. With a suitable definition of the canonical identification used to map 2-adic mass squared values to real numbers this is possible, and the differences between 2-adic and p-adic thermodynamics are extremely small for large values of for p≈ 2k. 2-adic temperature must be chosen to be T2=1/k whereas p-adic temperature is Tp= 1 for fermions. If the canonical identification is defined as

    n≥ 0 bn 2n→ ∑m ≥1 2-m+10≤ n< k bn+(k-1)m2n ,

    it maps all 2-adic integers n<2k to themselves and the predictions are essentially same as for p-adic thermodynamics. For large values of p≈ 2k 2-adic real thermodynamics with TR=1/k gives essentially the same results as the 2-adic one in the lowest order so that the interpretation in terms of effective 2-adic/p-adic topology is possible.

For background see the chapter Overall View about Quantum TGD.

To the ../index page