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 r_M=constant sphere of lightcone boundary, made local with respect to the light-like radial coordinate r_M 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 g_aa 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 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 F_kF_l = c_kl^mF_m). 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) = ó õ cklmf(A(s1,s2,s3))Fm(s)dms

   Here the coefficients c_kl^m 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 dm_s 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 òc_klf(A(s₁,s₂,s))Id dm_s so that one has
   
   

   áFk(s1)Fl(s2)ñ = ó õ cklf(A(s1,s2,s))dms.

   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) º 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
   
   

   á(Fk(s1)Fl(s2))Fm(s3)ñ

   
   

   = cklr ó õ f(DA(s1,s2,s))áFr(s)Fm(s3)ñdms

   
   

   =cklrcrm ó õ f(DA(s1,s2,s)) f(DA(s,s3,t))dmsdmt.

   Associativity requires that this expression equals to áF_k(s₁)(F_l(s₂)F_m(s₃))ñ 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
   
   

   áFk(s1)Fl(s2)ñ = ckl ó õ 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 Î {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.
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 n_int 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=n_out+n_int+n_in 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 Ì 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 Construction of Quantum Theory: Symmetries of "Towards S-matrix" and the article Topological Geometrodynamics: What Might Be the First Principles?.