Could operads allow the formulation of the generalized Feynman rules?

The previous discussion of symplectic fusion rules leaves open many questions.

1. How to combine symplectic and conformal fields to what might be called symplecto-conformal fields?

2. The previous discussion applies only in super-canonical degrees of freedom and the question is how to generalize the discussion to super Kac-Moody degrees of freedom.

3. How four-momentum and its conservation in the limits of measurement resolution enters this picture?

4. At least two operads related to measurement resolution seem to be present: the operads formed by the symplecto-conformal fields and by generalized Feynman diagrams. For generalized Feynman diagrams causal diamond (CD) is the basic object whereas disks of S² are the basic objects in the case of symplecto-conformal QFT with a finite measurement resolution. These two different views about finite measurement resolution should be more or less equivalent and one should understand this equivalence at the level of details.

5. Is it possible to formulate generalized Feynman diagrammatics and improved measurement resolution algebraically?

1. How to combine conformal fields with symplectic fields?

The conformal fields of conformal field theory should be somehow combined with symplectic scalar field to form what might be called symplecto-conformal fields.

1. The simplest thing to do is to multiply ordinary conformal fields by a symplectic scalar field so that the fields would be restricted to a discrete set of points for a given realization of N-dimensional fusion algebra. The products of these symplecto-conformal fields at different points would define a finite-dimensional algebra and the products of these fields at same point could be assumed to vanish.

2. There is a continuum of geometric realizations of the symplectic fusion algebra since the edges of symplectic triangles can be selected rather freely. The integrations over the coordinates z_k (most naturally the complex coordinate of S² transforming linearly under rotations around quantization axes of angular momentum) restricted to the circle appearing in the definition of simplest stringy amplitudes would thus correspond to the integration over various geometric realizations of a given N-dimensional symplectic algebra.

Fusion algebra realizes the notion of finite measurement resolution. One implication is that all n-point functions vanish for n > N. Second implication could be that the points appearing in the geometric realizations of N-dimensional symplectic fusion algebra have some minimal distance. This would imply a cutoff to the multiple integrals over complex coordinates z_k varying along circle giving the analogs of stringy amplitudes. This cutoff is not absolutely necessary since the integrals defining stringy amplitudes are well-defined despite the singular behavior of n-point functions. One can also ask whether it is wise to introduce a cutoff that is not necessary and whether fusion algebra provides only a justification for the 1+ie prescription to avoid poles used to obtain finite integrals.

The fixed values for the quantities òA_mdx^m along the edges of the symplectic triangles could indeed pose a lower limit on the distance between the vertices of symplectic triangles. Whether this occurs depends on what one precisely means with symplectic triangle.

1. The condition that the angles between the edges at vertices are smaller than p for triangle and larger than p for its conjugate is not enough to exclude loopy edges and one would obtain ordinary stringy amplitudes multiplied by the symplectic phase factors. The outcome would be an integral over arguments z₁,z₂,..z_n for standard stringy n-point amplitude multiplied by a symplectic phase factor which is piecewise constant in the integration domain.

2. The condition that the points at different edges of the symplectic triangle can be connected by a geodesic segment belonging to the interior of the triangle is much stronger and would induce a length scale cutoff. How to realize this cutoff at the level of calculations is not clear. One could argue that this problem need not have any nice solution and since finite measurement resolution requires only finite calculational resolution, the approximation allowing loopy edges is acceptable.

Symplecto-conformal should form an operad. This means that the improvement of measurement resolution should correspond also to an algebra homomorphism in which super-canonical symplecto-conformal fields in the original resolution are mapped by algebra homomorphism into fields which contain sum over products of conformal fields at different points: for the symplectic parts of field the products reduces always to a sum over the values of field. For instance, if the field at point s is mapped to an average of fields at points s_k, nilpotency condition x²=0 is satisfied.

2. Symplecto-conformal fields in Super-Kac-Moody sector

The picture described above is an over-simplification since it applies only in super-canonical degrees of freedom. The vertices of generalized Feynman diagrams are absent from the description and CP₂ Kähler form induced to space-time surface which is absolutely essential part of quantum TGD is nowhere visible in the treatment.

How should one bring in Super Kac-Moody (SKM) algebra representing the stringy degrees of freedom in the conventional sense of the world? The condition that the basic building bricks are same for the treatment of these degrees of freedom is a valuable guideline.

1. In the transition from super-canonical to SKM degrees of freedom the light-cone boundary is replaced with the light-like 3-surface X³ representing the light-like random orbit of parton and serving as the basic dynamical object of quantum TGD. The sphere S² of light-cone boundary is in turn replaced with a partonic 2-surface X². This suggests how to proceed.

2. In the case of SKM algebra the symplectic fusion algebra is represented geometrically as points of partonic 2-surface X² by replacing the symplectic form of S² with the induced CP₂ symplectic form at the partonic 2-surface and defining U(1) gauge field. This gives similar hierarchy of symplecto-conformal fields as in the super-canonical case. This also realizes the crucial aspects of the classical dynamics defined by Kähler action. In particular, for vacuum 2-surfaces symplectic fusion algebra trivializes since Kähler magnetic fluxes vanish identically and 2-surfaces near vacua require a large value of N for the dimension of the fusion algebra since the available Kähler magnetic fluxes are small.

3. In super-canonical case the projection along light-like ray allows to map the points at the light-cone boundaries of CD to points of same sphere S². In the case of light-like 3-surfaces light-like geodesics representing braid strands allow to map the points of the partonic two-surfaces at the future and past light-cone boundaries to the partonic 2-surface representing the vertex. The earlier proposal was that the ends of strands meet at the partonic 2-surface so that braids would replicate at vertices. The properties of symplectic fields would however force identical vanishing of the vertices if this were the case. There is actually no reason to assume this condition and with this assumption vertices involving total number N of incoming and outgoing strands correspond to symplecto-conformal N-point function as is indeed natural. Also now Kähler magnetic flux induces cutoff distance.

4. SKM braids reside at light-like 3-surfaces representing lines of generalized Feynman diagrams. If super-canonical braids are needed at all, they must be assigned to the two light-like boundaries of CD meeting each other at the sphere S² at which future and past directed light-cones meet.

3. The treatment of four-momentum and other quantum numbers

Four-momentum enjoys a special role in super-canonical and SKM representations in that it does not correspond to a quantum number assignable to the generators of these algebras. It would be nice if the somewhat mysterious phase factors associated with the representation of the symplectic algebra could code for the four-momentum - or rather the analogs of plane waves representing eigenstates of four-momentum at the points associated with the geometric representation of the symplectic fusion algebra. The situation is more complex as the following considerations show.

3.1 The representation of longitudinal momentum in terms of phase factors

1. The generalized coset representation for super-canonical and SKM algebras implies Equivalence Principle in the generalized sense that the differences of the generators of two super Virasoro algebras annihilate the physical states. In particular, the four-momenta associated with super-canonical resp. SKM degrees of freedom are identified as inertial resp. gravitational four- momenta and are equal by Equivalence Principle. The question is whether four-momentum could be coded in both algebras in terms of non-integrable phase factors appearing in the representations of the symplectic fusion algebras.

2. Four different phase factors are needed if all components of four-momentum are to be coded. Both number theoretical vision about quantum TGD and the realization of the hierarchy of Planck constants assign to each point of space-time surface the same plane M² Ì M⁴ having as the plane of non-physical polarizations. This condition allows to assign to a given light-like partonic 3-surface unique extremal of Kähler action defining the Kähler function as the value of Kähler action. Also p-adic mass calculations support the view that the physical states correspond to eigen states for the components of longitudinal momentum only (also the parton model for hadrons assumes this). This encourages to think that only M² part of four-momentum is coded by the phase factors. Transversal momentum squared would be a well defined quantum number and determined from mass shell conditions for the representations of super-canonical (or equivalently SKM) conformal algebra much like in string model.

3. The phase factors associated with the symplectic fusion algebra mean a deviation from conformal n-point functions, and the innocent question is whether these phase factors could be identified as plane-wave phase factors associated with the transversal part of the four-momentum so that the n-point functions would be strictly analogous with stringy amplitudes. In fact, the identification of the phase factors exp(iòA_mdx^m/(^h/_2p)) along a path as a phase factors exp(ip_L,kDm^k) defined by the ends of the path and associated with the longitudinal part of four-momentum would correspond to an integral form of covariant constancy condition [(dx^m)/ds](¶_m-iA_m)Y = 0 along the edge of the symplectic triangle of more general path. Second phase factor would come from the integral along the (most naturally) light-like curve defining braid strand associated with the point in question. A geometric representation for the two projections of the gravitational four-momentum would thus result in SKM degrees of freedom and apart from the non-uniqueness related to the multiples of 2p the components of M² momentum could be deduced from the phase factors. If one is satisfied with the projection of momentum in M², this is enough.

4. The phase factors assignable to CP₂ Kähler gauge potential are Lorentz invariant unlike the phase factors assignable to four-momentum. One can try to resolve the problem by noticing an important delicacy involved with the formulation of quantum TGD as almost topological QFT. In order to have a non-vanishing four-momentum it is necessary to assume that CP₂ Kähler form has Kähler gauge potential having M⁴ projection, which is Lorentz invariant constant vector in the direction of the vector field defined by light-cone proper time. One cannot eliminate this part of Kähler gauge potential by a gauge transformation since the symplectic transformations of CP₂ do not induce genuine gauge transformations but only symmetries of vacuum extremals of Kähler action. The presence of the M⁴ projection is necessary for having a non-vanishing gravitational mass in the fundamental theory relying on Chern-Simons action for light-like 3-surface and the magnitude of this vector brings gravitational constant into TGD as a fundamental constant and its value is dictated by quantum criticality.

5. Since the phase of the time-like phase factor is proportional to the increment of the proper time coordinate of light-cone, it is also Lorentz invariant! Since the selection of S² fixes a rest frame, one can however argue that the representation in terms of phases is only for the rest energy in the case of massive particle. Also number theoretic approach selects a preferred rest frame by assigning time direction to the hyper-quaternionic real unit. In the case of massless particle this interpretation does not work since the vanishing of the rest mass implies that light-like 3-surface is piece of light-cone boundary and thus vacuum extremal. p-Adic thermodynamics predicting small mass even for massless particles can save the situation. Second possibility is that the phase factor defined by Kähler gauge potential is proportional to the Kähler charge of the particle and vanishes for massless particles.

6. This picture would mean that the phase factors assignable to the symplectic triangles have nothing to do with momentum. Because the space-like phase factor exp(iS_zDf/(^h/_2p)) associated with the edge of the symplectic triangle is completely analogous to that for momentum, one can argue that the symplectic triangulation should define a kind of spin network utilized in discretized approaches to quantum gravity. The interpretation raises the question about the interpretation of the quantum numbers assignable to the Lorentz invariant phase factors defined by the CP₂ part of CP₂ Kähler gauge potential.

7. By generalized Equivalence Principle one should have two phase factors also in super-canonical degrees of freedom in order to characterize inertial four-momentum and spin. The inclusion of the phase factor defined by the radial integral along light-like radial direction of the light-cone boundary gives an additional phase factor if the gauge potential of the symplectic form of the light-cone boundary contains a gradient of the radial coordinate r_M varying along light-rays. Gravitational constant would characterize the scale of the "gauge parts" of Kähler gauge potentials both in M⁴ and CP₂ degrees of freedom. The identity of inertial and gravitational four-momenta means that super-canonical and SKM algebras represent one and same symplectic field in S² and X².

8. Equivalence Principle in the generalized form requires that also the super-canonical representation allows two additional Lorentz invariant phase factors. These phase factors are obtained if the Kähler gauge potential of the light-cone boundary has a gauge part also in CP₂. The invariance under U(2) Ì SU(3) fixes the choice the gauge part to be proportional to the gradient of the U(2) invariant radial distance from the origin of CP₂ characterizing the radii of 3-spheres around the origin. Thus M⁴×CP₂ would deviate from a pure Cartesian product in a very delicate manner making possible to talk about almost topological QFT instead of only topological QFT.

3.2 The quantum numbers associated with phase factors for CP₂ parts of Kähler gauge potentials

Suppose that it is possible to assign two independent and different phase factors to the same geometric representation, in other words have two independent symplectic fields with the same geometric representation. The product of two symplectic fields indeed makes sense and satisfies the defining conditions. One can define prime symplectic algebras and decompose symplectic algebras to prime factors. Since one can allow permutations of elements in the products it becomes possible to detect the presence of product structure experimentally by detecting different combinations for products of phases caused by permutations realized as different combinations of quantum numbers assigned with the factors. The geometric representation for the product of n symplectic fields would correspond to the assignment of n edges to any pair of points. The question concerns the interpretation of the phase factors assignable to the CP₂ parts of Kähler gauge potentials of S² and CP₂ Kähler form.

1. The only reasonable interpretation for the two additional phase factors would be in terms of two quantum numbers having both gravitational and inertial variants and identical by Equivalence Principle. These quantum numbers should be Lorentz invariant since they are associated with the CP₂ projection of the Kähler gauge potential of CP₂ Kähler form.

2. Color hyper charge and isospin are mathematically completely analogous to the components of four-momentum so that a possible identification of the phase factors is as a representation of these quantum numbers. The representation of plane waves as phase factors exp(ip_kDm^k/(^h/_2p)) generalizes to the representation exp(iQ_ADF^A/(^h/_2p)), where F_A are the angle variables conjugate to the Hamiltonians representing color hyper charge and isospin. This representation depends on end points only so that the crucial symplectic invariance with respect to the symplectic transformations respecting the end points of the edge is not lost (U(1) gauge transformation is induced by the scalar j^kA_k, where j^k is the symplectic vector field in question).

3. One must be cautious with the interpretation of the phase factors as a representation for color hyper charge and isospin since a breaking of color gauge symmetry would result since the phase factors associated with different values of color isospin and hypercharge would be different and could not correspond to same edge of symplectic triangle. This is questionable since color group itself represents symplectic transformations. The construction of CP₂ as a coset space SU(3)/U(2) identifies U(2) as the holonomy group of spinor connection having interpretation as electro-weak group. Therefore also the interpretation of the phase factors in terms of em charge and weak charge can be considered. In TGD framework electro-weak gauge potential indeed suffer a non-trivial gauge transformation under color rotations so that the correlation between electro-weak quantum numbers and non-integrable phase factors in Cartan algebra of the color group could make sense. Electro-weak symmetry breaking would have a geometric correlate in the sense that different values of weak isospin cannot correspond to paths with same values of phase angles DF^A between end points.

4. If the phase factors associated with the M⁴ and CP₂ are assumed to be identical, the existence of geometric representation is guaranteed. This however gives constraints between rest mass, spin, and color (or electro-weak) quantum numbers.

3.3 Some general comments

Some further comments about phase factors are in order.

1. By number theoretical universality the plane wave factors associated with four-momentum must have values coming as roots of unity (just as for a particle in box consisting of discrete lattice of points). At light-like boundary the quantization conditions reduce to the condition that the value of light-like coordinate is rational of form m/N, if N:th roots of unity are allowed.

2. In accordance with the finite measurement resolution of four-momentum, four-momentum conservation is replaced by a weaker condition stating that the products of phase factors representing incoming and outgoing four-momenta are identical. This means that positive and negative energy states at opposite boundaries of CD would correspond to complex conjugate representations of the fusion algebra. In particular, the product of phase factors in the decomposition of the conformal field to a product of conformal fields should correspond to the original field value. This would give constraints on the trees physically possible in the operad formed by the fusion algebras. Quite generally, the phases expressible as products of phases exp(inp/p), where p £ N is prime must be allowed in a given resolution and this suggests that the hierarchy of p-adic primes is involved. At the limit of very large N exact momentum conservation should emerge.

3. Super-conformal invariance gives rise to mass shell conditions relating longitudinal and transversal momentum squared. The massivation of massless particles by Higgs mechanism and p-adic thermodynamics pose additional constraints to these phase factors.

4. What does the improvement of measurement resolution really mean?

To proceed one must give a more precise meaning for the notion of measurement resolution. Two different views about the improvement of measurement resolution emerge. The first one relies on the replacement of braid strands with braids applies in SKM degrees of freedom and the homomorphism maps symplectic fields into their products. The homomorphism based on the averaging of symplectic fields over added points consistent with the extension of fusion algebra described in previous section is very natural in super-canonical degrees of freedom. The directions of these two algebra homomorphisms are different. The question is whether both can be involved with both super-canonical and SKM case. Since the end points of SKM braid strands correspond to both super-canonical and SKM degrees of freedom, it seems that division of labor is the only reasonable option.

1. Quantum classical correspondence requires that measurement resolution has a purely geometric meaning. A purely geometric manner to interpret the increase of the measurement resolution is as a replacement of a braid strand with a braid in the improved resolution. If one assigns the phase factor assigned with the fusion algebra element with four-momentum, the conservation of the phase factor in the associated homomorphism is a natural constraint. The mapping of a fusion algebra element (strand) to a product of fusion algebra elements (braid) allows to realize this condition. Similar mapping of field value to a product of field values should hold true for conformal parts of the fields. There exists a large number equivalent geometric representations for a given symplectic field value so that one obtains automatically an averaging in conformal degrees of freedom. This interpretation for the improvement of measurement resolution looks especially natural for SKM degrees of freedom for which braids emerge naturally.

2. One can also consider the replacement of symplecto-conformal field with an average over the points becoming visible in the improved resolution. In super-canonical degrees of freedom this looks especially natural since the assignment of a braid with light-cone boundary is not so natural as with light-like 3-surface. This map does not conserve the phase factor but this could be interpreted as reflecting the fact that the values of the light-like radial coordinate are different for points involved. The proposed extension of the symplectic algebra conforms with this interpretation.

3. In the super-canonical case the improvement of measurement resolution means improvement of angular resolution at sphere S². In SKM sector it means improved resolution for the position at partonic 2-surface. For SKM algebra the increase of the measurement resolution related to the braiding takes place inside light-like 3-surface. This operation corresponds naturally to an addition of sub-CD inside which braid strands are replaced with braids. This is like looking with a microscope a particular part of line of generalized Feynman graph inside CD and corresponds to a genuine physical process inside parton. In super-canonical case the replacement of a braid strand with braid (at light-cone boundary) is induced by the replacement of the projection of a point of a partonic 2-surface to S² with a a collection of points coming from several partonic 2-surfaces. This replaces the point s of S² associated with CD with a set of points s_k of S² associated with sub-CD. Note that the solid angle spanned by these points can be rather larger so that zoom-up is in question.

4. The improved measurement resolution means that a point of S² (X²) at boundary of CD is replaced with a point set of S² (X²) assignable to sub-CD. The task is to map the point set to a small disk around the point. Light-like geodesics along light-like X³ defines this map naturally in both cases. In super-canonical case this map means scaling down of the solid angle spanned by the points of S² associated with sub-CD.

5. How do the operads formed by generalized Feynman diagrams and symplecto-conformal fields relate?

The discussion above leads to following overall view about the situation. The basic operation for both symplectic and Feynman graph operads corresponds to an improvement of measurement resolution. In the case of planar disk operad this means to a replacement of a white region of a map with smaller white regions. In the case of Feynman graph operad this means better space-time resolution leading to a replacement of generalized Feynman graph with a new one containing new sub-CD bringing new vertices into daylight. For braid operad the basic operation means looking a braid strand with a microscope so that it can resolve into a braid: braid becomes a braid of braids. The latter two views are equivalent if sub-CD contains the braid of braids.

The disks D² of the planar disk operad has natural counterparts in both super-canonical and SKM sector.

1. For the geometric representations of the symplectic algebra the image points vary in continuous regions of S² (X²) since the symplectic area of the symplectic triangle is a highly flexible constraint. Posing the condition that any point at the edges of symplectic triangle can be connected to any another edge excludes symplectic triangles with loopy sides so that constraint becomes non-trivial. In fact, since two different elements of the symplectic algebra cannot correspond to the same point for a given geometric representation, each element must correspond to a connected region of S² (X²). This allows a huge number of representations related by the symplectic transformations S² in super-canonical case and by the symplectic transformations of CP₂ in SKM case. In the case of planar disk operad different representations are related by isotopies of plane.

   This decomposition to disjoint regions naturally correspond to the decomposition of the disk to disjoint regions in the case of planar disk operad and Feynman graph operad (allowing zero energy insertions). Perhaps one might say that N-dimensional elementary symplectic algebra defines an N-coloring of S² (S²) which is however not the same thing as the 2-coloring possible for the planar operad. TGD based view about Higgs mechanism leads to a decomposition of partonic 2-surface X² (its light-like orbit X³) into conformal patches. Since also these decompositions correspond to effective discretizations of X² (X³), these two decompositions would naturally correspond to each other.

2. In SKM sector disk D² of the planar disk operad is replaced with the partonic 2-surface X² and since measurement resolution is a local notion, the topology of X² does not matter. The improvement of measurement resolution corresponds to the replacement of braid strand with braid and homomorphism is to the direction of improved spatial resolution.

3. In super-canonical case D² is replaced with the sphere S² of light-cone boundary. The improvement of measurement resolution corresponds to introducing points near the original point and the homomorphism maps field to its average. For the operad of generalized Feynman diagrams CD defined by future and past directed light-cones is the basic object. Given CD can be indeed mapped to sphere S² in a natural manner. The light-like boundaries of CDs are metrically spheres S². The points of light-cone boundaries can be projected to any sphere at light-cone boundary. Since the symplectic area of the sphere corresponds to solid angle, the choice of the representative for S² does not matter. The sphere defined by the intersection of future and past light-cones of CD however provides a natural identification of points associated with positive and negative energy parts of the state as points of the same sphere. The points of S² appearing in n-point function are replaced by point sets in a small disks around the n points.

4. In both super-canonical and SKM sectors light-like geodesic along X³ mediate the analog of the map gluing smaller disk to a hole of a disk in the case of planar disk operad defining the decomposition of planar tangles. In super-canonical sector the set of points at the sphere corresponding to a sub-CD is mapped by SKM braid to the larger CD and for a typical braid corresponds to a larger angular span at sub-CD. This corresponds to the gluing of D² along its boundaries to a hole in D² in disk operad. A scaling transformation allowed by the conformal invariance is in question. This scaling can have a non-trivial effect if the conformal fields have anomalous scaling dimensions.

5. Homomorphisms between the algebraic structures assignable to the basic structures of the operad (say tangles in the case of planar tangle operad) are an essential part of the power of the operad. These homomorphisms associated with super-canonical and SKM sector code for two views about improvement of measurement resolution and might lead to a highly unique construction of M-matrix elements.

The operad picture gives good hopes of understanding how M-matrices corresponding to a hierarchy of measurement resolutions can be constructed using only discrete data.

1. In this process the n-point function defining M-matrix element is replaced with a superposition of n-point functions for which the number of points is larger: n® å_k=1,...,m n_k. The numbers n_k vary in the superposition. The points are also obtained by downwards scaling from those of smaller S². Similar scaling accompanies the composition of tangles in the case of planar disk operad. Algebra homomorphism property gives constraints on the compositeness and should govern to a high degree how the improved measurement resolution affects the amplitude. In the lowest order approximation the M-matrix element is just an n-point function for conformal fields of positive and negative energy parts of the state at this sphere and one would obtain ordinary stringy amplitude in this approximation.

2. Zero energy ontology means also that each addition in principle brings in a new zero energy insertion as the resolution is improved. Zero energy insertions describe actual physical processes in shorter scales in principle affecting the outcome of the experiment in longer time scales. Since zero energy states can interact with positive (negative) energy particles, zero energy insertions are not completely analogous to vacuum bubbles and cannot be neglected. In an idealized experiment these zero energy states can be assumed to be absent. The homomorphism property must hold true also in the presence of the zero energy insertions. Note that the Feynman graph operad reduces to planar disk operad in absence of zero energy insertions.

The article Category Theory and Quantum TGD gives a summary of the most recent ideas about applications of category theory in TGD framework. See also the new chapter Category Theory and TGD of "Towards S-matrix".