### Category theory and quantum TGD: generalized Feyman diagrams and generalized planar operads

In the sequel the idea that planar operads [5,6,7,13] or their appropriate generalization might allow to formulate generalized Feynman diagrammatics in zero energy ontology will be considered. Also a description of measurement resolution and arrow of geometric time in terms of operads is discussed.

** 1. Zeroth order heuristics about zero energy states**

Consider now the existing heuristic picture about the zero energy states and coupling constant evolution provided by causal diamonds (CDs), which can be regarded as regions bounded by future directed light-cone and past directed light-cone inside it [2,3].

- The tentative description for the increase of the measurement resolution in terms CDs is that one inserts to the upper and/or lower light-like boundary of CD smaller CDs by gluing them along light-like radial ray from the tip of CD. It is also possible that the vertices of generalized Feynman diagrams belong inside smaller CDs and it turns out that these CDs must be allowed.
- The considerations related to the arrow of geometric time suggest that there is asymmetry between upper and lower boundaries of CD. The minimum requirement is that the measurement resolution is better at upper light-like boundary.
- In zero energy ontology communications to the direction of geometric past are possible and phase conjugate laser photons represent one example of this.
- Second law of thermodynamics must be generalized in such a manner that it holds with respect to subjective time identified as sequence of quantum jumps. The arrow of geometric time can however vary so that apparent breaking of second law is possible in shorter time scales at least. One must however understand why second law holds true in so good an approximation.
- One must understand also why the contents of sensory experience is concentrated around a narrow time interval whereas the time scale of memories and anticipation are much longer. The proposed mechanism is that the resolution of conscious experience is higher at the upper boundary of CD. Since zero energy states correspond to light-like 3-surfaces, this could be a result of self-organization rather than a fundamental physical law.
- CDs define the perceptive field for self. Selves are curious about the space-time sheets outside their perceptive field in the geometric future of the imbedding space and perform quantum jumps tending to shift the superposition of the space-time sheets to the direction of geometric past (past defined as the direction of shift!). This creates the illusion that there is a time=snapshot front of consciousness moving to geometric future in fixed background space-time as an analog of train illusion.
- The fact that news come from the upper boundary of CD implies that self concentrates its attention to this region and improves the resolutions of sensory experience and quantum measurement here. The sub-CDs generated in this manner correspond to mental images with contents about this region. As a consequence, the contents of conscious experience, in particular sensory experience, tend to be about the region near the upper boundary.
- This mechanism in principle allows the arrow of the geometric time to vary and depend on p-adic length scale and the level of dark matter hierarchy. The occurrence of phase transitions forcing the arrow of geometric time to be same everywhere are however plausible for the reason that the lower and upper boundaries of given CD must possess the same arrow of geometric time.
- If this is the mechanism behind the arrow of time, planar operads can provide a description of the arrow of time but not its explanation.

- CDs define the perceptive field for self. Selves are curious about the space-time sheets outside their perceptive field in the geometric future of the imbedding space and perform quantum jumps tending to shift the superposition of the space-time sheets to the direction of geometric past (past defined as the direction of shift!). This creates the illusion that there is a time=snapshot front of consciousness moving to geometric future in fixed background space-time as an analog of train illusion.

This picture is certainly not general enough, can be wrong at the level of details, and at best relates to the the whole like single particle wave mechanics to quantum field theory.

** 2. Planar operads**

The geometric definition of planar operads without using the category theoretical jargon goes as follows.

- There is an external disk and some internal disks and a collection of disjoint lines connecting disk boundaries.
- To each disk one attaches a non-negative integer k, called the color of disk. The disk with color k has k points at each boundary with the labeling 1,2,...k running clockwise and starting from a distinguished marked point, decorated by '*'. A more restrictive definition is that disk colors are correspond to even numbers so that there are k=2n points lines leaving the disk boundary boundary. The planar tangles with k=2n correspond to inclusions of HFFs.
- Each curve is either closed (no common points with disk boundaries) or joins a marked point to another marked point. Each marked point is the end point of exactly one curve.
- The picture is planar meaning that the curves cannot intersect and diks cannot overlap.
- Disks differing by isotopies preserving *'s are equivalent.

Given a planar k-tangle-one of whose internal disks has color k_{i}- and a k_{i}-tangle S, one can define the tangle T _{i}S by isotoping S so that its boundary, together with the marked points and the '*'s co-indices with that of D_{i} and after that erase the boundary of D_{i}. The collection of planar tangle together with the the composition defined in this manner- is called the colored operad of planar tangles.

One can consider also generalizations of planar operads.

- The composition law is not affected if the lines of operads branch outside the disks. Branching could be allowed even at the boundaries of the disks although this does not correspond to a generic situation. One might call these operads branched operads.
- The composition law could be generalized to allow additional lines connecting the points at the boundary of the added disk so that each composition would bring in something genuinely new. Zero energy insertion could correspond to this kind of insertions.
- TGD picture suggests also the replacement of lines with braids. In category theoretical terms this means that besides association one allows also permutations of the points at the boundaries of the disks.

The question is whether planar operads or their appropriate generalizations could allow a characterization of the generalized Feynman diagrams representing the combinatorics of zero energy states in zero energy ontology and whether also the emergence of arrow of time could be described (but probably not explained) in this framework.

** 3. Planar operads are not enough for zero energy states**

Are planar operads sufficiently powerful to code the vision about the geometric correlates for the increase of the measurement resolution and coupling constant evolution formulated in terms of CDs? Or perhaps more realistically, could one improve this formulation by assuming that zero energy states correspond to wave functions in the space of planar tangles or of appropriate modifications of them? It seems that the answer to the first question is almost affirmative.

- Disks are analogous to the white regions of a map whose details are not visible in the measurement resolution used. Disks correspond to causal diamonds (CDs) in zero energy ontology. Physically the white regions relate to the vertices of the generalized Feynman diagrams and possibly also to the initial and final states (strictly speaking, the initial and final states correspond to the legs of generalized Feynman diagrams rather than their ends).
- The composition of tangles means addition of previously unknown details to a given white region of the map and thus to an increase of the measurement resolution. This conforms with the interpretation of inclusions of HFFs as a characterization of finite measurement resolution and raises the hope that planar operads or their appropriate generalization could provide the proper language to describe coupling constant evolution and their perhaps even generalized Feynman diagrams.
- For planar operad there is an asymmetry between the outer disk and inner disks. One might hope that this asymmetry could explain or at least allow to describe the arrow of time. This is not the case. If the disks correspond to causal diamonds (CDs) carrying positive
*resp.*negative energy part of zero energy state at upper*resp.*lower light-cone boundary, the TGD counterpart of the planar tangle is CD containing smaller CDs inside it. The smaller CDs contain negative energy particles at their upper boundary and positive energy particles at their lower boundary. In the ideal resolution vertices represented 2-dimensional partonic at which light-like 3-surfaces meet become visible. There is no inherent asymmetry between positive and negative energies and no inherent arrow of geometric time at the fundamental level. It is however possible to model the arrow of time by the distribution of sub-CDs. By previous arguments self-organization of selves can lead to zero energy states for which the measurement resolution is better near the upper boundary of the CD. - If the lines carry fermion or anti-fermion number, the number of lines entering to a given CD must be even as in the case of planar operads as the following argument shows.
- In TGD framework elementary fermions correspond to single wormhole throat associated with topologically condensed CP
_{2}type extremal and the signature of the induced metric changes at the throat. - Elementary bosons correspond to pairs of wormhole throats associated with wormhole contacts connecting two space-time sheets of opposite time orientation and modellable as a piece of CP
_{2}type extremal. Each boson therefore corresponds to 2 lines within CP_{2}radius. - As a consequence the total number of lines associated with given CD is even and the generalized Feynman diagrams can correspond to a planar algebra associated with an inclusion of HFFs.

- In TGD framework elementary fermions correspond to single wormhole throat associated with topologically condensed CP
- This picture does not yet describe zero energy insertions.
- The addition of zero energy insertions corresponds intuitively to the allowance of new lines inside the smaller CD:s not coming from the exterior. The addition of lines connecting points at the boundary of disk is possible without losing the basic geometric composition of operads. In particular one does not lose the possibility to color the added tangle using two colors (colors correspond to two groups G and H which characterize an inclusion of HFFs [5]).
- There is however a problem. One cannot remove the boundaries of sub-CD after the composition of CDs since this would give lines beginning from and ending to the interior of disk and they are invisible only in the original resolution. Physically this is of course what one wants but the inclusion of planar tangles is expected to fail in its original form, and one must generalize the composition of tangles to that of CD:s so that the boundaries of sub-CD:s are not thrown away in the process.
- It is easy to see that zero energy insertions are inconsistent with the composition of planar tangles. In the inclusion defining the composition of tangles both sub-tangle and tangle induce a color to a given segment of the inner disk. If these colors are identical, one can forget the presence of the boundary of the added tangle. When zero energy insertions are allowed, situation changes as is easy to see by adding a line connecting points in a segment of given color at the boundary of the included tangle. There exists no consistent coloring of the resulting structure by using only two colors. Coloring is however possible using four colors, which by four-color theorem is the minimum number of colors needed for a coloring of planar map: this however requires that the color can change as one moves through the boundary of the included disk - this is in accordance with the physical picture.
- Physical intuition suggests that zero energy insertion as an improvement of measurement resolution maps to an improved color resolution and that the composition of tangles generalizes by requiring that the included disk is colored by using new nuances of the original colors. The role of groups in the definition of inclusions of HFFs is consistent with idea that G and H describe color resolution in the sense that the colors obtained by their action cannot be resolved. If so, the improved resolution means that G and H are replaced by their subgroups G
_{1}Þ G and H_{1}Þ H. Since the elements of a subgroup have interpretation as elements of group, there are good hopes that by representing the inclusion of tangles as inclusion of groups, one can generalize the composition of tangles.

- The addition of zero energy insertions corresponds intuitively to the allowance of new lines inside the smaller CD:s not coming from the exterior. The addition of lines connecting points at the boundary of disk is possible without losing the basic geometric composition of operads. In particular one does not lose the possibility to color the added tangle using two colors (colors correspond to two groups G and H which characterize an inclusion of HFFs [5]).
- Also CDs glued along light-like ray to the upper and lower boundaries of CD are possible in principle and -according the original proposal- correspond to zero energy insertions according. These CDs might be associated with the phase transitions changing the value of (
^{h}/_{2p}) leading to different pages of the book like structure defined by the generalized imbedding space. - p-Adic length scale hypothesis is realized if the hierarchy of CDs corresponds to a hierarchy of temporal distances between tips of CDs given as a=T
_{n}=2^{-n}T_{0}using light-cone proper time. - How this description relates to braiding? Each line corresponds to an orbit of a partonic boundary component and in principle one must allow internal states containing arbitrarily high fermion and antifermion numbers. Thus the lines decompose into braids and one must allow also braids of braids hierarchy so that each line corresponds to a braid operad in improved resolution.

** 4. Relationship to ordinary Feynman diagrammatics**

The proposed description is not equivalent with the description based on ordinary Feynman diagrams.

- In standard physics framework the resolution scale at the level of vertices of Feynman diagrams is something which one is forced to pose in practical calculations but cannot pose at will as opposed to the measurement resolution. Light-like 3-surfaces can be however regarded only locally orbits of partonic 2-surfaces since generalized conformal invariance is true only in 3-D patches of the light-like 3-surface. This means that light-like 3-surfaces are in principle the fundamental objects so that zero energy states can be regarded only locally as a time evolutions. Therefore measurement resolution can be applied also to the distances between vertices of generalized Feynman diagrams and calculational resolution corresponds to physical resolution. Also the resolution can be better towards upper boundary of CD so that the arrow of geometric time can be understood. This is a definite prediction which can in principle kill the proposed scenario.
- A further counter argument is that generalized Feynman diagrams are identified as light-like 3-surfaces for which Kähler function defined by a preferred extremal of Kähler action is maximum. Therefore one cannot pose any ad hoc rules on the positions of the vertices. One can of course insist that maximum of Kähler function with the constraint that posed by T
_{n}=2^{n}T_{0}hierarchy is in question.

It would be too optimistic to believe that the details of the proposal are correct. However, if the proposal is on correct track, zero energy states could be seen as wave functions in the operad of generalized tangles (zero energy insertions and braiding) as far as combinatorics is involved and the coherence rules for these operads would give strong constraints on the zero energy state and fix the general structure of coupling constant evolution.

## References

[1] The chapter Nuclear String Model of "p-Adic length scale Hypothesis and Hierarchy of Planck constants". [2] M. Pitkänen (2008), Quantum TGD: What Might be the General Principles?. [3] M. Pitkänen (2008), About the Nature of Time. [4] C. Rogers and A. Hoffnung (2008), Categorified symplectic geometry and the classical string. [5] Operad theory. [6] Planar algebra. [7] D. Bisch, P. Das, and S. K. Gosh (2008), Planar algebra of group-type subfactors. [8] Multicategories. [9] J. Baez (2007), Quantum Quandaries. [10] J. Baez and M. Stay (2008), Physics, topology, logic and computation: a Rosetta Stone. [11] J. Baez (2008), Categorifying Fundamental Physics. [12]Planar Algebras, TFTs with Defects. [13] M. D. Sheppeard (2007), Gluon Phenomenology and a Linear Topos, thesis.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".

## 2 Comments:

From my point of view, even planar operads are not general enough to discuss the planck scale hierarchy, because (a) they are only two dimensional (categorically speaking) (b) a dual notion (comonads) needs to be incorporated as well to define the causal diamond and (c) they are not very topos theoretic.

I am actually forced to generalize and modify planar operads in several manners.

a) I must replace lines with braids and this brings in braid operad (braid strands replaced with braids) as a substructure meaning that also lines become "white regions" of map whose resolution can be gradually improved.

b) Second modification is the replacement of disks with causal diamonds which means that one must assign to lines arrow pointing to future or to past. What was a surprise for me that already the structure of planar operad allows particle number changing reactions in TGD framework (it is essential that gauge bosons and Higgs are bound states of fermion and antifermion located at opposite light-like throats of wormhole contact, gravitons are pairs of these).

b) The third modification is the allowance of the addition of zero energy insertions in composition meaning that besides continuing the lines from the boundary of CD to interior, one can add also new lines going from the upper boundary to the lower boundary of CD.

It would be interesting to know whether any interpretation analogous to function composition exists for this geometric operation.

About your comments.

a) Braiding makes the lines genuinely 3-D. CDs are actually 4+4 dimensional as regions of the generalized imbedding space and infinite-D if regarded as sectors of world of classical worlds. The fact that 2-D Minkowski space M^2 subset M^4 plays a pivotal role in quantum TGD (number theoretic realization of gauge invariance among other things) might allow to interpret CD also as a genuinely 2-D structure.

b) About comonads I cannot say anything precise. My intuition is that zero energy ontology fuses structure and co-structure to a kind of tensor product since Hilbert space and its dual appear as a pair. My guess is that the presence of time arrow for lines corresponds to this.

The replacement of two disks in tensor product with single CD could correspond to Connes tensor product instead of ordinary tensor product leading to loss of degrees of freedom and M-matrix.

c) I understand topos as a generalization of topological space. Imbedding space H=M^4xCP_2 is locally very classical. The hierarchy of Planck constants forces to generalize H to a book like structure. Books are important: maybe some-one has built a topos theoretic model of book;-).

Also the fusion of p-adic and real imbedding spaces to single coherent whole by gluing these spaces along common rationals might be seen as topos. This structure has locally classical topology just like manifold is locally Euclidian space.

In the vertices of generalized Feynman diagrams light-like 3-surfaces meet and braid replication occurs. These surfaces are singular as 4-manifolds just like Feynman diagrams are singular as 1-manifolds. Maybe topos theory might help also here: I indeed propose that generalized Feynman diagram could be seen as a category.

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