Tuesday, March 20, 2007

Planar algebras and generalized Feynman diagrams

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

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

1. Planar algebra very briefly

First a brief definition of planar algebra.

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

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

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

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

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

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

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

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

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

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

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

3. A more detailed view

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

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

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

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

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

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

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

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

5 Comments:

At 2:50 PM, Blogger Mahndisa S. Rigmaiden said...

03 22 07

Matti:
Come by my blog. I have introductory references to category theory that may assist. If I can get them, then I know you can!

Regarding planar algebra, certainly should do more work in this area. The beauty of planar algebra pops up when discussing anyons in Chern Simons gauge in two dimensions...The dynamical braids that show up in two dimensions are interesting too because they are mystery braids.

What I mean is that since the braids encode information about particles, and since particles are anyonic, then the braids themselves are dynamical objects. How can we discuss dynamical braid s in two dimensions? You seem to have the idea.

 
At 5:54 PM, Blogger CarlBrannen said...

Matti,

I saw your comments on quantum theory over on Kea's blog. I figured that your blog would be a better place to talk. Wish I'd found it earlier.

The question is about the uniqueness of the world and the nature of quantum measurement. I follow Schwinger's measurement algebra where QM is defined entirely through the measurement process itself. A website of mine that gives links to the founding papers (from the 1950s) is at www.MeasurementAlgebra.com .

If you eliminate measurement from the quantum theory, you end up with something that can be written on a single copy of spacetime and satisfies a classical wave equation. This is not well known, but a hint on the derivation comes from the derivation of density functional theory used in condensed matter.

If one is to describe or model an event as in relativity, in the quantum theory one is faced with the fact that the event can have alternative descriptions as a wave or particle and these are ontologically incompatible. To get around this impass, one must split events according to whether they are in the past or future of the experimenter / modeler.

Events in the future take wave function descriptions. In the past they take particle. The transition between them corresponds to the mysterious present. This transformation will allow QM to be described without the requirement that the universe be split. Instead, the events of the universe are split.

Another way of saying the same thing is to say that the future already exists just as much as space far away exists. But the future is modified by the passage of time experienced by the modeler.

The idea here is to add an extra coordinate to an event. We can call it the proper age of the universe "s", at the time the event is modeled. If t < s, then one uses a wave function. If t > s, then one uses a particle description.

To get these two descriptions to be compatible, one must put the particle description into a wave form. One does this with delta functions. And one must rewrite the QM wave description to live on a single copy of space-time. Then the transition from wave to particle form is the measurement process.

Carl

 
At 6:02 PM, Blogger Kea said...

Hi everyone. Thanks for writing a post on this Matti. I like Carl's description of future/past transitions. Carl, it is these observer dependent 'split universes' that make us talk about 'multiverses' although, by definition, there is only one universe.

 
At 6:11 PM, Blogger Kea said...

Regarding: the number of lines associated with disks is even in the case of k-tangles. Don't worry, Matti. One can relax this condition on the operad. See, eg. Morrison and Nieh.

 
At 11:48 PM, Blogger Matti Pitkanen said...

Thanks in advance for Mahndisa for references. What I mean with dynamics of braids is that braid strands go through the 2-D section of light-like 3-surface. Both dance metaphor and knot metaphor for braiding apply since light-like 3-surface is intermediate between the time evolution of space-like 2-surface and and 3-D space-like surface. Since induced metric is in question light-like 3-surface can look completely stationary when seen from imbedding space. In fact, the outer boundary of any physical object can be a light-like 3-surface (note also that black-hole horizon in induced metric is also a light-like 3-surface). %%%%%%%%

Carl, your view brings in my mind Cramer's transactional interpretation. What you suggest is essentially that both particle and wave aspects have descriptions at space-time level, or more generally, the entire quantum measurement process has space-time description. Say waves in past and particules in future. The recent experiment of Afshar challenging Copenhagen interpretation (you probably know about this experiment) is an interesting test for this view. The experiment would suggest that in the spacetime region behind double slit wave picture applies and in the region after than particle picture.


My own view resembles your view except that I accept non-determinism. By quantum-classical correspondence classical dynamics of the space-time surface (which includes the dynamics of the classical gauge fields as induced gauge fields) should provide a description of the entire quantum measurement: both the situation before state function reduction and after that.

First this looks impossible since usually classical field equations are deterministic. In TGD the failure of a complete classical determinism comes in rescue so that not only quantum states but also sequences of quantum jumps have space-time correlates as sequences of fully deterministic space-time regions.


The classical non-determinism emerges in the following manner. The dynamics of 4-surface dictated by Kahler action (Maxwell action for induced Kahler form) implies that 6-D sub-theories of full 8-D theory defined in M^4xY^2, where Y^2 is any Lagrangian sub-manifold of CP_2 having by definition vanishing induced Kahler form, have only classical completely non-deterministic vacua. The induced Kahler form vanishes so that one has U(1) gauge field which is pure gauge and Kahler action vanishes and one has a vacuum extremal. Any symplectic transformation of CP_2 acts as a gauge transformation and gives a new Y^2 and new vacuum sector.

This degeneracy does not correspond to a gauge symmetry since canonical transformations are not symmetries of non-vacua: classical gravitational fields (induced metric) break the symmetry. These vacua are completely non-deterministic within Y^2 and their small deformations would give rise to classical time evolutions representing quantum jump sequences.

This also implies 4-D spin glass degeneracy of TGD Universe for which 1/f noise and quantum criticality of high T_c superconductors could be one correlate for it. %%%%%%%%%%%%


Kea, thank you for a comment on k-tangles. I would be happy to understand what even number 2k for strands means. %%%%%

Thank you for very interesting discussions! By the way, tomorrow I will leave to spend weekend at the cottage of friend so that I will not respond for few days.

 

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