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TGD: Physics as Infinite-Dimensional Geometry
Note: Newest contributions are at the top!
Ulla send me a link to an article by Sam Nelson about very interesting new-to-me notion known as algebraic knots, which has initiated a revolution in knot theory. This notion was introduced 1996 by Louis Kauffmann so that it is already 15 year old concept. While reading the article I realized that this notion fits perfectly the needs of TGD and leads to a progress in attempts to articulate more precisely what generalized Feynman diagrams are.
The challenge is to understand how the notion of algebraic knot could be applied to generalized Feynman diagrams. The algebraic structrures kei, quandle, rack, and biquandle and their algebraic modifications as such are not enough. The lines of Feynman graphs are replaced by braids and in vertices braid strands redistribute. This poses several challenges: the crossing associated with braiding and crossing occurring in non-planar Feynman diagrams should be integrated to a more general notion; braids are replaced with sub-manifold braids; braids of braids ....of braids are possible; the redistribution of braid strands in vertices should be algebraized. One challenge is to abstract the basic operations which should be algebraized in the case of generalized Feynman diagrams.
One should be also able to concretely identify braids and 2-braids (string world sheets) as well as partonic 2-surfaces and I have discussed several identifications during last years. Legendrian braids turn out to be very natural candidates for braids and their duals for the partonic 2-surfaces. String world sheets in turn could correspond to the analogs of Lagrangian sub-manifolds or to minimal surfaces of space-time surface satisfying the weak form of electric-magnetic duality. The latter option turns out to be more plausible. Finite measurement resolution would be realized as symplectic invariance with respect to the subgroup of the symplectic group leaving the end points of braid strands invariant. In accordance with the general vision TGD as almost topological QFT would mean symplectic QFT. The identification of braids, partonic 2-surfaces and string world sheets - if correct - would solve quantum TGD explicitly at string world sheet level in other words in finite measurement resolution.
Irrespective of whether the algebraic knots are needed, the natural question is what generalized Feynman diagrams are. It seems that the basic building bricks can be identified so that one can write rather explicit Feynman rules already now. Of course, the rules are still far from something to be burned into the spine of the first year graduate student.
For details and background see the article Algebraic braids, sub-manifold braid theory, and generalized Feynman diagrams or the chapter Knots and TGD.
The general form of ansatz for preferred extremals implies that the Coulombic term in Kähler action vanishes so that it reduces to 3-dimensional surface terms in accordance with general coordinate invariance and holography. The weak form of electric-magnetic duality in turn reduces this term to Chern-Simons terms.
The strong form of General Coordinate Invariance implies effective 2-dimensionality (holding true in finite measurement resolution) so that also a strong form of holography emerges. The expectation is that Chern-Simons terms in turn reduces to 2-dimensional surface terms.
The only physically interesting possibility is that these 2-D surface terms correspond to areas for minimal surfaces defined by string world sheets and partonic 2-surfaces appearing in the solution ansatz for the preferred extremals. String world sheets would give to Kähler action an imaginary contribution having interpretation as Morse function. This contribution would be proportional to their total area and assignable with the Minkowskian regions of the space-time surface. Similar but real string world sheet contribution defining Kähler function comes from the Euclidian space-time regions and should be equal to the contribution of the partonic 2-surfaces. A natural conjecture is that the absolute values of all three areas are identical: this would realize duality between string world sheets and partonic 2-surfaces and duality between Euclidian and Minkowskian space-time regions.
Zero energy ontology combined with the TGD analog of large Nc expansion inspires an educated guess about the coefficient of the minimal surface terms and a beautiful connection with p-adic physics and with the notion of finite measurement resolution emerges. The t'Thooft coupling λ should be proportional to p-adic prime p characterizing particle. This means extremely fast convergence of the counterpart of large Nc expansion in TGD since it becomes completely analogous to the pinary expansion of the partition function in p-adic thermodynamics. Also the twistor description and its dual have a nice interpretation in terms of zero energy ontology. This duality permutes massive wormhole contacts which can have off mass shell with wormhole throats which are always massive (also for the internal lines of the generalized Feynman graphs).
For details and background see the article Is Kähler action expressible in terms of areas of minimal surfaces? or the chapter Identification of configuration space Kähler function.
Weak form of electric magnetic duality and duality between Minkowskian and Euclidian space-time regions
The reduction of the Kähler action for the space-time sheets with Minkowskian signature of the induced metric follows from the assumption that Kähler current is proportional to instanton current and from the weak form of electric-magnetic duality. The first property implies a reduction to a 3-D term associated with wormhole throats and the latter property reduces this term to Abelian Chern-Simons term. I have not explicitly considered whether the same happens in the 4-D regions of Euclidian signature representing wormhole contacts.
If these assumptions are made also in the Euclidian region, the outcome is that one obtains a difference of two Chern-Simons terms coming from Minkowskian and Euclidian regions at light-like wormhole throats. This difference can be non-trivial since Kähler form for CP2 defines non-trivial U(1) bundle. This however suggests that the total Kähler action is quantized in integer multiples of the Kähler action for CP2 type vacuum extremal so that one would have effectively sum over n-instanton configurations.
If the Kähler function of the "world of classical worlds" (WCW) is identified as total Kähler action, this implies the vanishing of the Kähler metric of WCW, which is a catastrophe. Should one modify the definition of Kähler function by considering only the contribution from either Minkowskian or Euclidian regions? What about vacuum functional: should one identify it as the exponent of Kähler function or of Kähler action in this case?
The overall conclusion is that the only reasonable definitions of Kähler function of WCW and vacuum functional realize the conjecture duality between Minkowskian and Euclidian regions of space-time surfaces. The overall conclusion is that the only reasonable definitions of Kähler function of WCW and vacuum functional realize the conjecture duality between Minkowskian and Euclidian regions of space-time surfaces. This duality would have also number theoretical interpretation. Minkowskian regions of the space-time surface would correspond to hyper-quaternionic and Eulidian regions to quaternionic regions. In hyper-quaternionic regions the modified gamma matrices would span hyper-quaternionic plane of complexified octonions (imaginary units multiplied by commutative imaginary unit). In quaternionic regions the modified gamma matrices multiplied by a product of fixed octonionic imaginary unit and commutative imaginary unit would span a quaternionic plane of complexified octonions (see Does the Modified Dirac Equation Define the Fundamental Variational Principle.
The following argument suggests that the effective metric defined by the anti-commutators of the modified gamma matrices is effectively one- or two-dimensional. Effective one-dimensionality would conform with the observation that the solutions of the modified Dirac equations can be localized to one-dimensional world lines in accordance with the vision that finite measurement resolution implies discretization reducing partonic many-particle states to quantum superpositions of braids. This localization to 1-D curves occurs always at the 3-D orbits of the partonic 2-surfaces.
The argument is based on the following assumptions.
Khovanov homology generalizes the Jones polynomial as knot invariant. The challenge is to find a quantum physical construction of Khovanov homology analous to the topological QFT defined by Chern-Simons action allowing to interpret Jones polynomial as vacuum expectation value of Wilson loop in non-Abelian gauge theory.
Witten's approach to Khovanov homology relies on fivebranes as is natural if one tries to define 2-knot invariants in terms of their cobordisms involving violent un-knottings. Despite the difference in approaches it is very useful to try to find the counterparts of this approach in quantum TGD since this would allow to gain new insights to quantum TGD itself as almost topological QFT identified as symplectic theory for 2-knots, braids and braid cobordisms.
An essentially unique identification of string world sheets and therefore also of the braids whose ends carry quantum numbers of many particle states at partonic 2-surfaces emerges if one identifies the string word sheets as singular surfaces in the same manner as is done in Witten's approach. Even more, the conjectured slicings of preferred extremals by 3-D surfaces and string world sheets central for quantum TGD can be identified uniquely. The slicing by 3-surfaces would be interpreted in gauge theory in terms of Higgs= constant surfaces with radial coordinate of CP2 playing the role of Higgs. The slicing by string world sheets would be induced by different choices of U(2) subgroup of SU(3) leaving Higgs=constant surfaces invariant.
Also a physical interpretation of the operators Q, F, and P of Khovanov homology emerges. P would correspond to instanton number and F to the fermion number assignable to right handed neutrinos. The breaking of M4 chiral invariance makes possible to realize Q physically. The finding that the generalizations of Wilson loops can be identified in terms of the gerbe fluxes ∫ HA J supports the conjecture that TGD as almost topological QFT corresponds essentially to a symplectic theory for braids and 2-knots.
I do not bother to type the details but give a link to the article Could one generalize braid invariant defined by vacuum expectation of Wilson loop to an invariant of braid cobordisms and of 2-knots?. See also the new chapter Knots and TGD.
Could one generalize braid invariant defined by vacuum expectation of Wilson loop to an invariant of braid cobordisms and of 2-knots?
Lubos gave a link to a recent talk of Witten about knots and quantum physics. While listening the lecture one senses the enormous respect and -I dare say- love that the audience feels towards this silently talking genius completely free of all what might be called ego. Warmly recommended.
Witten manages to explain in rather comprehensible manner both the construction recipe of Jones polynomial and the idea about how Jones polynomial emerges from topological quantum field theory as a vacuum expectation of so called Wilson loop defined by path integral with weighting coming from Chern-Simons action. Witten also tells that during the last year he has been working with an attempt to understand in terms of quantum theory the so called Khovanov polynomial associated with a much more abstract link invariant whose interpretation and real understanding remains still open.
This kind of talks are extremely inspiring and lead to a series of questions unavoidably culminating to the frustrating "Why I do not have the brain of Witten making perhaps possible to answer these questions?". This one must just accept. In the following I summarize some thoughts inspired by the associations of the talk of Witten with quantum TGD and with the model of DNA as topological quantum computer. In my own childish manner I dare believe that these associations are interesting and dare also hope that some more brainy individual might take them seriously.
An idea inspired by TGD approach which also main streamer might find interesting is that the Jones invariant defined as vacuum expectation for a Wilson loop in 2+1-D space-time generalizes to a vacuum expectation for a collection of Wilson loops in 2+2-D space-time and could define an invariant for 2-D knots and for cobordisms of braids analogous to Jones polynomial. As a matter fact, it turns out that a generalization of gauge field known as gerbe is needed and that in TGD framework classical color gauge fields defined the gauge potentials of this field. Also topological string theory in 4-D space-time could define this kind of invariants. Of course, it might well be that this kind of ideas have been already discussed in literature. As reader have noticed, the posting has gradually evolved during last days as I have noticed elementary errors and inaccuracies. My apologies for possible inconvenience.
1. Some TGD background
What makes quantum TGD interesting concerning the description of braids and braid cobordisms is that braids and braid cobordisms emerge both at the level of generalized Feynman diagrams and in the model of DNA as a topological quantum computer.
1.1 Time-like and space-like braidings for generalized Feynman diagrams
Time like braidings induces space-like braidings and one can speak of time-like or dynamical braiding and even duality of time-like and space-like braiding. What happens can be understood in terms of dance metaphor.
1.3 DNA as topological quantum computer
The model for topological quantum computation is based on the idea that time-like braidings defining topological quantum computer programs. These programs are robust since the topology of braiding is not affected by small deformations.
2. Could braid cobordisms define more general braid invariants?
Witten says that one should somehow generalize the notion of knot invariant. The above described framework indeed suggests a very natural generalization of braid invariants to those of braid cobordisms reducing to braid invariants when the braid at the other end is trivial. This description is especially natural in TGD but allows a generalization in which Wilson loops in 4-D sense describe invariants of braid cobordisms.
2.1 Difference between knotting and linking
Before my modest proposal of a more general invariant some comments about knotting and linking are in order.
2.2 Topological strings in 4-D space-time define knot cobordisms
What makes the 4-D braid cobordisms interesting is following.
3. Invariants 2-knots as vacuum expectations of Wilson loops in 4-D space-time?
The interpretation of string world sheets in terms of Wilson loops in 4-dimensional space-time is very natural. This raises the question whether Witten's a original identification of the Jones polynomial as vacuum expectation for a Wilson loop in 2+1-D space might be replaced with a vacuum expectation for a collection of Wilson loops in 3+1-D space-time and would characterize in the general case (multi-)braid cobordism rather than braid. If the braid at the lower or upped boundary is trivial, braid invariant is obtained. The intersections of the Wilson loops would correspond to the violent un-knotting operations and the boundaries of the resulting holes give an additional Wilson loop. An alternative interpretation would be as the analog of Jones polynomial for 2-D knots in 4-D space-time generalizing Witten's theory. This description looks completely general and does not require TGD at all.
The following considerations suggest that Wilson loops are not enough for the description of general 2-knots and that Wilson loops must be replaced with 2-D fluxes. This requires a generalization of gauge field concept so that it corresponds to a 3-form instead of 2-form is needed. In TGD framework this kind of generalized gauge fields exist and their gauge potentials correspond to classical color gauge fields.
3.1 What 2-knottedness means concretely?
It is easy to imagine what ordinary knottedness means. One has circle imbedded in 3-space. One projects it in some plane and looks for crossings. If there are no crossings one knows that un-knot is in question. One can modify a given crossing by forcing the strands to go through each other and this either generates or removes knottedness. One can also destroy crossing by reconnection and this always reduces knottedness. Since knotting reduces to linking in 3-D case, one can find a simple interpretation for knottedness in terms of linking of two circles. For 2-knots linking is not what gives rise to knotting.
One might hope to find something similar in the case of 2-knots. Can one imagine some simple local operations which either increase of reduce 2-knottedness?
Whether all possible 2-knots are allowed for stringy world sheets, is not clear. In particular, if they are dynamically determined it might happen that many possibilities are not realized. For instance, the condition that the signature of the induced metric is Minkowskian could be an effective killer of 2-knottedness not reducing to braid cobordism.
Suppose that the space-like braid strands connecting partonic 2-surfaces at given boundary of CD and light-like braids connecting partonic 2-surfaces belonging to opposite boundaries of CD form connected closed strands. The collection of closed loops can be identified as boundaries of Wilson loops and the expectation value is defined as the product of traces assignable to the loops. The definition is exactly the same as in 2+1-D case.
Is this generalization of Wilson loops enough to describe 2-knots? In the spirit of the proposed philosophy one could ask whether there exist two-knots not reducible to cobordisms of 1-knots whose knot invariants require cobordisms of 2-knots and therefore 2-braids in 5-D space-time. Could it be that dimension D=4 is somehow very special so that there is no need to go to D=5? This might be the case since for ordinary knots Jones polynomial is very faithful invariant.
Innocent novice could try to answer the question in the following manner. Let us study what happens locally as the 2-D closed surface in 4-D space gets knotted.
In the sequel the considerations are restricted to TGD and to a comparison of Witten's ideas with those emerging in TGD framework.
4.1 Weak form of electric-magnetic duality and duality of space-like and time-like braidings
Witten notices that much of his work in physics relies on the assumption that magnetic charges exist and that rather frustratingly, cosmic inflation implies that all traces of them disappear. In TGD Universe the non-trivial topology of CP2makes possible Kähler magnetic charge and inflation is replaced with quantum criticality. The recent view about elementary particles is that they correspond to string like objects with length of order electro-weak scale with Kähler magnetically charged wormhole throats at their ends. Therefore magnetic charges would be there and LHC might be able to detect their signatures if LHC would get the idea of trying to do this.
Witten mentions also electric-magnetic duality. If I understood correctly, Witten believes that it might provide interesting new insights to the knot invariants. In TGD framework one speaks about weak form of elecric magnetic duality. This duality states that Kähler electric fluxes at space-like ends of the space-time sheets inside CDs and at wormhole throats are proportional to Kähler magneic fluxes so that the quantization of Kähler electric charge quantization reduces to purely homological quantization of Kähler magnetic charge.
The weak form of electric-magnetic duality fixes the boundary conditions of field equations at the light-like and space-like 3-surfaces. Together with the conjecture that the Kähler current is proportional to the corresponding instanton current this implies that Kähler action for the preferred extremal sof Kähler action reduces to 3-D Chern-Simons term so that TGD reduces to almost topological QFT. This means an enormous mathematical simplification of the theory and gives hopes about the solvability of the theory. Since knot invariants are defined in terms of Abelian Chern-Simons action for induced Kähler gauge potential, one might hope that TGD could as a by-product define invariants of braid cobordisms in terms of the unitary U-matrix of the theory between zero energy states and having as its rows the non-unitary M-matrices analogous to thermal S-matrices.
Electric magnetic duality is 4-D phenomenon as is also the duality between space-like and time like braidings essential also for the model of topological quantum computation. Also this suggests that some kind of topological string theory for the space-time sheets inside CDs could allow to define the braid cobordism invariants.
4.2 Could Kähler magnetic fluxes define invariants of braid cobordisms?
Can one imagine of defining knot invariants or more generally, invariants of knot cobordism in this framework? As a matter fact, also Jones polynomial describes the process of unknotting and the replacement of unknotting with a general cobordism would define a more general invariant. Whether the Khovanov invariants might be understood in this more general framework is an interesting question.
4.3 Classical color gauge fields and their generalizations define gerbe gauge potentials allowing to replace Wilson loops with Wilson sheets
As already noticed, the description of 2-knots seems to necessitate the generalization of gauge field to 3-form and the introduction of a gerbe structure. This seems to be possible in TGD framework.
This picture is very speculative and sounds too good to be true but follows if one consistently applies holography.
5. Summing up
Let us summarize the ideas discussed above.
What is interesting that twistorial considerations lead to a conjecture that 4-D space-time surfaces in 8-D imbedding space have a dual description in terms of certain 6-D homomorphic surfaces which are sphere bundles in 12-D CP3× CP3 and effectively 4-D. This suggests a connection between descriptions based on topological strings in 6-D space and Wilson loops in 4-D space-time. Could it really be that these completely trivial observations of a mad Finnish scientist are not a standard part of knot theory?
Addition. I found from web an article by Dror Bar-Natan with title Khovanov's homology for tangles and cobordisms. The article states that the Khovanov Homology theory for knots and links generalizes to tangles, cobordisms and 2-knots. The articles says nothing explicit about Wilson loops but talks about topological QFTs.
Addition. An article of Witten about his physical approach to Khovanov homology has appeared in arXiv. The article is more or less abracadabra for anyone not working with M-theory but the basic idea is simple. Witten reformulates 3-D Chern-Simons theory as a path integral for N=4 super YM theory in the 4-D half space W×R. This allows him to use dualities and bring in the machinery of M-theory and branes. The basic structure of TGD forces a highly analogous appproach: replace 3-surfaces with 4-surfaces, consider knot cobordisms and also 2-knots, introduce gerbes, and be happy with symplectic instead of topological QFT, which might more or less be synonymous with TGD as almost topological QFT. Symplectic QFT would obviously make possible much more refined description of knots.
This posting can be found also as a more organized article Could one generalize braid invariant defined by vacuum expectation of Wilson loop to an invariant of braid cobordisms and of 2-knots?. See also the new chapter Knots and TGD.