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considerable progress in generalized Feynman diagrammatics
by Dr. Matti Pitkänen / May 22, 2010
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The following is expanded and somewhat edited response in Kea's blog. For reasons that should become obvious, the response deserves to be published also here although I have done this implicitly via links to pdf files in earlier postings. My sincere hope is that at least one single really intelligent reader might realize what is involved;-). This might be enough.
I have been working with twistor program inspired ideas in TGD framework for a couple of years. The basic conceptual elements are following.
1. The notion of generalized Feyman diagram defined by replacing lines of ordinary Feynman diagram with light-like 3-surfaces (elementary particle sized wormhole contacts with throats carrying quantum numbers) and vertices identified as their 2-D ends (I call them partonic 2-surfaces). Speaking somewhat loosely, generalized Feynman diagrams plus background space-time sheets define the "World of Classical Worlds" (WCW).
2. Zero Energy Ontology (ZEO) and causal diamonds (intersections of Future- and Past-directed lightcones). The crucial observation is that in ZEO it is possible to identify off-mass shell particles as pairs of on mass shell particles at throats of wormhole contact since both positive and negative signs of energy are possible.
The propagator defined by modified Dirac action does not diverge (except for incoming lines) although the fermions at throats are on mass shell. In other words, the generalized eigenvalue of the modified Dirac operator containing a term linear in momentum is non-vanishing and propagator reduces to G=i/λγwhere γ is modified gamma matrix in the direction of stringy coordinate. This means opening of the black box of off-mass shell particle-something which for some reason has not occurred to anyone fighting with the divergences of QFTs.
3. Representation of 8-D gamma matrices in terms of octonionic units and 2-D sigma matrices. Modified gamma matrices at space-time surfaces are quaternionic/associative and allow a genuine matrix representation. As a matter of fact, TGD and WCW can be formulated as study of associative local sub-algebras of the local Clifford algebra of 8-D imbedding space parameterized by quaternionic space-time surfaces. Central conjecture is that quaternionic 4-surfaces correspond to preferred extremals of Kähler action identified as critical ones (second variation of Kähler action vanishes for infinite number of deformations defining super-conformal algebra) and allow a slicing to string worldsheets parametrized by points of partonic 2-surfaces.
4. Number theoretic universality requiring the existence of Feynman amplitudes in all number fields when one allows suitable algebraic extensions: roots of unity are certainly required in order to realize plane waves. Also imbedding space, partonic 2-surfaces, and WCW must exist in all number fields and their extensions. These constraints are enormously powerful and the attempts to realize this vision have dominated Quantum-TGD for last 20 years.
5. As far as twistors are considered, the first key element is the reduction of the octonionic twistor structure to quaternionic one at space-time surfaces and giving effectively 4-D spinor and twistor structure for quaternionic surfaces.
Recently, quite a dramatic progress took place in this approach. It was just the simple observation (I should have made if for already half year ago!) that on-mass shell property puts enormously strong kinematic restrictions on the loop integrations. With mild restrictions on the number of parallel fermion lines appearing in vertices (there can be several since fermionic oscillator operator algebra defining SUSY algebra generates the parton states) -- all loops are manifestly finite and if particles has always mass (say small p-adic thermal mass also in case of massless particles and due to IR cutoff due to the presence largest CD) -- the number of diagrams is finite. Unitarity reduces to Cutkosky rules automatically satisfied as in the case of ordinary Feynman diagrams.
This is about momentum space aspects of Feynman diagrams but not yet about the functional (not path) integral over small deformations of the partonic 2-surfaces. It took some time to see that also the functional integrals over WCW can be carried out at general level both in real and p-adic context.
1. The p-adic generalization of Fourier analysis allows to algebraize integration (the horrible looking technical challenge of p-adic physics) for symmetric spaces for functions allowing the analog of discrete Fourier decomposion. Symmetric space property is indeed essential also for the existence of Kähler geometry for infinite-D spaces as was learned already from the case of loop spaces.
Plane waves and exponential functions expressible as roots of unity and powers of p multiplied by the direct analogs of corresponding exponent functions are the basic building bricks and key functions in harmonic analysis in symmetric spaces. The physically unavoidable finite measurement resolution corresponds to algebraically unavoidable finite algebraic dimension of algebraic extension of p-adics (at least some roots of unity are needed). The cutoff in roots of unity is very reminiscent to that occurring for the representations of quantum groups and is certainly very closely related to these as also to the inclusions of hyper-finite factors of Type II1 defining the finite measurement resolution.
2. WCW geometrization reduces to that for a single line of the generalized Feynman diagram defining the basic building brick for WCW. Kähler function decomposes to a sum of "kinetic" terms associated with its ends and interaction term associated with the line itself. p-Adicization boils down to the condition that Kähler function, matrix elements of Kähler form, WCW Hamiltonians and their super counterparts are rational functions of complex WCW coordinates just as they are for those symmetric spaces that I know of. This allows straightforward continuation to p-adic context. Incredibly simple!
3. As far as diagrams are considered, everything is manifestly finite as the general arguments (non-locality of Kähler function as functional of 3-surface) developed two decades ago indeed allow to expect. General conditions on the holomorphy properties of the generalized eigenvalues λ of the modified Dirac operator can be deduced from the conditions that propagator decomposes to a sum of products of harmonics associated with the ends of the line and that similar decomposition takes place for exponent of Kähler action identified as Dirac determinant.
This guarantees that the convolutions of propagators and vertices give rise to products of harmonic functions which can be Glebsch-Gordanized to harmonics and only the singlet contributes to the WCW integral in given vertex. The still unproven central conjecture is that Dirac determinant equals the exponent of Kähler function.
Ironically, twistors which stimulated all these development do not seem to be absolutely necessary in this approach although they are of course possible. The situation changes if one does not assumes small p-adically thermal mass due to the presence of massless particles and one must sum infinite number of diagrams. Here a potential problem is whether the infinite sum respects the algebraic extension in question.
For a more detailed representation of generalized Feynman diagrammatics, see the last section of the pdf article "Weak form of Electric-Magnetic Duality, Electroweak Massivation, and Color Confinement". For Feynman diagrams and WCW integration, see the article "How to define Generalized Feynman diagrams" summarizing the basic formulas. See also the chapter "Does the Modified Dirac Equation Define the Fundamental Action Principle?"
Comments
1. At 7:41 PM, Matti Pitkanen said...
To Ulla:
Twistor ideas were only the starting point leading to the progress in the understanding what generalized Feynman diagrams could be. The really great news about which I am desperately shouting at the noisy market place of ideas are following.
a) These individual diagrams are free of infinities for a reason which even a child able to integrate can understand;-). Cutkosky rules garanteeing unitarity apply.
b) For a given process their number is finite by the p-adic thermal massivation (and assuming that generalized SUSY cancels self energy loops).
c) This approach strongly suggests the solution of the really horrible looking problem of p-adicizing the real worlds of classical worlds.
All this shouting is about something which sounds rather technical and begins to make sense only with basic background in QFTs and after one has understood the underlying motivations which come from ideas like physics as infinite-D geometry of WCW and number theoretical universality.
Generalized Feynman diagrams are not identical to ordinary ones and standard twistor approach as such does not work. The notion of "twistor" must be modified.
All these results are physicist's very humble and very non-rigorous counterparts for what mathematician would call existence theorems.
Twistor approach to N=4 SUSY is about a system in which one has only gluons and their superpartners. As such, this system is not physically very interesting. Iit is an extremely useful product of theoretical experimentation. Nothing to do with Chemistry.
2. At 7:59 PM, Matti Pitkanen said...
This idea about Classically-connected space-time as quantum superpositions of disconnected space-times is interesting. One of the ideas which have been in TGD for decades albeit in different and more realistic form and has generated no fuss (possibly because the author does not speak native American English and even worse -speaks native Finnish;-)).
In TGD framework, the idea is slightly different. I speak of partonic 2-surfaces (or by holography space-like 3-surfaces) at the ends of generalized Feynman diagrams. They are indeed disjoint and quantum states associated with them are entangled. In particular, time-like entanglement is in question for M-matrix and for generalized Feynman diagrams associated with it. Four to two! This is the first difference.
Here comes second difference: 3-D light-like-surfaces/4-D space-time surfaces connecting these disjoint partonic 2-surfaces/space-like 3-surfaces provide the space-time correlate for this entanglement. Entanglement has space-time correlate. This is the second central notion which does not seem to be present in the proposal (on basis of the citation), which as such is from my point of view incorrect.
The disjointness of partonic 2-surfaces is absolutely essential for the integration over WCW to work.
Here is extreme simplification for entanglement geometry correlation: take two points and entangle the corresponding states and represent entanglement by connecting them by line.
Why these fellows do not introduce geometric entanglement in the manner I do it (as I bravely speculate without time to read the article!), is brutally simple. They speak about space-times as abstract manifolds and have not yet quite realized the space-time--Feynman diagram connection;-). When one starts to babble about space-times as 4-surfaces, the TGD based view pops up within few decades;-)
3. At 1:33 PM, Ulla said...
Can you give your view on emergent space? Seen in the light of Feynman diagrams, it seems nonsense talk to me. Also particles not seen or measured must have some kind of (wave?)structure. Measurement, also kinematic interference from the environment, must relax that structure, tough very slightly?
To talk of abstract, thought vectors or spinors (not baryonic or non-baryonic) is the same as to say Feynman diagrams are fantasies?
Kea said the emergent space was no creation but interference is the same as measurements and that is entanglements and creation. Also Verlinde talked of emergent space.
The pattern (clumped) of dark matter also talks for this view? Am I completely sailing out in the blue?
4. At 7:55 PM, Matti Pitkanen said...
I do not believe in emergent space. The notion is hopelessly poorly defined and means giving up the vision about geometrization of Physics which has been enormously successful.
This emergence idea is to some degree present already in string theory and M-theory in the sense that gravitation is assigned to strings in 10-/11-D space-time rather than 4-D space-time at fundamental level and the mysterious spontaneous compactification is hoped to lead to the observed 4-D physics. It did not work as we know now. The extreme elegance and beauty of Einstein's theory is replaced with endless construction of ad hoc models and the great question is now whether this theory (I should actually talk in plural) can make at least single clear-cut prediction.
TGD is based on generalization of the geometrization program. Geometrize not only Classical physical but also Quantum physics and do this in terms of the geometry of "World of Classical Worlds". There is also a generalization of the concepts of space-time and imbedding space. Many-sheeted space-time, p-adic variants of imbedding space, p-adic space-time sheets, book-like structure of the imbedding space to describe dark matter in terms of a hierarchy of Planck constants. The basic observation is that infinite-D geometry is extremely unique from the mere requirement of mathematical existence. Unfortunately, a wrong person discovered this so that colleagues prefer to wander in M-theory landscape!
One particular fascinating aspect of geometrization of quantum physics is the reduction of Feynman graphs to space-time geometry and the power of geometrization becomes manifest when combined with Zero Energy Ontology inspired view about virtual particles. The solution to the divergence problem of QFT theories is the dream of any young theoretician and this approach realizes this dream.
Discreteness often claimed to be the nature of space-time in Planck length scales and it is claimed that continuum somehow emerges from discreteness. This is self deception. In TGD, discreteness is not fundamental but serves as a space-time correlate for a finite measurement resolution. The resulting theory is much more interesting manner than misty attempts postulates about discrete fundamental structures. Number theoretical Quantum Field Theory emerges as a new discipline (something which I leave for younger ones when the time is mature).
5. At 12:42 AM, Ulla said...
One thing more. When I read about leptons and baryons, I see nowhere the gluons although they must be perhaps more important for the condensation. Gluons are the force that keep everything together. Are something forgotten here?
6. At 1:12 AM, Matti Pitkanen said...
Leptons and quarks are the really fundamental particles in TGD. Bosonic emergence means that bosons emerge as wormhole contacts with fermion and antifermion at opposite light-like throats. That is bound states of fermion and antifermion. This leads to a formulation of supersymmetry QFT limit using only the analog of Dirac action. No needed for Yang-Mills part of the action and all couplings follow as predictions.
State function collapse is badly chosen term and creates a lot of misinterpretation. State function reduction takes place at the level of the space of quantum states (Zero Energy states in TGD framework). The idea that something collapses at space-time levels leads to astray.
Planck length scale as such has nothing to do with state function reduction.
7. At 12:07 PM, Ulla said...
Did you see this?
-- the first direct observation of a tau particle in a muon neutrino beam
In the theories that physicists use to explain the behavior of fundamental particles (which is known as the Standard Model), neutrinos have no mass. For neutrinos to be able to oscillate, however, they must have mass. Therefore something must be missing from the Standard Model. Physicists have long known that there is much the Standard Model does not explain.
Lubos also has a story. Different particles are only different states (families?) of the same particle is said in the video. Like phases, or scaling :)
8. At 7:01 PM, Matti Pitkanen said...
Mixing of neutrinos involves new physics. This is known for a long time. In TGD, this physics means the reduction of the mixing to the mixing of topologies assignable to the partonic 2-surfaces of leptons and quarks. Quantum superpositions of sphere, torus, and sphere with 2 handles. Number theoretic constraints and experimental input lead to quite restrictive model for CKM mixing of quarks. In the case of leptons, one knows less but one obtains predictions also now.
It is almost 2 decades from this discovery of the fundamental mechanism behind the mixing. It is not recognized and particle physics continues in a state of stagnation. To me ,this is too high a price paid for the human vanity.