About the Nature of Contemporary Mathematics

I found from Not Even Wrong a link to a very nice popular article The Nature of Contemporary Core Mathematics by topologist and topological quantum field theorist Frank Quinn. To my own surprise I had several times the impression of understanding (perhaps the aging brain of scientists learns to generate this experience when required;-)). Core mathematics differs from pure and applied mathematics in Quinn's classification in that it can but need not yet have direct applications.

I have been following the work of John Baez and others based on category theoretical inspirations and I admit my frustration because I am unable to follow the technicalities and fail to see the connection to quantum physics. Quinn represents interesting comments about this work which I could use as an excuse for giving up totally my attempts to understand what they are doing;-).

First of all Quinn notices the potential fatal consequences coming from the lack of a direct connection with physics. It is easy to agree: the lack of a concrete connection makes it possible for a brilliant mathematician to generate endlessly formal structures as generalizations of existing ones and to generate theorems as analogs of existing ones. This is not wrong as such but the lack of the evolutionary pressures leads to an inflation of these mathematical life forms and eventually the mass extinction is unavoidable since the academic metabolic resources are finite. I also think that something genuinely new and irreducible is required. Deformations of existing structures - what most of recent day theoretical physics is - are not enough.

The second point of Quinn is more technical. He claims that the decomposition structure of categories (composition of arrows) cannot hold true in the physically interesting 4-D case even in topological QFTs. What decomposition means here is the assumption that S-matrices - the important arrows now- are parametrized by the value of a continuous time parameter telling the duration of time evolution between initial and final states. This implies that one can decompose S-matrix to a product in many manners (arrow as product of arrays). One can do the same also for path integral (but only formally). Quinn interprets this as locality. One can indeed imagine of decomposing the S-matrix to a product of infinite number of S-matrices associated with infinitesimal values of time parameterm and to obtain representations as an exponent of interaction Hamiltonian. The relationship between what happens by two points separated by a wall can be reconstructed from what happens at the wall. This is how Quinn states it.

For a short time I was enthusiastic about cobordism category and S-matrices as as representation of this category and talked about TGD as almost topological QFT but soon realized that this does not lead to anywhere. My own objection against the decomposition is based on the same observation as Quinn's but formulated differently. The decomposition means that the counterpart of unitary time evolution in the sense of exponentiation of interaction Hamiltonian) is assumed. This is impossible in TGD framework as became clear during the first months of TGD when I learned that Hamiltonian formalism makes no sense in TGD because of the extreme non-linearity of any general coordinate invariant variational principle. Within about 7 years this led to the vision about physics as infinite-dimensional geometry of world of classical worlds (or configuration space as I used to say then) and the last five years I have been talking about zero energy ontology formulated in terms of causal diamonds (CDs) defined as intersections of future and past directed lightcones.

1. The temporal distance between the tips of CD is quantized in powers of two (p-adic length scale hypothesis) and also the relative position of tips forms a discrete lattice like set in 3-D hyperboloid of future light-cone. Unions of CDs are allowed and they can also intersect and there are CDs within CDs so that one can speak about fractal hierarchy. Most of this is very relevant for the notions like radiative correction, coupling constant evolution, virtual particle, finite measurement resolution, second law, and even quantum cosmology (when one considers very large CDs). I feel that these fundamental quantum physical notions must be feeded explicitly to the mathematics at the fundamental level. I would be astonished if an approach starting from existing mathematical notions such as categories having motivations coming from the internal structure of mathematics of a particular era- in particular the idea to of transferring the results from one branch of mathematics to another one by using functors- could miraculously reproduce these notions.

2. One can assign M-matrix ("complex" square root of density matrix decomposing to a product of positive square root of diagonal density matrix and unitary S-matrix) to any CD as a representation of zero energy state. The collection of M-matrices orthonormal as zero energy states defines unitary U-matrix. There is a fractal hierarchy of M-matrices (again a physical input) but the product property fails. One cannot arrange CDs to a sequences of CDs so that product decomposition fails and one cannot speak of cobordism category.

3. Product property implying a representation of S-matrix as an exponential of interaction Hamiltonians is also inconsistent with number theoretical universality stating that M-matrices make sense in all number fields. In p-adic context the continuous unitary time evolution in non-sensical as follows from the fact that p-adic exponent function does not have the physical properties of real exponent function. To define the counterparts of the exponent of Hamiltonian one must introduce the counterparts of phases exp(iEt) as roots of unity appearing in algebraic extension of p-adic numbers. Discretization and number theoretical quantization are unvoidable. One has chronons and the classical Hamiltonian flow picture fails.

Quinn says many other things but just these points stimulated sufficient interest in me to write these comments. I recommend the article warmly for physicists willing to understand how mathematicians see the science.

Considerable progress in generalized Feynman diagrammatics

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 single really intelligent reader might realize what is 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/λγ , where 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 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.

Quite 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. 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?.

Strong CP breaking observed in BBbar system?

Both Lubos Motl, Tommaso Dorigo, and Jester comment the strong breaking of CP symmetry in B-Bbar system claimed by D0 collaboration. The claimed symmetry breaking is about 50 times larger than the breaking predicted by the standard model, and manifests itself as an asymmetry in the production of μ⁺μ⁺ and μ^-μ^- pairs in the decays producing B⁰Bbar⁰ pairs. The asymmetry is due to the oscillations between almost mass degenerate states B⁰ and Bbar⁰. It is the mass difference which is much larger than predicted one.

In TGD framework one can imagine a very general CP breaking mechanism due to the presence of imaginary exponent of the instanton density associated with Kähler action. A second very geometric mechanism is suggested by zero energy ontology. The basic notion is causal diamond (CD) defined as the intersection of future and past directed lightcones of Minkowski space (times CP₂ to be precise). The moduli space for CDs within given CD defining a preferred rest system consists of translations of the lower tip of sub-CD, which are continuous plus discrete moduli space for the relative position of the upper tip of the sub-CD relative to lower tip. This dicretization has turned out to be necessary for the p-adicization program and it could resolve also some cosmological mysteries as I have explained in earlier postings.

One could argue that the asymmetry between CDs could induce CP breaking. One could also argue that the overall cm degrees of freedom are associated with entire CD rather than lower tip so that no CP breaking should results. I am not sure. It might be that the rest system of larger CD is the natural one and indeed induces a spontaneous CP breaking within it.

Loops involving a decay to W bosons by top quark exchange are responsible for the CP breaking in standard model and the obvious guess is that there are new heavy fermions which contribute to the loops inducing 50 times larger CP breaking as expected. In TGD framework it is easy to imagine this kind of new fermions. One possibility is p-adically scaled up variant of top: I told about this in earlier posting. The new M₈₉ hadronic physics would provide new fermions which could also contribute to the CP breaking. In principle everything is calculable since only weak interactions are involved unless the M₈₉ hadrons couple to M₆₁ weak bosons rather than the ordinary M₈₉ ones. In this case they would have virtually no weak interactions. This possibility cannot be excluded.

How to perform WCW integrations in generalized Feynman diagrams?

The formidable looking challenge of quantum TGD is to calculate the M-matrix elements defined by the generalized Feynman diagrams. Zero energy ontology (ZEO) has provided profound understanding about how generalized Feynman diagrams differ from the ordinary ones. The most dramatic prediction is that loop momenta correspond to on mass shall momenta for the two throats of the wormhole contact defining virtual particles: the energies of the energies of on mass shell throats can have both signs in ZEO. This predicts finiteness of Feynman diagrams in the fermionic sector. Even more: the number of Feynman diagrams for a given process is finite if also massless particles receive a small mass by p-adic thermodynamics. The mass would be due to IR cutoff provided by the largest CD (causal diamond) involved.

The basic challenges are following.

1. One should perform the functional integral over world of classical worlds (WCW) for fixed values of on mass shell momenta appearing in internal lines. After this one must perform integral or summation over loop momenta.

2. One must achieve this also in the p-adic context. p-Adic Fourier analysis relying on algebraic continuation raises hopes in this respect. p-Adicity suggests strongly that the loop momenta are discretized and ZEO predicts this kind of discretization naturally.

The realization that p-adic integrals could be defined if the manifold is symmetric space as the world of classical world (WCW) is proposed to be raises the hope that the WCW integration for Feynman amplitudes could be carried at the general level using Fourier analysis for symmetric spaces. Even more, the possibility to define p-adic intergrals for symmetric spaces suggests that the theory could allow elegant p-adicization. This indeed seems to be the case. It seems that the dream of transforming TGD to a practical calculational machinery does not look non-realistic at all.

I do not bother to type more but give a link to a short article summarizing the basic formulas. For more background see also the article Weak form of electric-magnetic duality, electroweak massivation, and color confinement and the chapter Does the Modified Dirac Equation Define the Fundamental Action Principle?.

Weak form of electric-magnetic duality, particle concept, and Feynman diagrammatics

The notion of electric magnetic duality emerged already two decades ago in attempts to formulate the Kähler geometric of world of classical worlds. Quite recently a considerable step of progress took place in the understanding of this notion. Every new idea must be of course taken with a grain of salt but the good sign is that this concept leads to an identification of the physical particles as string like objects defined by magnetic charged wormhole throats connected by magnetic flux tubes. The second end of the string contains particle having electroweak isospin neutralizing that of elementary fermion and the size scale of the string is electro-weak scale would be in question. Hence the screening of electro-weak force takes place via weak confinement. This picture generalizes to magnetic color confinement.

Zero energy ontology in turn inspires the idea that virtual particles correspond to pairs of on mass shell states assignable to the opposite throats of wormhole contacts: in TGD framework the propagators do not diverge although particles are on mass shell in standard sense. This assumption leads to powerful constraints on the generalized Feynman diagrams giving excellent hopes about the finiteness of loops. Finiteness has been obvious on basis of general arguments but has been very difficult to demonstrate convincingly in the fermionic sector of the theory. In fact, there are good arguments supporting that only a finite number of diagrams contributes to a given reaction: something inspired by the vision about algebraic physics (infinite sums lead out of the algebraic extension used). The reason is that the on mass shell conditions on states at wormhole throats reduce the phase space dramatically, and already in the case of four-vertex loops leave only a discrete set of points under consideration. This implies also finiteness. This wisdom can be combined with the new stringy view about particles to build a very concrete stringy view about generalized Feynman diagrams.

The coutcome of the opening of the black box of virtual particle -an idea forced by the twistorial approach and made possible by zero energy ontology- is something which I dare to regard as a fulfillment of 32 year old dream.

For a more detailed representation of weak electric-magnetic duality 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?.

Does top quark has scaled up variant?

Tommaso Dorigo writes in Quantum Diaries Survivor about The 450 GeV Quark That Wouldn't Go Away. What has been observed in CDF is indication for what looks like a scaled up copy of top quark with mass of 450 GeV. It is fun to test various new physics prediction of TGD. As I have told again and again, p-adic length scale hypothesis allows scaled up variants of quarks with masses coming as half octaves. The mass obtained by multiplying top quark mass 170 GeV with a factor 2^3/2 is 480 GeV and and 6 per cent higher than the rough experimental estimate.