Authors: Dan Visser
Comments: 7 Pages.
In this version 3 the formulations are given for the existence of a force smaller than the smallest gravity. This is a new dark energy force, which affects neutrinos differently than is assumed according to current physics. The formulations also imply a different look on the Higgs-mass and dark matter-mass. A deeper analysis became important, because a new cosmological hypothesis is involved. The CERN-experiments on these issues are far from criticism. My set of equations mentioned in my paper “A New Dark Energy Force Theoretically Calculates Faster-than-light-neutrinos” and “Duonistic Neutrinos Violate Relativity” reveal such a criticism. However, until now my formulations withstand a hurricane, even after a Director of the OPERA-project had to resign. My set of equations theoretically proves the neutrino-faster-than-light experiments had to be investigated to the bottom.
Category: Mathematical Physics
Authors: Matti Pitkänen
Comments: 24 Pages.
The arguments of this article support the view that in TGD Universe number theoretic and geometric Langlands conjectures could be understood very naturally. The basic notions are following.
There are several steps leading from G to its twisted quantum Yangian. The first step replaces point like particles with partonic 2-surfaces: this brings in Kac-Moody character. The second step brings in finite measurement resolution meaning that Kac-Moody type algebra is replaced with its quantum version. The third step brings in zero energy ontology: one cannot treat single partonic surface or string world sheet as independent unit: always the collection of partonic 2-surfaces and corresponding string worlds sheets defines the geometric structure so that multilocality and therefore quantum Yangian algebra with multilocal generators is unavoidable.
In finite measurement resolution geometric Langlands duality and number theoretic Langlands duality are very closely related since partonic 2-surface is effectively replaced with the punctures representing the ends of braid strands and the orbit of this set under a discrete subgroup of G defines effectively a collection of "rational" 2-surfaces. The number of the "rational" surfaces in geometric Langlands conjecture replaces the number of rational points of partonic 2-surface in its number theoretic variant. The ability to compute both these numbers is very relevant for quantum TGD.
Authors: Matti Pitkänen
Comments: 16 Pages.
Infinite primes is a purely TGD inspired notion. The notion of infinity is number theoretical and infinite primes have well defined divisibility properties. One can partially order them by the real norm. p-Adic norms of infinite primes are well defined and finite. The construction of infinite primes is a hierarchical procedure structurally equivalent to a repeated second quantization of a supersymmetric arithmetic quantum field theory. At the lowest level bosons and fermions are labelled by ordinary primes. At the next level one obtains free Fock states plus states having interpretation as bound many particle states. The many particle states of a given level become the single particle states of the next level and one can repeat the construction ad infinitum. The analogy with quantum theory is intriguing and I have proposed that the quantum states in TGD Universe correspond to octonionic generalizations of infinite primes. It is interesting to compare infinite primes (and integers) to the Cantorian view about infinite ordinals and cardinals. The basic problems of Cantor's approach which relate to the axiom of choice, continuum hypothesis, and Russell's antinomy: all these problems relate to the definition of ordinals as sets. In TGD framework infinite primes, integers, and rationals are defined purely algebraically so that these problems are avoided. It is not surprising that these approaches are not equivalent. For instance, sum and product for Cantorian ordinals are not commutative unlike for infinite integers defined in terms of infinite primes.
Set theory defines the foundations of modern mathematics. Set theory relies strongly on classical physics, and the obvious question is whether one should reconsider the foundations of mathematics in light of quantum physics. Is set theory really the correct approach to axiomatization?
Authors: Matti Pitkänen
Comments: 80 Pages.
In this article the goal is to find whether the general mathematical structures associated with twistor approach, superstring models and M-theory could have a generalization or a modification in TGD framework. The contents of the chapter is an outcome of a rather spontaneous process, and represents rather unexpected new insights about TGD resulting as outcome of the comparisons.
1. Infinite primes, Galois groups, algebraic geometry, and TGD
In algebraic geometry the notion of variety defined by algebraic equation is very general: all number fields are allowed. One of the challenges is to define the counterparts of homology and cohomology groups for them. The notion of cohomology giving rise also to homology if Poincare duality holds true is central. The number of various cohomology theories has inflated and one of the basic challenges to find a sufficiently general approach allowing to interpret various cohomology theories as variations of the same motive as Grothendieck, who is the pioneer of the field responsible for many of the basic notions and visions, expressed it.
Cohomology requires a definition of integral for forms for all number fields. In p-adic context the lack of well-ordering of p-adic numbers implies difficulties both in homology and cohomology since the notion of boundary does not exist in topological sense. The notion of definite integral is problematic for the same reason. This has led to a proposal of reducing integration to Fourier analysis working for symmetric spaces but requiring algebraic extensions of p-adic numbers and an appropriate definition of the p-adic symmetric space. The definition is not unique and the interpretation is in terms of the varying measurement resolution.
The notion of infinite has gradually turned out to be more and more important for quantum TGD. Infinite primes, integers, and rationals form a hierarchy completely analogous to a hierarchy of second quantization for a super-symmetric arithmetic quantum field theory. The simplest infinite primes representing elementary particles at given level are in one-one correspondence with many-particle states of the previous level. More complex infinite primes have interpretation in terms of bound states.
This construction would realize the number theoretical, algebraic geometrical, and topological content in the construction of quantum states in TGD framework in accordance with TGD as almost TQFT philosophy, TGD as infinite-D geometry, and TGD as generalized number theory visions.
2. p-Adic integration and cohomology
This picture leads also to a proposal how p-adic integrals could be defined in TGD framework.
3. Floer homology, Gromov-Witten invariants, and TGD
Floer homology defines a generalization of Morse theory allowing to deduce symplectic homology groups by studying Morse theory in loop space of the symplectic manifold. Since the symplectic transformations of the boundary of δ M4+/-× CP2 define isometry group of WCW, it is very natural to expect that Kähler action defines a generalization of the Floer homology allowing to understand the symplectic aspects of quantum TGD. The hierarchy of Planck constants implied by the one-to-many correspondence between canonical momentum densities and time derivatives of the imbedding space coordinates leads naturally to singular coverings of the imbedding space and the resulting symplectic Morse theory could characterize the homology of these coverings.
One ends up to a more precise definition of vacuum functional: Kähler action reduces Chern-Simons terms (imaginary in Minkowskian regions and real in Euclidian regions) so that it has both phase and real exponent which makes the functional integral well-defined. Both the phase factor and its conjugate must be allowed and the resulting degeneracy of ground state could allow to understand qualitatively the delicacies of CP breaking and its sensitivity to the parameters of the system. The critical points with respect to zero modes correspond to those for Kähler function. The critical points with respect to complex coordinates associated with quantum fluctuating degrees of freedom are not allowed by the positive definiteness of Kähler metric of WCW. One can say that Kähler and Morse functions define the real and imaginary parts of the exponent of vacuum functional.
The generalization of Floer homology inspires several new insights. In particular, space-time surface as hyper-quaternionic surface could define the 4-D counterpart for pseudo-holomorphic 2-surfaces in Floer homology. Holomorphic partonic 2-surfaces could in turn correspond to the extrema of Kähler function with respect to zero modes and holomorphy would be accompanied by super-symmetry.
Gromov-Witten invariants appear in Floer homology and topological string theories and this inspires the attempt to build an overall view about their role in TGD. Generalization of topological string theories of type A and B to TGD framework is proposed. The TGD counterpart of the mirror symmetry would be the equivalence of formulations of TGD in H=M4× CP2 and in CP3× CP3 with space-time surfaces replaced with 6-D sphere bundles.
4. K-theory, branes, and TGD
K-theory and its generalizations play a fundamental role in super-string models and M-theory since they allow a topological classification of branes. After representing some physical objections against the notion of brane more technical problems of this approach are discussed briefly and it is proposed how TGD allows to overcome these problems. A more precise formulation of the weak form of electric-magnetic duality emerges: the original formulation was not quite correct for space-time regions with Euclidian signature of the induced metric. The question about possible TGD counterparts of R-R and NS-NS fields and S, T, and U dualities is discussed.
5. p-Adic space-time sheets as correlates for Boolean cognition
p-Adic physics is interpreted as physical correlate for cognition. The so called Stone spaces are in one-one correspondence with Boolean algebras and have typically 2-adic topologies. A generalization to p-adic case with the interpretation of p pinary digits as physically representable Boolean statements of a Boolean algebra with 2n>p>pn-1 statements is encouraged by p-adic length scale hypothesis. Stone spaces are synonymous with profinite spaces about which both finite and infinite Galois groups represent basic examples. This provides a strong support for the connection between Boolean cognition and p-adic space-time physics. The Stone space character of Galois groups suggests also a deep connection between number theory and cognition and some arguments providing support for this vision are discussed.
Authors: Matti Pitkänen
Comments: 17 Pages.
Witten was awarded by Fields medal from a construction recipe of Jones polynomial based on topological QFT assigned with braids and based on Chern-Simons action. Recently Witten 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.
The attempts to understand Witten's thoughts 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 this article 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.
Authors: Matti Pitkänen
Comments: 4 pages.
The vanishing of ordinary determinant tells that a group of linear equations possesses non-trivial solutions. Hyperdeterminant generalizes this notion to a situation in which one has homogenous multilinear equations. The notion has applications to the description of quantum entanglement and has stimulated interest in physics blogs. Hyperdeterminant applies to hyper-matrices with n matrix indices defined for an n-fold tensor power of vector space - or more generally - for a tensor product of vector spaces with varying dimensions. Hyper determinant is an n-linear function of the arguments in the tensor factors with the property that all partial derivatives of the hyper determinant vanish at the point, which corresponds to a non-trivial solution of the equation. A simple example is potential function of n arguments linear in each argument.
Why the notion of hyperdeterminant- or rather its infinite-dimensional generalization- might be interesting in TGD framework relates to the quantum criticality of TGD stating that TGD Universe involves a fractal hierarchy of criticalities: phase transitions inside phase transitions inside... At classical level the lowest order criticality means that the extremal of Kähler action possesses non-trivial second variations for which the action is not affected. The system is critical. In QFT context one speaks about zero modes. The vanishing of the so called Gaussian (of functional) determinant associated with second variations is the condition for the existence of critical deformations. In QFT context this situation corresponds to the presence of zero modes.
The simplest physical model for a critical system is cusp catastrophe defined by a potential function V(x) which is fourth order polynomial. At the edges of cusp two extrema of potential function stable and unstable extrema co-incide and the rank of the matrix defined by the potential function vanishes. This means vanishing of its determinant. At the tip of the cusp the also the third derivative vanishes of potential function vanishes. This situation is however not describable in terms of hyperdeterminant since it is genuinely non-linear rather than only multilinear.
In a complete analogy, one can consider also the vanishing of n:th variations in TGD framework as higher order criticality so that the vanishing of hyperdeterminant might serve as a criterion for the higher order critical point and occurrence of phase transition. Why multilinearity might replace non-linearity in TGD framework could be due to the non-locality. Multilinearty with respect to imbedding space-coordinates at different space-time points would imply also the vanishing of the standard local divergences of quantum field theory known to be absent in TGD framework on basis of very general arguments. In this article an attempt to concretize this idea is made. The challenge is highly non-trivial since in finite measurement resolution one must work with infinite-dimensional system.
Authors: Matti Pitkänen
Comments: 61 Pages.
There have been impressive steps in the understanding of N=4 maximally sypersymmetric YM theory possessing 4-D super-conformal symmetry. This theory is related by AdS/CFT duality to certain string theory in AdS5× S5 background. Second stringy representation was discovered by Witten and is based on 6-D Calabi-Yau manifold defined by twistors. The unifying proposal is that so called Yangian symmetry is behind the mathematical miracles involved.
In the following I will discuss briefly the notion of Yangian symmetry and suggest its generalization in TGD framework by replacing conformal algebra with appropriate super-conformal algebras. Also a possible realization of twistor approach and the construction of scattering amplitudes in terms of Yangian invariants defined by Grassmannian integrals is considered in TGD framework and based on the idea that in zero energy ontology one can represent massive states as bound states of massless particles. There is also a proposal for a physical interpretation of the Cartan algebra of Yangian algebra allowing to understand at the fundamental level how the mass spectrum of n-particle bound states could be understood in terms of the n-local charges of the Yangian algebra.
Twistors were originally introduced by Penrose to characterize the solutions of Maxwell's equations. Kähler action is Maxwell action for the induced Kähler form of CP2. The preferred extremals allow a very concrete interpretation in terms of modes of massless non-linear field. Both conformally compactified Minkowski space identifiable as so called causal diamond and CP2 allow a description in terms of twistors. These observations inspire the proposal that a generalization of Witten's twistor string theory relying on the identification of twistor string world sheets with certain holomorphic surfaces assigned with Feynman diagrams could allow a formulation of quantum TGD in terms of 3-dimensional holomorphic surfaces of CP3× CP3 mapped to 6-surfaces dual CP3× CP3, which are sphere bundles so that they are projected in a natural manner to 4-D space-time surfaces. Very general physical and mathematical arguments lead to a highly unique proposal for the holomorphic differential equations defining the complex 3-surfaces conjectured to correspond to the preferred extremals of Kähler action.
Authors: Matti Pitkänen
Comments: 17 Pages.
Shnoll and collaborators have discovered strange repeating patterns of random fluctuations of physical observables such as the number n of nuclear decays in a given time interval. Periodically occurring peaks for the distribution of the number N(n) of measurements producing n events in a series of measurements as a function of n is observed instead of a single peak. The positions of the peaks are not random and the patterns depend on position and time varying periodically in time scales possibly assignable to Earth-Sun and Earth-Moon gravitational interaction.
These observations suggest a modification of the expected probability distributions but it is very difficult to imagine any physical mechanism in the standard physics framework. Rather, a universal deformation of predicted probability distributions would be in question requiring something analogous to the transition from classical physics to quantum physics.
The hint about the nature of the modification comes from the TGD inspired quantum measurement theory proposing a description of the notion of finite measurement resolution in terms of inclusions of so called hyper-finite factors of type II1 (HFFs) and closely related quantum groups. Also p-adic physics -another key element of TGD- is expected to be involved. A modification of a given probability distribution P(nkenovert λi) for a positive integer valued variable n characterized by rational-valued parameters λi is obtained by replacing n and the integers characterizing λi with so called quantum integers depending on the quantum phase qm=exp(i2π/m). Quantum integer nq must be defined as the product of quantum counterparts pq of the primes p appearing in the prime decomposition of n. One has pq= sin(2π p/m)/sin(2π/m) for p≠ P and pq=P for p=P. m must satisfy m≥ 3, m≠ p, and m≠ 2p.
The quantum counterparts of positive integers can be negative. Therefore quantum distribution is defined first as p-adic valued distribution and then mapped by so called canonical identification I to a real distribution by the map taking p-adic -1 to P and powers Pn to P-n and other quantum primes to themselves and requiring that the mean value of n is for distribution and its quantum variant. The map I satisfies I(∑ Pn)=∑ I(Pn). The resulting distribution has peaks located periodically with periods coming as powers of P. Also periodicities with peaks corresponding to n=n+n-, n+q>0 with fixed n-q<0, are predicted. These predictions are universal and easily testable. The prime P and integer m characterizing the quantum variant of distribution can be identified from data. The shapes of the distributions obtained are qualitatively consistent with the findings of Shnoll but detailed tests are required to see whether the number theoretic predictions are correct.
The periodic dependence of the distributions would be most naturally assignable to the gravitational interaction of Earth with Sun and Moon and therefore to the periodic variation of Earth-Sun and Earth-Moon distances. The TGD inspired proposal is that the p-dic prime P and integer m characterizing the quantum distribution are determined by a process analogous to a state function reduction and their most probably values depend on the deviation of the distance R through the formulas Δ p/p≈ kpΔ R/R and Δ m/m≈ kmΔ R/R. The p-adic primes assignable to elementary particles are very large unlike the primes which could characterize the empirical distributions. The hierarchy of Planck constants allows the gravitational Planck constant assignable to the space-time sheets mediating gravitational interactions to have gigantic values and this allows p-adicity with small values of the p-adic prime P.
Authors: Malcolm Macleod
Comments: 3 Pages.
In this essay I propose a geometrical formula for a dimensionless magnetic monopole sigma constructed from Planck time, elementary charge, -c and alpha, the fine structure constant. A formula for the electron may be constructed from these monopoles, this formula suggests that the charge distribution of the electron is perfectly symmetrical. As this monopole comprises a 1/3rd part of electron charge, it is analogous to the quark. If all charges therefore emanate from this monopole, even if charges have opposite polarity, their magnitudes must be equivalent, the differences between the proton and electron for example being geometrical rather than physical. Furthermore, these formulas suggest a Planck universe, wave-particle duality becoming a wave-state to Planck-state oscillation.
Category: Mathematical Physics