Mathematical Physics

Comparison of Infinitely Large and Infinitely Small Quantities.

Authors: Victor Katyushchik
Comments: 4 Pages.

The three-dimensional space formatting allows to operate on infinitely large and infinitely small quantities confidently, without any contradictions, paradoxes and uncertainties.
Category: Mathematical Physics

Monopole Positioning in Closed Space, Trefoil Conversion & Energy Calculation.

Authors: Prakhar bhatnagar
Comments: 5 Pages.

Magnetic monopole is a hypothetical particle with a single pole. In this paper a new mathematical structure for Dirac string has been proposed .Quantitative aspects and qualitative aspects of a monopole represented as a Dirac string have been highlighted. Magnetic bundle has been defined in a complex form and pole has been defined as composition of that bundle for single pole the pole variable is single. The pole can be considered as a group of threads or a point lying on a monopole magnet bundle for one pole the thread is single and one dimensional. The function is in complex form and defines magnetic bundle.
Category: Mathematical Physics

New Dark Energy and Letter to the Nobel Committee.

Authors: Dan Visser
Comments: 10 Pages.

This paper describes what New Dark Energy is, as well as my comment “Nobel Nominations in a New World” to the Nobel Committee. These seemingly different looking subjects are importantly connected in a plead to change the way Nobel-nominations are performed in a changing world with internet and alternative archives, such as viXra. The point is, that institutional established physics and cosmology block new ideas from the public domain. Therefore an appendix has been added to this paper to summarize in a nutshell, relativity, space-expansion, accelerated space-expansion and the one time-direction in the Big Bang Cosmology, in order to explain better what new dark energy is and what its meaning is for physics and new cosmology. That message is meant for the Nobel Committee. Furthermore three-why’s are answered: Why are two extra time-directions needed in a new cosmology? Why is the introduction of dark mass, squared? Why would there be a Double Torus for New Cosmology? Also the basic-formulas are summarized, referring to my papers, such as the dark energy force formula from my thought-experiment, followed by its transformed-version: the force smaller than the smallest gravity. In the end New Dark Energy is formulated as a product of new dark mass (squared) and two extra time-directions from below the Planck scale. This marks that New Dark Energy could recalculate the established dual unitary-entangled quantum-spaces faster backwards in time than time goes forward in the Big Bang Cosmology ! This affects the established quantum dynamics and classical reality towards the past and future differently than Einstein’s General Relativity is prescribing. The letter to the Nobel Committee is send by me, because no other institutions is doing that for me.
Category: Mathematical Physics

Superluminal Physics & Instantaneous Physics as New Trends in Research

Authors: Florentin Smarandache
Comments: 4 Pages.

In a similar way as passing from Euclidean Geometry to Non-Euclidean Geometry, we can pass from Subluminal Physics to Superluminal Physics, and further to Instantaneous Physics. In the lights of two consecutive successful CERN experiments with superluminal particles in the Fall of 2011, we believe that these two new fields of research should begin developing.
Category: Mathematical Physics

The New Inverse-Coordinate Transformation of the New Accelerated System

Authors: sangwha Yi
Comments: 10 Pages. It is submission.

This article’s purpose is that discover the new inverse-coordinate transformation of the new accelerated theory in the article “The changed coordinate transformation of the constant accelerated coordinate system by tetrad”(vixra:1106,0054).
Category: Mathematical Physics

Particle Mass Ratios

Authors: DT Froedge
Comments: 12 Pages.

This paper presents straightforward explanation of particle mass ratios, and the specific values for some well known particles. The additional nuclear modes postulated, are similar to the Schrödinger modes in the atom, and, though speculative, the mass ratios calculated for elementary particles are very close the observed mass ratios.
Category: Mathematical Physics

Higgs-Limited Boson Surface (Higgs-LBS) is a Mass-Surface – Thus not a Singular Higgs-Boson and is Related to Faster-Than-Light-Neutrinos.

Authors: Dan Visser
Comments: 4 Pages. Avalable only for arXiv and informing me by email.

In this paper I give a formulation for why a new Higgs-mass is considered otherwise than is usually done. This leads to an additional explanation of my earlier posted papers about neutrinos-faster-than-light-in-vacuum; I also refer to my earlier calculated new Higgs-mass. The addition is a deeper analysis, which does not change the result in the former paper. It is extending the Higgs-energy. The first prediction is the Higgs-particle is a Limited Boson-Surface, or Higgs-LBS. This means: It is not a singular Higgs-mass boson. The second prediction is: The occurrence of an Higgs-LBS may be about 60 nanoseconds before a proton-proton collision in an accelerator, such as the LHC in CERN. This is possible because the Higgs-energy is related to a new dark energy force squared. These predictions are supposed to fit in a new cosmological hypothesis, called the Double Torus of dark energy and dark matter, wherein a new dark energy force is defined as a force smaller than the smallest gravity, which could act beyond General Relativity.
Category: Mathematical Physics

A New Force Smaller Than the Smallest Gravity.

Authors: Dan Visser
Comments: 7 Pages.

In the formulations of this 'paper' speaks 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. The 'paper' is also is an overview of recent 'papers' [1] , which already described these issues, but a deeper analysis became important, because a new cosmological hypothesis is involved. The CERN-experiments on these issues are the falsification for my formulations, but until now my formulations withstand several experimental results, and in this case the match with the latest neutrino-faster-than-light experiments is very convincing (more attention might be given to this 'paper' towards institutional disciplines by the Arxiv or Nature).
Category: Mathematical Physics

Langlands Conjectures in TGD Framework

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.

1. Zero energy ontology (ZEO) and the related notion of causal diamond CD (CD is short hand for the cartesian product of causal diamond of M⁴ and of CP₂). ZEO leads to the notion of partonic 2-surfaces at the light-like boundaries of CD and to the notion of string world sheet. These notions are central in the recent view about TGD. One can assign to the partonic 2-surfaces a conformal moduli space having as additional coordinates the positions of braid strand ends (punctures). By electric-magnetic duality this moduli space must correspond closely to the moduli space of string world sheets.

2. Electric-magnetic duality realized in terms of string world sheets and partonic 2-surfaces. The group G and its Langlands dual ^LG would correspond to the time-like and space-like braidings. Duality predicts that the moduli space of string world sheets is very closely related to that for the partonic 2-surfaces. The strong form of 4-D general coordinate invariance implying electric-magnetic duality and S-duality as well as strong form of holography indeed predicts that the collection of string world sheets is fixed once the collection of partonic 2-surfaces at light-like boundaries of CD and its sub-CDs is known.

3. The proposal is that finite measurement resolution is realized in terms of inclusions of hyperfinite factors of type II₁ at quantum level and represented in terms of confining effective gauge group. This effective gauge group could be some associate of G: gauge group, Kac-Moody group or its quantum counterpart, or so called twisted quantum Yangian strongly suggested by twistor considerations. At space-time level the finite measurement resolution would be represented in terms of braids at space-time level which come in two varieties correspond to braids assignable to space-like surfaces at the two light-like boundaries of CD and with light-like 3-surfaces at which the signature of the induced metric changes and which are identified as orbits of partonic 2-surfaces connecting the future and past boundaries of CDs.

   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.

4. The natural identification of the associate of G is as quantum Yangian of Kac-Moody type group associated with Minkowskian open string model assignable to string world sheet representing a string moving in the moduli space of partonic 2-surface. The dual group corresponds to Euclidian string model with partonic 2-surface representing string orbit in the moduli space of the string world sheets. The Kac-Moody algebra assigned with simply laced G is obtained using the standard tachyonic free field representation obtained as ordered exponentials of Cartan algebra generators identified as transversal parts of M⁴ coordinates for the braid strands. The importance of the free field representation generalizing to the case of non-simply laced groups in the realization of finite measurement resolution in terms of Kac-Moody algebra cannot be over-emphasized.

5. Langlands duality involves besides harmonic analysis side also the number theoretic side. Galois groups (collections of them) defined by infinite primes and integers having representation as symplectic flows defining braidings. I have earlier proposed that the hierarchy of these Galois groups define what might be regarded as a non-commutative homology and cohomology. Also G has this kind of representation which explains why the representations of these two kinds of groups are so intimately related. This relationship could be seen as a generalization of the MacKay correspondence between finite subgroups of SU(2) and simply laced Lie groups.

6. Symplectic group of the light-cone boundary acting as isometries of the WCW geometry kenociteallb/compl1 allowing to represent projectively both Galois groups and symmetry groups as symplectic flows so that the non-commutative cohomology would have braided representation. This leads to braided counterparts for both Galois group and effective symmetry group.

7. The moduli space for Higgs bundle playing central role in the approach of Witten and Kapustin to geometric Landlands program is in TGD framework replaced with the conformal moduli space for partonic 2-surfaces. It is not however possible to speak about Higgs field although moduli defined the analog of Higgs vacuum expectation value. Note that in TGD Universe the most natural assumption is that all Higgs like states are "eaten" by gauge bosons so that also photon and gluons become massive. This mechanism would be very general and mean that massless representations of Poincare group organize to massive ones via the formation of bound states. It might be however possible to see the contribution of p-adic thermodynamics depending on genus as analogous to Higgs contribution since the conformal moduli are analogous to vacuum expectation of Higgs field.

How Infinite Primes Relate to Other Views About Mathematical Infinity?

Authors: Matti Pitkänen
Comments: 18 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?

1. Quantum view about consciousness and cognition leads to a proposal that p-adic physics serves as a correlate for cognition. Together with the notion of infinite primes this suggests that number theory should play a key role in the axiomatics.
2. Algebraic geometry allows algebraization of the set theory and this kind of approach suggests itself strongly in physics inspired approach to the foundations of mathematics. This means powerful limitations on the notion of set.
3. Finite measurement resolution and finite resolution of cognition could have implications also for the foundations of mathematics and relate directly to the fact that all numerical approaches reduce to an approximation using rationals with a cutoff on the number of binary digits.
4. The TGD inspired vision about consciousness implies evolution by quantum jumps meaning that also evolution of mathematics so that no fixed system of axioms can ever catch all the mathematical truths for the simple reason that mathematicians themselves evolve with mathematics.

Motives and Infinite Primes

Authors: Matti Pitkänen
Comments: 79 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.

1. What makes infinite primes interesting from the point of view of algebraic geometry is that infinite primes, integers and rationals at the n:th level of the hierarchy are in 1-1 correspondence with rational functions of n arguments. One can solve the roots of associated polynomials and perform a root decomposition of infinite primes at various levels of the hierarchy and assign to them Galois groups acting as automorphisms of the field extensions of polynomials defined by the roots coming as restrictions of the basic polynomial to planes x_n=0, x_n=x_n-1=0, etc...

2. These Galois groups are suggested to define non-commutative generalization of homotopy and homology theories and non-linear boundary operation for which a geometric interpretation in terms of the restriction to lower-dimensional plane is proposed. The Galois group G_k would be analogous to the relative homology group relative to the plane x_k-1=0 representing boundary and makes sense for all number fields also geometrically. One can ask whether the invariance of the complex of groups under the permutations of the orders of variables in the reduction process is necessary. Physical interpretation suggests that this is not the case and that all the groups obtained by the permutations are needed for a full description.

3. The algebraic counterpart of boundary map would map the elements of G_k identified as analog of homotopy group to the commutator group [G_k-2,G_k-2] and therefore to the unit element of the abelianized group defining cohomology group. In order to obtains something analogous to the ordinary homology and cohomology groups one must however replaces Galois groups by their group algebras with values in some field or ring. This allows to define the analogs of homotopy and homology groups as their abelianizations. Cohomotopy, and cohomology would emerge as duals of homotopy and homology in the dual of the group algebra.

4. That the algebraic representation of the boundary operation is not expected to be unique turns into blessing when on keeps the TGD as almost topological QFT vision as the guide line. One can include all boundary homomorphisms subject to the condition that the anticommutator δⁱ_kδ^j_k-1+δ^j_kδⁱ_k-1 maps to the group algebra of the commutator group [G_k-2,G_k-2]. By adding dual generators one obtains what looks like a generalization of anticommutative fermionic algebra and what comes in mind is the spectrum of quantum states of a SUSY algebra spanned by bosonic states realized as group algebra elements and fermionic states realized in terms of homotopy and cohomotopy and in abelianized version in terms of homology and cohomology. Galois group action allows to organize quantum states into multiplets of Galois groups acting as symmetry groups of physics. Poincare duality would map the analogs of fermionic creation operators to annihilation operators and vice versa and the counterpart of pairing of k:th and n-k:th homology groups would be inner product analogous to that given by Grassmann integration. The interpretation in terms of fermions turns however to be wrong and the more appropriate interpretation is in terms of Dolbeault cohomology applying to forms with homomorphic and antiholomorphic indices.

5. The intuitive idea that the Galois group is analogous to 1-D homotopy group which is the only non-commutative homotopy group, the structure of infinite primes analogous to the braids of braids of braids of ... structure, the fact that Galois group is a subgroup of permutation group, and the possibility to lift permutation group to a braid group suggests a representation as flows of 2-D plane with punctures giving a direct connection with topological quantum field theories for braids, knots and links. The natural assumption is that the flows are induced from transformations of the symplectic group acting on δ M²_+/-× CP₂ representing quantum fluctuating degrees of freedom associated with WCW ("world of classical worlds"). Discretization of WCW and cutoff in the number of modes would be due to the finite measurement resolution. The outcome would be rather far reaching: finite measurement resolution would allow to construct WCW spinor fields explicitly using the machinery of number theory and algebraic geometry.

6. A connection with operads is highly suggestive. What is nice from TGD perspective is that the non-commutative generalization homology and homotopy has direct connection to the basic structure of quantum TGD almost topological quantum theory where braids are basic objects and also to hyper-finite factors of type II₁. This notion of Galois group makes sense only for the algebraic varieties for which coefficient field is algebraic extension of some number field. Braid group approach however allows to generalize the approach to completely general polynomials since the braid group make sense also when the ends points for the braid are not algebraic points (roots of the polynomial).

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.

1. The calculation of twistorial amplitudes reduces to multi-dimensional residue calculus. Motivic integration gives excellent hopes for the p-adic existence of this calculus and braid representation would give space-time representation for the residue integrals in terms of the braid points representing poles of the integrand: this would conform with quantum classical correspondence. The power of 2π appearing in multiple residue integral is problematic unless it disappears from scattering amplitudes. Otherwise one must allow an extension of p-adic numbers to a ring containing powers of 2π.

2. Weak form of electric-magnetic duality and the general solution ansatz for preferred extremals reduce the Kähler action defining the Kähler function for WCW to the integral of Chern-Simons 3-form. Hence the reduction to cohomology takes places at space-time level and since p-adic cohomology exists there are excellent hopes about the existence of p-adic variant of Kähler action. The existence of the exponent of Kähler gives additional powerful constraints on the value of the Kähler fuction in the intersection of real and p-adic worlds consisting of algebraic partonic 2-surfaces and allows to guess the general form of the Kähler action in p-adic context.

3. One also should define p-adic integration for vacuum functional at the level of WCW. p-Adic thermodynamics serves as a guideline leading to the condition that in p-adic sector exponent of Kähler action is of form (m/n)^r, where m/n is divisible by a positive power of p-adic prime p. This implies that one has sum over contributions coming as powers of p and the challenge is to calculate the integral for K= constant surfaces using the integration measure defined by an infinite power of Kähler form of WCW reducing the integral to cohomology which should make sense also p-adically. The p-adicization of the WCW integrals has been discussed already earlier using an approach based on harmonic analysis in symmetric spaces and these two approaches should be equivalent. One could also consider a more general quantization of Kähler action as sum K=K₁+K₂ where K₁=rlog(m/n) and K₂=n, with n divisible by p since exp(n) exists in this case and one has exp(K)= (m/n)^r × exp(n). Also transcendental extensions of p-adic numbers involving n+p-2 powers of e^1/n can be considered.

4. If the Galois group algebras indeed define a representation for WCW spinor fields in finite measurement resolution, also WCW integration would reduce to summations over the Galois groups involved so that integrals would be well-defined in all number fields.

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>p^n-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.

Could One Generalize Braid Invariant Defined by Vacuum Expecation of Wilson Loop to and Invariant of Braid Cobordisms and of 2-Knots?

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.

Could the Notion of Hyperdeterminant be Useful in TGD Framework?

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.

What Could be the Generalization of Yangian Symmetry of N=4 Susy in TGD Framework?

Authors: Matti Pitkänen
Comments: 45 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.

A Possible Explanation for Shnoll Effect

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 λ_i) for a positive integer valued variable n characterized by rational-valued parameters λ_i is obtained by replacing n and the integers characterizing λ_i with so called quantum integers depending on the quantum phase q_m=exp(i2π/m). Quantum integer n_q must be defined as the product of quantum counterparts p_q of the primes p appearing in the prime decomposition of n. One has p_q= sin(2π p/m)/sin(2π/m) for p≠ P and p_q=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(∑ P_n)=∑ I(P_n). 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≈ k_pΔ R/R and Δ m/m≈ k_mΔ 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.

A New Dark Energy Force Theoretically Calculates Faster-Than-Light-Neutrinos.

Authors: Dan Visser
Comments: 7 pages

A theoretical calculation with a new dark energy force formula discloses the correctness of the experimental faster-than-light-neutrinos in the CERN-San Grasso experiment. The formulation in this paper theoretically confirms that Einstein's Relativity could be violated. This introduces the obligation to accept a new cosmological model, called the Double Torus hypothesis . The theoretical calculation in this paper is based on a new momentum of dark energy force, formulated by its new force and two extra time dimensions below the Planck scale. In detail this completely new perspective shows that the dark energy force starts to dominate the lowest limit of the Newton-force-acceleration under specific conditions of neutrino-oscillations. This paper theoretically calculates 62.8 nanosecond for the experimental detected early-arrival of muon-neutrinos related to how light-in-vacuum would have arrived. This is a marvelous close match compared to the ((60.7 ± 6.9 (stat.) ± 7.4 (sys.)) nanosecond found during the 'neutrino-flight path' from CERN to San Grasso.
Category: Mathematical Physics

Zeno's Paradoxe and the Nature of Points in Quantized Euclidean Universe

Authors: Markos Georgallides
Comments: 9 pages

This article explains the correlation between Euclidean Geometry , Complex Numbers and Physics . A Straight line AB is continuous in Points between A and B [ i.e. all points between AB are the elements which fill AB ] , which Points are also , Nothing , Everything , and maybe Anywhere , without any Dimension , and one has to pass the infinite points between A and B . A point C is on line AB only when exists CA+ CB = AB , or the whole AB is equal to the parts CA , CB , and this is an equation , which differentiates geometries . Since points have not any dimension and since only AB has dimension ( the length AB and for ÃC the length AC ) and since on ÃB exist infinite AC → AB , which have infinite Spaces , Anti-Spaces and Sub-Spaces , then 1. Straight line AB is continuous with points as filling ( Infinitively divisible ) . 2. Straight line AB is discontinuous (discrete) with dimensional Units , ds =AB as filling ( that is made up of finite divisible or indivisible parts the Monads ds ) or ds → AB / n , where n = 1 , 2 , → ∞ ) , and for n = ∞ then ds = 0 . 3. Straight line AB is discontinuous (discrete) with dimensional Units ds , or ds = quantum = AB / n [ where n = 1,2,3 → ∞ , = ( a + b.i ) / n , Infinitively divisible and keeping always the conservation of properties at end points A , B ] as filling , and continuous with points as filling ( for n = ∞ then ds = 0 i.e. a point ) . This is the Vector relation of Monads , ds , ( or , as Complex Numbers in their general form , ds = a + b. i ) , which is the Dual Nature of lines AB , ( discrete and continuous ) . So travelling on Points ( ds = 0 ) between AB one never comes to B , on the contrary travelling with ds > 0 one comes in finite time . 4 . Achilles has to pass every point of line AB which is then as passing from the starting point A , ds =0 , where Velocity of Achilles is v(A) = ds/dt = 0 . The same happens for Tortoise at point B where Velocity v(T) = ds/dt = 0 . On the contrary , Achilles passing AB on dimensional Units , ds , then Achilles velocity v(A) = ds/dt(A) is greater than that of Tortoise v(T) = ds / dt(T) . Since in PNS , v = ∞ , T = 0 , meaning infinite velocity and Time not existing , then Arrow AB in [PNS] is constant because AB = ds = Constant = u . 0 = ∞ . 0 Straight line AB is discontinuous (discrete) with dimensional Units ds = AB / n where n = 1 → ∞ and continuous with points [ n = ∞ ] . Continuously on AB happens also with all discrete ds , ( This is the Dual Nature of lines ( Geometry ), discrete and continuous ) . Monads ds = 0 → ∞ are Simultaneously , actual infinite ( because for n = ∞ then ds = [ AB / n = ∞ ] = 0 i.e. a point ) , and potential infinite , ( because for n = 0 then ds = [ AB / n=0 ] = ∞ i.e. the straight line through AB .
Category: Mathematical Physics

Can Electromagnetic Scalar Waves be Radiated by a Metal Sphere?

Authors: Giuliano Bettini
Comments: 24 pages.

There is a lot of chattering on the Internet about Tesla waves, vacuum energy, scalar waves and so on. Professor Meyl says he has a complete theory, experimental evidence and apparatus on these waves. In a theoretical paper Van Vlaenderen introduced a generalization of classical electrodynamics for the prediction of scalar field effects. It is said the Monstein has demonstrated the physical existence of such scalar waves. NASA in a report seems to consider such waves as a promising item to be studied. Some other papers appeared in arXiv. I've already showed that such waves are a consequence of "generalized" Maxwell fields which simply mean space time analytic functions not limited by the Lorenz gauge condition, but accepted instead in a wide sense. In this paper I remember my ideas on these waves, together with my doubts about their physical existence. In fact, the deduction of the scalar waves equations, together with their physical interpretation, in my opinion demonstrates nothing about the physical existence of scalar waves. I discuss the experiment of Monstein, and suggest some other experiment. Obviously I think that the lack of demonstration of the existence doesn't mean the demonstration of inexistence.
Category: Mathematical Physics

New Value for the Higgs Mass.

Authors: Dan Visser
Comments: 4 pages. Publication is free for any institute or magazine

The Higgs mass is approximately 0.5 TeV/c2 . This is a new predicted value. This mass value for Higgs is calculated with a new dark energy force formula, which is performing in the hypothesis of the Double Torus, a new cosmological "model" for the universe. This "model" embeds the Big Bang framework. The new Higgs mass is theoretically disclosed by implementing the dimensional features of relativistic spacetime into the dimensions of the dark energy force formula and an equalization to the lowest limit for acceleration in Newton force, being a transition where Newton breaks down and the dark energy force takes over. The dark energy force generates gravitational movement and thus mass. This enables to calculate a Higgs mass differently than in currently used theories.
Category: Mathematical Physics

Electrostatics and Fluid Flow with 3D Analytic Functions

Authors: Giuliano Bettini
Comments: 21 pages.

I present examples of applications of 3-dimensional analytic functions to electrostatics and potential flows, mainly devoted to engineers and physicists. Of course, the paper only suggests areas of future development, despite that a persistent idea, from Sommerfeld on, seems to be "The powerful tool of the theory of complex functions cannot be used in three-dimensional potential theory" (Sommerfeld, "Mechanics of Deformable Bodies", Academic Press, 1950) I summarize here unpublished manuscripts dated 1994.
Category: Mathematical Physics

Majorana Neutrino: Chirality and Helicity

Authors: Valeriy V. Dvoeglazov
Comments: 19 Pages

We introduce the Majorana spinors in the momentum representation. They obey the Dirac-like equation with eight components, which has been first introduced by Markov. Thus, the Fock space for corresponding quantum fields is doubled (as shown by Ziino). Particular attention has been paid to the questions of chirality and helicity (two concepts which frequently are confused in the literature) for Dirac and Majorana states.
Category: Mathematical Physics

Black Hole Vs. Variable Rest Mass Neutron Star

Authors: D.T. Froedge
Comments: 14 pages

In a previous paper we have discussed the conjecture of a variable particle rest mass, as a function of gravitational potential (Scalar Gravitational Theory with Variable Rest Mass) This paper discuses the implications of that theory in regard to a large neutron star, and contrast the difference between the predicted phenomena, and Black Hole theory as put fourth by standard GR. The validity of this theory would be established by the finding of a neutron star having a mass greater than allowed by Black Hole theory.
Category: Mathematical Physics

Scalar Gravitational Theory with Variable Rest Mass

Authors: D.T. Froedge
Comments: 16 pages

The purpose of this paper is to present a non-tensor theory of gravitation that provides the equivalent equations of motion, but does not result in the issue of black holes, non-localizable energy, or spacetime singularities. The prime assumption is the notion that the rest mass of a particle entering a gravitational potential is reduced in proportion to the energy gained by the velocity increases. One could designate this development as a "catalytic" theory in that gravitation is a vector catalyst, that converts rest energy into kinetic energy. The total mass energy will be considered localized with the individual mass particles, and defined relative to a given observer. No energy will be ascribed to the field, thus there is no stress energy tensor. We will develop the energy mass relation, and show that it can result in the proper orbital precession as demonstrated by GR. The rest mass is not significantly different from that of a particle defined in a stationary asymptotically flat GR space-time, when the defining point particles via the Komar mass. Since rest mass of a particle goes to zero on approaching a Schwarzschild boundary, the formation of black holes becomes problematic.
Category: Mathematical Physics

Matter Over Anti-Matter Explained in a Double Torus.

Authors: Dan Visser
Comments: 4 pages

September 1 2009 a new cosmological hypothesis was proposed to imagine the universe otherwise. Many institutional scientists still try to implement their results into the commonly accepted Big Bang theory. I am one of the two scientists who propose a Double Torus as a new shape for the universe. Several 'papers' are hosted at 'vixra' about this subject. This paper adds an explanaition to these afore published papers and in particular explains the 'dominance of matter over anti-matter', deduced from the earlier derived 'dark energy force formula' and the perspective of the Double Torus.
Category: Mathematical Physics

Recalculation-Mechanism of the Big Bang in a Double Torus Universe.

Authors: Dan Visser
Comments: 5 pages

Here in this 'paper' a deeper analysis of the 'dark energy force formula' in the Double Torus Cosmology is made. Earlier 'papers' were posted as 'pre-papers' in the vixra-archive. These are subject to a new cosmological hypothesis, being further described in higher order mathematics in the near future by Christopher Forbes. I have found evidence through an unexpected interpretation of CMB results, that the 'dark energy force formula' appears to be a functional 'recalculation-mechanism' for quantum gravity.
Category: Mathematical Physics

The Gauge Theory's Expansion in the Electro-Magnetic Field

Authors: Sangwha-Yi
Comments: 6 pages.

In the special relativity theory, study the gauge theory in the Electro-magnetic field theory.Using that the Electro-magnetic potential is 4-vector, treat invariant potential. And the Electro-Magnetic field theory's the gauge theory expand.
Category: Mathematical Physics

The Changed Coordinate Transformation of the Constant Accelerated Coordinate System by Tetrad

Authors: Sangwha-Yi
Comments: 12 pages.

In the general relativity theory, instead of in the present accelerated system theory, the coordinate transformation that the inertial coordinate system and the accelerated coordinate system, find the new relation's coordinate transformation that used the tetrad on the new method.And using the new relation's coordinate transformation of the new accelerated system, organize the expansive accelerated system, the relation of the inertial coordinate system and an accelerated system that accelerated constantly on an inertial coordinate system.
Category: Mathematical Physics

Exact Solution of Viscous-Plastic Flow Equations for Glacier Dynamics in 2-Dimensional Case.

Authors: Sergey V. Ershkov
Comments: 6 pages

Here is presented a new exact solution of Ice dynamics in Glaciers in terms of viscousplastic theory of movements, for 2-dimensional case: x (t) = y (t). In general case, 2-D solution of Ice dynamics could be classified as Riccati's type. Due to a very special character of Riccati's type equation, it's general solution is proved to have a proper gap of components of such a solution.
Category: Mathematical Physics

A First-Order Phase Transition for Pulsar Surface

Authors: Giuseppe Iurato
Comments: 14 pages

In this paper, we explain the existence of a possible first-order phase transition that may occur over the external solid crust of a pulsar, by means of an interpretation of a particular formal model (drew from Theoretical Astrophysics) whose thermodynamical phenomenology shows a possible first-order phase transition (according to Landau's Phenomenological Theory).
Category: Mathematical Physics

Discussion Needed About Three (New) Cosmological Models Based on Mathematics and Physics

Authors: Dan Visser
Comments: 5 pages

Three new cosmological models are 'circling' the science-community: a 'Bouncing Universe', a 'Conformal Cyclic Cosmology and a 'Double Torus Cosmology'. All new ideas, commented anonymously and discussed institutionally. This paper wants more cosmologists and physicists to involve the discussion openly in the media, without walking save roads. The authors and readers are invited to involve this discussion. Which model is true and can the mathematics be matched?
Category: Mathematical Physics

Hard Theoretical Evidence for the Dark Energy Force Formula in a Double Torus Universe.

Authors: Dan Visser
Comments: 6 pages

This paper shows how a 'dark energy force formula' emerges five more space- and two more time-dimensions in nature. The 'formula' is earlier described in vixra-papers, announcing the universe is a double torus of dark energy and dark matter. The 'formula' is a completely different force than the cosmological constant of Einstein used to explain accelerated expansion in the big bang. With this in mind, two independent experimental investigations has given additional proof, that the 'dark energy force formula' correlates the independent investigations, because: 1) Five extra space-dimensions have to exist, according to how electrons behave in graphene-experiments. 2) Computer-simulation shows a double torus geometry, that emerges from two colliding blackholes involving a third torus during pulsation. These results match with the postulate that the universe exists of a double torus of dark energy and dark matter, including a 'dark energy force formula'.
Category: Mathematical Physics

A Few Implications of the Laws of Transactions, from the Abstraction Theory.

Authors: Subhajit Ganguly
Comments: 10 pages.

Considering transport of light through space-time, following the laws of physical transactions (viXra:1101.0035) it may be said that there must be a spreading effect on it. Over suitable distances from a source of light, an observer's perception is bound to be affected due to this spreading. In the following paper, these effects on the reception of a signal, due to the spreading of light are studied. Experimental set-ups are desired to verify the actual angles of spread with their theoretical values. An experiment regarding the minimum distance between two disturbances for them to be distinguishable is also carried out. The energy quantum is also studied in a new light.
Category: Mathematical Physics

Dark Matter Formula for Fundamental Calculation of Satelite Flybys in Hyperbolic Orbits.

Authors: Dan Visser
Comments: 8 pages

For the first time an announcement is made in this paper to have found fundamental evidence for the flyby-anomalies of six satelites earlier investigated by John Anderson and coworkers of the Jet Propulsion Laboratory (JPL) in Pasadena USA. The central part of this theoretical evidence exists of a 'dark matter impuls flow' being the cause of a velocity-change for satelites during their 'flyby' along the earth. A formula is given to calculate the velocity-change caused by dark matter. Also a dark matter constant is suggested. The origin of the evidence is related to a 'dark energy force formula', which is a new force in a new proposed cosmological model describing dark energy and dark matter in a double torus geometry (TTM). Originally the 'dark energy force formula' is discovered by an independent cosmologist and E-ingenieur, Dan Visser from Almere, the Netherlands. Then afterwards a few 'papers' have been published in the vixra-archive since September 1 2009 in colaboration with mathematician and physicist Christopher Forbes (UK). These 'papers' could be considered as 'pre-exercises' in awaiting for a more robust mathematical framework for the new proposed double torus cosmological model.
Category: Mathematical Physics

A Note on the Action at a Distance

Authors: José Francisco García Juliá
Comments: 3 pages

We consider that the action at a distance is carried out by virtual carriers of the force, which are created and annihilated in the vacuum by the field in a time less than that of the Heisenberg's uncertainty.
Category: Mathematical Physics

The Finite Element Method (Fem)to Finding the Reverberation Times of Irregular Rooms

Authors: Jalil Olia, Vahid Afshinmehr
Comments: 20 pages

In this paper we applied a finite element method to finding the effects on the reverberation times of common irregularities like curved surfaces, non-parallel walls and large open-walled ante-rooms, found in auditoria. The number of modes having a reverberation time in a specified time interval is expressed as a function of the total allowed degrees of freedom and it is shown that even when the number of degrees of freedom of the model is large there is, in general, no one dominant group. Curved surfaces in particular lead to a situation where some modes have very long reverberation times, leading to bad acoustics. In such situations a knowledge of the offending mode shapes give an indication on where to position absorptive material for optimum effect.
Category: Mathematical Physics

Double Torus Cosmology Reveals Cosmic Backround to Measure Dark Energy.

Authors: Dan Visser, Chris Forbes
Comments: 7 pages.

Particularly this paper announces dark energy could be measured as a cosmic backround (CMB)-frame related to a specific quantumstate of dark energy and dark matter conform a double torus cosmology (TTM[1,2,3,4,5,6,7]). In addition this paper also refers to a planned dark energy interferometerproject expected to be operational in 2014[8]. Both aspects can be combined in order to get a better expectation and interpretation of the detection of dark energy. This paper shows dark energy to interact differently than in the planned experiments of the dark energy interferometer-project. That is the motive to publish this paper. Benchmark is dark energy and dark matter are not considered in a big bang cosmology, but in a double torus universe of one torus of dark energy, which encloses and intertwines a second torus of dark matter (TTM-cosmology). In derivations is shown that dark energy will affect falling (super positioned) Cesium atoms in the dark energy interferometerproject unexpectedly: 1. It will touch the super-positioned atoms twice! ; the cause will be the torus-geometry of the TTM-derived CMB-frame. 2. The 'Twice-touching' will vary subsequently; the cause will be a dependency on 'expansion or contraction' of the CMB-torus geometry through dynamics that are caused by the "+" and "-" strength of the dark energy force in the TTM and which is produced by the dark energy torus. This has motivated me to calculate a specific value for TTM-dark energy on about 4 x 10^-114 [X.s] in 6.4 x 10^-48 [m²] as a new sort of spin-quantumstate [X.s], which drives the expansion of big bang cosmology analogue to how elementary spin is a 'generator for rotation' in conventional quantumphysics. Probably this paper might be of interest to the dark energy interferometer-project at least.
Category: Mathematical Physics

A Beautiful Theory of Everything: How Simplexity Leads to Reality!

Authors: Ayind T Mahamba
Comments: 8 Pages. Submissions for FQXi essay contest.

The quest to explain the true nature of reality is one of the great scientific goals. In fact, this essay contest asks: is Nature fundamentally continuous or discrete and how can these two different but very useful concepts be fully reconciled? Physical science is vast, complex and remains mysterious [10]. Since long ago, the great thinkers and scholars have dedicated their lives to the attempted comprehension1 of this reality that has become so abstract. Throughout the centuries and through experimentation, they have established numerous laws, concepts, theories, and principles concerning the fundamental notions of reality (centered on matter-energy and spacetime). I propose a central theory (MIT), based on the information of, and compatible with, the contemporary scientific knowledge; the existing fundamental relation between the "physical entities" passes through the determined quantitative transmission (quantity) of this preserved transcendent greatness (quality). In addition to a "formal" relationship (existence) which creates an informal description of what is real, there is a causal relationship between "phenomena" (relativity). My informational approach has been productive in several domains where many enigma persist; solutions for these problems must be envisaged globally, using ideas and concepts from numerous different fields, with a coherent schema... The "Theory of Universal Relativity" (TUR as a ToE) proposed here lays bridges between domains which, at first glance, have nothing to do with each other; it also provides insight into how we can improve our knowledge by understanding the interplay of complexity and simplicity. Therefore emerging from simplexity (contraction of simplicity and complexity), reality is both digital and analogue (and between) and also more! We know there is a strange and mysterious world that surrounds us, a world largely hidden from our senses with extra dimensions and as a mathematical concept of reality, MIT may confirm that we are part of a cosmic hologram (a paradigm shift). My theory has the advantage of being extremely simple, not limited to scientists because everyone can understand it (I = 1 ± i). So, in this essay, I will try to explain why and how [1][13][48][51].
Category: Mathematical Physics

Quintessence-Momentum as Link Between Mass and Charge

Authors: Malcolm Macleod
Comments: 3 pages.

This paper suggests a 'quantity of momentum', a square root of Planck momentum, here referred to as Quintessence-momentum, as a natural unit that is common to both mass and charge. In terms of this Quintessence momentum Q, alpha (Sommerfeld fine structure constant) and c; geometrical formulas for the natural physical constants and the electron mass are proposed. Results are consistent with CODATA 2006.
Category: Mathematical Physics

Scale Dimension as the Fifth Dimension of Spacetime

Authors: Sergey G. Fedosin
Comments: 5 pages. In Russian

The scale dimension which is discovered in the theory of infinite nesting of matter is studied from the perspective of the physical implementation of well-studied four-and n-dimensional geometric objects. Adding of the scale dimension to Minkowski space means the need to use the five-dimensional spacetime.
Category: Mathematical Physics

On the Stability of Linear Systems

Authors: Daniele Sasso
Comments: 7 pages, 5 figures.

The criteria of stability defined in the standard theory of linear systems aren't exhaustive and show some inconsistencies. In this article we define new criteria of stability more consistent with real physical situations. In particular we distinguish between static stability and dynamic stability in order to analyse the stability of systems in the time domain and in Laplace's equivalent domain. Let introduce then the frequency stability in order to analyse the stability of systems in the Fourier domain.
Category: Mathematical Physics

Entanglement Related to Cosmology-TTM

Authors: Dan Visser, Christopher Forbes
Comments: 10 pages

This paper postulates a theoretical structure within entangled photons. The postulate is introduced within the framework of the cosmological Twin-Tori Model (TTM). Related papers are to be found in viXra[1,2,3,4,5,6]. After generally derived equations and interpretations, a mass (m_t) per 2π is calculated on ~ 2.6 x 10^-34 [(J.s) m² / s]. Such a tiny spinning-forward surface per second (torus geometry) has an energy much smaller than the Planckenergy in Joule, suggesting a subdivision of 10⁴³ surfaces below the Planckscale: If one photon changes spin, the entangled photon could follow by means of the spinning-forward tiny surface-structure within the torus geometry (per 2π), being an information-flux for entanglement in general below the Planckscale.
Category: Mathematical Physics

Hidden Mathematical Symmetries in the 32 Crystal Point Groups?

Authors: Giuliano Bettini
Comments: 10 pages

In a preceding paper we introduced a conjecture: the classification of the 32 crystal classes with 5 bits. In the present paper we will review our preceding result, and continue showing some further interesting issues. In the paper, it is argued that bits should be identified with five basic unknown symmetries generating these 32 groups. Probably it is not merely a coincidence that 32 means 5 bits. And probably is it not merely a coincidence that each complete subset of bits (properties) means the holohedry of a crystal system; and each new bit means a new crystal system. The purpose of this article was of course not to draw a conclusive theory, but to suggest ideas that, we hope, will be useful for researchers in mathematics, group theory and crystallography.
Category: Mathematical Physics

Fine Structure Constant α ~ 1/137.036 and Blackbody Radiation Constant α_R ~ 1/157.555

Authors: Ke Xiao
Comments: 5 pages

The fine structure constant α = e²/hc ~ 1/137.036 and the blackbody radiation constant α_R = e²(a_R/k⁴_B)^1/3 ~ 1/157.555 are linked by prime numbers. The blackbody radiation constant is a new method to measure the fine structure constant. It also links the fine structure constant to the Boltzmann constant.
Category: Mathematical Physics

Abstraction in Theory Laws of Physical Transactions

Authors: Subhajit Ganguly
Comments: 15 pages

Considering transport or tendency ... (see paper)
Category: Mathematical Physics

32 Point Groups of Three Dimensional Crystal Cells Described by 5 Bits

Authors: Giuliano Bettini
Comments: 9 pages, in Italian

There are 32 possible combinations of symmetry operations that define the external symmetry of crystals. These 32 possible combinations result in the 32 crystal classes. But for a radar engineer it is inevitable to associate "32" to "5 bits". I submit a tentative classification of the 32 crystal classes with a 5 bit classification, obviously with a (tentative) physical meaning of each bit. Each bit means a physical property.
Category: Mathematical Physics

Further on Non-Cartesian Systems

Authors: Elemér E Rosinger
Comments: 9 pages

A class of non-Cartesian physical systems, [7], are those whose composite state spaces are given by significantly extended tensor products. A more detailed presentation of the way such extended tensor products are constructed is offered, based on a step by step comparison with the construction of usual tensor products. This presentation clarifies the extent to which the extended tensor products are indeed more general than the usual ones.
Category: Mathematical Physics

Non-Cartesian Systems :An Open Problem

Authors: Elemer E Rosinger
Comments: 6 pages

The following open problem is presented and motivated : Are there physical systems whose state spaces do not compose according to either the Cartesian product, as classical systems do, or the usual tensor product, as quantum systems do ?
Category: Mathematical Physics

Four Departures in Mathematics and Physics

Authors: Elemer E Rosinger
Comments: 28 pages

Much of Mathematics, and therefore Physics as well, have been limited by four rather consequential restrictions. Two of them are ancient taboos, one is an ancient and no longer felt as such bondage, and the fourth is a surprising omission in Algebra. The paper brings to the attention of those interested these four restrictions, as well as the fact that each of them has by now ways, even if hardly yet known ones, to overcome them.
Category: Mathematical Physics

Some Comments on Projective Quadrics Subordinate to Pseudo-Hermitian Spaces

Authors: Arkadiusz Jadczyk
Comments: 7 pages, To appear in Advances in Applied Clifford Algebras

We study in some detail the structure of the projective quadric Q' obtained by taking the quotient of the isotropic cone in a standard pseudohermitian space H_p,q with respect to the positive real numbers R⁺ and, further, by taking the quotient ~Q = Q'/U(1). The case of signature (1. 1) serves as an illustration. ~Q is studied as a compactification of RxH_p-1,q-1
Category: Mathematical Physics

Some Orbital and Other Properties of the 'special Gravitating Annulus'

Authors: Guy Moore, Richard Moore
Comments: 40 pages

Our obtaining the analytical equations for the gravitation of a particular type of mathematical annulus, which we called a 'Special Gravitating Annulus' (SGA), greatly facilitates studying its orbital properties by computer programming. This includes isomorphism, periodic and chaotic polar orbits, and orbits in three dimensions. We provide further insights into the gravitational properties of this annulus and describe our computer algorithms and programs. We study a number of periodic orbits, giving them names to aid identification. 'Ellipses extraordinaires' which are bisected by the annulus, have no gravitating matter at either focus and represent a fundamental departure from the normal association of elliptical orbits with Keplerian motion. We describe how we came across this type of orbit and the analysis we performed. We present the simultaneous differential equations of motion of 'ellipses extraordinaires' and other orbits as a mathematical challenge. The 'St.Louis Gateway Arch' orbit contains two 'instantaneous static points' (ISP). Polar elliptical orbits can wander considerably without tending to form other kinds of orbit. If this type of orbit is favoured then this gives a similarity to spiral galaxies containing polar orbiting material. Annular oscillatory orbits and rotating polar elliptical orbits are computed in isometric projection. A 'daisy' orbit is computed in stereo-isometric projection. The singularity at the centre of the SGA is discussed in relation to mechanics and computing, and it appears mathematically different from a black hole. In the Appendix, we prove by a mathematical method that a thin plane self-gravitating Newtonian annulus, free from external influence, exhibiting radial gravitation that varies inversely with the radius in the annular plane, must have an area mass density which also varies inversely with the radius and this exact solution is the only exact solution.
Category: Mathematical Physics

Dark Matter and Visible Matter Fundamentally Related in New Cosmological Model and Recalculated.

Authors: Dan Visser
Comments: 7 pages

A new surface energy-value for dark matter is calculated, derivated from a perspective of a "higher order universe". A universe of dark energy, dark- and visible matter, and a dark energy force. A fundamental connection between dark matter and visible matter is related to dark energy (viXra-paper 1010.0014 in particular is the reference for this novum [5]. The surface energy-density of dark matter seems to be a factor 5 to 20 times higher than earlier predicted value-ranges by the CDMS-project and the Fermi-satelite. Also the produced energies through particle-collissions by LHC CERN will not be enough to achieve the dark matter surface energy-value. The "fact" some of these projects have announced some vaque "bliebs" might be due to a new phenomenon in the search of dark matter. It could be caused by "three dimensional time", which is embedded as a "new duality" in the new model, the "Twin-Tori cosmological Model (TTM)"[1,2,3,4]. The "three dimensional time" might cause dark matter taking unknown paths before detection. This paper has calculated the surface energy-density value for dark matter on 1 TeV in a surface of 6.4 x 10 ^-48 [m2].
Category: Mathematical Physics

On Using Quaternion Operators in Quantum and Microphysics.

Authors: Debayan Dasgupta
Comments: 10 pages

The main purpose of this paper is to show the application of mathematical system of quaternions in physics. We show a basic way of generating a quaternion operator and a method of obtaining the Schrodinger's equation using this operator matrix. Then we investigate the various probable uses of the coefficient matrix to scale relativity and spacetime quantization.
Category: Mathematical Physics

Deeper Properties Through Dark and Visible-Matter in a New Cosmological Twin-Tori Model (TTM).

Authors: Dan Visser
Comments: 5 pages

A new cosmological model, named the Twin-Tori Model (TTM)[1], postulates a dark energy force F_de , which empowers the dynamic of a lower order universe, well known as the big bang. In this paper I introduce the 1st derivative F'_de of this dark energy force to reveal deeper properties of the TTM, such as: why quantummechanics exists in the big bang, why dark matter and visible matter are equally responsible for gravity in galaxies for 1/4 of the density of dark matter at a specific length, why the big bang universe is recalculated by subquantumlevel-information below the Plancklength, and why the impression of space-expansion is due to the higher order cosmological model TTM.
Category: Mathematical Physics

A Multiple Particle System Equation Underlying the Klein-Gordon-Dirac-Schrödinger Equations

Authors: D.T. Froedge
Comments: 16 pages 38 equations 98kb

The purpose of this paper is to illustrate a fundamental, multiple particle, system equation for which the Klein-Gordon-Dirac-Schrödinger equations are single particle special cases. In the same manner that eigenvalues of the Schrödinger equation represents energy levels of an interacting atomic system, eigenvalues represent particle energies in an interacting system of particles. An equation is proposed that has vector solutions defined in Dirac, or Clifford algebra, that treats a collection of particles as a single system..The proposed solution is a descriptor of a symmetric, light speed expanding group of interacting particles having real, as well as the familiar QM constituents.
Category: Mathematical Physics

A New Generic Class of Beltrami "Force-Free" Fields. Part-I: Theoretical Considerations

Authors: T. E. Raptis
Comments: 13 pages.

We report on a new general class of solutions of the Beltrami equation, with special characteristics. We also provide examples of solutions that also satisfy Maxwell equations. A subset of these solutions can be isolated which corresponds to "gauge" fields. A special projective geometry of vacuum fields is also revealed and discussed.
Category: Mathematical Physics

Causal Set Theory and the Origin of Mass-Ratio

Authors: Carey R Carlson
Comments: 16 pages.

Quantum theory is reconstructed using standalone causal sets. The frequency ratios inherent in causal sets are used to define energy-ratios, implicating the causal link as the quantum of action. Space-time and its particle-like sequences are then constructed from causal links. A 4-D time-lattice structure is defined and then used to model neutrinos and electron clouds, which together constitute a 4-D manifold. A 6-D time-lattice is used to model the nucleons. The integration of the nucleus with its electron cloud affords calculation of the mass-ratio of the proton (or the neutron) with respect to the electron. Arrow diagrams, along with several ball-and-stick models, are used to streamline the presentation.
Category: Mathematical Physics

Handbook of Functions Errata

Authors: Fredy Zypman
Comments: 2 pages

Formulas connecting toroidal functions and elliptical functions are useful in various areas of physics. In solving a problem in electrostatics we run across an error in the Handbook of mathematical functions of Abramowitz and Stegun. In this paper we report the details.
Category: Mathematical Physics

The Geometry of CP₂ and Its Relationship to Standard Model

Authors: Matti Pitkänen
Comments: 13 Pages.

This appendix contains basic facts about CP₂ as a symmetric space and Kähler manifold. The coding of the standard model symmetries to the geometry of CP₂, the physical interpretation of the induced spinor connection in terms of electro-weak gauge potentials, and basic facts about induced gauge fields are discussed
Category: Mathematical Physics

Could the Dynamics of Kähler Action Predict the Hierarchy of Planck Constants?

Authors: Matti Pitkänen
Comments: 5 Pages.

The original justification for the hierarchy of Planck constants came from the indications that Planck constant could have large values in both astrophysical systems involving dark matter and also in biology. The realization of the hierarchy in terms of the singular coverings and possibly also factor spaces of CD and CP₂ emerged from consistency conditions. It however seems that TGD actually predicts this hierarchy of covering spaces. The extreme non-linearity of the field equations defined by Kähler action means that the correspondence between canonical momentum densities and time derivatives of the imbedding space coordinates is 1-to-many. This leads naturally to the introduction of the covering space of CD x CP₂, where CD denotes causal diamond defined as intersection of future and past directed light-cones.
Category: Mathematical Physics

Weak Form of Electric-Magnetic Duality, Electroweak Massivation, and Color Confinement

Authors: Matti Pitkänen
Comments: 13 Pages.

The notion of electric magnetic duality emerged already two decades ago in the attempts to formulate the Kähler geometry of the "world of classical worlds". Quite recently a considerable step of progress took place in the understanding of this notion. This concept leads to the identification of the physical particles as string like objects defined by magnetic charged wormhole throats connected by magnetic ux 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. Electric-magnetic duality leads also to a detailed understanding of how TGD reduces to almost topological quantum field theory. A surprising outcome is the necessity to replace CP₂ Kähler form in Kähler action with its sum with S² Kähler form.
Category: Mathematical Physics

How to Define Generalized Feynman Diagrams?

Authors: Matti Pitkänen
Comments: 16 Pages.

Generalized Feynman diagrams have become the central notion of quantum TGD and one might even say that space-time surfaces can be identified as generalized Feynman diagrams. The challenge is to assign a precise mathematical content for this notion, show their mathematical existence, and develop a machinery for calculating them. Zero energy ontology has led to a dramatic progress in the understanding of generalized Feynman diagrams at the level of fermionic degrees of freedom. In particular, manifest finiteness in these degrees of freedom follows trivially from the basic identifications as does also unitarity and non-trivial coupling constant evolution. There are however several formidable looking challenges left.

1. One should perform the functional integral over WCW degrees of freedom for fixed values of on mass shell momenta appearing in the internal lines. After this one must perform integral or summation over loop momenta.
2. One must define the functional integral 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.

Physics as Generalized Number Theory: Infinite Primes

Authors: Matti Pitkänen
Comments: 37 Pages.

Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields, and infinite primes discussed in this article. The construction of infinite primes is formally analogous to a repeated second quantization of an arithmetic quantum field theory by taking the many particle states of previous level elementary particles at the new level. Besides free many particle states also the analogs of bound states appear. In the representation in terms of polynomials the free states correspond to products of first order polynomials with rational zeros. Bound states correspond to nth order polynomials with non-rational but algebraic zeros. The construction can be generalized to classical number fields and their complexifications obtained by adding a commuting imaginary unit. Special class corresponds to hyper-octonionic primes for which the imaginary part of ordinary octonion is multiplied by the commuting imaginary unit so that one obtains a sub-space M⁸ with Minkowski signature of metric. Also in this case the basic construction reduces to that for rational or complex rational primes and more complex primes are obtained by acting using elements of the octonionic automorphism group which preserve the complex octonionic integer property. Can one map infinite primes/integers/rationals to quantum states? Do they have space-time surfaces as correlates? Quantum classical correspondence realized in terms of modified Dirac operator implies that if infinite rationals can be mapped to quantum states then the mapping of quantum states to space-time surfaces automatically gives the map to space-time surfaces. The question is therefore whether the mapping to quantum states defined by WCW spinor fields is possible. A natural hypothesis is that number theoretic fermions can be mapped to real fermions and number theoretic bosons to WCW ("world of classical worlds") Hamiltonians. The crucial observation is that one can construct infinite hierarchy of hyper-octonionic units by forming ratios of infinite integers such that their ratio equals to one in real sense: the integers have interpretation as positive and negative energy parts of zero energy states. One can construct also sums of these units with complex coefficients using commuting imaginary unit and these sums can be normalized to unity and have interpretation as states in Hilbert space. These units can be assumed to possess well defined standard model quantum numbers. It is possible to map the quantum number combinations of WCW spinor fields to these states. Hence the points of M⁸ can be said to have infinitely complex number theoretic anatomy so that quantum states of the universe can be mapped to this anatomy. One could talk about algebraic holography or number theoretic Brahman=Atman identity. One can also ask how infinite primes relate to the p-adicization program and to the hierarchy of Planck constants. The key observation is that infinite primes are in one-one correspondence with rational numbers at the lower level of hierarchy. At the first level of hierarchy the p-adic norm with respect to p-adic prime for this rational gives power p^-n so that one has two powers of p - pⁿ⁺ and p^n- since two infinite primes corresponding to fermionic vacua X±1, where X is the product of all primes at given level of hierarchy, characterize the partonic 2-surface. The proposal inspired by the p-adicization program is that Δφ = 2π/pⁿ defines angle measurement resolution crucial in the construction of p-adic variants of WCW ("world of classical world") as a union of symmetric coset spaces by starting from discrete variants of the real counterpart of symmetric space having common points with tis p-adic variant. The two measurement resolutions correspond to CD and CP₂ degrees of freedom. The hierarchy of Planck constants generalizes imbedding space to a book like structure with pages identified in terms of singular coverings and factor spaces of CD and CP₂. There are good arguments suggesting that only coverings characterized by integers n_a and n_bare realized. The identifications na = n₊ and nb = n- lead to highly non-trivial physical predictions and allow sharpen the view about the hierarchy of Planck constants. Therefore the notion of finite measurement resolution becomes the common denominator for the three threads of the number theoretic vision and give also a connection with the physics as infinite-dimensional geometry program and with the inclusions of hyper-finite factors defined by WCW spinor fields and proposed to characterize finite measurement resolution at quantum level.
Category: Mathematical Physics

Physics as Generalized Number Theory: Classical Number Fields

Authors: Matti Pitkänen
Comments: 28 Pages.

Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields discussed in this article, and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article the connection between standard model symmetries and classical number fields is discussed. The basis vision is that the geometry of the infinite-dimensional WCW ("world of classical worlds") is unique from its mere existence. This leads to its identification as union of symmetric spaces whose Kähler geometries are fixed by generalized conformal symmetries. This fixes space-time dimension and the decomposition M⁴ x S and the idea is that the symmetries of the Kähler manifold S make it somehow unique. The motivating observations are that the dimensions of classical number fields are the dimensions of partonic 2-surfaces, space-time surfaces, and imbedding space and M⁸ can be identified as hyper-octonions- a sub-space of complexified octonions obtained by adding a commuting imaginary unit. This stimulates some questions. Could one understand S = CP₂ number theoretically in the sense that M⁸ and H = M⁴ x CP₂ be in some deep sense equivalent ("number theoretical compactification" or M⁸ - H duality)? Could associativity define the fundamental dynamical principle so that space-time surfaces could be regarded as associative or co-associative (defined properly) sub-manifolds of M⁸ or equivalently of H. One can indeed define the associativite (co-associative) 4-surfaces using octonionic representation of gamma matrices of 8-D spaces as surfaces for which the modified gamma matrices span an associate (co-associative) sub-space at each point of space-time surface. Also M⁸ - H duality holds true if one assumes that this associative sub-space at each point contains preferred plane of M⁸ identifiable as a preferred commutative or co-commutative plane (this condition generalizes to an integral distribution of commutative planes in M⁸). These planes are parametrized by CP₂ and this leads to M⁸ - H duality. WCW itself can be identified as the space of 4-D local sub-algebras of the local Clifford algebra of M⁸ or H which are associative or co-associative. An open conjecture is that this characterization of the space-time surfaces is equivalent with the preferred extremal property of Kähler action with preferred extremal identified as a critical extremal allowing infinite-dimensional algebra of vanishing second variations.
Category: Mathematical Physics

Physics as Generalized Number Theory: P-Adic Physics and Number Theoretic Universality

Authors: Matti Pitkänen
Comments: 51 Pages.

Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields (in particular, identifying associativity condition as the basic dynamical principle), and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article p-adic physics and the technical problems relates to the fusion of p-adic physics and real physics to a larger structure are discussed. The basic technical problems relate to the notion of definite integral both at space-time level, imbedding space level and the level of WCW (the "world of classical worlds"). The expressibility of WCW as a union of symmetric spacesleads to a proposal that harmonic analysis of symmetric spaces can be used to define various integrals as sums over Fourier components. This leads to the proposal the p-adic variant of symmetric space is obtained by a algebraic continuation through a common intersection of these spaces, which basically reduces to an algebraic variant of coset space involving algebraic extension of rationals by roots of unity. This brings in the notion of angle measurement resolution coming as Δφ = 2π/pⁿ for given p-adic prime p. Also a proposal how one can complete the discrete version of symmetric space to a continuous p-adic versions emerges and means that each point is effectively replaced with the p-adic variant of the symmetric space identifiable as a p-adic counterpart of the real discretization volume so that a fractal p-adic variant of symmetric space results. If the Kähler geometry of WCW is expressible in terms of rational or algebraic functions, it can in principle be continued the p-adic context. One can however consider the possibility that that the integrals over partonic 2-surfaces defining ux Hamiltonians exist p-adically as Riemann sums. This requires that the geometries of the partonic 2-surfaces effectively reduce to finite sub-manifold geometries in the discretized version of δM₊⁴. If Kähler action is required to exist p-adically same kind of condition applies to the space-time surfaces themselves. These strong conditions might make sense in the intersection of the real and p-adic worlds assumed to characterized living matter.
Category: Mathematical Physics

Construction of Configuration Space Spinor Structure

Authors: Matti Pitkänen
Comments: 34 Pages.

There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first approach relies on a direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach discussed in this article relies on the construction of spinor structure based on the hypothesis that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for the second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of spinor structure. This implies a geometrization of fermionic statistics. The basic philosophy is that at fundamental level the construction of WCW geometry reduces to the second quantization of the induced spinor fields using Dirac action. This assumption is parallel with the bosonic emergence stating that all gauge bosons are pairs of fermion and antifermion at opposite throats of wormhole contact. Vacuum function is identified as Dirac determinant and the conjecture is that it reduces to the exponent of Kähler function. In order to achieve internal consistency induced gamma matrices appearing in Dirac operator must be replaced by the modified gamma matrices defined uniquely by Kähler action and one must also assume that extremals of Kähler action are in question so that the classical space-time dynamics reduces to a consistency condition. This implies also super-symmetries and the fermionic oscillator algebra at partonic 2-surfaces has intepretation as N = 1 generalization of space-time supersymmetry algebra different however from standard SUSY algebra in that Majorana spinors are not needed. This algebra serves as a building brick of various super-conformal algebras involved. The requirement that there exist deformations giving rise to conserved Noether charges requires that the preferred extremals are critical in the sense that the second variation of the Kähler action vanishes for these deformations. Thus Bohr orbit property could correspond to criticality or at least involve it. Quantum classical correspondence demands that quantum numbers are coded to the properties of the preferred extremals given by the Dirac determinant and this requires a linear coupling to the conserved quantum charges in Cartan algebra. Effective 2-dimensionality allows a measurement interaction term only in 3-D Chern-Simons Dirac action assignable to the wormhole throats and the ends of the space-time surfaces at the boundaries of CD. This allows also to have physical propagators reducing to Dirac propagator not possible without the measurement interaction term. An essential point is that the measurement interaction corresponds formally to a gauge transformation for the induced Kähler gauge potential. If one accepts the weak form of electric-magnetic duality Kähler function reduces to a generalized Chern-Simons term and the effect of measurement interaction term to Kähler function reduces effectively to the same gauge transformation. The basic vision is that WCW gamma matrices are expressible as super-symplectic charges at the boundaries of CD. The basic building brick of WCW is the product of infinite-D symmetric spaces assignable to the ends of the propagator line of the generalized Feynman diagram. WCW Kähler metric has in this case "kinetic" parts associated with the ends and "interaction" part between the ends. General expressions for the super-counterparts of WCW ux Hamiltoniansand for the matrix elements of WCW metric in terms of their anticommutators are proposed on basis of this picture.
Category: Mathematical Physics

Construction of Configuration Space Geometry from Symmetry Principles

Authors: Matti Pitkänen
Comments: 26 Pages.

There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first one relies on a direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach relies on the construction of spinor structure assuming that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for the second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of spinor structure. In this article the construction of Kähler form and metric based on symmetries is discussed. The basic vision is that WCW can be regarded as the space of generalized Feynman diagrams with lines thickned to light-like 3-surfaces and vertices identified as partonic 2-surfaces. In zero energy ontology the strong form of General Coordinate Invariance (GCI) implies effective 2-dimensionality and the basic objects are pairs partonic 2-surfaces X² at opposite light-like boundaries of causal diamonds (CDs). The hypothesis is that WCW can be regarded as a union of infinite-dimensional symmetric spaces G/H labeled by zero modes having an interpretation as classical, non-quantum uctuating variables. A crucial role is played by the metric 2-dimensionality of the light-cone boundary δM₊⁴ + and of light-like 3-surfaces implying a generalization of conformal invariance. The group G acting as isometries of WCW is tentatively identified as the symplectic group of δM₊⁴ x CP₂ localized with respect to X². H is identified as Kac-Moody type group associated with isometries of H = M₊⁴ x CP₂ acting on light-like 3-surfaces and thus on X². An explicit construction for the Hamiltonians of WCW isometry algebra as so called ux Hamiltonians is proposed and also the elements of Kähler form can be constructed in terms of these. Explicit expressions for WCW ux Hamiltonians as functionals of complex coordinates of the Cartesisian product of the infinite-dimensional symmetric spaces having as points the partonic 2-surfaces defining the ends of the the light 3-surface (line of generalized Feynman diagram) are proposed.
Category: Mathematical Physics

Identification of the Configuration Space Kähler Function

Authors: Matti Pitkänen
Comments: 38 Pages.

There are two basic approaches to quantum TGD. The first approach, which is discussed in this article, is a generalization of Einstein's geometrization program of physics to an infinitedimensional context. Second approach is based on the identification of physics as a generalized number theory. The first approach relies on the vision of quantum physics as infinite-dimensional Kähler geometry for the "world of classical worlds" (WCW) identified as the space of 3-surfaces in in certain 8-dimensional space. There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first approach relies on direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach relies on the construction of spinor structure based on the hypothesis that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of WCW spinor structure. In this article the proposal for Kähler function based on the requirement of 4-dimensional General Coordinate Invariance implying that its definition must assign to a given 3-surface a unique space-time surface. Quantum classical correspondence requires that this surface is a preferred extremal of some some general coordinate invariant action, and so called Kähler action is a unique candidate in this respect. The preferred extremal has intepretation as an analog of Bohr orbit so that classical physics becomes and exact part of WCW geometry and therefore also quantum physics. The basic challenge is the explicit identification of WCW Kähler function K. Two assumptions lead to the identification of K as a sum of Chern-Simons type terms associated with the ends of causal diamond and with the light-like wormhole throats at which the signature of the induced metric changes. The first assumption is the weak form of electric magnetic duality. Second assumption is that the Kähler current for preferred extremals satisfies the condition jK ^ djK = 0 implying that the ow parameter of the ow lines of jK defines a global space-time coordinate. This would mean that the vision about reduction to almost topological QFT would be realized. Second challenge is the understanding of the space-time correlates of quantum criticality. Electric-magnetic duality helps considerably here. The realization that the hierarchy of Planck constant realized in terms of coverings of the imbedding space follows from basic quantum TGD leads to a further understanding. The extreme non-linearity of canonical momentum densities as functions of time derivatives of the imbedding space coordinates implies that the correspondence between these two variables is not 1-1 so that it is natural to introduce coverings of CD x CP₂. This leads also to a precise geometric characterization of the criticality of the preferred extremals.
Category: Mathematical Physics

Physics as Infinite-Dimensional Geometry and Generalized Number Theory: Basic Visions

Authors: Matti Pitkänen
Comments: 32 Pages.

There are two basic approaches to the construction of quantum TGD. The first approach relies on the vision of quantum physics as infinite-dimensional Kähler geometry for the "world of classical worlds" identified as the space of 3-surfaces in in certain 8-dimensional space. Essentially a generalization of the Einstein's geometrization of physics program is in question. The second vision is the identification of physics as a generalized number theory. This program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields (in particular, identifying associativity condition as the basic dynamical principle), and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article brief summaries of physics as infinite-dimensional geometry and generalized number theory are given to be followed by more detailed articles.
Category: Mathematical Physics

Unfolding the Labyrinth: Open Problems in Physics, Mathematics, Astrophysics, and Other Areas of Science

Authors: Florentin Smarandache, V. Christianto, Fu Yuhua, Radi I. Khrapko, J. Hutchison
Comments: 147 pages

The reader will find herein a collection of unsolved problems in mathematics and the physical sciences. Theoretical and experimental domains have each been given consideration. The authors have taken a liberal approach in their selection of problems and questions, and have not shied away from what might otherwise be called speculative, in order to enhance the opportunities for scientific discovery. Progress and development in our knowledge of the structure, form and function of the Universe, in the true sense of the word, its beauty and power, and its timeless presence and mystery, before which even the greatest intellect is awed and humbled, can spring forth only from an unshackled mind combined with a willingness to imagine beyond the boundaries imposed by that ossified authority by which science inevitably becomes, as history teaches us, barren and decrepit. Revealing the secrets of Nature, so that we truly see 'the sunlit plains extended, and at night the wondrous glory of the everlasting stars', requires far more than mere technical ability and mechanical dexterity learnt form books and consensus. The dustbin of scientific history is replete with discredited consensus and the grand reputations of erudite reactionaries. Only by boldly asking questions, fearlessly, despite opposition, and searching for answers where most have not looked for want of courage and independence of thought, can one hope to discover for one's self. From nothing else can creativity blossom and grow, and without which the garden of science can only aspire to an overpopulation of weeds.
Category: Mathematical Physics

A 'planck-Like' Characterization of Exponential Function

Authors: Constantinos Ragazas
Comments: 8 pages

We derive a characterization of simple exponential functions that has the exact mathematical form to Planck's Formula for blackbody radiation in Quantum Physics.
Category: Mathematical Physics

Mathematical Model of Information

Authors: Elemér E Rosinger
Comments: 17 pages.

A simple and rather general mathematical model of the phenomenon of information is presented, followed by several examples and comments.
Category: Mathematical Physics

Invariants Relative to Change of Value of the Independent Variable and Their Role in the Physics.

Authors: Vladimir I. Smirnov
Comments: 34 pages, Russian and English versions included

It is identified the new class of invariants which values are constant at change of value of an inde-pendent variable. Their properties and a deriving method are shown on already known and still un-known instances, concerning to various areas of physics. In particular, new invariants (50), (55) and (57) for the straight lines intersected in one point on a plane have been discovered. Besides, the re-quest for detection of the third (not dependent on two already known) an invariant (31) electromag-netic fields for a special case of the special theory of relativity is made.
Category: Mathematical Physics

Mathematical and Phenomenological Elements of the Twin-Tori Model of Physics and Cosmology.

Authors: Chris Forbes
Comments: 10 pages

In this, a follow up to a previous paper 'A Short Article On A Newly Proposed Model Of Cosmology' (viXra:0909.0005), some of the basic mathematical structures to be used in the formulation of the model are shown, and several advantages are discussed. The paper then takes a more phenomenological approach and several simple (1+1) dimensional models are explored.
Category: Mathematical Physics

A First Order Singular Perturbation Solution to a Simple One-Phase Stefan Problem with Finite Neumann Boundary Conditions

Authors: Bruce Rout
Comments: 13 pages

This paper examines the difference between infinite and finite domains of a Stefan Problem. It is pointed out that attributes of solutions to the Diffusion Equation suggest assumptions of an infinite domain are invalid during initial times for finite domain Stefan Problems. The paper provides a solution for initial and early times from an analytical approach using a perturbation. This solution can then easily be applied to numerical models for later times. The differences of the two domains are examined and discussed.
Category: Mathematical Physics

Cylindrical Wave, Wave Equation, and Method of Separation of Variables

Authors: Hamid V. Ansari
Comments: 7 pages

It is shown that the wave equation cannot be solved for the general spreading of the cylindrical wave using the method of separation of variables. But an equation is presented in case of its solving the above act will have occurred. Also using this equation the above-mentioned general spreading of the cylindrical wave for large distances is obtained which contrary to what is believed consists of arbitrary functions.
Category: Mathematical Physics

New Calculuses

Authors: Vladislav Konovalov
Comments: 12 pages

In the article the new calculuses are offered similar differential and integral, but differing, that in them the analysis of the previous and subsequent values of a function is made. The new calculuses allow to decide problems, the solution which one with usage customary differential and integral calculus is impossible.
Category: Mathematical Physics

3x3 Unitary to Magic Matrix Transformations

Authors: Philip Gibbs
Comments: 5 pages, published in Prespacetime Journal, V5

We prove that any 3x3 unitary matrix can be transformed to a magic matrix by multiplying its rows and columns by phase factors. A magic matrix is defined as one for which the sum of the elements in any row or column add to the same value. This result is relevant to recent observations on particle mixing matrices.
Category: Mathematical Physics

Topological Maxwell Field Theory and Symmetry Breaking.

Authors: R. M. Kiehn
Comments: recovered from sciprint.org

Finally, I have found time to think about, and the incentive to study, how the field theory of Topological thermodynamics, electrodynamics, and hydrodynamics can be compared to field theory concepts that have been developed by Lagrangian methods, for both the classic and quantum mechanical varieties. For more than 30 years I have known that Cartan's topological methods could be applied to dissipative systems; the methods based on diffeomorphic-invariant Lagrangian field theories can not. The incentive came when I realized that the topological methods of Cartan gave dynamical results that can explain "symmetry breaking" and quantization in terms of continuous topological evolution.
Category: Mathematical Physics

A Hamiltonian Operator Whose Energies Are the Roots of the Riemann XI-Function

Authors: Jose Javier garcia
Comments: 25 Pages.

We give a possible interpretation of the Xi-function of Riemann as the Functional determinant for a certain Hamiltonian quantum operator in one dimension for a real-valued function V(x) , this potential V is related to the half-integral of the logarithmic derivative for the Riemann Xi-function, through the paper we will assume that the reduced Planck constant is defined in units where and that the mass is .In this case the Energies of the Hamiltonian operator will be the square of the imaginary part of the Riemann Zeros Also trhough this paper we may refer to the Hamiltonian Operator whose Energies are the square of the imaginary part of the Riemann Zeros as or (square) in the same case we will refer to the potential inside this Hamiltonian either as or to simplify notation. Keywords: = Riemann Hypothesis, Functional determinant, WKB semiclassical Approximation , Trace formula , Quantum chaos.
Category: Mathematical Physics

A Hamiltonian Operator Whose Roots Are the Roots of the Riemann XI-Function

Authors: Jose Javier Garcia Moreta
Comments: 23 Pages. there is an ERROR, whenever it says 'Whose zeros' should we put 'Whose Energies' (A Hamiltonian has no zeros)H

We give a possible interpretation of the Xi-function of Riemann as the Functional determinant for a certain Hamiltonian quantum operator in one dimension for a real-valued function V(x) , this potential V is related to the half-integral of the logarithmic derivative for the Riemann Xi-function, through the paper we will assume that the reduced Planck constant is defined in units where and that the mass is .In this case the Energies of the Hamiltonian operator will be the square of the imaginary part of the Riemann Zeros Also trhough this paper we may refer to the Hamiltonian Operator whose Energies are the square of the imaginary part of the Riemann Zeros as or (square) in the same case we will refer to the potential inside this Hamiltonian either as or to simplify notation.
Category: Mathematical Physics

A Hamiltonian Operator Whose Zeros Are the Roots of the Riemann XI-Function

Authors: Jose Javier Garcia Moreta
Comments: 23 Pages.

We give a possible interpretation of the Xi-function of Riemann as the Functional determinant for a certain Hamiltonian quantum operator in one dimension for a real-valued function V(x) , this potential V is related to the half-integral of the logarithmic derivative for the Riemann Xi-function, through the paper we will assume that the reduced Planck constant is defined in units where and that the mass is
Category: Mathematical Physics

Langlands Conjectures in TGD Framework

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.

1. Zero energy ontology (ZEO) and the related notion of causal diamond CD (CD is short hand for the cartesian product of causal diamond of M⁴ and of CP₂). ZEO leads to the notion of partonic 2-surfaces at the light-like boundaries of CD and to the notion of string world sheet. These notions are central in the recent view about TGD. One can assign to the partonic 2-surfaces a conformal moduli space having as additional coordinates the positions of braid strand ends (punctures). By electric-magnetic duality this moduli space must correspond closely to the moduli space of string world sheets.

2. Electric-magnetic duality realized in terms of string world sheets and partonic 2-surfaces. The group G and its Langlands dual ^LG would correspond to the time-like and space-like braidings. Duality predicts that the moduli space of string world sheets is very closely related to that for the partonic 2-surfaces. The strong form of 4-D general coordinate invariance implying electric-magnetic duality and S-duality as well as strong form of holography indeed predicts that the collection of string world sheets is fixed once the collection of partonic 2-surfaces at light-like boundaries of CD and its sub-CDs is known.

3. The proposal is that finite measurement resolution is realized in terms of inclusions of hyperfinite factors of type II₁ at quantum level and represented in terms of confining effective gauge group. This effective gauge group could be some associate of G: gauge group, Kac-Moody group or its quantum counterpart, or so called twisted quantum Yangian strongly suggested by twistor considerations. At space-time level the finite measurement resolution would be represented in terms of braids at space-time level which come in two varieties correspond to braids assignable to space-like surfaces at the two light-like boundaries of CD and with light-like 3-surfaces at which the signature of the induced metric changes and which are identified as orbits of partonic 2-surfaces connecting the future and past boundaries of CDs.

   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.

4. The natural identification of the associate of G is as quantum Yangian of Kac-Moody type group associated with Minkowskian open string model assignable to string world sheet representing a string moving in the moduli space of partonic 2-surface. The dual group corresponds to Euclidian string model with partonic 2-surface representing string orbit in the moduli space of the string world sheets. The Kac-Moody algebra assigned with simply laced G is obtained using the standard tachyonic free field representation obtained as ordered exponentials of Cartan algebra generators identified as transversal parts of M⁴ coordinates for the braid strands. The importance of the free field representation generalizing to the case of non-simply laced groups in the realization of finite measurement resolution in terms of Kac-Moody algebra cannot be over-emphasized.

5. Langlands duality involves besides harmonic analysis side also the number theoretic side. Galois groups (collections of them) defined by infinite primes and integers having representation as symplectic flows defining braidings. I have earlier proposed that the hierarchy of these Galois groups define what might be regarded as a non-commutative homology and cohomology. Also G has this kind of representation which explains why the representations of these two kinds of groups are so intimately related. This relationship could be seen as a generalization of the MacKay correspondence between finite subgroups of SU(2) and simply laced Lie groups.

6. Symplectic group of the light-cone boundary acting as isometries of the WCW geometry kenociteallb/compl1 allowing to represent projectively both Galois groups and symmetry groups as symplectic flows so that the non-commutative cohomology would have braided representation. This leads to braided counterparts for both Galois group and effective symmetry group.

7. The moduli space for Higgs bundle playing central role in the approach of Witten and Kapustin to geometric Landlands program is in TGD framework replaced with the conformal moduli space for partonic 2-surfaces. It is not however possible to speak about Higgs field although moduli defined the analog of Higgs vacuum expectation value. Note that in TGD Universe the most natural assumption is that all Higgs like states are "eaten" by gauge bosons so that also photon and gluons become massive. This mechanism would be very general and mean that massless representations of Poincare group organize to massive ones via the formation of bound states. It might be however possible to see the contribution of p-adic thermodynamics depending on genus as analogous to Higgs contribution since the conformal moduli are analogous to vacuum expectation of Higgs field.

How Infinite Primes Relate to Other Views About Mathematical Infinity?

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?

1. Quantum view about consciousness and cognition leads to a proposal that p-adic physics serves as a correlate for cognition. Together with the notion of infinite primes this suggests that number theory should play a key role in the axiomatics.
2. Algebraic geometry allows algebraization of the set theory and this kind of approach suggests itself strongly in physics inspired approach to the foundations of mathematics. This means powerful limitations on the notion of set.
3. Finite measurement resolution and finite resolution of cognition could have implications also for the foundations of mathematics and relate directly to the fact that all numerical approaches reduce to an approximation using rationals with a cutoff on the number of binary digits.
4. The TGD inspired vision about consciousness implies evolution by quantum jumps meaning that also evolution of mathematics so that no fixed system of axioms can ever catch all the mathematical truths for the simple reason that mathematicians themselves evolve with mathematics.

Motives and Infinite Primes

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.

1. What makes infinite primes interesting from the point of view of algebraic geometry is that infinite primes, integers and rationals at the n:th level of the hierarchy are in 1-1 correspondence with rational functions of n arguments. One can solve the roots of associated polynomials and perform a root decomposition of infinite primes at various levels of the hierarchy and assign to them Galois groups acting as automorphisms of the field extensions of polynomials defined by the roots coming as restrictions of the basic polynomial to planes x_n=0, x_n=x_n-1=0, etc...

2. These Galois groups are suggested to define non-commutative generalization of homotopy and homology theories and non-linear boundary operation for which a geometric interpretation in terms of the restriction to lower-dimensional plane is proposed. The Galois group G_k would be analogous to the relative homology group relative to the plane x_k-1=0 representing boundary and makes sense for all number fields also geometrically. One can ask whether the invariance of the complex of groups under the permutations of the orders of variables in the reduction process is necessary. Physical interpretation suggests that this is not the case and that all the groups obtained by the permutations are needed for a full description.

3. The algebraic counterpart of boundary map would map the elements of G_k identified as analog of homotopy group to the commutator group [G_k-2,G_k-2] and therefore to the unit element of the abelianized group defining cohomology group. In order to obtains something analogous to the ordinary homology and cohomology groups one must however replaces Galois groups by their group algebras with values in some field or ring. This allows to define the analogs of homotopy and homology groups as their abelianizations. Cohomotopy, and cohomology would emerge as duals of homotopy and homology in the dual of the group algebra.

4. That the algebraic representation of the boundary operation is not expected to be unique turns into blessing when on keeps the TGD as almost topological QFT vision as the guide line. One can include all boundary homomorphisms subject to the condition that the anticommutator δⁱ_kδ^j_k-1+δ^j_kδⁱ_k-1 maps to the group algebra of the commutator group [G_k-2,G_k-2]. By adding dual generators one obtains what looks like a generalization of anticommutative fermionic algebra and what comes in mind is the spectrum of quantum states of a SUSY algebra spanned by bosonic states realized as group algebra elements and fermionic states realized in terms of homotopy and cohomotopy and in abelianized version in terms of homology and cohomology. Galois group action allows to organize quantum states into multiplets of Galois groups acting as symmetry groups of physics. Poincare duality would map the analogs of fermionic creation operators to annihilation operators and vice versa and the counterpart of pairing of k:th and n-k:th homology groups would be inner product analogous to that given by Grassmann integration. The interpretation in terms of fermions turns however to be wrong and the more appropriate interpretation is in terms of Dolbeault cohomology applying to forms with homomorphic and antiholomorphic indices.

5. The intuitive idea that the Galois group is analogous to 1-D homotopy group which is the only non-commutative homotopy group, the structure of infinite primes analogous to the braids of braids of braids of ... structure, the fact that Galois group is a subgroup of permutation group, and the possibility to lift permutation group to a braid group suggests a representation as flows of 2-D plane with punctures giving a direct connection with topological quantum field theories for braids, knots and links. The natural assumption is that the flows are induced from transformations of the symplectic group acting on δ M²_+/-× CP₂ representing quantum fluctuating degrees of freedom associated with WCW ("world of classical worlds"). Discretization of WCW and cutoff in the number of modes would be due to the finite measurement resolution. The outcome would be rather far reaching: finite measurement resolution would allow to construct WCW spinor fields explicitly using the machinery of number theory and algebraic geometry.

6. A connection with operads is highly suggestive. What is nice from TGD perspective is that the non-commutative generalization homology and homotopy has direct connection to the basic structure of quantum TGD almost topological quantum theory where braids are basic objects and also to hyper-finite factors of type II₁. This notion of Galois group makes sense only for the algebraic varieties for which coefficient field is algebraic extension of some number field. Braid group approach however allows to generalize the approach to completely general polynomials since the braid group make sense also when the ends points for the braid are not algebraic points (roots of the polynomial).

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.

1. The calculation of twistorial amplitudes reduces to multi-dimensional residue calculus. Motivic integration gives excellent hopes for the p-adic existence of this calculus and braid representation would give space-time representation for the residue integrals in terms of the braid points representing poles of the integrand: this would conform with quantum classical correspondence. The power of 2π appearing in multiple residue integral is problematic unless it disappears from scattering amplitudes. Otherwise one must allow an extension of p-adic numbers to a ring containing powers of 2π.

2. Weak form of electric-magnetic duality and the general solution ansatz for preferred extremals reduce the Kähler action defining the Kähler function for WCW to the integral of Chern-Simons 3-form. Hence the reduction to cohomology takes places at space-time level and since p-adic cohomology exists there are excellent hopes about the existence of p-adic variant of Kähler action. The existence of the exponent of Kähler gives additional powerful constraints on the value of the Kähler fuction in the intersection of real and p-adic worlds consisting of algebraic partonic 2-surfaces and allows to guess the general form of the Kähler action in p-adic context.

3. One also should define p-adic integration for vacuum functional at the level of WCW. p-Adic thermodynamics serves as a guideline leading to the condition that in p-adic sector exponent of Kähler action is of form (m/n)^r, where m/n is divisible by a positive power of p-adic prime p. This implies that one has sum over contributions coming as powers of p and the challenge is to calculate the integral for K= constant surfaces using the integration measure defined by an infinite power of Kähler form of WCW reducing the integral to cohomology which should make sense also p-adically. The p-adicization of the WCW integrals has been discussed already earlier using an approach based on harmonic analysis in symmetric spaces and these two approaches should be equivalent. One could also consider a more general quantization of Kähler action as sum K=K₁+K₂ where K₁=rlog(m/n) and K₂=n, with n divisible by p since exp(n) exists in this case and one has exp(K)= (m/n)^r × exp(n). Also transcendental extensions of p-adic numbers involving n+p-2 powers of e^1/n can be considered.

4. If the Galois group algebras indeed define a representation for WCW spinor fields in finite measurement resolution, also WCW integration would reduce to summations over the Galois groups involved so that integrals would be well-defined in all number fields.

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>p^n-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.

Could One Generalize Braid Invariant Defined by Vacuum Expecation of Wilson Loop to and Invariant of Braid Cobordisms and of 2-Knots?

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.

Yangian Symmetry, Twistors, and TGD

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.

A Possible Explanation for Shnoll Effect

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 λ_i) for a positive integer valued variable n characterized by rational-valued parameters λ_i is obtained by replacing n and the integers characterizing λ_i with so called quantum integers depending on the quantum phase q_m=exp(i2π/m). Quantum integer n_q must be defined as the product of quantum counterparts p_q of the primes p appearing in the prime decomposition of n. One has p_q= sin(2π p/m)/sin(2π/m) for p≠ P and p_q=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(∑ P_n)=∑ I(P_n). 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≈ k_pΔ R/R and Δ m/m≈ k_mΔ 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.

The Changed Coordinate Transformation of the Constant Accelerated Coordinate System by Tetrad

Authors: sangwha Yi
Comments: 15 Pages. It is replacement

In the general relativity theory, instead of in the present accelerated system theory, the coordinate transformation that the inertial coordinate system and the accelerated coordinate system, find the new relation’s coordinate transformation that used the tetrad on the new method.And using the new relation’s coordinate transformation of the new accelerated system, organize the expansive accelerated system, the relation of the inertial coordinate system and an accelerated system that accelerated constantly on an inertial coordinate system.
Category: Mathematical Physics

The Changed Coordinate Transformation of the Constant Accelerated Coordinate System by Tetrad

Authors: Sangwha Yi
Comments: 12 Pages. It is replacement

In the general relativity theory, instead of in the present accelerated system theory, the coordinate transformation that the inertial coordinate system and the accelerated coordinate system, find the new relation’s coordinate transformation that used the tetrad on the new method.And using the new relation’s coordinate transformation of the new accelerated system, organize the expansive accelerated system, the relation of the inertial coordinate system and an accelerated system that accelerated constantly on an inertial coordinate system.
Category: Mathematical Physics

The Changed Coordinate Transformation of the Constant Accelerated Coordinate System by Tetrad

Authors: Sangwha-Yi
Comments: 12 pages.

In the general relativity theory, instead of in the present accelerated system theory, the coordinate transformation that the inertial coordinate system and the accelerated coordinate system, find the new relation's coordinate transformation that used the tetrad on the new method.And using the new relation's coordinate transformation of the new accelerated system, organize the expansive accelerated system, the relation of the inertial coordinate system and an accelerated system that accelerated constantly on an inertial coordinate system.
Category: Mathematical Physics

The Changed Coordinate Transformation of the Constant Accelerated Coordinate System by Tetrad

Authors: Sangwha-Yi
Comments: 12 pages.

In the general relativity theory, instead of in the present accelerated system theory, the coordinate transformation that the inertial coordinate system and the accelerated coordinate system, find the new relation's coordinate transformation that used the tetrad on the new method.And using the new relation's coordinate transformation of the new accelerated system, organize the expansive accelerated system, the relation of the inertial coordinate system and an accelerated system that accelerated constantly on an inertial coordinate system.
Category: Mathematical Physics

Exact Solution of Viscous-Plastic Flow Equations for Glacier Dynamics in 2-Dimensional Case.

Authors: Sergey V. Ershkov
Comments: 9 pages

Here is presented a new exact solution of Ice dynamics in Glaciers in terms of viscousplastic theory of movements, for 2-dimensional case: x (t) = y (t). In general case, 2-D solution of Ice dynamics could be classified as Riccati's type. Due to a very special character of Riccati's type equation, it's general solution is proved to have a proper gap of components of such a solution.
Category: Mathematical Physics

Exact Solution of Viscous-Plastic Flow Equations for Glacier Dynamics in 2-Dimensional Case.

Authors: Sergey V. Ershkov
Comments: 7 pages

Here is presented a new exact solution of Ice dynamics in Glaciers in terms of viscousplastic theory of movements, for 2-dimensional case: x (t) = y (t). In general case, 2-D solution of Ice dynamics could be classified as Riccati's type. Due to a very special character of Riccati's type equation, it's general solution is proved to have a proper gap of components of such a solution.
Category: Mathematical Physics

A Note on the Action at a Distance

Authors: José Francisco García Juliá
Comments: 3 Pages.

We consider that the action at a distance is carried out by virtual carriers of the force, which are created and annihilated in the vacuum by the field in a time less than that of the Heisenberg's uncertainty.
Category: Mathematical Physics

Quintessence-Momentum as Link Between Mass and Charge

Authors: Malcolm Macleod
Comments: 3 pages. v1 in Russian, v4 in English

The natural constants, G; h; e, μ₀ and m_e are presented as geometrical shapes in terms of Planck momentum, α (Sommerfeld fine structure constant) and c. A square root solution of Planck momentum denoted Quintessence-momentum Q links the mass and charge constants. The electron formula describes a dimensionless magnetic monopole. The Rydberg constant R1, the most accurate of the natural constants, is used for crossreference, the solutions are consistent with CODATA 2010 precision.
Category: Mathematical Physics

Quintessence-Momentum as Link Between Mass and Charge

Authors: Malcolm Macleod
Comments: 3 pages. v1 in Russian, v3 in English

A mathematical description of the natural constants, G, h, e, μ₀ m_e R_∞, is presented in terms of momentum Q, alpha (Sommerfeld fine structure constant) and c. This momentum is referred to as Quintessence-momentum and is the square root of Planck momentum. The formulas describe geometrical forms, the units are consistent with corresponding SI units and the numerical values, including the Rydberg constant and the vacuum permeability, are consistent with CODATA 2006.
Category: Mathematical Physics

Quintessence-Momentum as Link Between Mass and Charge

Authors: Malcolm Macleod
Comments: 3 pages. v1 in Russian, v2 in English

This paper suggests a 'quantity of momentum', a square root of Planck momentum, here referred to as Quintessence-momentum, as a natural unit that is common to both mass and charge. In terms of this Quintessence momentum Q, alpha (Sommerfeld fine structure constant) and c; geometrical formulas for the natural physical constants and the electron mass are proposed. Results are consistent with CODATA 2006.
Category: Mathematical Physics

Scale Dimension as the Fifth Dimension of Spacetime

Authors: Sergey G. Fedosin
Comments: 4 pages. v1 in Russian, v2 in English

The scale dimension which is discovered in the theory of infinite nesting of matter is studied from the perspective of the physical implementation of well-studied four-and n-dimensional geometric objects. Adding of the scale dimension to Minkowski space means the need to use the five-dimensional spacetime.
Category: Mathematical Physics

Fine Structure Constant α ~ 1/137.036 and Blackbody Radiation Constant α_R ~ 1/157.555

Authors: Ke Xiao
Comments: 5 pages

The fine structure constant α = e²/hc ~ 1/137.036 and the blackbody radiation constant α_R = e²(a_R/k⁴_B)^1/3 ~ 1/157.555 are linked by prime numbers. The blackbody radiation constant is a new method to measure the fine structure constant. It also links the fine structure constant to the Boltzmann constant.
Category: Mathematical Physics

32 Point Groups of Three Dimensional Crystal Cells Described by 5 Bits

Authors: Giuliano Bettini
Comments: 9 pages, v3 in Italian, v2 in English, corrections to the tables, and a new table added.

There are 32 possible combinations of symmetry operations that define the external symmetry of crystals. These 32 possible combinations result in the 32 crystal classes. But for a radar engineer it is inevitable to associate "32" to "5 bits". I submit a tentative classification of the 32 crystal classes with a 5 bit classification, obviously with a (tentative) physical meaning of each bit. Each bit means a physical property.
Category: Mathematical Physics

32 Point Groups of Three Dimensional Crystal Cells Described by 5 Bits

Authors: Giuliano Bettini
Comments: 9 pages, v1 in Italian, v2 in English, corrections to the tables, and a new table added.

There are 32 possible combinations of symmetry operations that define the external symmetry of crystals. These 32 possible combinations result in the 32 crystal classes. But for a radar engineer it is inevitable to associate "32" to "5 bits". I submit a tentative classification of the 32 crystal classes with a 5 bit classification, obviously with a (tentative) physical meaning of each bit. Each bit means a physical property.
Category: Mathematical Physics

Further on Non-Cartesian Systems

Authors: Elemér E Rosinger
Comments: 9 pages

A class of non-Cartesian physical systems, [7], are those whose composite state spaces are given by significantly extended tensor products. A more detailed presentation of the way such extended tensor products are constructed is offered, based on a step by step comparison with the construction of usual tensor products. This presentation clarifies the extent to which the extended tensor products are indeed more general than the usual ones.
Category: Mathematical Physics

Deeper Properties Through Dark and Visible-Matter in a New Cosmological Twin-Tori Model (TTM).

Authors: Dan Visser
Comments: 5 pages

A new cosmological model, named the Twin-Tori Model (TTM)[1], postulates a dark energy force F_de , which empowers the dynamic of a lower order universe, well known as the big bang. In this paper I introduce the 1st derivative F'_de of this dark energy force to reveal deeper properties of the TTM, such as: why quantummechanics exists in the big bang, why dark matter and visible matter are equally responsible for gravity in galaxies for 1/4 of the density of dark matter at a specific length, why the big bang universe is recalculated by subquantumlevel-information below the Plancklength, and why the impression of space-expansion is due to the higher order cosmological model TTM.
Category: Mathematical Physics

A Multiple Particle System Equation Underlying the Klein-Gordon-Dirac-Schrödinger Equations

Authors: DT Froedge
Comments: 39 Pages.

The purpose of this paper is to illustrate a fundamental, multiple particle, system equation for which the Klein-Gordon-Dirac-Schrödinger equations are single particle special cases. The basic concept is that there is a broader picture, based on a more general equation that includes the entire system of particles. The first part will be to postulate an equation, and then, by modifying the methods of Path Integrals, develop a solution which describes the internal dynamics as well as particle interactions of quantum particles. The complete function has both real and imaginary, as well as timelike and spacelike parts, each of which are separable into independent expressions that define particle properties. In the same manner that eigenvalues of the Schrödinger equation represents energy levels of an atomic system, particle are eigenvalues in an interacting universe of particles. The Dirac massive and massless equation and solution will be shown as factorable independent components. A clear distinction between the classical and quantum properties of particles is made, increasing the scope of QM.
Category: Mathematical Physics

A Multiple Particle System Equation Underlying the Klein-Gordon-Dirac-Schrödinger Equations

Authors: D.T. Froedge
Comments: 30 pages.

The purpose of this paper is to illustrate a fundamental, multiple particle, system equation for which the Klein-Gordon-Dirac-Schrödinger equations are single particle special cases. In the same manner that eigenvalues of the Schrödinger equation represents energy levels of an interacting atomic system, eigenvalues represent particle energies in an interacting system of particles. An equation and a solution is proposed that treats all of the particles in the universe as a single system. The proposed solution is a descriptor of a symmetric, light speed expanding group of interacting particles having familiar constituents.
Category: Mathematical Physics

A Multiple Particle System Equation Underlying the Klein-Gordon-Dirac-Schrödinger Equations

Authors: D.T. Froedge
Comments: 17 pages 38 equations 98kb

The purpose of this paper is to illustrate a fundamental, multiple particle, system equation for which the Klein-Gordon-Dirac-Schrödinger equations are single particle special cases. In the same manner that eigenvalues of the Schrödinger equation represents energy levels of an interacting atomic system, eigenvalues represent particle energies in an interacting system of particles. An equation is proposed that has vector solutions defined in Dirac, or Clifford algebra, that treats all of the particles in the universe as a single system. The proposed solution is a descriptor of a symmetric, light speed expanding group of interacting particles having real, as well as the familiar QM constituents.
Category: Mathematical Physics

The Geometry of CP₂ and Its Relationship to Standard Model

Authors: Matti
Comments: 13 Pages.

This appendix contains basic facts about CP₂ as a symmetric space and Kähler manifold. The coding of the standard model symmetries to the geometry of CP₂, the physical interpretation of the induced spinor connection in terms of electro-weak gauge potentials, and basic facts about induced gauge fields are discussed
Category: Mathematical Physics

Could the Dynamics of Kähler Action Predict the Hierarchy of Planck Constants?

Authors: Matti Pitkänen
Comments: 5 Pages.

The original justification for the hierarchy of Planck constants came from the indications that Planck constant could have large values in both astrophysical systems involving dark matter and also in biology. The realization of the hierarchy in terms of the singular coverings and possibly also factor spaces of CD and CP₂ emerged from consistency conditions. It however seems that TGD actually predicts this hierarchy of covering spaces. The extreme non-linearity of the field equations defined by Kähler action means that the correspondence between canonical momentum densities and time derivatives of the imbedding space coordinates is 1-to-many. This leads naturally to the introduction of the covering space of CD x CP₂, where CD denotes causal diamond defined as intersection of future and past directed light-cones.
Category: Mathematical Physics

Weak Form of Electric-Magnetic Duality and Its Implications

Authors: Matti Pitkänen
Comments: 25 Pages.

The notion of electric magnetic duality emerged already two decades ago in the attempts to formulate the Kähler geometry of the "world of classical worlds". Quite recently a considerable step of progress took place in the understanding of this notion. This concept leads to the identification of the physical particles as string like objects defined by magnetic charged wormhole throats connected by magnetic ux 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. Electric-magnetic duality leads also to a detailed understanding of how TGD reduces to almost topological quantum field theory. A surprising outcome is the necessity to replace CP₂ Kähler form in Kähler action with its sum with S² Kähler form.
Category: Mathematical Physics

How to Define Generalized Feynman Diagrams?

Authors: Matti Pitkänen
Comments: 16 Pages.

Generalized Feynman diagrams have become the central notion of quantum TGD and one might even say that space-time surfaces can be identified as generalized Feynman diagrams. The challenge is to assign a precise mathematical content for this notion, show their mathematical existence, and develop a machinery for calculating them. Zero energy ontology has led to a dramatic progress in the understanding of generalized Feynman diagrams at the level of fermionic degrees of freedom. In particular, manifest finiteness in these degrees of freedom follows trivially from the basic identifications as does also unitarity and non-trivial coupling constant evolution. There are however several formidable looking challenges left.

1. One should perform the functional integral over WCW degrees of freedom for fixed values of on mass shell momenta appearing in the internal lines. After this one must perform integral or summation over loop momenta.
2. One must define the functional integral 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.

Physics as Generalized Number Theory: Infinite Primes

Authors: Matti Pitkänen
Comments: 38 Pages.

The focus of this book is the number theoretical vision about physics. This vision involves three loosely related parts.

1. The fusion of real physic and various p-adic physics to a single coherent whole by generalizing the number concept by fusing real numbers and various p-adic number fields along common rationals. Extensions of p-adic number fields can be introduced by gluing them along common algebraic numbers to reals. Algebraic continuation of the physics from rationals and their their extensions to various number fields (generalization of completion process for rationals) is the key idea, and the challenge is to understand whether how one could achieve this dream. A profound implication is that purely local p-adic physics would code for the p-adic fractality of long length length scale real physics and vice versa, and one could understand the origins of p-adic length scale hypothesis.
2. Second part of the vision involves hyper counterparts of the classical number fields defined as subspaces of their complexifications with Minkowskian signature of metric. Allowed space-time surfaces would correspond to what might be called hyper-quaternionic sub-manifolds of a hyper-octonionic space and mappable to M⁴× CP₂ in natural manner. One could assign to each point of space-time surface a hyper-quaternionic 4-plane which is the plane defined by the modified gamma matrices but not tangent plane in general. Hence the basic variational principle of TGD would have deep number theoretic content.
3. The third part of the vision involves infinite primes identifiable in terms of an infinite hierarchy of second quantized arithmetic quantum fields theories on one hand, and as having representations as space-time surfaces analogous to zero loci of polynomials on the other hand. Single space-time point would have an infinitely complex structure since real unity can be represented as a ratio of infinite numbers in infinitely many manners each having its own number theoretic anatomy. Single space-time point would be in principle able to represent in its structure the quantum state of the entire universe. This number theoretic variant of Brahman=Atman identity would make Universe an algebraic hologram.

   Number theoretical vision suggests that infinite hyper-octonionic or -quaternionic primes could could correspond directly to the quantum numbers of elementary particles and a detailed proposal for this correspondence is made. Furthermore, the generalized eigenvalue spectrum of the Chern-Simons Dirac operator could be expressed in terms of hyper-complex primes in turn defining basic building bricks of infinite hyper-complex primes from which hyper-octonionic primes are obtained by dicrete SU(3) rotations performed for finite hyper-complex primes.

Besides this holy trinity I will discuss loosely related topics. Included are possible applications of category theory in TGD framework; TGD inspired considerations related to Riemann hypothesis; topological quantum computation in TGD Universe; and TGD inspired approach to Langlands program.

Category: Mathematical Physics

Physics as Generalized Number Theory: Infinite Primes

Authors: Matti Pitkänen
Comments: 695 Pages.

The focus of this book is the number theoretical vision about physics. This vision involves three loosely related parts.

1. The fusion of real physic and various p-adic physics to a single coherent whole by generalizing the number concept by fusing real numbers and various p-adic number fields along common rationals. Extensions of p-adic number fields can be introduced by gluing them along common algebraic numbers to reals. Algebraic continuation of the physics from rationals and their their extensions to various number fields (generalization of completion process for rationals) is the key idea, and the challenge is to understand whether how one could achieve this dream. A profound implication is that purely local p-adic physics would code for the p-adic fractality of long length length scale real physics and vice versa, and one could understand the origins of p-adic length scale hypothesis.
2. Second part of the vision involves hyper counterparts of the classical number fields defined as subspaces of their complexifications with Minkowskian signature of metric. Allowed space-time surfaces would correspond to what might be called hyper-quaternionic sub-manifolds of a hyper-octonionic space and mappable to M⁴× CP₂ in natural manner. One could assign to each point of space-time surface a hyper-quaternionic 4-plane which is the plane defined by the modified gamma matrices but not tangent plane in general. Hence the basic variational principle of TGD would have deep number theoretic content.
3. The third part of the vision involves infinite primes identifiable in terms of an infinite hierarchy of second quantized arithmetic quantum fields theories on one hand, and as having representations as space-time surfaces analogous to zero loci of polynomials on the other hand. Single space-time point would have an infinitely complex structure since real unity can be represented as a ratio of infinite numbers in infinitely many manners each having its own number theoretic anatomy. Single space-time point would be in principle able to represent in its structure the quantum state of the entire universe. This number theoretic variant of Brahman=Atman identity would make Universe an algebraic hologram.

   Number theoretical vision suggests that infinite hyper-octonionic or -quaternionic primes could could correspond directly to the quantum numbers of elementary particles and a detailed proposal for this correspondence is made. Furthermore, the generalized eigenvalue spectrum of the Chern-Simons Dirac operator could be expressed in terms of hyper-complex primes in turn defining basic building bricks of infinite hyper-complex primes from which hyper-octonionic primes are obtained by dicrete SU(3) rotations performed for finite hyper-complex primes.

Besides this holy trinity I will discuss loosely related topics. Included are possible applications of category theory in TGD framework; TGD inspired considerations related to Riemann hypothesis; topological quantum computation in TGD Universe; and TGD inspired approach to Langlands program.

Category: Mathematical Physics

Physics as Generalized Number Theory: Classical Number Fields

Authors: Matti Pitkänen
Comments: 28 Pages.

Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields discussed in this article, and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article the connection between standard model symmetries and classical number fields is discussed. The basis vision is that the geometry of the infinite-dimensional WCW ("world of classical worlds") is unique from its mere existence. This leads to its identification as union of symmetric spaces whose Kähler geometries are fixed by generalized conformal symmetries. This fixes space-time dimension and the decomposition M⁴ x S and the idea is that the symmetries of the Kähler manifold S make it somehow unique. The motivating observations are that the dimensions of classical number fields are the dimensions of partonic 2-surfaces, space-time surfaces, and imbedding space and M⁸ can be identified as hyper-octonions- a sub-space of complexified octonions obtained by adding a commuting imaginary unit. This stimulates some questions. Could one understand S = CP₂ number theoretically in the sense that M⁸ and H = M⁴ x CP₂ be in some deep sense equivalent ("number theoretical compactification" or M⁸ - H duality)? Could associativity define the fundamental dynamical principle so that space-time surfaces could be regarded as associative or co-associative (defined properly) sub-manifolds of M⁸ or equivalently of H. One can indeed define the associativite (co-associative) 4-surfaces using octonionic representation of gamma matrices of 8-D spaces as surfaces for which the modified gamma matrices span an associate (co-associative) sub-space at each point of space-time surface. Also M⁸ - H duality holds true if one assumes that this associative sub-space at each point contains preferred plane of M⁸ identifiable as a preferred commutative or co-commutative plane (this condition generalizes to an integral distribution of commutative planes in M⁸). These planes are parametrized by CP₂ and this leads to M⁸ - H duality. WCW itself can be identified as the space of 4-D local sub-algebras of the local Clifford algebra of M⁸ or H which are associative or co-associative. An open conjecture is that this characterization of the space-time surfaces is equivalent with the preferred extremal property of Kähler action with preferred extremal identified as a critical extremal allowing infinite-dimensional algebra of vanishing second variations.
Category: Mathematical Physics

Physics as Generalized Number Theory: P-Adic Physics and Number Theoretic Universality

Authors: Matti Pitkänen
Comments: 51 Pages.

Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields (in particular, identifying associativity condition as the basic dynamical principle), and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article p-adic physics and the technical problems relates to the fusion of p-adic physics and real physics to a larger structure are discussed. The basic technical problems relate to the notion of definite integral both at space-time level, imbedding space level and the level of WCW (the "world of classical worlds"). The expressibility of WCW as a union of symmetric spacesleads to a proposal that harmonic analysis of symmetric spaces can be used to define various integrals as sums over Fourier components. This leads to the proposal the p-adic variant of symmetric space is obtained by a algebraic continuation through a common intersection of these spaces, which basically reduces to an algebraic variant of coset space involving algebraic extension of rationals by roots of unity. This brings in the notion of angle measurement resolution coming as Δφ = 2π/pⁿ for given p-adic prime p. Also a proposal how one can complete the discrete version of symmetric space to a continuous p-adic versions emerges and means that each point is effectively replaced with the p-adic variant of the symmetric space identifiable as a p-adic counterpart of the real discretization volume so that a fractal p-adic variant of symmetric space results. If the Kähler geometry of WCW is expressible in terms of rational or algebraic functions, it can in principle be continued the p-adic context. One can however consider the possibility that that the integrals over partonic 2-surfaces defining ux Hamiltonians exist p-adically as Riemann sums. This requires that the geometries of the partonic 2-surfaces effectively reduce to finite sub-manifold geometries in the discretized version of δM₊⁴. If Kähler action is required to exist p-adically same kind of condition applies to the space-time surfaces themselves. These strong conditions might make sense in the intersection of the real and p-adic worlds assumed to characterized living matter.
Category: Mathematical Physics

Construction of Configuration Space Spinor Structure

Authors: Matti Pitkänen
Comments: 95 Pages.

There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first approach relies on a direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach discussed in this article relies on the construction of spinor structure based on the hypothesis that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for the second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of spinor structure. This implies a geometrization of fermionic statistics. The basic philosophy is that at fundamental level the construction of WCW geometry reduces to the second quantization of the induced spinor fields using Dirac action. This assumption is parallel with the bosonic emergence stating that all gauge bosons are pairs of fermion and antifermion at opposite throats of wormhole contact. Vacuum function is identified as Dirac determinant and the conjecture is that it reduces to the exponent of Kähler function. In order to achieve internal consistency induced gamma matrices appearing in Dirac operator must be replaced by the modified gamma matrices defined uniquely by Kähler action and one must also assume that extremals of Kähler action are in question so that the classical space-time dynamics reduces to a consistency condition. This implies also super-symmetries and the fermionic oscillator algebra at partonic 2-surfaces has intepretation as N = 1 generalization of space-time supersymmetry algebra different however from standard SUSY algebra in that Majorana spinors are not needed. This algebra serves as a building brick of various super-conformal algebras involved. The requirement that there exist deformations giving rise to conserved Noether charges requires that the preferred extremals are critical in the sense that the second variation of the Kähler action vanishes for these deformations. Thus Bohr orbit property could correspond to criticality or at least involve it. Quantum classical correspondence demands that quantum numbers are coded to the properties of the preferred extremals given by the Dirac determinant and this requires a linear coupling to the conserved quantum charges in Cartan algebra. Effective 2-dimensionality allows a measurement interaction term only in 3-D Chern-Simons Dirac action assignable to the wormhole throats and the ends of the space-time surfaces at the boundaries of CD. This allows also to have physical propagators reducing to Dirac propagator not possible without the measurement interaction term. An essential point is that the measurement interaction corresponds formally to a gauge transformation for the induced Kähler gauge potential. If one accepts the weak form of electric-magnetic duality Kähler function reduces to a generalized Chern-Simons term and the effect of measurement interaction term to Kähler function reduces effectively to the same gauge transformation. The basic vision is that WCW gamma matrices are expressible as super-symplectic charges at the boundaries of CD. The basic building brick of WCW is the product of infinite-D symmetric spaces assignable to the ends of the propagator line of the generalized Feynman diagram. WCW Kähler metric has in this case "kinetic" parts associated with the ends and "interaction" part between the ends. General expressions for the super-counterparts of WCW ux Hamiltoniansand for the matrix elements of WCW metric in terms of their anticommutators are proposed on basis of this picture.
Category: Mathematical Physics

Construction of Configuration Space Geometry from Symmetry Principles

Authors: Matti Pitkänen
Comments: 26 Pages.

There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first one relies on a direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach relies on the construction of spinor structure assuming that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for the second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of spinor structure. In this article the construction of Kähler form and metric based on symmetries is discussed. The basic vision is that WCW can be regarded as the space of generalized Feynman diagrams with lines thickned to light-like 3-surfaces and vertices identified as partonic 2-surfaces. In zero energy ontology the strong form of General Coordinate Invariance (GCI) implies effective 2-dimensionality and the basic objects are pairs partonic 2-surfaces X² at opposite light-like boundaries of causal diamonds (CDs). The hypothesis is that WCW can be regarded as a union of infinite-dimensional symmetric spaces G/H labeled by zero modes having an interpretation as classical, non-quantum uctuating variables. A crucial role is played by the metric 2-dimensionality of the light-cone boundary δM₊⁴ + and of light-like 3-surfaces implying a generalization of conformal invariance. The group G acting as isometries of WCW is tentatively identified as the symplectic group of δM₊⁴ x CP₂ localized with respect to X². H is identified as Kac-Moody type group associated with isometries of H = M₊⁴ x CP₂ acting on light-like 3-surfaces and thus on X². An explicit construction for the Hamiltonians of WCW isometry algebra as so called ux Hamiltonians is proposed and also the elements of Kähler form can be constructed in terms of these. Explicit expressions for WCW ux Hamiltonians as functionals of complex coordinates of the Cartesisian product of the infinite-dimensional symmetric spaces having as points the partonic 2-surfaces defining the ends of the the light 3-surface (line of generalized Feynman diagram) are proposed.
Category: Mathematical Physics

Identification of the Configuration Space Kähler Function

Authors: Matti Pitkänen
Comments: 29 Pages.

There are two basic approaches to quantum TGD. The first approach, which is discussed in this article, is a generalization of Einstein's geometrization program of physics to an infinitedimensional context. Second approach is based on the identification of physics as a generalized number theory. The first approach relies on the vision of quantum physics as infinite-dimensional Kähler geometry for the "world of classical worlds" (WCW) identified as the space of 3-surfaces in in certain 8-dimensional space. There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first approach relies on direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach relies on the construction of spinor structure based on the hypothesis that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of WCW spinor structure. In this article the proposal for Kähler function based on the requirement of 4-dimensional General Coordinate Invariance implying that its definition must assign to a given 3-surface a unique space-time surface. Quantum classical correspondence requires that this surface is a preferred extremal of some some general coordinate invariant action, and so called Kähler action is a unique candidate in this respect. The preferred extremal has intepretation as an analog of Bohr orbit so that classical physics becomes and exact part of WCW geometry and therefore also quantum physics. The basic challenge is the explicit identification of WCW Kähler function K. Two assumptions lead to the identification of K as a sum of Chern-Simons type terms associated with the ends of causal diamond and with the light-like wormhole throats at which the signature of the induced metric changes. The first assumption is the weak form of electric magnetic duality. Second assumption is that the Kähler current for preferred extremals satisfies the condition jK ^ djK = 0 implying that the ow parameter of the ow lines of jK defines a global space-time coordinate. This would mean that the vision about reduction to almost topological QFT would be realized. Second challenge is the understanding of the space-time correlates of quantum criticality. Electric-magnetic duality helps considerably here. The realization that the hierarchy of Planck constant realized in terms of coverings of the imbedding space follows from basic quantum TGD leads to a further understanding. The extreme non-linearity of canonical momentum densities as functions of time derivatives of the imbedding space coordinates implies that the correspondence between these two variables is not 1-1 so that it is natural to introduce coverings of CD x CP₂. This leads also to a precise geometric characterization of the criticality of the preferred extremals.
Category: Mathematical Physics

Physics as Infinite-Dimensional Geometry and Generalized Number Theory: Basic Visions

Authors: Matti Pitkänen
Comments: 33 Pages.

There are two basic approaches to the construction of quantum TGD. The first approach relies on the vision of quantum physics as infinite-dimensional Kähler geometry for the "world of classical worlds" identified as the space of 3-surfaces in in certain 8-dimensional space. Essentially a generalization of the Einstein's geometrization of physics program is in question. The second vision is the identification of physics as a generalized number theory. This program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields (in particular, identifying associativity condition as the basic dynamical principle), and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article brief summaries of physics as infinite-dimensional geometry and generalized number theory are given to be followed by more detailed articles.
Category: Mathematical Physics

3x3 Unitary to Magic Matrix Transformations

Authors: Philip Gibbs
Comments: 5 pages

We prove that any 3x3 unitary matrix can be transformed to a magic matrix by multiplying its rows and columns by phase factors. A magic matrix is defined as one for which the sum of the elements in any row or column add to the same value. This result is relevant to recent observations on particle mixing matrices.
Category: Mathematical Physics