What's new in

Towards M-Matrix

Note: Newest contributions are at the top!



Year 2018



Are space-time surfaces minimal surfaces everywhere except at 2-D interaction vertices?

The action S determining space-time surfaces as preferred extremals follows from twistor lift and equals to the sum of volume term Vol and Kähler action SK. The field equation is a geometric generalization of d'Alembert (Laplace) equation in Minkowskian (Eucidian) regions of space-time surface coupled with induced Kähler form analogous to Maxwell field. Generalization of equations of motion for particle by replacing it with 3-D surface is in question and the orbit of particle defines a region of space-time surface.

  1. Zero energy ontology (ZEO) suggests that the external particles arriving to the boundaries of given causal diamond (CD) are like free massless particles and correspond to minimal surfaces as a generalization of light-like geodesic. This dynamic reduces to mere algebraic conditions and there is no dependence on the coupling parameters of S. In contrast to this, in the interaction regions inside CDs there could be a coupling between Vol and SK due to the non-vanishing divergences of energy momentum currents associated with the two terms in action cancelling each other.
  2. Similar algebraic picture emerges from M8-H duality at the level of M8 and from what is known about preferred extremals of S assumed to satisfy infinite number of super-symplectic gauge conditions at the 3-surfaces defining the ends of space-time surface at the opposite boundaries of CD.
  3. At M8 side of M8-H duality associativity is realized as quaternionicity of either tangent or normal space of the space-time surface. The condition that there is 2-D integral distribution of sub-spaces of tangent spaces defining a distribution of complex planes as subspaces of octonionic tangent space implies the map of the space-time surface in M8 to that of H. Given point m8 of M8 is mapped to a point of M4× CP2 as a pair of points (m4,s) formed by M4 ⊂ M8 projection m4 of m8 point and by CP2 point s parameterizing the tangent space or the normal space of X4⊂ M8.
  4. If associativity or even the condition about the existence of the integrable distribution of 2-planes fails, the map to M4× CP2 is lost. One could cope with the situation since the gauge conditions at the boundaries of CD would allow to construct preferred extremal connecting the 3-surfaces at the boundaries of CD if this kind of surface exists at all. One can however wonder whether giving up the map M8→ H is necessary.
  5. Number theoretic dynamics in M8 involves no action principle and no coupling constants, just the associativity and the integrable distribution of complex planes M2(x) of complexified octonions. This suggests that also the dynamics at the level of H involves coupling constants only via boundary conditions. This is the case for the minimal surface solutions suggesting that M8-H duality maps the surfaces satisfying the above mentioned conditions to minimal surfaces. The universal dynamics conforms also with quantum criticality.
  6. One can argue that the dependence of field equations on coupling parameters in interactions leading to a perturbative series in coupling parameters in the interior of the space-time surface spoils the extremely beautiful purely algebraic picture about the construction of solutions of field equations using conformal invariance assignable to quantum criticality. Classical perturbation series is also in conflict with the vision that the TGD counterparts twistorial Grassmannian amplitudes do not involve any loop contributions coming as powers of coupling constant parameters.
Thus both M8-H duality, number theoretic vision, quantum criticality, twistor lift of TGD reducing dynamics to the condition about the existence of induced twistor structure, and the proposal for the construction of twistor scattering amplitudes suggest an extremely simple picture about the situation. The divergences of the energy momentum currents of Vol and SK would be non-vanishing only at discrete points at partonic 2-surfaces defining generalized vertices so that minimal surface equations would hold almost everywhere as the original proposal indeed stated.
  1. The fact that all the known extremals of field equations for S are minimal surfaces conforms with the idea. This might be due to the fact that these extremals are especially easy to construct but could be also true quite generally apart from singular points. The divergences of the energy momentum currents associated with SK and Vol vanish separately: this follows from the analog of holomorphy reducing the field equations to purely algebraic conditions.

    It is essential that Kähler current jK vanishes or is light-like so that its contraction with the gradients of the imbedding space coordinates vanishes. Second condition is that in transversal degrees of freedom energy momentum tensor is tensor of form (1,1) in the complex sense and second fundamental form consists of parts of type (1,1) and (-1-1). In longitudinal degrees of freedom the trace Hk of the second fundamental form Hkαβ= Dβαhk vanishes.

  2. Minimal surface equations are an analog of massless field equation but one would like to have also the analog of massless particle. The 3-D light-like boundaries between Minkowskian and Euclidian space-time regions are indeed analogs of massless particles as are also the string like word sheets, whose exact identification is not yet fully settled. In any case, they are crucial for the construction of scattering amplitudes in TGD based generalization of twistor Grassmannian approach. At M8 side these points could correspond to singularities at which Galois group of the extension of rationals has a subgroup leaving the point invariant. The points at which roots of polynomial as function of parameters co-incide would serve as an analog.

    The intersections of string world sheets with the orbits of partonic 2-surface are 1-D light-like curves X1L defining fermion lines. The twistor Grassmannian proposal is that the ends of the fermion lines at partonic 2-surfaces defining vertices provide the information needed to construct scattering amplitudes so that information theoretically the construction of scattering amplitudes would reduce to an analog of quantum field theory for point-like particles.

  3. Number theoretic vision reduces coupling constant evolution to a discrete evolution. This implies that twistor scattering amplitudes for given values of discretized coupling constants involve no radiative corrections. The cuts for the scattering amplitudes would be replaced by sequences of poles. This is unavoidable also because there is number theoretical discretization of momenta from the condition that their components belong to an extension of rationals defining the adele.
What could the reduction of cuts to poles for twistorial scattering amplitudes at the level of momentum space mean at space-time level?
  1. Poles of an analytic function are co-dimension 2 objects. d'Alembert/Laplace equations holding true in Minkowskian/Euclidian signatures express the analogs of analyticity in 4-D case. Co-dimension 2 rule forces to ask whether partonic 2-surfaces defining the vertices and string world sheets could serve analogs of poles at space-time level? In fact, the light-like orbits X3L of partonic 2-surfaces allow a generalization of 2-D conformal invariance since they are metrically 2-D so that X3L and string world sheets could serve in the role of poles.

    X3L could be seen as analogs of orbits of bubbles in hydrodynamical flow in accordance with the hydrodynamical interpretations. Particle reactions would correspond to fusions and decays of these bubbles. Strings would connect these bubbles and give rise to tensor networks and serve as space-time correlates for entanglement. Reaction vertices would correspond to common ends for the incoming and outgoing bubbles. They would be analogous to the lines of Feynman diagram meeting at vertex: now vertex would be however 2-D partonic 2-surface.

  2. What can one say about the singularities associated with the light-like orbits of partonic 2-surfaces? The divergence of the Kähler part TK of energy momentum current T is proportional to a sum of contractions of Kähler current jK with gradients ∇ hk of H coordinates. jK need not be vanishing: it is enough that its contraction with ∇ hk vanishes and this is true if jK is light-like. This is the case for so called massless extremals (MEs). For the other known extremals jK vanishes.

    Could the Kähler current jK be light-like and non-vanishing and singular at X3L and at string world sheets? This condition would provide the long sought-for precise physical identification of string world sheets. Minimal surface equations would hold true also at these surface. Even more: jK could be non-vanishing and thus also singular only at the 1-D intersections X1L of string world sheets with X3L - I have called these curves fermionic lines?

    What it means that jK is singular - that is has 2-D delta function singularity at string world sheets? jK is defined as divergence of the induced Kähler form J so that one can use the standard definition of derivative to define jK at string world sheet as the limiting value jKα= (Div+- J)α = limΔ xn→ 0 (J+α n- J-α n)/Δ xn, where xn is a coordinate normal to the string world sheet. If J is not light-like, it gives rise to isometry currents with non-vanishing divergence at string world sheet. This current should be light like to guarantee that energy momentum currents are divergenceless. This is guaranteed if the isometry currents T&n; A are continuous through the string world sheet.

  3. If the light-like jK at partonic orbits is localized at fermionic lines X1L, the divergences of energy momentum currents could be non-vanishing and singular only at the vertices defined at partonic 2-surfaces at which fermionic lines X1L meet. The divergences of energy momentum tensors TK of SK and TVol of Vol would be non-vanishing only at these vertices. They should of course cancel each other: Div TK=-Div TVol.
  4. Div TK should be non-vanishing and singular only at the intersections of string world sheets and partonic 2-surfaces defining the vertices as the ends of fermion lines. How to translate this statement to a more precise mathematical form? How to precisely define the notions of divergence at the singularity?

    The physical picture is that there is a sharing of conserved isometry charges of the incoming partonic orbit i=1 determined TK between 2 outgoing partonic orbits labelled by j=2,3 . This implies charge transfer from i=1 to the partonic orbits j=2,3 such that the sum of transfers sum up to to the total incoming charge. This must correspond to a non-vanishing divergence proportional to delta function. The transfer of the isometry charge for given pair i,j of partonic orbits that is Divi→ j TK must be determined as the limiting value of the quantity Δi→ j TKα,A/Δ xα as Δ xα approaches zero. Here Δi→ j TKα,A is the difference of the components of the isometry currents between partonic orbits i and j at the vertex. The outcome is proportional delta function.

  5. Similar description applies also to the volume term. Now the trace of the second fundamental form would have delta function singularity coming from Div TK. The condition Div TK= -Div TVol would bring in the dependence of the boundary conditions on coupling parameters so that space-time surface would depend on the coupling constants in accordance with quantum-classical correspondence. The manner how the coupling constants make themselves visible in the properties of space-time surface would be extremely delicate.
This picture conforms with the vision about scattering amplitudes at both M8 and H sides of M8-H duality.
  1. M8 dynamics based on algebraic equations for space-time surfaces leads to the proposal that scattering amplitudes can be constructed using the data only at the points of space-time surface with M8 coordinates in the extension of the rationals defining the adele. I call this discrete set of points cognitive representation.
  2. At H side the information theoretic interpretation would be that all information needed to construct scattering amplitudes would come from points at which the divergences of the energy momentum tensors of SK and Vol are non-vanishing and singular.
Both pictures would realize extremely strong form of holography, much stronger than the strong form of holography that stated that only partonic 2-surfaces and string world sheets are needed.

See the chapter The Recent View about Twistorialization in TGD Framework or the article with the same title.



Three dualities at the level of field equations

The basic field equations of TGD allow several dualities. There are 3 of them at the level of basic field equations (and several other dualities such as M8-M4× CP2 duality).

  1. The first duality is the analog of particle-field duality. The spacetime surface describing the particle (3-surface of M4× CP2 instead of point-like particle) corresponds to the particle aspect and the fields inside it geometrized in terms of sub-manifold geometry in terms of quantities characterizing geometry of M4× CP2 to the field aspect. Particle orbit serves as wave guide for field, one might say.
  2. Second duality is particle-spacetime duality. Particle identified as 3-D surface means that particle orbit is space-time surface glued to a larger space-time surface by topological sum contacts. It depends on the scale used, whether it is more appropriate to talk about particle or of space-time.
  3. The third duality is hydrodynamics-massless field theory duality Hydrodynamical equations state local conservation of Noether currents. Field equations indeed reduce to local conservation conditions of Noether currents associated with isometries of M4× CP2. One the other hand, these equations have interpretation as non-linear geometrization of massless wave equation with coupling to Maxwell fields. This realizes the ultimate dream of theoretician: symmetries dictate the dynamics completely. This is expected to be realized also at the level of scattering amplitudes and the generalization of twistor Grassmannian amplitudes could realize this in terms of Yangian symmetry.

    Hydrodynamics-wave equations duality generalizes to the fermionic sector and involves superconformal symmetry.

  4. What I call modified gamma matrices are obtained as contractions of the partial derivatives of the action defining space-time surface with respect to the gradients of imbedding space coordinate with imbedding space gamma matrices. Their divergences vanish by field equations for the space-time surface and this is necessary for the internal consistency the Dirac equation. The modified gamma matrices reduces to ordinary ones if space-time surface is M4 and one obtains ordinary massless Dirac equation.
  5. Modified Dirac equation expresses conservation of super current and actually infinite number of super currents obtained by contracting second quantized induced spinor field with the solutions of modified Dirac. This corresponds to the super-hydrodynamic aspect. On the other hand, modified Dirac equation corresponds to fermionic analog of massless wave equation as super-counterpart of the non-linear massless field equation determining space-time surface.
See the chapter The Recent View about Twistorialization in TGD Framework or the article with the same title.



New insights about quantum criticality for twistor lift inspired by analogy with ordinary criticality

Quantum criticality (QC) is one of the basic ideas of TGD. Zero energy ontology (ZEO) is second key notion and leads to a theory of consciousness as a formulation of quantum measurement theory solving the basic paradox of standard quantum measurement theory, which is usualy tried to avoid by introducing some "interpretation".

ZEO allows to see quantum theory could be seen as "square root" of thermodynamics. It occurred to me that it would be interesting to apply this vision in the case of quantum criticality to perhaps gain additional insights about its meaning. We have a picture about criticality in the framework of thermodynamics: what would be the analogy in ZEO based interpretation of Quantum TGD? Could it help to understand more clearly the somewhat poorly understood views about the notion of self, which as a quantum physical counterpart of observer becomes in ZEO a key concept of fundamental physics?

The basic ingredients involved are discrete coupling constant evolution, zero energy ontology (ZEO) implying that quantum theory is analogous to "square root" of thermodynamics, self as generalized Zeno effect as counterpart of observer made part of the quantum physical system, M8--M4× CP2 duality, and quantum criticality. A further idea is that vacuum functional is analogous to a thermodynamical partition function as exponent of energy E= TS-PV.

The correspondence rules are simple. The mixture of phases with different 3-volumes per particle in a critical region of thermodynamical system is replaced with a superposition of space-time surfaces of different 4-volumes assignable to causal diamonds (CDs) with different sizes. Energy E is replaced with action S for preferred extremals defining Kähler function in the "world of classical worlds" (WCW). S is sum of Kähler action and 4-volume term, and these terms correspond to entropy and volume in the generalization E= TS-PV → S. P resp. T corresponds to the inverse of Kähler coupling strength αK resp. cosmological constant Λ. Both have discrete spectrum of values determined by number theoretically determined discrete coupling constant evolution. Number theoretical constraints force the analog of micro-canonical ensemble so that S as the analog of E is constant for all 4-surfaces appearing in the quantum superposition. This implies quantization rules for Kähler action and volume, which are very strong since αK is complex.

This kind of quantum critical zero energy state is created in unitary evolution created in single step in the process defining self as a generalized Zeno effect. This unitary process implying time delocalization is followed by a weak measurement reducing the state to a fixed CD so that the clock time idenfified as the distance between its tips is well-defined. The condition that the action is same for all space-time surfaces in the superposition poses strong quantization conditions between the value of Kähler action (Kähler coupling strength is complex) and volume term proportional to cosmological constant. The outcome is that after sufficiently large number of steps no space-time surfaces satisfying the conditions can be found, and the first reduction to the opposite boundary of CD must occur - self dies. This is the classical counterpart for the fact that eventually all state function reduction leaving the members of state pairs at the passive boundary of CD invariant are made and the first reduction to the opposite boundary remains the only option.

The generation of magnetic flux tubes provides a manner to satisfy the constancy conditions for the action so that the existing phenomenology as well as TGD counterpart of cyclic cosmology as re-incarnations of cosmic self follows as a prediction.

This picture generalizes to the twistor lift of TGD and cosmology provides an interesting application. One ends up with a precise model for the p-adic coupling constant evolution of the cosmological constant Λ explaining the positive sign and smallness of Λ in long length scales as a cancellation effect for M4 and CP2 parts of the Kähler action for the sphere of twistor bundle in dimensional reduction, a prediction for the radius of the sphere of M4 twistor bundle as Compton length associated with Planck mass (2π times Planck length), and a prediction for the p-adic coupling constant evolution for Λ and coupling strength of M4 part of Kähler action giving also insights to the CP breaking and matter antimatter asymmetry. The observed two values of Λ could correspond to two different p-adic length scales differing by a factor of 21/2.

See the chapter The Recent View about Twistorialization in TGD Framework or the article with the same title or the shorter article New insights about quantum criticality for twistor lift inspired by analogy with ordinary criticality.



Complex 8-momenta are necessary for the realization of massless many-particle states implying unitary without loops

I have proposed a realization of unitarity in twistor approach without loops and with discrete coupling constant evolution dictated by number theory (see this). The proposal relies crucially on the identification quantum numbers in M8 picture as like-light quaternionic 8-momenta and the assumption that also many-particle states are massles. The 8-momenta are also complex already at classical level with corresponding imaginary unit i commuting with octonionic imaginary units Ik of M8.

The essential assumption was that the 8-momenta of also many-particle states are light-like. It is easy to see that this cannot make sense if single particle states have light-like 8-momenta unless they are also parallel. For a moment I thought that complexification of single particle 8-momenta might help but it did not.

Next came the realization that BCFW construction actually gives analogs of zero energy states having complex light-like momenta. The single particle momenta are not however light-like anymore. In TGD these states can be assigned with the interior regions of causal diamonds and have interpretation as resonances/bound states with complex momenta. The following tries to articular this more precisely.

  1. In BCFW approach the expression of residue integral as sum of poles in the variable z associated with the amplitude obtained by the deformation pi→ pi+zri of momenta (∑ ri=0, ri• rj=0) leads to a decomposition of the tree scattering amplitude to a sum of products of amplitudes in resonance channels with complex momenta at poles. The products involve 1/P2 factor giving pole and the analog of cut in unitary condition. Proof of tree level unitarity is achieved by using complexified momenta as a mere formal trick and complex momenta are an auxiliary notion. The complex massless poles are associated with groups I of particles whereas the momenta of particles inside I are complex and non-light-like.
  2. Could BCFW deformation give a description of massless bound states massless particles so that the complexification of the momenta would describe the effect of bound state formation on the single particle states by making them non-light-like? This makes sense if one assumes that all 8-momenta - also external - are complex. The classical charges are indeed complex already classically since Kähler coupling strength is complex (see this). A possible interpretation for the imaginary part is in terms of decay width characterizing the life-time of the particle and defining a length of four-vector.
  3. The basic question in the construction of scattering amplitudes is what happens inside CD for the external particles with light-like momenta. The BCFW deformation leading to factorization suggests an answer to the question. The factorized channel pair corresponds to two CDs inside which analogs of M and N-M particle bound states of external massless particles would be formed by the deformation pi→ pi+zri making particle momenta non-light-like. The allowed values of z would correspond to the physical poles. The factorization of BCFW scattering amplitude would correspond to a decomposition to products of bound state amplitudes for pairs of CDs. The analogs of bound states for zero energy states would be in question. BCFW factorization could be continued down to the lowest level below which no factorization is possible.
  4. One can of course worry about the non-uniqueness of the BCFW deformation. For instance, the light-like momenta ri must be parallel (rii r) but the direction of r is free. Also the choice of λi is free to a high extent. BCFW expression for the amplitude as a residue integral over z is however unique. What could this non-uniqueness mean?

    Suppose one accepts the number theoretic vision that scattering amplitudes are representations for sequences of algebraic manipulations. These representations are bound to be highly non-unique since very many sequences can connect the same initial and final expressions. The space-time surface associated with given representation of the scattering amplitude is not unique since each computation corresponds to different space-time surface. There however exists a representation with maximal simplicity.

    Could these two kinds of non-uniqueness relate?

It is indeed easy to see that many-particle states with light-like single particle momenta cannot have light-like momenta unless the single-particle momenta are parallel so that in non-parallel case one must give up light-likeness condition also in complex sense.
  1. The condition of light-likeness in complex sense allows the vanishing of real and imaginary mass squared for individual particles

    Im(pi)= λi Re(pi) ,

    (Re(pi))2=(Im(pi))2=0 .

    Real and imaginary parts are parallel and light-like in 8-D sense.

  2. The remaining two conditions come from the vanishing of the real and imaginary parts of the total mass squared:

    i≠ j Re(pi)• Re(pj)-Im(pi)• Im(pj) =0 ,

    i≠ j Re(pi)• Im(pj)=0 .

    By using proportionality of Re(pi) and Im(pi) one can express the conditions in terms of the real momenta

    i≠ j (1-λiλj) Re(pi)• Re(pj) =0 ,

    i≠ j λj Re(pi)• Re(pj)=0 .

    For positive/negative energy part of zero energy state the sign of time component of momentum is fixed and therefore λi have fixed sign. Suppose that λi have fixed sign. Since the inner products pi• pj of time-like vectors with fixed sign of time compomemet are all positive or negative the second term can vanish only if one has pi•pj=0. If the sign of λi can vary, one can satisfy the condition linear in λi but not the first condition as is easy to see in 2-particle case.

  3. States with light-like parallel 8-momenta are allowed and one can ask whether this kind of states might be realized inside magnetic flux tubes identified as carriers of dark matter in TGD sense. The parallel light-like momenta in 8-D sense would give rise to a state analogous to super-conductivity. Could this be true also for quarks inside hadrons assumed to move in parallel in QCD based model. This also brings in mind the earlier intuitive proposal that the momenta of fermions and antifermions associated with partonic 2-surfaces must be parallel so that the propagators for the states containing altogether n fermions and antifermions would behave like 1/(p2)n/2 and would not correspond to ordinary particles.
See the chapter The Recent View about Twistorialization in TGD Framework or the article with the same title.



Connection between quaternionicity and causality

The notion of quaternionicity is a central element of M8-H duality. At the level of momentum space it means that 8-momenta -, which by M8-H-duality correspond to 4-momenta at level of M4 and color quantum numbers at the level of CP2 - are quaternionic. Quaternionicity means that the time component of 8-momentum, which is parallel to real octonion unit, is non-vanishing. The 8-momentum itself must be time-like, in fact light-like. In this case one can always regard the momentum as momentum in some quaternionic sub-space. Causality requires a fixed sign for the time component of the momentum.

It must be however noticed that 8-momentum can be complex: also the 4-momentum can be complex at the level of M× CP2 already classically. A possible interpretation is in terms of decay width as part of momentum as it indeed is in phenomenological description of unstable particles.

Remark: At space-time level either the tangent space or normal space of space-time surface in M8 is quaternionic (equivalently associative) in the regions having interpretation as external particles arriving inside causal diamond (CD). Inside CD this assumption is not made. The two options correspond to space-time regions with Minkowskian and Euclidian signatures of the induced metric.

Could one require that the quaternionic momenta form a linear space with respect to octonionic sum? This is the case if the energy - that is the time-like part parallel to the real octonionic unit - has a fixed sign. The sum of the momenta is quaternionic in this case since the sum of light-like momenta is in general time-like and in special case light-like. If momenta with opposite signs of energy are allowed, the sum can become space-like and the sum of momenta is co-quaternionic.

This result is technically completely trivial as such but has a deep physical meaning. Quaternionicity at the level of 8-momenta implies standard view about causality: only time-like or at most light-like momenta and fixed sign of time-component of momentum.

Remark: The twistorial construction of S-matrix in TGD framework based on generalization of twistors leads to a proposal allowing to have unitary S-matrix with vanishing loop corrections and number theoretically determined discrete coupling constant evolution. Also the problems caused by non-planar diagrams disappear and one can have particles, which are massive in M4 sense.

The proposal boils down to the condition that the 8-momenta of many-particle states are light-like (in complex sense). One has however a superposition over states with different directions of the projection of light-like 8-momentum to E4 in M8= M4× E4). At the level of CP2 one has massive state but in color representation for which color spin and hypercharge vanish but color Casimir operator can have value of the order of the mass squared for the state. This prediction sharply distinguishes TGD from QCD.

See the chapter The Recent View about Twistorialization in TGD Framework or the article with the same title.



Could functional equation and Riemann hypothesis generalize?

Number theoretical considerations lead to the modification of zeta function by replacing the powers n-s with= exp(-log(n)s) with powers exp(-Log(n)s), where rational valued number theoretic logarithm Log(n) is defined as sumpp p/π(p) corresponding to the decomposion of n to a product of powers of prime. For large primes Log(p) equals in good approximation to log(p). The point of the replacement is that Log(n) carriers number theoretical information so that the definition is very natural. This number theoretical zeta will denoted with Ζ to distinguish it from ordinary zeta function denoted by ζ.

It is interesting to list the elementary properties of the Ζ before trying to see whether functional equation for ζ and Riemann hypothesis generalize.

  1. The replacement log(n)→ Log(n)== sump kpLog(p) implies that Ζ codes explicitly number theoretic information. Note that Log(n) satisfies the crucial identity Log(mn)= Log(m)+ Log(n). Ζ is an analog of partition function with rational number valued Log(n) taking the role of energy and 1/s that of a complex temperature. In ZEO this partition function like entity could be associated with zero energy state as a "square root" of thermodynamical partition function: in this case complex temperatures are possible.|Ζ|2 would be the analog of ordinary partition function.
  2. Reduction of Ζ to a product of "prime factors" 1/[1-exp(-Log(p)s)] holds true by Log(n)== sump kpLog(p), Log(p) =p/π(p).
  3. Ζ is a combination of exponentials exp(-Log(n)s), which converge for Re(s)>0. For ζ one has exponentials exp(-log(n)s), which also converge for Re(s)>0: the sum ∑ n-s does not however converge in the region Re(s)<1. Presumably Ζ fails to converge for Re(s)≤ 1. The behavior of terms exp(-Log(n)s) for large values of n is very similar to that in ζ.
  4. One can express ζ o in terms of η function defined as

    η(s)= ∑ (-1)n n-s .

    The powers (-1)n guarantee that η converges (albeit not absolutely) inside the critical strip 0<s<1.

    By using a decomposition of integers to odd and even ones, one can express ζ in terms of η:

    ζ = η(s)/(-1+2-s+1) .

    This definition converges inside critical strip. Note the pole at s=1 coming from the factor.

    One can define also Η as counterpart of η:

    Η(s)= ∑ (-1)n e-Log(n)s) .

    The formula relating Ζ and Η generalizes: 2-s is replaced with exp(-2s) (Log(2)=2):

    Ζ = Η(s)/(-1+2e-2s) .

    This definition Ζ converges in the critical strip Re(s) ∈ (0,1) and also for Re(s)>1. Ζ(1-s) converges for Re(s)<1 so that in Η representation both converge.

    Note however that the poles of ζ at s=1 has shifted to that at s=log(2)/2 and is below Re(s)=1/2 line. If a symmetrically posioned pole at s= 1-log(2)/2 is not present in Η, functional equation cannot be true.

  5. Log(n) approaches log(n) for integers n not containing small prime factors p for which π(n) differs strongly from p/log(p). This suggests that allowing only terms exp(-Log(n)s) in the sum defining Ζ not divisible by primes p<pmax might give a cutoff Ζcut,pmax behaving very much like ζ from which "prime factors" 1/(1-exp(-Log(p)s) , p<pmax are dropped of. This is just division of Ζ by these factors and at least formally, this does not affect the zeros of Ζ. Arbitrary number of factors can be droped. Could this mean that Ζcut has same or very nearly same zeros as ζ at critical line? This sounds paradoxical and might reflect my sloppy thinking: maybe the lack of the absolute implies that the conclusion is incorrect.
The key questions are whether Ζ allows a generalization of the functional equation ξ(s)= ξ(1-s) with ξ(s)= (1/2) s(s-1) Γ(s/2) π-s/2 ζ(s) and whether Riemann hypothesis generalizes. The derivation of the functional equation is quite a tricky task and involves integral representation of ζ .
  1. One can start from the integral representation of ζ true for s>0.

    ζ(s)=[1/(1-21-s)Γ(s)]∫0[ts-1/(et+1)] dt , Re(s)>0 .

    deducible from the expression in terms of η(s). The factor 1/(1+et) can be expanded in geometric series 1/(1+et)=∑ (-1)n exp(nt) converning inside the critical strip. One formally performs the integrations by taking nt as an integration variable. The integral gives the result ∑ (-1)n/nz)Γ(s).

    The generalization of this would be obtained by a generalization of geometric series:

    1/(1+et)=∑ (-1)n exp(nt)→ ∑ (-1)n eexp(Log(n))t

    in the integral representation. This would formally give Ζ: the only difference is that one takes u= exp(Log(n))t as integration variable.

    One could try to prove the functional equation by using this representation. One proof (see this) starts from the alternative expression of ζ as

    ζ(s)=[1/Γ(s)]∫1[ ts-1/(et-1)]dt , Re(s)>1 .

    One modifies the integration contour to a contour C coming from +∞ above positive real axis, circling the origin and returning back to +∞ below the real axes to get a modified representation of ζ:

    ζ(s)=1/[2isin(π s)Γ(s)]∫1[(-w)s-1/(ew-1)] dw , Re(s)>1 .

    One modifies C further so that the origin is circle d around a square with vertices at +/- (2n+1)π and +/- i(2n+1)π.

    One calculates the integral the integral along C as a residue integral. The poles of the integrand proportional to 1/(1-et) are at imaginary axis and correspond to w= ir2π, r∈ Z. The residue integral gives the other side of the functional equation.

  2. Could one generalize this representation to the recent case? One must generalize the geometric series defined by 1/(ew-1) to -∑ eexp(Log(n))w. The problem is that one has only a generalization of the geometric series and not closed form for the counterpart of 1/(exp(w)-1) so that one does not know what the poles are. The naive guess is that one could compute the residue integrals term by term in the sum over n. An equally naive guess would be that for the poles the factors in the sum are equal to unity as they would be for Riemann zeta. This would give for the poles of n:th term the guess wn,r=r2π/exp(Log(n), r∈ Z. This does not however allow to deduce the residue at poles.Note that the pole of Η at s= log(2)/2 suggests that functional equation is not true.
There is however no need for a functional equation if one is only interested in F(s)== Ζ(s)+Ζ(1-s) at the critical line! Also the analog of Riemann hypothesis follows naturally!
  1. In the representation using Η F(s) converges at critical striple and is real(!) at the critical line Re(s)=1/2 as follows from the fact that 1-s= s* for Re(s)=1/2! Hence F(s) is expected to have a large number of zeros at critical line. Presumably their number is infinite, since F(s)cut,pmax approaches 2ζcut,pmax for large enough pmax at critical line.
  2. One can define a different kind of cutoff of Ζ for given nmax: n<nmax in the sum over e-Log(n)s. Call this cutoff Ζcut,nmax. This cutoff must be distinguished from the cutoff Ζcut,pmax obtained by dropping the "prime factors" with p<pmax. The terms in the cutoff are of the form u∑ kpp/π(p), u = exp(-s). It is analogous to a polymomial but with fractional powers of u. It can be made a polynomial by a change of variable u→ v=exp(-s/a), where a is the product of all π(p):s associated with all the primes involved with the integers n<nmax.

    One could solve numerically the zeros of Ζ(s)+Ζ(1-s) using program modules calculating π(p) for a given p and roots of a complex polynomial in given order. One can check whether also all zeros of Ζ(s)+Ζ(1-s) might reside at critical line.

  3. One an define also F(s)cut,nmax to be distinguished from F(s)cut,pmax. It reduces to a sum of terms exp(-Log(n)/2) cos(-Log(n)y) at critical line, n<nmax. Cosines come from roots of unity. F(s) function is not sum of rational powers of exp(-iy) unlike Ζ(s). The existence of zero could be shown by showing that the sign of this function varies as function of y. The functions cos(-Log(n)y) have period Δ y = 2π/Log(n). For small values of n the exponential terms exp(-Log(n)/2) are largest so that they dominate. For them the periods Δ y are smallest so that one expected that the sign of both F(s) and F(s)cut,nmax varies and forces the presence of zeros.

    One could perhaps interpret the system as quantum critical system. The rather large rapidly varying oscillatory terms with n<nmax with small Log(n) give a periodic infinite set of approximate roots and the exponentially smaller slowly varying higher terms induce small perturbations of this periodic structure. The slowly varying terms with large Log(n) become however large near the Im(s)=0 so that here the there effect is large and destroys the period structure badly for small root of Ζ.

To sum up, the definition of modified zeta and eta functions makes sense, as also the analog of Riemann Hypothesis. It however seems that the counterpart of functional equation does not hold true. This is however not a problem since one can define symmetrized zeta so that it is well-defined in critical strip.

For background see the chapter The Recent View about Twistorialization in TGD Framework or the article with the same title.



Considerations related to coupling constant evolution and Riemann zeta

I have made several number theoretic peculations related to the possible role of zeros of Riemann zeta in coupling constant evolution. The basic problem is that it is not even known whether the zeros of zeta are rationals, algebraic numbers or genuine transcendentals or belong to all these categories. Also the question whether number theoretic analogs of ζ defined for p-adic number fields could make sense in some sense is interesting.

1. Is number theoretic analog of ζ possible using Log(p) instead of log(p)?

The definition of Log(n) based on factorization Log(n)==∑pkpLog(p) allows to define the number theoretic version of Riemann Zeta ζ(s)=∑ n-s via the replacement n-s=exp(-log(n)s)→ exp(-Log(n)s).

  1. In suitable region of plane number-theoretic Zeta would have the usual decomposition to factors via the replacement 1/(1-p-s)→ 1/(1-exp(-Log(p)s). p-Adically this makes sense for s= O(p) and thus only for a finite number of primes p for positive integer valued s: one obtains kind of cut-off zeta. Number theoretic zeta would be sensitive only to a finite number of prime factors of integer n.
  2. This might relate to the strong physical indications that only a finite number of cognitive representations characterized by p-adic primes are present in given quantum state: the ramified primes for the extension are excellent candidates for these p-adic primes. The size scale n of CD could also have decomposition to a product of powers of ramified primes. The finiteness of cognition conforms with the cutoff: for given CD size n and extension of rationals the p-adic primes labelling cognitive representations would be fixed.
  3. One can expand the regions of converge to larger p-adic norms by introducing an extension of p-adics containing e and some of its roots (ep is automatically a p-adic number). By introducing roots of unity, one can define the phase factor exp(-iLog(n)Im(s)) for suitable values of Im(s). Clearly, exp(-ipIm(s))/π(p)) must be in the extension used for all primes p involved. One must therefore introduce prime roots exp(i/π(p)) for primes appearing in cutoff. To define the number theoretic zeta for all p-adic integer values of Re(s) and all integer values of Im(s), one should allow all roots of unity (ep(i2π/n)) and all roots e1/n: this requires infinite-dimensional extension.
  4. One can thus define a hierarchy of cutoffs of zeta: for this the factorization of Zeta to a finite number of "prime factors" takes place in genuine sense, and the points Im(s)= ikπ(p) give rise to poles of the cutoff zeta as poles of prime factors. Cutoff zeta converges to zero for Re(s)→ ∞ and exists along angles corresponding to allowed roots of unity. Cutoff zeta diverges for (Re(s)=0, Im(s)= ik π(p)) for the primes p appearing in it.
Remark: One could modify also the definition of ζ for complex numbers by replacing exp(log(n)s) with exp(Log(n)s) with Log(n)= ∑p kpLog(p) to get the prime factorization formula. I will refer to this variant of zeta as modified zeta below.

2. Could the values of 1/αK be given as zeros of ζ or of modified ζ

I have discussed the possibility that the zeros s=1/2+iy of Riemann zeta at critical line correspond to the values of complex valued Kähler coupling strength αK: s=i/αK (see this). The assumption that piy is root of unity for some combinations of p and y [log(p)y =(r/s)2π] was made. This does not allow s to be complex rational. If the exponent of Kähler action disappears from the scattering amplitudes as M8-H duality requires, one could assume that s has rational values but also algebraic values are allowed.

  1. If one combines the proposed idea about the Log-arithmic dependence of the coupling constants on the size of CD and algebraic extension with s=i/αK hypothesis, one cannot avoid the conjecture that the zeros of zeta are complex rationals. It is not known whether this is the case or not. The rationality would not have any strong implications for number theory but the existence irrational roots would have (see this). Interestingly, the rationality of the roots would have very powerful physical implications if TGD inspired number theoretical conjectures are accepted.

    The argument discussed below however shows that complex rational roots of zeta are not favored by the observations about the Fourier transform for the characteristic function for the zeros of zeta. Rather, the findings suggest that the imaginary parts (see this) should be rational multiples of 2π, which does not conform with the vision that 1/αK is algebraic number. The replacement of log(p) with Log(p) and of 2π with is natural p-adic approximation in an extension allowing roots of unity however allows 1/αK to be an algebraic number. Could the spectrum of 1/αK correspond to the roots of ζ or of modified ζ?

  2. A further conjecture discussed was that there is 1-1 correspondence between primes p≈ 2k, k prime, and zeros of zeta so that there would be an order preserving map k→ sk. The support for the conjecture was the predicted rather reasonable coupling constant evolution for αK. Primes near powers of 2 could be physically special because Log(n) decomposes to sum of Log(p):s and would increase dramatically at n=2k slightly above them.

    In an attempt to understand why just prime values of k are physically special, I have proposed that k-adic length scales correspond to the size scales of wormhole contacts whereas particle space-time sheets would correspond to p≈ 2k. Could the logarithmic relation between Lp and Lk correspond to logarithmic relation between p and π(p) in case that π(p) is prime and could this condition select the preferred p-adic primes p?

3. The argument of Dyson for the Fourier transform of the characteristic function for the set of zeros of ζ

Consider now the argument suggesting that the roots of zeta cannot be complex rationals. On basis of numerical evidence Dyson (see this) has conjectured that the Fourier transform for the characteristic function for the critical zeros of zeta consists of multiples of logarithms log(p) of primes so that one could regard zeros as one-dimensional quasi-crystal.

This hypothesis makes sense if the zeros of zeta decompose into disjoint sets such that each set corresponds to its own prime (and its powers) and one has piy= Um/n=exp(i2π m/n) (see the appendix of this). This hypothesis is also motivated by number theoretical universality (see this).

  1. One can re-write the discrete Fourier transform over zeros of ζ at critical line as

    f(x)= ∑y exp(ixy)) , y=Im(s) .

    The alternative form reads as

    f(u) =∑s uiy , u=exp(x) .

    f(u) is located at powers pn of primes defining ideals in the set of integers.

    For y=pn one would have piny=exp(inlog(p)y). Note that k=nlog(p) is analogous to a wave vector. If exp(inlog(p)y) is root of unity as proposed earlier for some combinations of p and y, the Fourier transform becomes a sum over roots of unity for these combinations: this could make possible constructive interference for the roots of unity, which are same or at least have the same sign. For given p there should be several values of y(p) with nearly the same value of exp(inlog(p)y(p)) whereas other values of y would interfere deconstructively.

    For general values y= xn x≠ p the sum would not be over roots of unity and constructive interference is not expected. Therefore the peaking at powers of p could take place. This picture does not support the hypothesis that zeros of zeta are complex rational numbers so that the values of 1/αK correspond to zeros of zeta and would be therefore complex rationals as the simplest view about coupling constant evolution would suggest.

  2. What if one replaces log(p) with Log(p) =p/π(p), which is rational and thus ζ with modified ζ? For large enough values of p Log(p)≈ log(p) finite computational accuracy does not allow distinguish Log(p) from log(p). For Log(p) one could thus understand the finding in terms of constructive interference for the roots of unity if the roots of zeta are of form s= 1/2+i(m/n)2π. The value of y cannot be rational number and 1/αK would have real part equal to y proportional to 2π which would require infinite-D extension of rationals. In p-adic sectors infinite-D extension does not conform with the finiteness of cognition.
  3. Numerical calculations have however finite accuracy, and allow also the possibility that y is algebraic number approximating rational multiple of 2π in some natural manner. In p-adic sectors would obtain the spectrum of y and 1/αK as algebraic numbers by replacing 2π in the formula is= αK= i/2+ q× 2π, q=r/s, with its approximate value:

    2π→ sin(2π/n)n= in/2(exp(i2π/n)- exp(-i2π/n))

    for an extension of rationals containing n:th of unity. Maximum value of n would give the best approximation. This approximation performed by fundamental physics should appear in the number theoretic scattering amplitudes in the expressions for 1/αK to make it algebraic number.

    y can be approximated in the same manner in p-adic sectors and a natural guess is that n=p defines the maximal root of unity as exp(i2π/p). The phase exp(ilog(p)y) for y= q sin(2π/n(y)), q=r/s, is replaced with the approximation induced by log(p)→ Log(p) and 2π→ sin(2π/n)n giving

    exp(ilog(p)y) → exp(iq(y) sin(2π/n(y))p/π(p)) .

    If s in q=r/s does not contain higher powers of p, the exponent exists p-adically for this extension and can can be expanded in positive powers of p as

    n inqn sin(2π/p)n (p/π(p))n .

    This makes sense p-adically.

    Also the actual complex roots of ζ could be algebraic numbers:

    s= i/2+ q× sin(2π/n(y))n(y) .

    If the proposed correlation between p-adic primes p≈ 2k, k prime and zeros of zeta predicting a reasonable coupling constant evolution for 1/αK is true, one can have naturally, n(y)=p(y), where p is the p-adic prime associated with y: the accuracy in angle measurement would increase with the size scale of CD. For given p there could be several roots y with same p(y) but different q(y) giving same phases or at least phases with same sign of real part.

    Whether the roots of modified ζ are algebraic numbers and at critical line Re(s)=1/2 is an interesting question.

Remark: This picture allows many variants. For instance, if one assumes standard zeta, one could consider the possibility that the roots yp associated with p and giving rise to constructive interference are of form y= q×(Log(p)/log(p))× sin(2π/p)p, q=r/s.

For background see the chapter The Recent View about Twistorialization in TGD Framework or the article with the same title.



General ideas about coupling constant evolution

The discrete coupling constant evolution would be associated with the scale hierarchy for CDs and the hierarchy of extensions of rationals.

  1. Discrete p-adic coupling constant evolution would naturally correspond to the dependence of coupling constants on the size of CD. For instance, I have considered a concrete but rather ad hoc proposal for the evolution of Kähler couplings strength based on the zeros of Riemann zeta (see this). Number theoretical universality suggests that the size scale of CD identified as the temporal distance between the tips of CD using suitable multiple of CP2 length scale as a length unit is integer, call it l. The prime factors of the integer could correspond to preferred p-adic primes for given CD.
  2. I have also proposed that the so called ramified primes of the extension of rationals correspond to the physically preferred primes. Ramification is algebraically analogous to criticality in the sense that two roots understood in very general sense co-incide at criticality. Could the primes appearing as factors of l be ramified primes of extension? This would give strong correlation between the algebraic extension and the size scale of CD.
In quantum field theories coupling constants depend in good approximation logarithmically on mass scale, which would be in the case of p-adic coupling constant evolution replaced with an integer n characterizing the size scale of CD or perhaps the collection of prime factors of n (note that one cannot exclude rational numbers as size scales). Coupling constant evolution could also depend on the size of extension of rationals characterized by its order and Galois group.

In both cases one expects approximate logarithmic dependence and the challenge is to define "number theoretic logarithm" as a rational number valued function making thus sense also for p-adic number fields as required by the number theoretical universality.

Consider first the coupling constant as a function of the length scale lCD(n)/lCD(1)=n.

  1. The number π(n) of primes p≤ n behaves approximately as π(n)= n/log(n). This suggests the definition of what might be called "number theoretic logarithm" as Log(n)== n/π(n). Also iterated logarithms such log(log(x)) appearing in coupling constant evolution would have number theoretic generalization.
  2. If the p-adic variant of Log(n) is mapped to its real counterpart by canonical identification involving the replacement p→ 1/p, the behavior can very different from the ordinary logarithm. Log(n) increases however very slowly so that in the generic case one can expect Log(n)<pmax, where pmax is the largest prime factor of n, so that there would be no dependence on p for pmax and the image under canonical identification would be number theoretically universal.

    For n=pk, where p is small prime the situation changes since Log(n) can be larger than small prime p. Primes p near primes powers of 2 and perhaps also primes near powers of 3 and 5 - at least - seem to be physically special. For instance, for Mersenne prime Mk=2k-1 there would be dramatic change in the step Mk→ Mk+1=2k, which might relate to its special physical role.

  3. One can consider also the analog of Log(n) as

    Log(n)= ∑p kpLog(p) ,

    where pki is a factor of n. Log(n) would be sum of number theoretic analogs for primes factors and carry information about them.

    One can extend the definition of Log(x) to the rational values x=m/n of the argument. The logarithm Logb(n) in base b=r/s can be defined as Logb(x)= Log(x)/Log(b).

    One can wonder whether one could replace the log(p) appearing as a unit in p-adic negentropy with a rational unit Log(p)= p/π(p) to gain number theoretical universality? One could therefore interpret the p-adic negentropy as real or p-adic number for some prime. Interestingly, |Log(p)|p=1/p approaches zero for large primes p (eye cannot see itself!) whereas |Log(p)|q=1/|π(p)|q has large values for the prime power factors qr of π(p).

    For p∈ {2,3,5} one has Log(p)>log(p), where for larger primes one has Log(p)<log(p). One has Log(2)=2>log(2)=.693..., Log(3)= 3k/2> log(3)= 1.099, Log(5)= 5/3=1.666..>log(5)= 1.609. For p=7 one has Log(7)= 7/4≈ 1.75<log(7)≈ 1.946. Hence these primes and CD size scales n involving large powers of p∈ {2,3,5} ought to be physically special as indeed conjectured on basis of p-adic calculations and some observations related to music and biological evolution (see this).

    In particular, for Mersenne primes Mk=2k-1 one would have Log(Mk) ≈ k log(2) for large enough k. For Log(2k) one would have k × Log(2)=2k>log(2k)=klog(2): there would be sudden increase in the value of Log(n) at n=Mk. This jump in p-adic length scale evolution might relate to the very special physical role of Mersenne primes strongly suggested by p-adic mass calculations (see this).

  4. The definition of Log(n) based on factorization Log(n)== ∑pkpLog(p) allows to define the number theoretic version of Riemann Zeta ζ(s)=∑ n-s via the replacement n-s=exp(-log(n)s)→ exp(-Log(n)s). In suitable region of plane number-theoretic Zeta would have the usual decomposition to factors via the replacement 1/(1-p-s)→ 1/(1-exp(-Log(p)s). p-Adically this makes sense for s= O(p) and thus only for a finite number of primes p for positive integer valued s: one obtains kind of cut-off zeta. Number theoretic zeta would be sensitive only to a finite number of prime factors of integer n.

    This might relate to the strong physical indications that only a finite number of cognitive representations characterized by p-adic primes are present in given quantum state: the ramified primes for the extension are excellent candidates for these p-adic primes. The size scale n of CD could also have decomposition to a product of powers of ramified primes. The finiteness of cognition conforms with the cutoff: for given CD size n and extension of rationals the p-adic primes labelling cognitive representations would be fixed.

    One can expand the regions of converge to larger p-adic norms by introducing an extension of p-adics containing e and some of its roots (ep is automatically a p-adic number). By introducing roots of unity, one can define the phase factor exp(-iLog(n)Im(s)) for suitable values of Im(s). Clearly, exp(-ipIm(s))/π(p)) must be in the extension used for all primes p involved. One must therefore introduce prime roots exp(i/π(p)) for primes appearing in cutoff. To define the number theoretic zeta for all p-adic integer values of Re(s) and all integer values of Im(s), one should allow all roots of unity (ep(i2π/n)) and all roots e1/n: this requires infinite-dimensional extension.

    One can thus define a hierarchy of cutoffs of zeta: for this the factorization of Zeta to a finite number of "prime factors" takes place in genuine sense, and the points Im(s)= ikπ(p) give rise to poles of the cutoff zeta as poles of prime factors. Cutoff zeta converges to zero for Re(s)→ ∞ and exists along angles corresponding to allowed roots of unity. Cutoff zeta diverges for (Re(s)=0, Im(s)= ik π(p)) for the primes p appearing in it.

  5. I have discussed the possibility that the zeros s=1/2+iy of Riemann zeta at critical line correspond to the values of complex valued Kähler coupling strength αK: s=i/αK (see this). The assumption that piy is root of unity for some combinations of p and y [log(p)y =(r/s)2π] was made. This does not allow s to be complex rational. If the exponent of Kähler action disappears from the scattering amplitudes as M8-H duality requires, one can assume that s has rational values.

    If one combines the proposed idea about the Log-arithmic dependence of the coupling constants on the size of CD and algebraic extension with s=i/αK hypothesis, one cannot avoid the conjecture that the zeros of zeta are complex rationals. It is not known whether this is the case or not. The rationality would not have any strong implications for number theory but the existence irrational roots would have (see this). Interestingly, the rationality of the roots would have very powerful physical implications if TGD inspired number theoretical conjectures are accepted.

    A further conjecture discussed was that there is 1-1 correspondence between primes p≈ 2k, k prime, and zeros of zeta so that there would be an order preserving map k→ sk. The support for the conjecture was the predicted rather reasonable coupling constant evolution for αK. Primes near powers of 2 could be physically special because Log(n) decomposes to sum of Log(p):s and would increase dramatically at n=2k slightly above them. In an attempt to understand why just prime values of k are physically special, I have proposed that k-adic length scales correspond to the size scales of wormhole contacts whereas particle space-time sheets would correspond to p≈ 2k. Could the logarithmic relation of Lp and Lk correspond to logarithmic relation of p and π(p)?

Consider next the dependence on the extension of rationals. The natural algebraization of the problem is to consider the Galois group of the extension.
  1. Consider first the counterparts of primes and prime factorization for groups. The counterparts of primes are simple groups, which do not have normal subgroups H satisfying gH=Hg implying invariance under automorphisms of G. Simple groups have no decomposition to a product of sub-groups. If the group has normal subgroup H, it can be decomposed to a product H× G/H and any finite group can be decomposed to a product of simple groups.

    All simple finite groups have been classified (see this). There are cyclic groups, alternating groups, 16 families of simple groups of Lie type, 26 sporadic groups. This includes 20 quotients G/H by a normal subgroup of monster group and 6 groups which for some reason are referred to as pariahs.

  2. Suppose that finite groups can be ordered so that one can assign number N(G) to group G. The roughest ordering criterion is based on ord(G). For given order ord(G)=n one has all groups, which are products of cyclic groups associated with prime factors of n plus products involving non-Abelian groups for which the order is not prime. N(G)>ord(G) thus holds true. For groups with the same order one should have additional ordering criteria, which could relate to the complexity of the group. The number of simple factors would serve as an additional ordering criterion.

    If its possible to define N(G) in a natural manner then for given G one can define the number π1(N(G)) of simple groups (analogs of primes) not larger than G. The first guess is that that the number π1(N(G)) varies slowly as a function of G. Since Zi is simple group, one has π1(N(G)) ≥ π(N(G)).

  3. One can consider two definitions of number theoretic logarithm, call it Log1.

    a) Log1(N(G))= N(G)/π1(N(G)) ,

    b) Log1(G)= ∑i ki Log1(N(Gi)) , Log1(N(Gi)) = N(Gi)/π1(N(Gi)) .

    Option a) does not provide information about the decomposition of G to a product of simple factors. For Option b) one decomposes G to a product of simple groups Gi: G= ∏i Giki and defines the logarithm as Option b) so that it carries information about the simple factors of G.

  4. One could organize the groups with the same order to same equivalence class. In this case the above definitions would give

    a) Log1(ord(G))= ord(G)/π1(ord(G)) < Log(ord(G)) ,

    b) Log1(ord(G))= ∑i ki Log(ord(Gi)) , Log1(ord(Gi)) = ord(Gi)/π1(ord(Gi)) .

    Besides groups with prime orders there are non-Abelian groups with non-prime orders. The occurrence of same order for two non-isomorphic finite simple groups is very rare (see this). This would suggests that one has π1(ord(G)) <ord(G) so that Log1(ord(G))/ord(G)<1 would be true.

  5. For orders n(G)∈ {2,3,5} one has Log1(n(G))=Log(n(G))>log(n(G)) so that the ordes n(G) involving large factors of p∈ {2,3,5} would be special also for the extensions of rationals. S3 with order 6 is the first non-abelian simple group. One has π(S3)=4 giving Log(6)= 6/4=1.5<log(6)=1.79 so that S3 is different from the simple groups below it.
To sum up, number theoretic logarithm could provide answer to the long-standing question what makes Mersenne primes and also other small primes so special.

For background see the chapter The Recent View about Twistorialization in TGD Framework or the article with the same title.



Summary about twistorialization in TGD framework

Since the contribution means in well-defined sense a breakthrough in the understanding of TGD counterparts of scattering amplitudes, it is useful to summarize the basic results deduced above as a polished answer to a Facebook question.

There are two diagrammatics: Feynman diagrammatics and twistor diagrammatics.

  1. Virtual state is an auxiliary mathematical notion related to Feynman diagrammatics coding for the perturbation theory. Virtual particles in Feynman diagrammatics are off-mass-shell.
  2. In standard twistor diagrammatics one obtains counterparts of loop diagrams. Loops are replaced with diagrams in which particles in general have complex four-momenta, which however light-like: on-mass-shell in this sense. BCFW recursion formula provides a powerful tool to calculate the loop corrections recursively.
  3. Grassmannian approach in which Grassmannians Gr(k,n) consisting of k-planes in n-D space are in a central role, gives additional insights to the calculation and hints about the possible interpretation.
  4. There are two problems. The twistor counterparts of non-planar diagrams are not yet understood and physical particles are not massless in 4-D sense.
In TGD framework twistor approach generalizes.
  1. Massless particles in 8-D sense can be massive in 4-D sense so that one can describe also massive particles. If loop diagrams are not present, also the problems produced by non-planarity disappear.
  2. There are no loop diagrams- radiative corrections vanish. ZEO does not allow to define them and they would spoil the number theoretical vision, which allows only scattering amplitudes, which are rational functions of data about external particles. Coupling constant evolution - something very real - is now discrete and dictated to a high degree by number theoretical constraints.
  3. This is nice but in conflict with unitarity if momenta are 4-D. But momenta are 8-D in M8 picture (and satisfy quaternionicity as an additional constraint) and the problem disappears! There is single pole at zero mass but in 8-D sense and also many-particle states have vanishing mass in 8-D sense: this gives all the cuts in 4-D mass squared for all many-particle state. For many-particle states not satisfying this condition scattering rates vanish: these states do not exist in any operational sense! This is certainly the most significant new discovery in the recent contribution.

    BCFW recursion formula for the calculation of amplitudes trivializes and one obtains only tree diagrams. No recursion is needed. A finite number of steps are needed for the calculation and these steps are well-understood at least in 4-D case - even I might be able to calculate them in Grassmannian approach!

  4. To calculate the amplitudes one must be able to explicitly formulate the twistorialization in 8-D case for amplitudes. I have made explicit proposals but have no clear understanding yet. In fact, BCFW makes sense also in higher dimensions unlike Grassmannian approach and it might be that the one can calculate the tree diagrams in TGD framework using 8-D BCFW at M8 level and then transform the results to M4× CP2.
What I said above does yet contain anything about Grassmannians.
  1. The mysterious Grassmannians Gr(k,n) might have a beautiful interpretation in TGD: they could correspond at M8 level to reduced WCWs which is a highly natural notion at M4× CP2 level obtained by fixing the numbers of external particles in diagrams and performing number theoretical discretization for the space-time surface in terms of cognitive representation consisting of a finite number of space-time points.

    Besides Grassmannians also other flag manifolds - having Kähler structure and maximal symmetries and thus having structure of homogenous space G/H - can be considered and might be associated with the dynamical symmetries as remnants of super-symplectic isometries of WCW.

  2. Grassmannian residue integration is somewhat frustrating procedure: it gives the amplitude as a sum of contributions from a finite number of residues. Why this work when outcome is given by something at finite number of points of Grassmannian?!

    In M8 picture in TGD cognitive representations at space-time level as finite sets of points of space-time determining it completely as zero locus of real or imaginary part of octonionic polynomial would actually give WCW coordinates of the space-time surface in finite resolution.

    The residue integrals in twistor diagrams would be the manner to realize quantum classical correspondence by associating a space-time surface to a given scattering amplitude by fixing the cognitive representation determining it. This would also give the scattering amplitude.

    Cognitive representation would be highly unique: perhaps modulo the action of Galois group of extension of rationals. Symmetry breaking for Galois representation would give rise to supersymmetry breaking. The interpretation of supersymmetry would be however different: many-fermion states created by fermionic oscillator operators at partonic 2-surface give rise to a representation of supersymmetry in TGD sense.

For background see the chapter The Recent View about Twistorialization in TGD Framework or the article with the same title.



The Recent View about Twistorialization in TGD Framework

The recent view about the twistorialization in TGD framework is discussed.

  1. A proposal made already earlier is that scattering diagrams as analogs of twistor diagrams are constructible as tree diagrams for CDs connected by free particle lines. Loop contributions are not even well-defined in zero energy ontology (ZEO) and are in conflict with number theoretic vision. The coupling constant evolution would be discrete and associated with the scale of CDs (p-adic coupling constant evolution) and with the hierarchy of extensions of rationals defining the hierarchy of adelic physics.
  2. Logarithms appear in the coupling constant evolution in QFTs. The identification of their number theoretic versions as rational number valued functions required by number-theoretical universality for both the integer characterizing the size scale of CD and for the hierarchy of Galois groups leads to an answer to a long-standing question what makes small primes and primes near powers of them physically special. The primes p =2,3,5 indeed turn out to be special from the point of view of number theoretic logarithm.
  3. The reduction of the scattering amplitudes to tree diagrams is in conflict with unitarity in 4-D situation. The imaginary part of the scattering amplitude would have discontinuity proportional to the scattering rate only for many-particle states with light-like total momenta. Scattering rates would vanish identically for the physical momenta for many-particle states.

    In TGD framework the states would be however massless in 8-D sense. Massless pole corresponds now to a continuum for M4 mass squared and one would obtain the unitary cuts from a pole at P2=0! Scattering rates would be non-vanishing only for many-particle states having light-like 8-momentum, which would pose a powerful condition on the construction of many-particle states. This strong form of conformal symmetry has highly non-trivial implications concerning color confinement.

  4. The key idea is number theoretical discretization in terms of "cognitive representations" as space-time time points with M8-coordinates in an extension of rationals and therefore shared by both real and various p-adic sectors of the adele. Discretization realizes measurement resolution, which becomes an inherent aspect of physics rather than something forced by observed as outsider. This fixes the space-time surface completely as a zero locus of real or imaginary part of octonionic polynomial.

    This must imply the reduction of "world of classical worlds" (WCW) corresponding to a fixed number of points in the extension of rationals to a finite-dimensional discretized space with maximal symmetries and Kähler structure.

    The simplest identification for the reduced WCW would be as complex Grassmannian - a more general identification would be as a flag manifold. More complex options can of course be considered. The Yangian symmetries of the twistor Grassmann approach known to act as diffeomorphisms respecting the positivity of Grassmannian and emerging also in its TGD variant would have an interpretation as general coordinate invariance for the reduced WCW. This would give a completely unexpected connection with supersymmetric gauge theories and TGD.

  5. M8 picture implies the analog of SUSY realized in terms of polynomials of super-octonions whereas H picture suggests that supersymmetry is broken in the sense that many-fermion states as analogs of components of super-field at partonic 2-surfaces are not local. This requires breaking of SUSY. At M8 level the breaking could be due to the reduction of Galois group to its subgroup G/H, where H is normal subgroup leaving the point of cognitive representation defining space-time surface invariant. As a consequence, local many-fermion composite in M8 would be mapped to a non-local one in H by M8-H correspondence.
For background see the chapter The Recent View about Twistorialization in TGD Framework or the article with the same title.



To the index page