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Physics as a Generalized Number Theory

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Year 2015



Why the non-trivial zeros of Riemann zeta should reside at critical line?

Riemann Hypothess (RH) states that the non-trivial (critical) zeros of zeta lie at critical line s=1/2. It would be interesting to know how many physical justifications for why this should be the case has been proposed during years. Probably this number is finite, but very large it certainly is. In Zero Energy Ontology (ZEO) forming one of the cornerstones of the ontology of quantum TGD, the following justification emerges naturally. I represented it in the answer to previous posting, but there was stupid error in the answer so that I represent the corrected argument here.

  1. The "World of Classical Worlds" (WCW) consisting of space-time surfaces having ends at the boundaries of causal diamond (CD), the intersection of future and past directed light-cones times CP2 (recall that CDs form a fractal hierarchy). WCW thus decomposes to sub-WCWs and conscious experience for the self associated with CD is only about space-time surfaces in the interior of CD: this is a trong restriction to epistemology, would philosopher say.

    Also the light-like orbits of the partonic 2-surfaces define boundary like entities but as surfaces at which the signature of the induced metric changes from Euclidian to Minkowskian. By holography either kinds of 3-surfaces can be taken as basic objects, and if one accepts strong form of holography, partonic 2-surfaces defined by their intersections plus string world sheets become the basic entities.

  2. One must construct tangent space basis for WCW if one wants to define WCW Kähler metric and gamma matrices. Tangent space consists of allowed deformations of 3-surfaces at the ends of space-time surface at boundaries of CD, and also at light-like parton orbits extended by field equations to deformations of the entire space-time surface. By strong form of holography only very few deformations are allowed since they must respect the vanishing of the elements of a sub-algebra of the classical symplectic charges isomorphic with the entire algebra. One has almost 2-dimensionality: most deformations lead outside WCW and have zero norm in WCW metric.
  3. One can express the deformations of the space-like 3-surface at the ends of space-time using a suitable function basis. For CP2 degrees of freedom color partial waves with well defined color quantum numbers are natural. For light-cone boundary S2× R+, where R+ corresponds to the light-like radial coordinate, spherical harmonics with well defined spin are natural choice for S2 and for R+ analogs of plane waves are natural. By scaling invariance in the light-like radial direction they look like plane waves ψs(r)= rs= exp(us), u=log(r/r0), s= x+iy. Clearly, u is the natural coordinate since it replaces R+ with R natural for ordinary plane waves.
  4. One can understand why Re[s]=1/2 is the only possible option by using a simple argument. One has super-symplectic symmetry and conformal invariance extended from 2-D Riemann surface to metrically 2-dimensional light-cone boundary. The natural scaling invariant integration measure defining inner product for plane waves in R+ is du= dr/r =dlog(r/r0) with u varying from -∞ to +∞ so that R+ is effectively replaced with R. The inner product must be same as for the ordinary plane waves and indeed is for ψs(r) with s=1/2+iy since the inner product reads as

    ⟨ s1,s2⟩ == ∫0 (ψ*s1) ψs2dr= ∫0 exp(i(y1-y2)r-x1-x2 dr .

    For x1+x2=1 one obtains standard delta function normalization for ordinary plane waves:

    ⟨s1,s2⟩ = ∫-∞+∞exp[i(y1-y2)u] du∝ δ (y1-y2) .

    If one requires that this holds true for all pairs (s1,s2), one obtains xi=1/2 for all si. Preferred extremal condition gives extremely powerful additional constraints and leads to a quantisation of s=-x-iy: the first guess is that non-trivial zeros of zeta are obtained: s=1/2+iy. This identification would be natural by generalised conformal invariance. Thus RH is physically extremely well motivated but this of course does not prove it.

  5. The presence of the real part Re[s]=1/2 in eigenvalues of scaling operator apparently breaks hermiticity of the scaling operator. There is however a compensating breaking of hermiticity coming from the fact that real axis is replaced with half-line and origin is pathological. What happens that real part 1/2 effectively replaces half-line with real axis and obtains standard plane wave basis. The integration measure becomes scaling invariant - something very essential for the representations of super-symplectic algebra. For Re[s]=1/2 the hermicity conditions for the scaling generator rd/dr in R+ for s=1/2+iy reduce to those for the translation generator d/du in R but with Re[s] dropped away.
This relates also to the number theoretical universality and mathematical existence of WCW in an interesting manner.
  1. If one assumes that p-adic primes p correspond to zeros s=1/2+y of zeta in 1-1 manner in the sense that piy(p) is root of unity existing in all number fields (algebraic extension of p-adics) one obtains that the plane wave exists for p at points r= pn. p-Adically wave function is discretized to a delta function distribution concentrated at (r/r0)= pn- a logarithmic lattice. This can be seen as space-time correlate for p-adicity for light-like momenta to be distinguished from that for massive states where length scales come as powers of p1/2. Something very similar is obtained from the Fourier transform of the distribution of zeros at critical line (Dyson's argument), which led to a the TGD inspired vision about number theoretical universality .
  2. My article Strategy for Proving Riemann Hypothesis (for a slightly improved version see this written for 12 years ago relies on coherent states instead of eigenstates of Hamiltonian. The above approach in turn absorbs the problematic 1/2 to the integration measure at light cone boundary and conformal invariance is also now central.
  3. Quite generally, I believe that conformal invariance in the extended form applying at metrically 2-D light-cone boundary (and at light-like orbits of partonic 2-surfaces) could be central for understanding why physics requires RH and maybe even for proving RH assuming it is provable at all in existing standard axiomatic system. For instance, the number of generating elements of the extended supersymplectic algebra is infinite (rather than finite as for ordinary conformal algebras) and generators are labelled by conformal weights defined by zeros of zeta (perhaps also the trivial conformal weights). s=1/2+iy guarantees that the real parts of conformal weights for all states are integers. By conformal confinement the sum of ys vanishes for physical states. If some weight is not at critical line the situation changes. One obtains as net conformal weights all multiples of x shifted by all half odd integer values. And of course, the realisation as plane waves at boundary of light-cone fails and the resulting loss of unitary makes things too pathological and the mathematical existence of WCW is threatened.
  4. The existence of non-trivial zeros outside the critical line could thus spole the representations of super-symplectic algebra and destroy WCW geometry. RH would be crucial for the mathematical existence of the physical world! And physical worlds exist only as mathematical objects in TGD based ontology: there are no physical realities behind the mathematical objects (WCW spinor fields) representing the quantum states. TGD inspired theory of consciousness tells that quantum jumps between the zero energy states give rise to conscious experience, and this is in principle all that is needed to understand what we experience.
See the chapter Number Theoretical Vision or the article Why the non-trivial zeros of Riemann zeta should reside at critical line?.



Does Riemann Zeta Code for Generic Coupling Constant Evolution?

The understanding of coupling constant evolution and predicting it is one of the greatest challenges of TGD. During years I have made several attempts to understand coupling evolution.

  1. The first idea dates back to the discovery of WCW Kähler geometry defined by Kähler function defined by Kähler action (this happened around 1990) (see this). The only free parameter of the theory is Kähler coupling strength αK analogous to temperature parameter αK postulated to be is analogous to critical temperature. Whether only single value or entire spectrum of of values αK is possible, remained an open question.

    About decade ago I realized that Kähler action is complex receiving a real contribution from space-time regions of Euclidian signature of metric and imaginary contribution from the Minkoswkian regions. Euclidian region would give Kähler function and Minkowskian regions analog of QFT action of path integral approach defining also Morse function. Zero energy ontology (ZEO) (see this) led to the interpretation of quantum TGD as complex square root of thermodynamics so that the vacuum functional as exponent of Kähler action could be identified as a complex square root of the ordinary partition function. Kähler function would correspond to the real contribution Kähler action from Euclidian space-time regions. This led to ask whether also Kähler coupling strength might be complex: in analogy with the complexification of gauge coupling strength in theories allowing magnetic monopoles. Complex αK could allow to explain CP breaking. I proposed that instanton term also reducing to Chern-Simons term could be behind CP breaking

  2. p-Adic mass calculations for 2 decades ago (see this) inspired the idea that length scale evolution is discretized so that the real version of p-adic coupling constant would have discrete set of values labelled by p-adic primes. The simple working hypothesis was that Kähler coupling strength is renormalization group (RG) invariant and only the weak and color coupling strengths depend on the p-adic length scale. The alternative ad hoc hypothesis considered was that gravitational constant is RG invariant. I made several number theoretically motivated ad hoc guesses about coupling constant evolution, in particular a guess for the formula for gravitational coupling in terms of Kähler coupling strength, action for CP2 type vacuum extremal, p-adic length scale as dimensional quantity (see this). Needless to say these attempts were premature and a hoc.
  3. The vision about hierarchy of Planck constants heff=n× h and the connection heff= hgr= GMm/v0, where v0<c=1 has dimensions of velocity (see this>) forced to consider very seriously the hypothesis that Kähler coupling strength has a spectrum of values in one-one correspondence with p-adic length scales. A separate coupling constant evolution associated with heff induced by αK∝ 1/hbareff ∝ 1/n looks natural and was motivated by the idea that Nature is theoretician friendly: when the situation becomes non-perturbative, Mother Nature comes in rescue and an heff increasing phase transition makes the situation perturbative again.

    Quite recently the number theoretic interpretation of coupling constant evolution (see this> or this in terms of a hierarchy of algebraic extensions of rational numbers inducing those of p-adic number fields encouraged to think that 1/αK has spectrum labelled by primes and values of heff. Two coupling constant evolutions suggest themselves: they could be assigned to length scales and angles which are in p-adic sectors necessarily discretized and describable using only algebraic extensions involve roots of unity replacing angles with discrete phases.

  4. Few years ago the relationship of TGD and GRT was finally understood (see this>) . GRT space-time is obtained as an approximation as the sheets of the many-sheeted space-time of TGD are replaced with single region of space-time. The gravitational and gauge potential of sheets add together so that linear superposition corresponds to set theoretic union geometrically. This forced to consider the possibility that gauge coupling evolution takes place only at the level of the QFT approximation and αK has only single value. This is nice but if true, one does not have much to say about the evolution of gauge coupling strengths.
  5. The analogy of Riemann zeta function with the partition function of complex square root of thermodynamics suggests that the zeros of zeta have interpretation as inverses of complex temperatures s=1/β. Also 1/αK is analogous to temperature. This led to a radical idea to be discussed in detail in the sequel.

    Could the spectrum of 1/αK reduce to that for the zeros of Riemann zeta or - more plausibly - to the spectrum of poles of fermionic zeta ζF(ks)= ζ(ks)/ζ(2ks) giving for k=1/2 poles as zeros of zeta and as point s=2? ζF is motivated by the fact that fermions are the only fundamental particles in TGD and by the fact that poles of the partition function are naturally associated with quantum criticality whereas the vanishing of ζ and varying sign allow no natural physical interpretation.

    The poles of ζF(s/2) define the spectrum of 1/αK and correspond to zeros of ζ(s) and to the pole of ζ(s/2) at s=2. The trivial poles for s=2n, n=1,2,.. correspond naturally to the values of 1/αK for different values of heff=n× h with n even integer. Complex poles would correspond to ordinary QFT coupling constant evolution. The zeros of zeta in increasing order would correspond to p-adic primes in increasing order and UV limit to smallest value of poles at critical line. One can distinguish the pole s=2 as extreme UV limit at which QFT approximation fails totally. CP2 length scale indeed corresponds to GUT scale.

  6. One can test this hypothesis. 1/αK corresponds to the electroweak U(1) coupling strength so that the identification 1/αK= 1/αU(1) makes sense. One also knows a lot about the evolutions of 1/αU(1) and of electromagnetic coupling strength 1/αem= 1/[cos2WU(1). What does this predict?

    It turns out that at p-adic length scale k=131 (p≈ 2k by p-adic length scale hypothesis, which now can be understood number theoretically (see this ) fine structure constant is predicted with .7 per cent accuracy if Weinberg angle is assumed to have its value at atomic scale! It is difficult to believe that this could be a mere accident because also the prediction evolution of αU(1) is correct qualitatively. Note however that for k=127 labelling electron one can reproduce fine structure constant with Weinberg angle deviating about 10 per cent from the measured value of Weinberg angle. Both models will be considered.

  7. What about the evolution of weak, color and gravitational coupling strengths? Quantum criticality suggests that the evolution of these couplings strengths is universal and independent of the details of the dynamics. Since one must be able to compare various evolutions and combine them together, the only possibility seems to be that the spectra of gauge coupling strengths are given by the poles of ζF(w) but with argument w=w(s) obtained by a global conformal transformation of upper half plane - that is Möbius transformation (see this) with real coefficients (element of GL(2,R)) so that one as ζF((as+b)/(cs+d)). Rather general arguments force it to be and element of GL(2,Q), GL(2,Z) or maybe even SL(2,Z) (ad-bc=1) satisfying additional constraints. Since TGD predicts several scaled variants of weak and color interactions, these copies could be perhaps parameterized by some elements of SL(2,Z) and by a scaling factor K.

    Could one understand the general qualitative features of color and weak coupling contant evolutions from the properties of corresponding Möbius transformation? At the critical line there can be no poles or zeros but could asymptotic freedom be assigned with a pole of cs+d and color confinement with the zero of as+b at real axes? Pole makes sense only if Kähler action for the preferred extremal vanishes. Vanishing can occur and does so for massless extremals characterizing conformally invariant phase. For zero of as+b vacuum function would be equal to one unless Kähler action is allowed to be infinite: does this make sense?. One can however hope that the values of parameters allow to distinguish between weak and color interactions. It is certainly possible to get an idea about the values of the parameters of the transformation and one ends up with a general model predicting the entire electroweak coupling constant evolution successfully.

To sum up, the big idea is the identification of the spectra of coupling constant strengths as poles of ζF((as+b/)(cs+d)) identified as a complex square root of partition function with motivation coming from ZEO, quantum criticality, and super-conformal symmetry; the discretization of the RG flow made possible by the p-adic length scale hypothesis p≈ kk, k prime; and the assignment of complex zeros of ζ with p-adic primes in increasing order. These assumptions reduce the coupling constant evolution to four real rational or integer valued parameters (a,b,c,d). One can say that one of the greatest challenges of TGD has been overcome.

See the new chapter Does Riemann Zeta Code for Generic Coupling Constant Evolution? or the article Does Riemann Zeta Code for Generic Coupling Constant Evolution?.



Some applications of Number Theoretic Universality

Number Theoretic Universality (NTU) in the strongest form says that all numbers involved at "basic level" (whatever this means!) of adelic TGD are products of roots of unity and of power of a root of e defining finite-dimensional extensions of p-adic numbers (ep is ordinary p-adic number). This is extremely powerful physics inspired conjecture with a wide range of possible mathematical applications.

  1. For instance, vacuum functional defined as an exponent of Kähler action for preferred externals would be number of this kind. One could define functional integral as adelic operation in all number fields: essentially as sum of exponents of Kähler action for stationary preferred extremals since Gaussian and metric determinants potentially spoiling NTU would cancel each other leaving only the exponent.
  2. The implications of NTU for the zeros of Riemann zeta expected to be closely related to super-symplectic conformal weights will be discussed below.
  3. NTU generalises to all Lie groups. Exponents exp(iniJi/n) of lie-algebra generators define generalisations of number theoretically universal group elements and generate a discrete subgroup of compact Lie group. Also hyperbolic "phases" based on the roots em/n are possible and make possible discretized NTU versions of all Lie-groups expected to play a key role in adelization of TGD.

    NTU generalises also to quaternions and octonions and allows to define them as number theoretically universal entities. Note that ordinary p-adic variants of quaternions and octonions do not give rise to a number field: inverse of quaternion can have vanishing p-adic variant of norm squared satisfying ∑n xn2=0.

    NTU allows to define also the notion of Hilbert space as an adelic notion. The exponents of angles characterising unit vector of Hilbert space would correspond to roots of unity.

Super-symplectic conformal weights and Riemann zeta

The existence of WCW geometry highly nontrivial already in the case of loop spaces. Maximal group of isometries required and is infinite-dimensional. Super-symplectic algebra is excellent candidate for the isometry algebra. There is also extended conformal algebra associated with δ CD. These algebras have fractal structure. Conformal weights for isomorphic subalgebra n-multiples of those for entire algebra. Infinite hierarchy labelled by integer n>0. Generating conformal weights could be poles of fermionic zeta ζF. This demands n>0. Infinite number of generators with different non-vanishing conformal weight with other quantum numbers fixed. For ordinary conformal algebras there are only finite number of generating elements (n=1).

If the radial conformal weights for the generators of g consist of poles of ζF, the situation changes. ζF is suggested by the observation that fermions are the only fundamental particles in TGD.

  1. Riemann Zeta ζ(s)= ∏p(1/(1-p-s) identifiable formally as a partition function ζB(s) of arithmetic boson gas with bosons with energy log(p) and temperature 1/s= 1/(1/2+iy) should be replaced with that of arithmetic fermionic gas given in the product representation by ζF(s) =∏p (1+p-s) so that the identity ζB(s))/ζF(s) =ζB(2s) follows. This gives

    ζB(s)/ζB(2s) .

    ζF(s) has zeros at zeros sn of ζ (s) and at the pole s=1/2 of zeta(2s). ζF(s) has poles at zeros sn/2 of ζ(2s) and at pole s=1 of ζ(s).

    The spectrum of 1/T would be for the generators of algebra {(-1/2+iy)/2, n>0, -1}. In p-adic thermodynamics the p-adic temperature is 1/T=1/n and corresponds to "trivial" poles of ζF. Complex values of temperature does not make sense in ordinary thermodynamics. In ZEO quantum theory can be regarded as a square root of thermodynamics and complex temperature parameter makes sense.

  2. If the spectrum of conformal weights of generators of algebra (not the entire algebra!) corresponds to poles serving as analogs of propagator poles, it consists of the "trivial" conformal h=n>0- the standard spectrum with h=0 assignable to massless particles excluded - and "non-trivial" h=-1/4+iy/2. There is also a pole at h=-1.

    Both the non-trivial pole with real part hR= -1/4 and the pole h=-1 correspond to tachyons. I have earlier proposed conformal confinement meaning that the total conformal weight for the state is real. If so, one obtains for a conformally confined two-particle states corresponding to conjugate non-trivial zeros in minimal situation hR= -1/2 assignable to N-S representation.

    In p-adic mass calculations ground state conformal weight must be -5/2. The negative fermion ground state weight could explain why the ground state conformal weight must be tachyonic -5/2. With the required 5 tensor factors one would indeed obtain this with minimal conformal confinement. In fact, arbitrarily large tachyonic conformal weight is possible but physical state should always have conformal weights h>0.

  3. h=0 is not possible for generators, which reminds of Higgs mechanism for which the naive ground states corresponds to tachyonic Higgs. h=0 conformally confined massless states are necessarily composites obtained by applying the generators of Kac-Moody algebra or super-symplectic algebra to the ground state. This is the case according to p-adic mass calculations, and would suggest that the negative ground state conformal weight can be associated with super-symplectic algebra and the remaining contribution comes from ordinary super-conformal generators. Hadronic masses whose origin is poorly understood could come from super-symplectic degrees of freedom. There is no need for p-adic thermodynamics in super-symplectic degrees of freedom.

Are the zeros of Riemann zeta number theoretically universal?

Dyson's comment about Fourier transform of Riemann Zeta is very interesting concerning NTU for Riemann zeta.

  1. The numerical calculation of Fourier transform for the distribution of the imaginary parts iy of zeros s=1/2+iy of zeta shows that it is concentrated at discrete set of frequencies coming as log(pn), p prime. This translates to the statement that the zeros of zeta form a 1-dimensional quasicrystal, a discrete structure Fourier spectrum by definition is also discrete (this of course holds for ordinary crystals as a special case). Also the logarithms of powers of primes would form a quasicrystal, which is very interesting from the point of view of p-adic length scale hypothesis. Primes label the "energies" of elementary fermions and bosons in arithmetic number theory, whose repeated second quantization gives rise to the hierarchy of infinite primes. The energies for general states are logarithms of integers.
  2. Powers pn label the points of quasicrystal defined by points log(pn) and Riemann zeta has interpretation as partition function for boson case with this spectrum. Could pn label also the points of the dual lattice defined by iy?
  3. The existence of Fourier transform for points log(pin) for any vector ya requires piiya to be a root of unity. This could define the sense in which zeros of zeta are universal. This condition also guarantees that the factor n-1/2-iy appearing in zeta at critical line are number theoretically universal (p1/2 is problematic for Qp: the problem might be solved by eliminating from p-adic analog of zeta the factor 1-p-s.

    1. One obtains for the pair (pi,sa) the condition log(pi)ya= qia2π, where qia is a rational number. Dividing the conditions for (i,a) and (j,a) gives

      pi= pjqia/qja

      for every zero sa so that the ratios qia/qja do not depend on sa. Since the exponent is rational number one obtains piM= pjN for some integers, which cannot be true.

    2. Dividing the conditions for (i,a) and (i,b) one obtains

      ya/yb= qia/qib

      so that the ratios qia/qib do not depend on pi. The ratios of the imaginary parts of zeta would be therefore rational number which is very strong prediction and zeros could be mapped by scaling ya/y1 where y1 is the zero which smallest imaginary part to rationals.

    3. The impossible consistency conditions for (i,a) and (j,a) can be avoided if each prime and its powers correspond to its own subset of zeros and these subsets of zeros are disjoint: one would have infinite union of sub-quasicrystals labelled by primes and each p-adic number field would correspond to its own subset of zeros: this might be seen as an abstract analog for the decomposition of rational to powers of primes. This decomposition would be natural if for ordinary complex numbers the contibution in the complement of this set to the Fourier trasform vanishes. The conditions (i,a) and (i,b) require now that the ratios of zeros are rationals only in the subset associated with pi.
For the general option the Fourier transform can be delta function for x=log(pk) and the set {ya(p)} contains Np zeros. The following argument inspires the conjecture that for each p there is an infinite number Np of zeros ya(p) satisfying

piya(p)=u(p)=e(r(p)/m(p))i2π ,

where u(p) is a root of unity that is ya(p)=2π (m(a)+r(p))/log(p) and forming a subset of a lattice with a lattice constant y0=2π/log(p), which itself need not be a zero.

In terms of stationary phase approximation the zeros ya(p) associated with p would have constant stationary phase whereas for ya(pi≠ p)) the phase piya(pi) would fail to be stationary. The phase eixy would be non-stationary also for x≠ log(pk) as function of y.

  1. Assume that for x =qlog(p), q not a rational, the phases eixy fail to be roots of unity and are random implying the vanishing/smallness of F(x) .
  2. Assume that for a given p all powers piy for y not in {ya(p)} fail to be roots of unity and are also random so that the contribution of the set y not in {ya(p)} to F(p) vanishes/is small.
  3. For x= log(pk/m) the Fourier transform should vanish or be small for m different from 1 (rational roots of primes) and give a non-vanishing contribution for m=1. One has

    F(x= log(pk/m ) =∑1≤ n≤ N(p) e[kM(n,p)/mN(n,p)]i2π .

    Obviously one can always choose N(n,p)=N(p).

  4. For the simplest option N(p)=1 one would obtain delta function distribution for x=log(pk). The sum of the phases associated with ya(p) and -ya(p) from the half axes of the critical line would give

    F(x= log(pn)) ∝ X(pn)==2cos(n× (r(p)/m(p))× 2π) .

    The sign of F would vary.

  5. The rational r(p)/m(p) would characterize given prime (one can require that r(p) and m(p) have no common divisors). F(x) is non-vanishing for all powers x=log(pn) for m(p) odd. For p=2, also m(2)=2 allows to have |X(2n)|=2. An interesting ad hoc ansatz is m(p)=p or ps(p). One has periodicity in n with period m(p) that is logarithmic wave. This periodicity serves as a test and in principle allows to deduce the value of r(p)/m(p) from the Fourier transform.
What could one conclude from the data (see this)?
  1. The first graph gives |F(x=log(pk| and second graph displays a zoomed up part of |F(x| for small powers of primes in the range [2,19]. For the first graph the eighth peak (p=11) is the largest one but in the zoomed graphs this is not the case. Hence something is wrong or the graphs correspond to different approximations suggesting that one should not take them too seriously.

    In any case, the modulus is not constant as function of pk. For small values of pk the envelope of the curve decreases and seems to approach constant for large values of pk (one has x< 15 (e15≈ 3.3× 106).

  2. According to the first graph | F(x)| decreases for x=klog(p)<8, is largest for small primes, and remains below a fixed maximum for 8<x<15. According to the second graph the amplitude decreases for powers of a given prime (say p=2). Clearly, the small primes and their powers have much larger | F(x)| than large primes.
There are many possible reasons for this behavior. Most plausible reason is that the sums involved converge slowly and the approximation used is not good. The inclusion of only 104 zeros would show the positions of peaks but would not allow reliable estimate for their intensities.
  1. The distribution of zeros could be such that for small primes and their powers the number of zeros is large in the set of 104 zeros considered. This would be the case if the distribution of zeros ya(p) is fractal and gets "thinner" with p so that the number of contributing zeros scales down with p as a power of p, say 1/p, as suggested by the envelope in the first figure.
  2. The infinite sum, which should vanish, converges only very slowly to zero. Consider the contribution Δ F(pk,p1) of zeros not belonging to the class p1≠ p to F(x=log(pk)) =∑pi Δ F(pk,pi), which includes also pi=p. Δ F(pk,pi), p≠ p1 should vanish in exact calculation.
    1. By the proposed hypothesis this contribution reads as

      l Δ F(p,p1)= ∑a cos[X(pk,p1)(M(a,p1)+ r(p1)/m(p1))2π)t] .

      X(pk,p1)=log(pk)/log(p1).

      Here a labels the zeros associated with p1. If pk is "approximately divisible" by p1 in other words, pk≈ np1, the sum over finite number of terms gives a large contribution since interference effects are small, and a large number of terms are needed to give a nearly vanishing contribution suggested by the non-stationarity of the phase. This happens in several situations.

    2. The number π(x) of primes smaller than x goes asymptotically like π(x) ≈ x/log(x) and prime density approximately like 1/log(x)-1/log(x)2 so that the problem is worst for the small primes. The problematic situation is encountered most often for powers pk of small primes p near larger prime and primes p (also large) near a power of small prime (the envelope of | F(x)| seems to become constant above x∼ 103).
    3. The worst situation is encountered for p=2 and p1=2k-1 - a Mersenne prime and p1= 22k+1, k≤ 4 - Fermat prime. For (p,p1)=(2k,Mk) one encounters X(2k,Mk)= (log(2k)/log(2k-1) factor very near to unity for large Mersennes primes. For (p,p1)=(Mk,2) one encounters X(Mk,2)= (log(2k-1)/log(2) ≈ k. Examples of Mersennes and Fermats are (3,2),(5,2),(7,2),(17,2),(31,2), (127,2),(257,2),... Powers 2k, k=2,3,4,5,7,8,.. are also problematic.
    4. Also twin primes are problematic since in this case one has factor X(p=p1+2,p1)=log(p1+2)/log(p1). The region of small primes contains many twin prime pairs: (3,5), (5,7), (11,13), (17,19), (29,31),....
    These observations suggest that the problems might be understood as resulting from including too small number of zeros.
  3. The predicted periodicity of the distribution with respect to the exponent k of pk is not consistent with the graph for small values of prime unless the periodic m(p) for small primes is large enough. The above mentioned effects can quite well mask the periodicity. If the first graph is taken at face value for small primes, r(p)/m(p) is near zero, and m(p) is so large that the periodicity does not become manifest for small primes. For p=2 this would require m(2)>21 since the largest power 2n≈ e15 corresponds to n∼ 21.
To summarize, the prediction is that for zeros of zeta should divide into disjoint classes {ya(p)\ labelled by primes such that within the class labelled by p one has piya(p)=e(r(p)/m(p))i2π so that has ya(p) = [M(a,p) +r(p)/m(p))] 2π/log(p).

What this speculative picture from the point of view of TGD?

  1. A possible formulation for number theoretic universality for the poles of fermionic Riemann zeta ζF(s)= ζ(s)/ζ(2s) could be as a condition that is that the exponents pksn(p)/2= pk/4pikyn(p)/2 exist in a number theoretically universal manner for the zeros sn(p) for given p-adic prime p and for some subset of integers k. If the proposed conditions hold true, exponent reduces pk/4 requiring that k is a multiple of 4. The number of the non-trivial generating elements of super-symplectic algebra in the monomial creating physical state would be a multiple of 4. These monomials would have real part of conformal weight -1. Conformal confinement suggests that these monomials are products of pairs of generators for which imaginary parts cancel. The conformal weights are however effectively real for the exponents automatically. Could the exponential formulation of the number theoretic universality effectively reduce the generating elements to those with conformal weight -1/4 and make the operators in question hermitian?
  2. Quasi-crystal property might have an application to TGD. The functions of light-like radial coordinate appearing in the generators of supersymplectic algebra could be of form rs, s zero of zeta or rather, its imaginary part. The eigenstate property with respect to the radial scaling rd/dr is natural by radial conformal invariance.

    The idea that arithmetic QFT assignable to infinite primes is behind the scenes in turn suggests light-like momenta assignable to the radial coordinate have energies with the dual spectrum log(pn). This is also suggested by the interpretation of ζ as square root of thermodynamical partition function for boson gas with momentum log(p) and analogous interpretation of ζF.

    The two spectra would be associated with radial scalings and with light-like translations of light-cone boundary respecting the direction and light-likeness of the light-like radial vector. log(pn) spectrum would be associated with light-like momenta whereas p-adic mass scales would characterize states with thermal mass. Note that generalization of p-adic length scale hypothesis raises the scales defined by pn to a special physical position: this might relate to ideal structure of adeles.

  3. Finite measurement resolution suggests that the approximations of Fourier transforms over the distribution of zeros taking into account only a finite number of zeros might have a physical meaning. This might provide additional understand about the origins of generalized p-adic length scale hypothesis stating that primes p≈ p1k, p1 small prime - say Mersenne primes - have a special physical role.
See the chapter Unified Number Theoretic Vision or the article Could one realize number theoretical universality for functional integral?.



More about physical interpretation of algebraic extensions of rationals

The number theoretic vision has begun to show its power. The basic hierarchies of quantum TGD would reduce to a hierarchy of algebraic extensions of rationals and the parameters - such as the degrees of the irreducible polynomials characterizing the extension and the set of ramified primes - would characterize quantum criticality and the physics of dark matter as large heff phases. The identification of preferred p-adic primes as remified primes of the extension and generalization of p-adic length scale hypothesis as prediction of NMP are basic victories of this vision (see this and this).

By strong form of holography the parameters characterizing string world sheets and partonic 2-surfaces serve as WCW coordinates. By various conformal invariances, one expects that the parameters correspond to conformal moduli, which means a huge simplification of quantum TGD since the mathematical apparatus of superstring theories becomes available and number theoretical vision can be realized. Scattering amplitudes can be constructed for a given algebraic extension and continued to various number fields by continuing the parameters which are conformal moduli and group invariants characterizing incoming particles.

There are many un-answered and even un-asked questions.

  1. How the new degrees of freedom assigned to the n-fold covering defined by the space-time surface pop up in the number theoretic picture? How the connection with preferred primes emerges?
  2. What are the precise physical correlates of the parameters characterizing the algebraic extension of rationals? Note that the most important extension parameters are the degree of the defining polynomial and ramified primes.

1. Some basic notions

Some basic facts about extensions are in order. I emphasize that I am not a specialist.

1.1. Basic facts

The algebraic extensions of rationals are determined by roots of polynomials. Polynomials be decomposed to products of irreducible polynomials, which by definition do not contain factors which are polynomials with rational coefficients. These polynomials are characterized by their degree n, which is the most important parameter characterizing the algebraic extension.

One can assign to the extension primes and integers - or more precisely, prime and integer ideals. Integer ideals correspond to roots of monic polynomials Pn(x)=xn+..a0 in the extension with integer coefficients. Clearly, for n=0 (trivial extension) one obtains ordinary integers. Primes as such are not a useful concept since roots of unity are possible and primes which differ by a multiplication by a root of unity are equivalent. It is better to speak about prime ideals rather than primes.

Rational prime p can be decomposed to product of powers of primes of extension and if some power is higher than one, the prime is said to be ramified and the exponent is called ramification index. Eisenstein's criterion states that any polynomial Pn(x)= anxn+an-1xn-1+...a1x+ a0 for which the coefficients ai, i<n are divisible by p and a0 is not divisible by p2 allows p as a maximally ramified prime. mThe corresponding prime ideal is n:th power of the prime ideal of the extensions (roughly n:th root of p). This allows to construct endless variety of algebraic extensions having given primes as ramified primes.

Ramification is analogous to criticality. When the gradient potential function V(x) depending on parameters has multiple roots, the potential function becomes proportional a higher power of x-x0. The appearance of power is analogous to appearance of higher power of prime of extension in ramification. This gives rise to cusp catastrophe. In fact, ramification is expected to be number theoretical correlate for the quantum criticality in TGD framework. What this precisely means at the level of space-time surfaces, is the question.

1.2 Galois group as symmetry group of algebraic physics

I have proposed long time ago that Galois group acts as fundamental symmetry group of quantum TGD and even made clumsy attempt to make this idea more precise in terms of the notion of number theoretic braid. It seems that this notion is too primitive: the action of Galois group must be realized at more abstract level and WCW provides this level.

First some facts (I am not a number theory professional, as the professional reader might have already noticed!).

  1. Galois group acting as automorphisms of the field extension (mapping products to products and sums to sums and preserves norm) characterizes the extension and its elements have maximal order equal to n by algebraic n-dimensionality. For instance, for complex numbers Galois group acs as complex conjugation. Galois group has natural action on prime ideals of extension mapping them to each other and preserving the norm determined by the determinant of the linear map defined by the multiplication with the prime of extension. For instance, for the quadratic extension Q(51/2) the norm is N(x+51/2y)=x2-5y2: not that number theory leads to Minkowkian metric signatures naturally. Prime ideals combine to form orbits of Galois group.
  2. Since Galois group leaves the rational prime p invariant, the action must permute the primes of extension in the product representation of p. For ramified primes the points of the orbit of ideal degenerate to single ideal. This means that primes and quite generally, the numbers of extension, define orbits of the Galois group.

Galois group acts in the space of integers or prime ideals of the algebraic extension of rationals and it is also physically attractive to consider the orbits defined by ideals as preferred geometric structures. If the numbers of the extension serve as parameters characterizing string world sheets and partonic 2-surfaces, then the ideals would naturally define subsets of the parameter space in which Galois group would act.

The action of Galois group would leave the space-time surface invariant if the sheets co-incide at ends but permute the sheets. Of course, the space-time sheets permuted by Galois group need not co-incide at ends. In this case the action need not be gauge action and one could have non-trivial representations of the Galois group. In Langlands correspondence these representation relate to the representations of Lie group and something similar might take place in TGD as I have indeed proposed.

Remark: Strong form of holography supports also the vision about quaternionic generalization of conformal invariance implying that the adelic space-time surface can be constructed from the data associated with functions of two complex variables, which in turn reduce to functions of single variable.

If this picture is correct, it is possible to talk about quantum amplitudes in the space defined by the numbers of extension and restrict the consideration to prime ideals or more general integer ideals.

  1. These number theoretical wave functions are physical if the parameters characterizing the 2-surface belong to this space. One could have purely number theoretical quantal degrees of freedom assignable to the hierarchy of algebraic extensions and these discrete degrees of freedom could be fundamental for living matter and understanding of consciousness.
  2. The simplest assumption that Galois group acts as a gauge group when the ends of sheets co-incide at boundaries of CD seems however to destroy hopes about non-trivial number theoretical physics but this need not be the case. Physical intuition suggests that ramification somehow saves the situation and that the non-trivial number theoretic physics could be associated with ramified primes assumed to define preferred p-adic primes.

2. How new degrees of freedom emerge for ramified primes?

How the new discrete degrees of freedom appear for ramified primes?

  1. The space-time surfaces defining singular coverings are n-sheeted in the interior. At the ends of the space-time surface at boundaries of CD however the ends co-incide. This looks very much like a critical phenomenon.

    Hence the idea would be that the end collapse can occur only for the ramified prime ideals of the parameter space - ramification is also a critical phenomenon - and means that some of the sheets or all of them co-incide. Thus the sheets would co-incide at ends only for the preferred p-adic primes and give rise to the singular covering and large heff. End-collapse would be the essence of criticality! This would occur, when the parameters defining the 2-surfaces are in a ramified prime ideal.

  2. Even for the ramified primes there would be n distinct space-time sheets, which are regarded as physically distinct. This would support the view that besides the space-like 3-surfaces at the ends the full 3-surface must include also the light-like portions connecting them so that one obtains a closed 3-surface. The conformal gauge equivalence classes of the light-like portions would give rise to additional degrees of freedom. In space-time interior and for string world sheets they would become visible.

    For ramified primes n distint 3-surfaces would collapse to single one but the n discrete degrees of freedom would be present and particle would obtain them. I have indeed proposed number theoretical second quantization assigning fermionic Clifford algebra to the sheets with n oscillator operators. Note that this option does not require Galois group to act as gauge group in the general case. This number theoretical second quantization might relate to the realization of Boolean algebra suggested by weak form of NMP (see this).

3. About the physical interpretation of the parameters characterizing algebraic extension of rationals in TGD framework

It seems that Galois group is naturally associated with the hierarchy heff/h=n of effective Planck constants defined by the hierarchy of quantum criticalities. n would naturally define the maximal order for the element of Galois group. The analog of singular covering with that of z1/n would suggest that Galois group is very closely related to the conformal symmetries and its action induces permutations of the sheets of the covering of space-time surface.

Without any additional assumptions the values of n and ramified primes are completely independent so that the conjecture that the magnetic flux tube connecting the wormhole contacts associated with elementary particles would not correspond to very large n having the p-adic prime p characterizing particle as factor (p=M127=2127-1 for electron). This would not induce any catastrophic changes.

TGD based physics could however change the situation and reduce number theoretical degrees of freedom: the intuitive hypothesis that p divides n might hold true after all.

  1. The strong form of GCI implies strong form of holography. One implication is that the WCW Kähler metric can be expressed either in terms of Kähler function or as anti-commutators of super-symplectic Noether super-charges defining WCW gamma matrices. This realizes what can be seen as an analog of Ads/CFT correspondence. This duality is much more general. The following argument supports this view.
    1. Since fermions are localized at string world sheets having ends at partonic 2-surfaces, one expects that also Kähler action can be expressed as an effective stringy action. It is natural to assume that string area action is replaced with the area defined by the effective metric of string world sheet expressible as anti-commutators of Kähler-Dirac gamma matrices defined by contractions of canonical momentum currents with imbedding space gamma matrices. It string tension is proportional to heff2, string length scales as heff.
    2. AdS/CFT analogy inspires the view that strings connecting partonic 2-surfaces serve as correlates for the formation of - at least gravitational - bound states. The distances between string ends would be of the order of Planck length in string models and one can argue that gravitational bound states are not possible in string models and this is the basic reason why one has ended to landscape and multiverse non-sense.
  2. In order to obtain reasonable sizes for astrophysical objects (that is sizes larger than Schwartschild radius rs=2GM) For heff=hgr=GMm/v0 one obtains reasonable sizes for astrophysical objects. Gravitation would mean quantum coherence in astrophysical length scales.
  3. In elementary particle length scales the value of heff must be such that the geometric size of elementary particle identified as the Minkowski distance between the wormhole contacts defining the length of the magnetic flux tube is of order Compton length - that is p-adic length scale proportional to p1/2. Note that dark physics would be an essential element already at elementary particle level if one accepts this picture also in elementary particle mass scales. This requires more precise specification of what darkness in TGD sense really means.

    One must however distinguish between two options.

    1. If one assumes n≈ p1/2, one obtains a large contribution to classical string energy as Δ ∼ mCP22Lp/hbar2eff ∼ mCP2/p1/2, which is of order particle mass. Dark mass of this size looks un-feasible since p-adic mass calculations assign the mass with the ends wormhole contacts. One must be however very cautious since the interpretations can change.
    2. Second option allows to understand why the minimal size scale associated with CD characterizing particle correspond to secondary p-adic length scale. The idea is that the string can be thought of as being obtained by a random walk so that the distance between its ends is proportional to the square root of the actual length of the string in the induced metric. This would give that the actual length of string is proportional to p and n is also proportional to p and defines minimal size scale of the CD associated with the particle. The dark contribution to the particle mass would be Δ m ∼ mCP22Lp/hbar2eff∼ mCP2/p, and completely negligible suggesting that it is not easy to make the dark side of elementary visible.
  4. If the latter interpretation is correct, elementary particles would have huge number of hidden degrees of freedom assignable to their CDs. For instance, electron would have p=n=2127-1 ≈ 1038 hidden discrete degrees of freedom and would be rather intelligent system - 127 bits is the estimate- and thus far from a point-like idiot of standard physics. Is it a mere accident that the secondary p-adic time scale of electron is .1 seconds - the fundamental biorhythm - and the size scale of the minimal CD is slightly large than the circumference of Earth?

    Note however, that the conservation option assuming that the magnetic flux tubes connecting the wormhole contacts representing elementary particle are in heff/h=1 phase can be considered as conservative option.

See the chapter Unified Number Theoretic Vision or the article More about physical interpretation of algebraic extensions of rationals.



What could be the origin of p-adic length scale hypothesis?

The argument would explain the existence of preferred p-adic primes. It does not yet explain p-adic length scale hypothesis stating that p-adic primes near powers of 2 are favored. A possible generalization of this hypothesis is that primes near powers of prime are favored. There indeed exists evidence for the realization of 3-adic time scale hierarchies in living matter (see this) and in music both 2-adicity and 3-adicity could be present, this is discussed in TGD inspired theory of music harmony and genetic code (see this).

The weak form of NMP might come in rescue here.

  1. Entanglement negentropy for a negentropic entanglement characterized by n-dimensional projection operator is the log(Np(n) for some p whose power divides n. The maximum negentropy is obtained if the power of p is the largest power of prime divisor of p, and this can be taken as definition of number theoretic entanglement negentropy. If the largest divisor is pk, one has N= k× log(p). The entanglement negentropy per entangled state is N/n=klog(p)/n and is maximal for n=pk. Hence powers of prime are favoured which means that p-adic length scale hierarchies with scales coming as powers of p are negentropically favored and should be generated by NMP. Note that n=pk would define a hierarchy of heff/h=pk. During the first years of heff hypothesis I believe that the preferred values obey heff=rk, r integer not far from r= 211. It seems that this belief was not totally wrong.
  2. If one accepts this argument, the remaining challenge is to explain why primes near powers of two (or more generally p) are favoured. n=2k gives large entanglement negentropy for the final state. Why primes p=n2= 2k-r would be favored? The reason could be following. n=2k corresponds to p=2, which corresponds to the lowest level in p-adic evolution since it is the simplest p-adic topology and farthest from the real topology and therefore gives the poorest cognitive representation of real preferred extremal as p-adic preferred extermal (Note that p=1 makes formally sense but for it the topology is discrete).
  3. Weak form of NMP suggests a more convincing explanation. The density matrix of the state to be reduced is a direct sum over contributions proportional to projection operators. Suppose that the projection operator with largest dimension has dimension n. Strong form of NMP would say that final state is characterized by n-dimensional projection operator. Weak form of NMP allows free will so that all dimensions n-k, k=0,1,...n-1 for final state projection operator are possible. 1-dimensional case corresponds to vanishing entanglement negentropy and ordinary state function reduction isolating the measured system from external world.
  4. The negentropy of the final state per state depends on the value of k. It is maximal if n-k is power of prime. For n=2k=Mk+1, where Mk is Mersenne prime n-1 gives the maximum negentropy and also maximal p-adic prime available so that this reduction is favoured by NMP. Mersenne primes would be indeed special. Also the primes n=2k-r near 2k produce large entanglement negentropy and would be favored by NMP.
  5. This argument suggests a generalization of p-adic length scale hypothesis so that p=2 can be replaced by any prime.
This argument together with the hypothesis that preferred prime is ramified would correlate the character of the irreducible extension and character of super-conformal symmetry breaking. The integer n characterizing super-symplectic conformal sub-algebra acting as gauge algebra would depends on the irreducible algebraic extension of rational involved so that the hierarchy of quantum criticalities would have number theoretical characterization. Ramified primes could appear as divisors of n and n would be essentially a characteristic of ramification known as discriminant. An interesting question is whether only the ramified primes allow the continuation of string world sheet and partonic 2-surface to a 4-D space-time surface. If this is the case, the assumptions behind p-adic mass calculations would have full first principle justification.

See the chapter Unified Number Theoretic Vision or the article The Origin of Preferred p-Adic Primes?.



What is the origin of the preferred p-adic primes?

A long-standing question has been the origin of preferred p-adic primes characterizing elementary particles. I have proposed several explanations and the most convincing hitherto is related to the algebraic extensions of rationals and p-adic numbers selecting naturally preferred primes as those which are ramified for the extension in question.

See the chapter Unified Number Theoretic Vision .



About the twistorial description of light-likeness in 8-D sense using octonionic spinors

The twistor approach to TGD require that the expression of light-likeness of M4 momenta in terms of twistors generalizes to 8-D case. The light-likeness condition for twistors states that the 2× 2 matrix representing M4 momentum annihilates a 2-spinor defining the second half of the twistor. The determinant of the matrix reduces to momentum squared and its vanishing implies the light-likeness. This should be generalized to a situation in one has M4 and CP2 twistor, which are not light-like separately but light-likeness in 8-D sense holds true (allowing massive particles in M4 sense and thus generalization of twistor approach for massive particles).

The case of M8=M4× E4

M8-H duality suggests that it might be useful to consider first the twistorialiation of 8-D light-likeness first the simpler case of M8 for which CP2 corresponds to E4. It turns out that octonionic representation of gamma matrices provide the most promising formulation.

In order to obtain quadratic dispersion relation, one must have 2× 2 matrix unless the determinant for the 4× 4 matrix reduces to the square of the generalized light-likeness condition.

  1. The first approach relies on the observation that the 2× 2 matrices characterizing four-momenta can be regarded as hyper-quaternions with imaginary units multiplied by a commuting imaginary unit. Why not identify space-like sigma matrices with hyper-octonion units?
  2. The square of hyper-octonionic norm would be defined as the determinant of 4× 4 matrix and reduce to the square of hyper-octonionic momentum. The light-likeness for pairs formed by M4 and E4 momenta would make sense.
  3. One can generalize the sigma matrices representing hyper-quaternion units so that they become the 8 hyper-octonion units. Hyper-octonionic representation of gamma matrices exists (γ0z× 1, γk= σy× Ik) but the octonionic sigma matrices represented by octonions span the Lie algebra of G2 rather than that of SO(1, 7). This dramatically modifies the physical picture and brings in also an additional source of non-associativity. Fortunately, the flatness of M8 saves the situation.
  4. One obtains the square of p2=0 condition from the massless octonionic Dirac equation as vanishing of the determinant much like in the 4-D case. Since the spinor connection is flat for M8 the hyper-octonionic generalization indeed works.
This is not the only possibility that I have considered.
  1. Is it enough to allow the four-momentum to be complex? One would still have 2× 2 matrix and vanishing of complex momentum squared meaning that the squares of real and imaginary parts are same (light-likeness in 8-D sense) and that real and imaginary parts are orthogonal to each other. Could E4 momentum correspond to the imaginary part of four-momentum?
  2. The signature causes the first problem: M8 must be replaced with complexified Minkowski space Mc4 for to make sense but this is not an attractive idea although Mc4 appears as sub-space of complexified octonions.
  3. For the extremals of Kähler action Euclidian regions (wormhole contacts identifiable as deformations of CP2 type vacuum extremals) give imaginary contribution to the four-momentum. Massless complex momenta and also color quantum numbers appear also in the standard twistor approach. Also this suggest that complexification occurs also in 8-D situation and is not the solution of the problem.

The case of M8=M4× CP2

What about twistorialization in the case of M4× CP2? The introduction of wave functions in the twistor space of CP2 seems to be enough to generalize Witten's construction to TGD framework and that algebraic variant of twistors might be needed only to realize quantum classical correspondence. It should correspond to tangent space counterpart of the induced twistor structure of space-time surface, which should reduce effectively to 4-D one by quaternionicity of the space-time surface.

  1. For H=M4× CP2 the spinor connection of CP2 is not trivial and the G2 sigma matrices are proportional to M4 sigma matrices and act in the normal space of CP2 and to M4 parts of octonionic imbedding space spinors, which brings in mind co-associativity. The octonionic charge matrices are also an additional potential source of non-associativity even when one has associativity for gamma matrices.

    Therefore the octonionic representation of gamma matrices in entire H cannot be physical. It is however equivalent with ordinary one at the boundaries of string world sheets, where induced gauge fields vanish. Gauge potentials are in general non-vanishing but can be gauge transformed away. Here one must be of course cautious since it can happen that gauge fields vanish but gauge potentials cannot be gauge transformed to zero globally: topological quantum field theories represent basic example of this.

  2. Clearly, the vanishing of the induced gauge fields is needed to obtain equivalence with ordinary induced Dirac equation. Therefore also string world sheets in Minkowskian regions should have 1-D CP2 projection rather than only having vanishing W fields if one requires that octonionic representation is equivalent with the ordinary one. For CP2 type vacuum extremals electroweak charge matrices correspond to quaternions, and one might hope that one can avoid problems due to non-associativity in the octonionic Dirac equation. Unless this is the case, one must assume that string world sheets are restricted to Minkowskian regions. Also imbedding space spinors can be regarded as octonionic (possibly quaternionic or co-quaternionic at space-time surfaces): this might force vanishing 1-D CP2 projection.
    1. Induced spinor fields would be localized at 2-surfaces at which they have no interaction with weak gauge fields: of course, also this is an interaction albeit very implicit one! This would not prevent the construction of non-trivial electroweak scattering amplitudes since boson emission vertices are essentially due to re-groupings of fermions and based on topology change.
    2. One could even consider the possibility that the projection of string world sheet to CP2 corresponds to CP2 geodesic circle at which also the induced gauge potentials vanish so that one could assign light-like 8-momentum to entire string world sheet, which would be minimal surface in M4× S1. This would mean enormous technical simplification in the structure of the theory. Whether the spinor harmonics of imbedding space with well-defined M4 and color quantum numbers can co-incide with the solutions of the induced Dirac operator at string world sheets defined by minimal surfaces remains an open problem.
    3. String world sheets cannot be present inside wormhole contacts, which have 4-D CP2 projection so that string world sheets cannot carry vanishing induced gauge fields. Therefore the strings in TGD are open.
Summarizing

To sum up, the generalization of the notion of twistor to 8-D context allows description of massive particles using twistors but requires that octonionic Dirac equation is introduced. If one requires that octonionic and ordinary description of Dirac equation are equivalent, the description is possible only at surfaces having at most 1-D CP2 projection - geodesic circle for the most stringent option. The boundaries of string world sheets are such surfaces and also string world sheets themselves if they have 1-D CP2 projection, which must be geodesic circle if also induce gauge potentials are required to vanish. In spirit with M8-H duality, string boundaries give rise to classical M8 twistorizalization analogous to the standard M4 twistorialization and generalize 4-momentum to massless 8-momentum whereas imbedding space spinor harmonics give description in terms of four-momentum and color quantum numbers. One has SO(4)-SU(3) duality: a wave function in the space of 8-momenta corresponds to SO(4) description of hadrons at low energies as opposed to that for quarks at high energies in terms of color. The M4 projection of the 8-D M8 momentum must by quantum classical correspondence be equal to the four-momentum assignable to imbedding space-spinor harmonics serving as building bricks for various super-conformal representations. This is nothing but Equivalence Principle (EP) in the most concrete form: gravitational four-momentum equals to inertial four-momentum. EP for internal quantum numbers is clearly more delicate. In twistorialization also helicity is brought and for CP2 degrees of freedom M8 helicity means that electroweak spin is described in terms of helicity.

Biologists have a principle known as "ontogeny recapitulates phylogeny" (ORP) stating that the morphogenesis of the individual reflects evolution of the species. The principle seems to be realized also in theoretical physics - at least in TGD Universe. ORP would now say that the evolution of theoretical physics via the emergence of increasingly complex notion of particle reflects the structure physics itself. Point like particles are really there as points at partonic 2-surfaces carrying fermion number: their 1-D orbits correspond to the boundaries of string world sheets; 2-D hyper-complex string world sheets in flat space (M4× S1) are there and carry induced spinors; also complex (or co-complex) partonic 2-surfaces (Euclidian string world sheets) and carry particle numbers; 3-D space-like surfaces at the ends of causal diamonds (CDs) and the 3-D light-like orbits of partonic 2-surfaces are there; 4-D space-time surfaces are there as quaternionic or co-quaternionic sub-manifolds of 8-D octonionic imbedding space: there the hierarchy ends since there are no higher-dimensional classical number fields. ORP would thus also realize evolution of mathematics at the level of physics.

The M4 projection of the 8-D M8 momentum must by quantum classical correspondence be equal to the four-momentum assignable to imbedding space-spinor harmonics serving as building bricks for various super-conformal representations. This is nothing but Equivalence Principle in the most concrete form: gravitational four-momentum equals to inertial four-momentum.

See the chapter TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts or the article Classical part of the twistor story.



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