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TGD: Physics as Infinite-Dimensional Geometry

Note: Newest contributions are at the top!

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?.

Field equations as conservation laws, Frobenius integrability conditions, and a connection with quaternion analyticity

The following represents qualitative picture of field equations of TGD trying to emphasize the physical aspects. What is new is the discussion of the possibility that Frobenius integrability conditions are satisfied and correspond to quaternion analyticity.

  1. Kähler action is Maxwell action for induced Kähler form and metric expressible in terms of imbedding space coordinates and their gradients. Field equations reduce to those for imbedding space coordinates defining the primary dynamical variables. By GCI only four of them are independent dynamical variables analogous to classical fields.
  2. The solution of field equations can be interpreted as a section in fiber bundle. In TGD the fiber bundle is just the Cartesian product X4× CD× CP2 of space-time surface X4 and causal diamond CD× CP2. CD is the intersection of future and past directed light-cones having two light-like boundaries, which are cone-like pieces of light-boundary δ M4+/-× CP2. Space-time surface serves as base space and CD× CP2 as fiber. Bundle projection Π is the projection to the factor X4. Section corresponds to the map x→ hk(x) giving imbedding space coordinates as functions of space-time coordinates. Bundle structure is now trivial and rather formal.

    By GCI one could also take suitably chosen 4 coordinates of CD× CP2 as space-time coordinates, and identify CD× CP2 as the fiber bundle. The choice of the base space depends on the character of space-time surface. For instance CD, CP2 or M2× S2 (S2 a geodesic sphere of CP2), could define the base space. The bundle projection would be projection from CD× CP2 to the base space. Now the fiber bundle structure can be non-trivial and make sense only in some space-time region with same base space.

  3. The field equations derived from Kähler action must be satisfied. Even more: one must have a preferred extremal of Kähler action. One poses boundary conditions at the 3-D ends of space-time surfaces and at the light-like boundaries of CD× CP2.

    One can fix the values of conserved Noether charges at the ends of CD (total charges are same) and require that the Noether charges associated with a sub-algebra of super-symplectic algebra isomorphic to it and having conformal weights coming as n-ples of those for the entire algebra, vanish. This would realize the effective 2-dimensionality required by SH. One must pose boundary conditions also at the light-like partonic orbits. So called weak form of electric-magnetic duality is at least part of these boundary conditions.

    It seems that one must restrict the conformal weights of the entire algebra to be non-negative r≥ 0 and those of subalgebra to be positive: mn>0. The condition that also the commutators of sub-algebra generators with those of the entire algebra give rise to vanishing Noether charges implies that all algebra generators with conformal weight m≥ n vanish so the dynamical algebra becomes effectively finite-dimensional. This condition generalizes to the action of super-symplectic algebra generators to physical states.

    M4 time coordinate cannot have vanishing time derivative dm0/dt so that four-momentum is non-vanishing for non-vacuum extremals. For CP2 coordinates time derivatives dsk/dt can vanish and for space-like Minkowski coordinates dmi/dt can be assumed to be non-vanishing if M4 projection is 4-dimensional. For CP2 coordinates dsk/dt=0 implies the vanishing of electric parts of induced gauge fields. The non-vacuum extremals with the largest conformal gauge symmetry (very small n) would correspond to cosmic string solutions for which induced gauge fields have only magnetic parts. As n increases, also electric parts are generated. Situation becomes increasingly dynamical as conformal gauge symmetry is reduced and dynamical conformal symmetry increases.

  4. The field equations involve besides imbedding space coordinates hk also their partial derivatives up to second order. Induced Kähler form and metric involve first partial derivatives ∂αhk and second fundamental form appearing in field equations involves second order partial derivatives ∂αβhk.

    Field equations are hydrodynamical, in other worlds represent conservation laws for the Noether currents associated with the isometries of M4× CP2. By GCI there are only 4 independent dynamical variables so that the conservation of m≤ 4 isometry currents is enough if chosen to be independent. The dimension m of the tangent space spanned by the conserved currents can be smaller than 4. For vacuum extremals one has m= 0 and for massless extremals (MEs) m= 1! The conservation of these currents can be also interpreted as an existence of m≤ 4 closed 3-forms defined by the duals of these currents.

  5. The hydrodynamical picture suggests that in some situations it might be possible to assign to the conserved currents flow lines of currents even globally. They would define m≤ 4 global coordinates for some subset of conserved currents (4+8 for four-momentum and color quantum numbers). Without additional conditions the individual flow lines are well-defined but do not organize to a coherent hydrodynamic flow but are more like orbits of randomly moving gas particles. To achieve global flow the flow lines must satisfy the condition dφA/dxμ= kABJBμ or dφA= kABJB so that one can special of 3-D family of flow lines parallel to kABJB at each point - I have considered this kind of possibility in detail earlier but the treatment is not so general as in the recent case.

    Frobenius integrability conditions follow from the condition d2φA=0= dkAB∧ JB+ kABdJB=0 and implies that dJB is in the ideal of exterior algebra generated by the JA appearing in kABJB. If Frobenius conditions are satisfied, the field equations can define coordinates for which the coordinate lines are along the basis elements for a sub-space of at most 4-D space defined by conserved currents. Of course, the possibility that for preferred extremals there exists m≤ 4 conserved currents satisfying integrability conditions is only a conjecture.

    It is quite possible to have m<4. For instance for vacuum extremals the currents vanish identically For MEs various currents are parallel and light-like so that only single light-like coordinate can be defined globally as flow lines. For cosmic strings (cartesian products of minimal surfaces X2 in M4 and geodesic spheres S2 in CP2 4 independent currents exist). This is expected to be true also for the deformations of cosmic strings defining magnetic flux tubes.

  6. Cauchy-Riemann conditions in 2-D situation represent a special case of Frobenius conditions. Now the gradients of real and imaginary parts of complex function w=w(z)= u+iv define two conserved currents by Laplace equations. In TGD isometry currents would be gradients apart from scalar function multipliers and one would have generalization of C-R conditions. In citeallb/prefextremals,twistorstory I have considered the possibility that the generalization of Cauchy-Riemann-Fuerter conditions could define quaternion analyticity - having many non-equivalent variants - as a defining property of preferred extremals. The integrability conditions for the isometry currents would be the natural physical formulation of CRF conditions. Different variants of CRF conditions would correspond to varying number of independent conserved isometry currents.
  7. The problem caused by GCI is that there is infinite number of coordinate choices. How to pick a physically preferred coordinate system? One possible manner to do this is to use coordinates for the projection of space-time surface to some preferred sub-space of imbedding - geodesic manifold is an excellent choice. Only M1× X3 geodesic manifolds are not possible but these correspond to vacuum extremals.

    One could also consider a philosophical principle behind integrability. The variational principle itself could give rise to at least some preferred space-time coordinates in the same manner as TGD based quantum physics would realize finite measurement resolution in terms of inclusions of HFFs in terms of hierarchy of quantum criticalities and fermionic strings connecting partonic 2-surfaces. Frobenius integrability of the isometry currents would define some preferred coordinates. Their number need not be the maximal four however.

    For instance, for massless extremals only light-like coordinate corresponding to the light-like momentum is obtained. To this one can however assign another local light-like coordinate uniquely to obtain integrable distribution of planes M2. The solution ansatz however defines directly an integrable choice of two pairs of coordinates at imbedding space level usable also as space-time coordinates - light-like local direction defining local plane M2 and polarization direction defining a local plane E2. These choices define integrable distributions of orthogonal planes and local hypercomplex and complex coordinates. Pair of analogs of C-R equations is the outcome. I have called these coordinates Hamilton-Jacobi coordinates for M4.

  8. This picture allows to consider a generalization of the notion of solution of field equation to that of integral manifold. If the number of independent isometry currents is smaller than 4 (possibly locally) and the integrability conditions hold true, lower-dimensional sub-manifolds of space-time surface define integral manifolds as kind of lower-dimensional effective solutions. Genuinely lower-dimensional solutions would of course have vanishing (g41/2) and vanishing Kähler action.

    String world sheets can be regarded as 2-D integral surfaces. Charged (possibly all) weak boson gauge fields vanish at them since otherwise the electromagnetic charge for spinors would not be well-defined. These conditions force string world sheets to be 2-D in the generic case. In special case 4-D space-time region as a whole can satisfy these conditions. Well-definedness of Kähler-Dirac equation demands that the isometry currents of Kähler action flow along these string world sheets so that one has integral manifold. The integrability conditions would allow 2<m≤ n integrable flows outside the string world sheets, and at string world sheets one or two isometry currents would vanish so that the flows would give rise 2-D independent sub-flow.

  9. The method of characteristics is used to solve hyperbolic partial differential equations by reducing them to ordinary differential equations. The (say 4-D) surface representing the solution in the field space has a foliation using 1-D characteristics. The method is especially simple for linear equations but can work also in the non-linear case. For instance, the expansion of wave front can be described in terms of characteristics representing light rays. It can happen that two characteristics intersect and a singularity results. This gives rise to physical phenomena like caustics and shock waves.

    In TGD framework the flow lines for a given isometry current in the case of an integrable flow would be analogous to characteristics, and one could also have purely geometric counterparts of shockwaves and caustics. The light-like orbits of partonic 2-surface at which the signature of the induced metric changes from Minkowskian to Euclidian might be seen as an example about the analog of wave front in induced geometry. These surfaces serve as carriers of fermion lines in generalized Feynman diagrams. Could one see the particle vertices at which the 4-D space-time surfaces intersect along their ends as analogs of intersections of characteristics - kind of caustics? At these 3-surfaces the isometry currents should be continuous although the space-time surface has "edge".

For details see the chapter Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds" or the article Could One Define Dynamical Homotopy Groups in WCW?.

Could one define dynamical homotopy groups in WCW?

I learned that Agostino Prastaro has done highly interesting work with partial differential equations, also those assignable to geometric variational principles such as Kähler action in TGD. I do not understand the mathematical details but the key idea is a simple and elegant generalization of Thom's cobordism theory, and it is difficult to avoid the idea that the application of Prastaro's idea might provide insights about the preferred extremals, whose identification is now on rather firm basis.

One could also consider a definition of what one might call dynamical homotopy groups as a genuine characteristics of WCW topology. The first prediction is that the values of conserved classical Noether charges correspond to disjoint components of WCW. Could the natural topology in the parameter space of Noether charges zero modes of WCW metric) be p-adic and realize adelic physics at the level of WCW? An analogous conjecture was made on basis of spin glass analogy long time ago. Second surprise is that the only the 6 lowest dynamical homotopy/homology groups of WCW would be non-trivial. The Kähler structure of WCW suggets that only Π0, Π2, and Π4 are non-trivial.

The interpretation of the analog of Π1 as deformations of generalized Feynman diagrams with elementary cobordism snipping away a loop as a move leaving scattering amplitude invariant conforms with the number theoretic vision about scattering amplitude as a representation for a sequence of algebraic operation can be always reduced to a tree diagram. TGD would be indeed topological QFT: only the dynamical topology would matter.

See the chapter Recent View about Kähler Geometry and Spin Structure of WCW or the article Could one define dynamical homotopy groups in WCW?.

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 .

Is the formation of gravitational bound states impossible in superstring models?

In the previous posting I told about the possibility that string world sheets with area action could be present in TGD at fundamental level with the ratio of hbar G/R2 of string tension to the square of CP2 radius fixed by quantum criticality. I however found that the assumption that gravitational binding has as correlates strings connecting the bound partonic 2-surfaces leads to grave difficulties: the sizes of the gravitationally bound states cannot be much longer than Planck length. This binding mechanism is strongly suggested by AdS/CFT correspondence but perturbative string theory does not allow it.

I proposed that the replacement of h with heff = n× h= hgr= GMm/v0 could resolve the problem. It does not. I soo noticed that the typical size scale of string world sheet scales as hgr1/2, not as hgr= GMm/v0 as one might expect. The only reasonable option is that string tension behave as 1/hgr2. In the following I demonstrate that TGD in its basic form and defined by super-symmetrized Kähler action indeed predicts this behavior if string world sheets emerge. They indeed do so number theoretically from the condition of associativity and also from the condition that electromagnetic charge for the spinor modes is well-defined. By the analog of AdS/CFT correspondence the string tension could characterize the action density of magnetic flux tubes associated with the strings and varying string tension would correspond to the effective string tension of the magnetic flux tubes as carriers of magnetic energy (dark energy is identified as magnetic energy in TGD Universe).

Therefore the visit of string theory to TGD Universe remained rather short but it had a purpose: it made completely clear why superstring are not the theory of gravitation and why TGD can be this theory.

Do associativty and commutativity define the laws of physics?

The dimensions of classical number fields appear as dimensions of basic objects in quantum TGD. Imbedding space has dimension 8, space-time has dimension 4, light-like 3-surfaces are orbits of 2-D partonic surfaces. If conformal QFT applies to 2-surfaces (this is questionable), one-dimensional structures would be the basic objects. The lowest level would correspond to discrete sets of points identifiable as intersections of real and p-adic space-time sheets. This suggests that besides p-adic number fields also classical number fields (reals, complex numbers, quaternions, octonions are involved and the notion of geometry generalizes considerably. In the recent view about quantum TGD the dimensional hierarchy defined by classical number field indeed plays a key role. H=M4× CP2 has a number theoretic interpretation and standard model symmetries can be understood number theoretically as symmetries of hyper-quaternionic planes of hyper-octonionic space.

The associativity condition A(BC)= (AB)C suggests itself as a fundamental physical law of both classical and quantum physics. Commutativity can be considered as an additional condition. In conformal field theories associativity condition indeed fixes the n-point functions of the theory. At the level of classical TGD space-time surfaces could be identified as maximal associative (hyper-quaternionic) sub-manifolds of the imbedding space whose points contain a preferred hyper-complex plane M2 in their tangent space and the hierarchy finite fields-rationals-reals-complex numbers-quaternions-octonions could have direct quantum physical counterpart. This leads to the notion of number theoretic compactification analogous to the dualities of M-theory: one can interpret space-time surfaces either as hyper-quaternionic 4-surfaces of M8 or as 4-surfaces in M4× CP2. As a matter fact, commutativity in number theoretic sense is a further natural condition and leads to the notion of number theoretic braid naturally as also to direct connection with super string models.

At the level of modified Dirac action the identification of space-time surface as a hyper-quaternionic sub-manifold of H means that the modified gamma matrices of the space-time surface defined in terms of canonical momentum currents of Kähler action using octonionic representation for the gamma matrices of H span a hyper-quaternionic sub-space of hyper-octonions at each point of space-time surface (hyper-octonions are the subspace of complexified octonions for which imaginary units are octonionic imaginary units multiplied by commutating imaginary unit). Hyper-octonionic representation leads to a proposal for how to extend twistor program to TGD framework .

How to achieve associativity in the fermionic sector?

In the fermionic sector an additional complication emerges. The associativity of the tangent- or normal space of the space-time surface need not be enough to guarantee the associativity at the level of Kähler-Dirac or Dirac equation. The reason is the presence of spinor connection. A possible cure could be the vanishing of the components of spinor connection for two conjugates of quaternionic coordinates combined with holomorphy of the modes.

  1. The induced spinor connection involves sigma matrices in CP2 degrees of freedom, which for the octonionic representation of gamma matrices are proportional to octonion units in Minkowski degrees of freedom. This corresponds to a reduction of tangent space group SO(1,7) to G2. Therefore octonionic Dirac equation identifying Dirac spinors as complexified octonions can lead to non-associativity even when space-time surface is associative or co-associative.
  2. The simplest manner to overcome these problems is to assume that spinors are localized at 2-D string world sheets with 1-D CP2 projection and thus possible only in Minkowskian regions. Induced gauge fields would vanish. String world sheets would be minimal surfaces in M4× D1⊂ M4× CP2 and the theory would simplify enormously. String area would give rise to an additional term in the action assigned to the Minkowskian space-time regions and for vacuum extremals one would have only strings in the first approximation, which conforms with the success of string models and with the intuitive view that vacuum extremals of Kähler action are basic building bricks of many-sheeted space-time. Note that string world sheets would be also symplectic covariants.

    Without further conditions gauge potentials would be non-vanishing but one can hope that one can gauge transform them away in associative manner. If not, one can also consider the possibility that CP2 projection is geodesic circle S1: symplectic invariance is considerably reduces for this option since symplectic transformations must reduce to rotations in S1.

  3. The fist heavy objection is that action would contain Newton's constant G as a fundamental dynamical parameter: this is a standard recipe for building a non-renormalizable theory. The very idea of TGD indeed is that there is only single dimensionless parameter analogous to critical temperature. One can of coure argue that the dimensionless parameter is hbarG/R2, R CP2 "radius".

    Second heavy objection is that the Euclidian variant of string action exponentially damps out all string world sheets with area larger than hbar G. Note also that the classical energy of Minkowskian string would be gigantic unless the length of string is of order Planck length. For Minkowskian signature the exponent is oscillatory and one can argue that wild oscillations have the same effect.

    The hierarchy of Planck constants would allow the replacement hbar→ hbareff but this is not enough. The area of typical string world sheet would scale as heff and the size of CD and gravitational Compton lengths of gravitationally bound objects would scale (heff)1/2 rather than heff = GMm/v0 which one wants. The only way out of problem is to assume T ∝ (hbar/heff)2. This is however un-natural for genuine area action. Hence it seems that the visit of the basic assumption of superstring theory to TGD remains very short. In any case, if one assumes that string connect gravitationally bound masses, super string models in perturbative description are definitely wrong as physical theories as has of course become clear already from landscape catastrophe.

Is super-symmetrized Kähler-Dirac action enough?

Could one do without string area in the action and use only K-D action, which is in any case forced by the super-conformal symmetry? This option I have indeed considered hitherto. K-D Dirac equation indeed tends to reduce to a lower-dimensional one: for massless extremals the K-D operator is effectively 1-dimensional. For cosmic strings this reduction does not however take place. In any case, this leads to ask whether in some cases the solutions of Kähler-Dirac equation are localized at lower-dimensional surfaces of space-time surface.

  1. The proposal has indeed been that string world sheets carry vanishing W and possibly even Z fields: in this manner the electromagnetic charge of spinor mode could be well-defined. The vanishing conditions force in the generic case 2-dimensionality.

    Besides this the canonical momentum currents for Kähler action defining 4 imbedding space vector fields must define an integrable distribution of two planes to give string world sheet. The four canonical momentum currents Πkα= ∂ LK/∂α hk identified as imbedding 1-forms can have only two linearly independent components parallel to the string world sheet. Also the Frobenius conditions stating that the two 1-forms are proportional to gradients of two imbedding space coordinates Φi defining also coordinates at string world sheet, must be satisfied. These conditions are rather strong and are expected to select some discrete set of string world sheets.

  2. To construct preferred extremal one should fix the partonic 2-surfaces, their light-like orbits defining boundaries of Euclidian and Minkowskian space-time regions, and string world sheets. At string world sheets the boundary condition would be that the normal components of canonical momentum currents for Kähler action vanish. This picture brings in mind strong form of holography and this suggests that might make sense and also solution of Einstein equations with point like sources.
  3. The localization of spinor modes at 2-D surfaces would would follow from the well-definedness of em charge and one could have situation is which the localization does not occur. For instance, covariantly constant right-handed neutrinos spinor modes at cosmic strings are completely de-localized and one can wonder whether one could give up the localization inside wormhole contacts.
  4. String tension is dynamical and physical intuition suggests that induced metric at string world sheet is replaced by the anti-commutator of the K-D gamma matrices and by conformal invariance only the conformal equivalence class of this metric would matter and it could be even equivalent with the induced metric. A possible interpretation is that the energy density of Kähler action has a singularity localized at the string world sheet.

    Another interpretation that I proposed for years ago but gave up is that in spirit with the TGD analog of AdS/CFT duality the Noether charges for Kähler action can be reduced to integrals over string world sheet having interpretation as area in effective metric. In the case of magnetic flux tubes carrying monopole fluxes and containing a string connecting partonic 2-surfaces at its ends this interpretation would be very natural, and string tension would characterize the density of Kähler magnetic energy. String model with dynamical string tension would certainly be a good approximation and string tension would depend on scale of CD.

  5. There is also an objection. For M4 type vacuum extremals one would not obtain any non-vacuum string world sheets carrying fermions but the successes of string model strongly suggest that string world sheets are there. String world sheets would represent a deformation of the vacuum extremal and far from string world sheets one would have vacuum extremal in an excellent approximation. Situation would be analogous to that in general relativity with point particles.
  6. The hierarchy of conformal symmetry breakings for K-D action should make string tension proportional to 1/heff2 with heff=hgr giving correct gravitational Compton length Λgr= GM/v0 defining the minimal size of CD associated with the system. Why the effective string tension of string world sheet should behave like (hbar/hbareff)2?

    The first point to notice is that the effective metric Gαβ defined as hklΠkαΠlβ, where the canonical momentum current Πkα=∂ LK/∂α hk has dimension 1/L2 as required. Kähler action density must be dimensionless and since the induced Kähler form is dimensionless the canonical momentum currents are proportional to 1/αK.

    Should one assume that αK is fundamental coupling strength fixed by quantum criticality to αK≈1/137? Or should one regard gK2 as fundamental parameter so that one would have 1/αK= hbareff/4π gK2 having spectrum coming as integer multiples (recall the analogy with inverse of critical temperature)?

    The latter option is the in spirit with the original idea stating that the increase of heff reduces the values of the gauge coupling strengths proportional to αK so that perturbation series converges (Universe is theoretician friendly). The non-perturbative states would be critical states. The non-determinism of Kähler action implying that the 3-surfaces at the boundaries of CD can be connected by large number of space-time sheets forming n conformal equivalence classes. The latter option would give Gαβ ∝ heff2 and det(G) ∝ 1/heff2 as required.

  7. It must be emphasized that the string tension has interpretation in terms of gravitational coupling on only at the GRT limit of TGD involving the replacement of many-sheeted space-time with single sheeted one. It can have also interpretation as hadronic string tension or effective string tension associated with magnetic flux tubes and telling the density of Kähler magnetic energy per unit length.

    Superstring models would describe only the perturbative Planck scale dynamics for emission and absorption of heff/h=1 on mass shell gravitons whereas the quantum description of bound states would require heff/n>1 when the masses. Also the effective gravitational constant associated with the strings would differ from G.

    The natural condition is that the size scale of string world sheet associated with the flux tube mediating gravitational binding is G(M+m)/v0, By expressing string tension in the form 1/T=n2 hbar G1, n=heff/h, this condition gives hbar G1= hbar2/Mred2, Mred= Mm/(M+m). The effective Planck length defined by the effective Newton's constant G1 analogous to that appearing in string tension is just the Compton length associated with the reduced mass of the system and string tension equals to T= [v0/G(M+m)]2 apart from a numerical constant (2G(M+m) is Schwartschild radius for the entire system). Hence the macroscopic stringy description of gravitation in terms of string differs dramatically from the perturbative one. Note that one can also understand why in the Bohr orbit model of Nottale for the planetary system and in its TGD version v0 must be by a factor 1/5 smaller for outer planets rather than inner planets.

Are 4-D spinor modes consistent with associativity?

The condition that octonionic spinors are equivalent with ordinary spinors looks rather natural but in the case of Kähler-Dirac action the non-associativity could leak in. One could of course give up the condition that octonionic and ordinary K-D equation are equivalent in 4-D case. If so, one could see K-D action as related to non-commutative and maybe even non-associative fermion dynamics. Suppose that one does not.

  1. K-D action vanishes by K-D equation. Could this save from non-associativity? If the spinors are localized to string world sheets, one obtains just the standard stringy construction of conformal modes of spinor field. The induce spinor connection would have only the holomorphic component Az. Spinor mode would depend only on z but K-D gamma matrix Γz would annihilate the spinor mode so that K-D equation would be satisfied. There are good hopes that the octonionic variant of K-D equation is equivalent with that based on ordinary gamma matrices since quaternionic coordinated reduces to complex coordinate, octonionic quaternionic gamma matrices reduce to complex gamma matrices, sigma matrices are effectively absent by holomorphy.
  2. One can consider also 4-D situation (maybe inside wormhole contacts). Could some form of quaternion holomorphy allow to realize the K-D equation just as in the case of super string models by replacing complex coordinate and its conjugate with quaternion and its 3 conjugates. Only two quaternion conjugates would appear in the spinor mode and the corresponding quaternionic gamma matrices would annihilate the spinor mode. It is essential that in a suitable gauge the spinor connection has non-vanishing components only for two quaternion conjugate coordinates. As a special case one would have a situation in which only one quaternion coordinate appears in the solution. Depending on the character of quaternionion holomorphy the modes would be labelled by one or two integers identifiable as conformal weights.

    Even if these octonionic 4-D modes exists (as one expects in the case of cosmic strings), it is far from clear whether the description in terms of them is equivalent with the description using K-D equation based ordinary gamma matrices. The algebraic structure however raises hopes about this. The quaternion coordinate can be represented as sum of two complex coordinates as q=z1+Jz2 and the dependence on two quaternion conjugates corresponds to the dependence on two complex coordinates z1,z2. The condition that two quaternion complexified gammas annihilate the spinors is equivalent with the corresponding condition for Dirac equation formulated using 2 complex coordinates. This for wormhole contacts. The possible generalization of this condition to Minkowskian regions would be in terms Hamilton-Jacobi structure.

    Note that for cosmic strings of form X2× Y2⊂ M4× CP2 the associativity condition for S2 sigma matrix and without assuming localization demands that the commutator of Y2 imaginary units is proportional to the imaginary unit assignable to X2 which however depends on point of X2. This condition seems to imply correlation between Y2 and S2 which does not look physical.


To summarize, the minimal and mathematically most optimistic conclusion is that Kähler-Dirac action is indeed enough to understand gravitational binding without giving up the associativity of the fermionic dynamics. Conformal spinor dynamics would be associative if the spinor modes are localized at string world sheets with vanishing W (and maybe also Z) fields guaranteeing well-definedness of em charge and carrying canonical momentum currents parallel to them. It is not quite clear whether string world sheets are present also inside wormhole contacts: for CP2 type vacuum extremals the Dirac equation would give only right-handed neutrino as a solution (could they give rise to N=2 SUSY?).

Associativity does not favor fermionic modes in the interior of space-time surface unless they represent right-handed neutrinos for which mixing with left-handed neutrinos does not occur: hence the idea about interior modes of fermions as giving rise to SUSY is dead whereas the original idea about partonic oscillator operator algebra as SUSY algebra is well and alive. Evolution can be seen as a generation of gravitationally bound states of increasing size demanding the gradual increase of h_eff implying generation of quantum coherence even in astrophysical scales.

The construction of preferred extremals would realize strong form of holography. By conformal symmetry the effective metric at string world sheet could be conformally equivalent with the induced metric at string world sheets. Dynamical string tension would be proportional to hbar/heff2 due to the proportionality αK∝ 1/heff and predict correctly the size scales of gravitationally bound states for hgr=heff=GMm/v0. Gravitational constant would be a prediction of the theory and be expressible in terms of αK and R2 and hbareff (G∝ R2/gK2).

In fact, all bound states - elementary particles as pairs of wormhole contacts, hadronic strings, nuclei, molecules, etc. - are described in the same manner quantum mechanically. This is of course nothing new since magnetic flux tubes associated with the strings provide a universal model for interactions in TGD Universe. This also conforms with the TGD counterpart of AdS/CFT duality.

See the chapter Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds" .

Is the formation of gravitational bound states impossible in superstring models?

I decided to take here from a previous posting an argument allowing to conclude that super string models are unable to describe macroscopic gravitation involving formation of gravitationally bound states. Therefore superstrings models cannot have desired macroscopic limit and are simply wrong. This is of course reflected also by the landscape catastrophe meaning that the theory ceases to be a theory in macroscopic scales. The failure is not only at the level of superstring models: it is at the level of quantum theory itself. Instead of single value of Planck constant one must allow a hierarchy of Planck constants predicted by TGD. My sincere hope is that this message could gradually leak through the iron curtain to the ears of the super string gurus.

Superstring action has bosonic part proportional to string area. The proportionality constant is string tension proportional to 1/hbar G and is gigantic. One expects only strings of length of order Planck length be of significance.

It is now clear that also in TGD the action in Minkowskian regions contains a string area. In Minkowskian regions of space-time strings dominate the dynamics in an excellent approximation and the naive expectation is that string theory should give an excellent description of the situation.

String tension would be proportional to 1/hbar G and this however raises a grave classical counter argument. In string model massless particles are regarded as strings, which have contracted to a point in excellent approximation and cannot have length longer than Planck length. How this can be consistent with the formation of gravitationally bound states is however not understood since the required non-perturbative formulation of string model required by the large valued of the coupling parameter GMm is not known.

In TGD framework strings would connect even objects with macroscopic distance and would obviously serve as correlates for the formation of bound states in quantum level description. The classical energy of string connecting say the two wormhole contacts defining elementary particle is gigantic for the ordinary value of hbar so that something goes wrong.

I have however proposed that gravitons - at least those mediating interaction between dark matter have large value of Planck constant. I talk about gravitational Planck constant and one has heff= hgr=GMm/v0, where v0/c<1 (v0 has dimensions of velocity). This makes possible perturbative approach to quantum gravity in the case of bound states having mass larger than Planck mass so that the parameter GMm analogous to coupling constant is very large. The velocity parameter v0/c becomes the dimensionless coupling parameter. This reduces the string tension so that for string world sheets connecting macroscopic objects one would have T ∝ v0/G2Mm. For v0= GMm/hbar, which remains below unity for Mm/mPl2 one would have hgr/h=1. Hence the action remains small and its imaginary exponent does not fluctuate wildly to make the bound state forming part of gravitational interaction short ranged. This is expected to hold true for ordinary matter in elementary particle scales. The objects with size scale of large neutron (100 μm in the density of water) - probably not an accident - would have mass above Planck mass so that dark gravitons and also life would emerge as massive enough gravitational bound states are formed. hgr=heff hypothesis is indeed central in TGD based view about living matter.

To conclude, it seems that superstring theory with single value of Planck constant cannot give rise to macroscopic gravitationally bound matter and would be therefore simply wrong much better than to be not-even-wrong.

See the chapter Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds" .

Updated view about Kähler geometry of WCW

TGD differs in several respects from quantum field theories and string models. The basic mathematical difference is that the mathematically poorly defined notion of path integral is replaced with the mathematically well-defined notion of functional integral defined by the Kähler function defining Kähler metric for WCW ("world of classical worlds"). Apart from quantum jump, quantum TGD is essentially theory of classical WCW spinor fields with WCW spinors represented as fermionic Fock states. One can say that Einstein's geometrization of physics program is generalized to the level of quantum theory.

It has been clear from the beginning that the gigantic super-conformal symmetries generalizing ordinary super-conformal symmetries are crucial for the existence of WCW Kähler metric. The detailed identification of Kähler function and WCW Kähler metric has however turned out to be a difficult problem. It is now clear that WCW geometry can be understood in terms of the analog of AdS/CFT duality between fermionic and space-time degrees of freedom (or between Minkowskian and Euclidian space-time regions) allowing to express Kähler metric either in terms of Kähler function or in terms of anti-commutators of WCW gamma matrices identifiable as super-conformal Noether super-charges for the symplectic algebra assignable to δ M4+/-× CP2. The string model description of gravitation emerges and also the TGD based view about dark matter becomes more precise.

Kähler function, Kähler action, and connection with string models

The definition of Kähler function in terms of Kähler action is possible because space-time regions can have also Euclidian signature of induced metric. Euclidian regions with 4-D CP2 projection - wormhole contacts - are identified as lines of generalized Feynman diagrams - space-time correlates for basic building bricks of elementary particles. Kähler action from Minkowskian regions is imaginary and gives to the functional integrand a phase factor crucial for quantum field theoretic interpretation. The basic challenges are the precise specification of Kähler function of "world of classical worlds" (WCW) and Kähler metric.

There are two approaches concerning the definition of Kähler metric: the conjecture analogous to AdS/CFT duality is that these approaches are mathematically equivalent.

  1. The Kähler function defining Kähler metric can be identified as Kähler action for space-time regions with Euclidian signature for a preferred extremal containing 3-surface as the ends of the space-time surfaces inside causal diamond (CD). Minkowskian space-time regions give to Kähler action an imaginary contribution interpreted as the counterpart of quantum field theoretic action. The exponent of Kähler function defines functional integral in WCW. WCW metric is dictated by the Euclidian regions of space-time with 4-D CP2 projection.

    The basic question concerns the attribute "preferred". Physically the preferred extremal is analogous to Bohr orbit. What is the mathematical meaning of preferred extremal of Kähler action? The latest step of progress is the realization that the vanishing of generalized conformal charges for the ends of the space-time surface fixes the preferred extremals to high extent and is nothing but classical counterpart for generalized Virasoro and Kac-Moody conditions.

  2. Fermions are also needed. The well-definedness of electromagnetic charge led to the hypothesis that spinors are restricted to string world sheets. It has become also clear that string world sheets are most naturally minimal surfaces with 1-D CP2 projection (this brings in gravitational constant) and that Kähler action in Minkowskian regions involves also the string area (, which does not contribute to Kähler function) giving the entire action in the case of M4 type vacuum extremals with vanishing Kähler form. Hence vacuum extremals might serve as an excellent approximation for the sheets of the many-sheeted space-time in Minkowskian space-time regions.
  3. Second manner to define Kähler metric is as anticommutators of WCW gamma matrices identified as super-symplectic Noether charges for the Dirac action for induced spinors with string tension proportional to the inverse of Newton's constant. These charges are associated with the 1-D space-like ends of string world sheets connecting the wormhole throats. WCW metric contains contributions from the spinor modes associated with various string world sheets connecting the partonic 2-surfaces associated with the 3-surface.

    It is clear that the information carried by WCW metric about 3-surface is rather limited and that the larger the number of string world sheets, the larger the information. This conforms with strong form of holography and the notion of measurement resolution as a property of quantums state. Clearly. Duality means that Kähler function is determined either by space-time dynamics inside Euclidian wormhole contacts or by the dynamics of fermionic strings in Minkowskian regions outside wormhole contacts. This duality brings strongly in mind AdS/CFT duality. One could also speak about fermionic emergence since Kähler function is dictated by the Kähler metric part from a real part of gradient of holomorphic function: a possible identification of the exponent of Kähler function is as Dirac determinant.

Realization of super-conformal symmetries

The detailed realization of various super-conformal symmetries has been also a long standing problem but recent progress leads to very beautiful overall view.

  1. Super-conformal symmetry requires that Dirac action for string world sheets is accompanied by string world sheet area as part of bosonic action. String world sheets are implied and can be present only in Minkowskian regions if one demands that octonionic and ordinary representations of induced spinor structure are equivalent (this requires vanishing of induced spinor curvature to achieve associativity in turn implying that CP2 projection is 1-D). Note that 1-dimensionality of CP2 projection is symplectically invariant property. Neither string world sheet area nor Kähler action is invariant under symplectic transformations. This is necessary for having non-trivial Kähler metric. Whether WCW really possesses super-symplectic isometries remains an open problem.
  2. Super-conformal symmetry also demands that Kähler action is accompanied by what I call Kähler-Dirac action with gamma matrices defined by the contractions of the canonical momentum currents with imbedding space-gamma matrices. Hence also induced spinor fields in the space-time interior must be present. Indeed, inside wormhole contacts Kähler-Dirac equation reducing to CP2 Dirac equation for CP2 vacuum extremals dictates the fermionic dynamics.

    Strong form of holography implied by strong form of general coordinate invariance strongly suggests that super-conformal invariance in the interior of the space-time surface is a broken gauge invariance in the sense that the super-conformal charges for a sub-algebra with conformal weights vanishing modulo some integer n vanish. The proposal is that n corresponds to the effective Planck constant as heff/h=n. For string world sheets super-conformal symmetries are not gauge symmetries and strings dominate in good approximation the fermionic dynamics.

Interior dynamics for fermions, the role of vacuum extremals, dark matter, and SUSY

The key role of CP2-type and M4-type vacuum extremals has been rather obvious from the beginning but the detailed understanding has been lacking. Both kinds of extremals are invariant under symplectic transformations of δ M4× CP2, which inspires the idea that they give rise to isometries of WCW. The deformations CP2-type extremals correspond to lines of generalized Feynman diagrams. M4 type vacuum extremals in turn are excellent candidates for the building bricks of many-sheeted space-time giving rise to GRT space-time as approximation. For M4 type vacuum extremals CP2 projection is (at most 2-D) Lagrangian manifold so that the induced Kähler form vanishes and the action is fourth-order in small deformations. This implies the breakdown of the path integral approach and of canonical quantization, which led to the notion of WCW.

If the action in Minkowskian regions contains also string area, the situation changes dramatically since strings dominate the dynamics in excellent approximation and string theory should give an excellent description of the situation: this of course conforms with the dominance of gravitation.

String tension would be proportional to 1/hbar G and this raises a grave classical counter argument. In string model massless particles are regarded as strings, which have contracted to a point in excellent approximation and cannot have length longer than Planck length. How this can be consistent with the formation of gravitationally bound states is however not understood since the required non-perturbative formulation of string model required by the large valued of the coupling parameter GMm is not known.

In TGD framework strings would connect even objects with macroscopic distance and would obviously serve as correlates for the formation of bound states in quantum level description. The classical energy of string connecting say the two wormhole contacts defining elementary particle is gigantic for the ordinary value of hbar so that something goes wrong.

I have however proposed that gravitons - at least those mediating interaction between dark matter have large value of Planck constant. I talk about gravitational Planck constant and one has heff= hgr=GMm/v0, where v0/c<1 (v0 has dimensions of velocity). This makes possible perturbative approach to quantum gravity in the case of bound states having mass larger than Planck mass so that the parameter GMm analogous to coupling constant is very large. The velocity parameter v0/c becomes the dimensionless coupling parameter. This reduces the string tension so that for string world sheets connecting macroscopic objects one would have T ∝ v0/G2Mm. For v0= GMm/hbar, which remains below unity for Mm/mPl2 one would have hgr/h=1. Hence the action remains small and its imaginary exponent does not fluctuate wildly to make the bound state forming part of gravitational interaction short ranged. This is expected to hold true for ordinary matter in elementary particle scales. The objects with size scale of large neutron (100 μm in the density of water) - probably not an accident - would have mass above Planck mass so that dark gravitons and also life would emerge as massive enough gravitational bound states are formed. hgr=heff hypothesis is indeed central in TGD based view about living matter. In this framework superstring theory with single value of Planck constant would not give rise to macroscopic gravitationally bound matter and would be thus simply wrong.

If one assumes that for non-standard values of Planck constant only n-multiples of super-conformal algebra in interior annihilate the physical states, interior conformal gauge degrees of freedom become partly dynamical. The identification of dark matter as macroscopic quantum phases labeled by heff/h=n conforms with this.

The emergence of dark matter corresponds to the emergence of interior dynamics via breaking of super-conformal symmetry. The induced spinor fields in the interior of flux tubes obeying Kähler Dirac action should be highly relevant for the understanding of dark matter. The assumption that dark particles have essentially same masses as ordinary particles suggests that dark fermions correspond to induced spinor fields at both string world sheets and in the space-time interior: the spinor fields in the interior would be responsible for the long range correlations characterizing heff/h=n. Magnetic flux tubes carrying dark matter are key entities in TGD inspired quantum biology. Massless extremals represent second class of M4 type non-vacuum extremals.

This view forces once again to ask whether space-time SUSY is present in TGD and how it is realized. With a motivation coming from the observation that the mass scales of particles and sparticles most naturally have the same p-adic mass scale as particles in TGD Universe I have proposed that sparticles might be dark in TGD sense. The above argument leads to ask whether the dark variants of particles correspond to states in which one has ordinary fermion at string world sheet and 4-D fermion in the space-time interior so that dark matter in TGD sense would almost by definition correspond to sparticles!

See the chapter Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds" .

Surface area as geometric representation of entanglement entropy?

In Thinking Allowed Original there was a link to a talk by James Sully and having the title Geometry of Compression. I must admit that I understood very little about the talk. My not so educated guess is however that information is compressed: UV or IR cutoff eliminating entanglement in short length scales and describing its presence in terms of density matrix - that is thermodynamically - is another manner to say it. The TGD inspired proposal for the interpretation of the inclusions of hyper-finite factors of type II1 (HFFs) is in spirit with this.

The space-time counterpart for the compression would be in TGD framework discretization. Discretizations using rational points (or points in algebraic extensions of rationals) make sense also p-adically and thus satisfy number theoretic universality. Discretization would be defined in terms of intersection (rational or in algebraic extension of rationals) of real and p-adic surfaces. At the level of "world of classical worlds" the discretization would correspond to - say - surfaces defined in terms of polynomials, whose coefficients are rational or in some algebraic extension of rationals. Pinary UV and IR cutoffs are involved too. The notion of p-adic manifold allows to interpret rthe p-adic variants of space-time surfaces as cognitive representations of real space-time surfaces.

Finite measurement resolution does not allow state function reduction reducing entanglement totally. In TGD framework also negentropic entanglement stable under Negentropy Maximixation Principle (NMP) is possible. For HFFs the projection into single ray of Hilbert space is indeed impossible: the reduction takes always to infinite-D sub-space.

The visit to the URL was however not in vain. There was a link to an article discussing the geometrization of entanglement entropy inspired by the AdS/CFT hypothesis.

Quantum classical correspondence is basic guiding principle of TGD and suggests that entanglement entropy should indeed have space-time correlate, which would be the analog of Hawking-Bekenstein entropy.

Generalization of AdS/CFT to TGD context

AdS/CFT generalizes to TGD context in non-trivial manner. There are two alternative interpretations, which both could make sense. These interpretations are not mutually exclusive. The first interpretation makes sense at the level of "world of classical worlds" (WCW) with symplectic algebra and extended conformal algebra associated with δ M4+/- replacing ordinary conformal and Kac-Moody algebras. Second interpretation at the level of space-time surface with the extended conformal algebras of the light-likes orbits of partonic 2-surfaces replacing the conformal algebra of boundary of AdSn.

1. First interpretation

For the first interpretation 2-D conformal invariance is generalised to 4-D conformal invariance relying crucially on the 4-dimensionality of space-time surfaces and Minkowski space.

  1. One has an extension of the conformal invariance provided by the symplectic transformations of δ CD× CP2 for which Lie algebra has the structure of conformal algebra with radial light-like coordinate of δ M4+ replacing complex coordinate z.
  2. One could see the counterpart of AdSn as imbedding space H=M4 × CP2 completely unique by twistorial considerations and from the condition that standard model symmetries are obtained and its causal diamonds defined as sub-sets CD× CP2, where CD is an intersection of future and past directed light-cones. I will use the shorthand CD for CD× CP2. Strings in AdS5× S5 are replaced with space-time surfaces inside 8-D CD.
  3. For this interpretation 8-D CD replaces the 10-D space-time AdS5× S5. 7-D light-like boundaries of CD correspond to the boundary of say AdS5, which is 4-D Minkowski space so that zero energy ontology (ZEO) allows rather natural formulation of the generalization of AdS/CFT correspondence since the positive and negative energy parts of zero energy states are localized at the boundaries of CD.

2. Second interpretation

For the second interpretation relies on the observation that string world sheets as carriers of induced spinor fields emerge in TGD framework from the condition that electromagnetic charge is well-defined for the modes of induced spinor field.

  1. One could see the 4-D space-time surfaces X4 as counterparts of AdS4. The boundary of AdS4 is replaced in this picture with 3-surfaces at the ends of space-time surface at opposite boundaries of CD and by strong form of holography the union of partonic 2-surfaces defining the intersections of the 3-D boundaries between Euclidian and Minkowskian regions of space-time surface with the boundaries of CD. Strong form of holography in TGD is very much like ordinary holography.
  2. Note that one has a dimensional hierarchy: the ends of the boundaries of string world sheets at boundaries of CD as pointlike partices, boundaries as fermion number carrying lines, string world sheets, light-like orbits of partonic 2-surfaces, 4-surfaces, imbedding space M4× CP2. Clearly the situation is more complex than for AdS/CFT correspondence.
  3. One can restrict the consideration to 3-D sub-manifolds X3 at either boundary of causal diamond (CD): the ends of space-time surface. In fact, the position of the other boundary is not well-defined since one has superposition of CDs with only one boundary fixed to be piece of light-cone boundary. The delocalization of the other boundary is essential for the understanding of the arrow of time. The state function reductions at fixed boundary leave positive energy part (say) of the zero energy state at that boundary invariant (in positive energy ontology entire state would remain unchanged) but affect the states associated with opposite boundaries forming a superposition which also changes between reduction: this is analog for unitary time evolution. The average for the distance between tips of CDs in the superposition increases and gives rise to the flow of time.
  4. One wants an expression for the entanglement entropy between X3 and its partner. Bekenstein area law allows to guess the general expression for the entanglement entropy: for the proposal discussed in the article the entropy would be the area of the boundary of X3 divided by gravitational constant: S= A/4G. In TGD framework gravitational constant might be replaced by the square of CP2 radius apart from numerical constant. How gravitational constant emerges in TGD framework is not completely understood although one can deduce for it an estimate using dimensional analyses. In any case, gravitational constant is a parameter which characterizes GRT limit of TGD in which many-sheeted space-time is in long scales replaced with a piece of Minkowski space such that the classical gravitational fields and gauge potentials for sheets are summed. The physics behind this relies on the generalization of linear superposition of fields: the effects of different space-time sheets particle touching them sum up rather than fields.
  5. The counterpart for the boundary of X3 appearing in the proposal for the geometrization of the entanglement entropy naturally corresponds to partonic 2-surface or a collection of them if strong form of holography holds true.

With what kind of systems 3-surfaces can entangle?

With what system X3 is entangled/can entangle? There are several options to consider and they could correspond to the two TGD variants for the AdS/CFT correspondence.

  1. X3 could correspond to a wormhole contact with Euclidian signature of induced metric. The entanglement would be between it and the exterior region with Minkowskian signature of the induced metric.
  2. X3 could correspond to single sheet of space-time surface connected by wormhole contacts to a larger space-time sheet defining its environment. More precisely, X3 and its complement would be obtained by throwing away the wormhole contacts with Euclidian signature of induce metric. Entanglement would be between these regions. In the generalization of the formula

    S= A/4hbar G

    area A would be replaced by the total area of partonic 2-surfaces and G perhaps with CP2 length scale squared.

  3. In ZEO the entanglement could also correspond to time-like entanglement between the 3-D ends of the space-time surface at opposite light-like boundaries of CD. M-matrix, which can be seen as the analog of thermal S-matrix, decomposes to a product of hermitian square root of density matrix and unitary S-matrix and this hermitian matrix could also define p-adic thermodynamics. Note that in ZEO quantum theory can be regarded as square root of thermodynamics.

Minimal surface property is not favored in TGD framework

Minimal surface property for the 3-surfaces X3 at the ends of space-time surface looks at first glance strange but a proper generalization of this condition makes sense if one assumes strong form of holography. Strong form of holography realizes General Coordinate Invariance (GCI) in strong sense meaning that light-like parton orbits and space-like 3-surfaces at the ends of space-time surfaces are equivalent physically. As a consequence, partonic 2-surfaces and their 4-D tangent space data must code for the quantum dynamics.

The mathematical realization is in terms of conformal symmetries accompanying the symplectic symmetries of δ M4+/-× CP2 and conformal transformations of the light-like partonic orbit. The generalizations of ordinary conformal algebras correspond to conformal algebra, Kac-Moody algebra at the light-like parton orbits and to symplectic transformations δ M4× CP2 acting as isometries of WCW and having conformal structure with respect to the light-like radial coordinate plus conformal transformations of δ M4+/-, which is metrically 2-dimensional and allows extended conformal symmetries.

  1. If the conformal realization of the strong form of holography works, conformal transformations act at quantum level as gauge symmetries in the sense that generators with no-vanishing conformal weight are zero or generate zero norm states. Conformal degeneracy can be eliminated by fixing the gauge somehow. Classical conformal gauge conditions analogous to Virasoro and Kac-Moody conditions satisfied by the 3-surfaces at the ends of CD are natural in this respect. Similar conditions would hold true for the light-like partonic orbits at which the signature of the induced metric changes.
  2. What is also completely new is the hierarchy of conformal symmetry breakings associated with the hierarchy of Planck constants heff/h=n. The deformations of the 3-surfaces which correspond to non-vanishing conformal weight in algebra or any sub-algebra with conformal weights vanishing modulo n give rise to vanishing classical charges and thus do not affect the value of the Kähler action.

    The inclusion hierarchies of conformal sub-algebras are assumed to correspond to those for hyper-finite factors. There is obviously a precise analogy with quantal conformal invariance conditions for Virasoro algebra and Kac-Moody algebra. There is also hierarchy of inclusions which corresponds to hierarchy of measurement resolutions. An attractive interpretation is that singular conformal transformations relate to each other the states for broken conformal symmetry. Infinitesimal transformations for symmetry broken phase would carry fractional conformal weights coming as multiples of 1/n.

  3. Conformal gauge conditions need not reduce to minimal surface conditions holding true for all variations.
  4. Note that Kähler action reduces to Chern-Simons term at the ends of CD if weak form of electric magnetic duality holds true. The conformal charges at the ends of CD cannot however reduce to Chern-Simons charges by this condition since only the charges associated with CP2 degrees of freedom would be non-trivial.


The generalisation of the conjecture about surface area proportionality of entropy to TGD context looks rather straightforward but is physically highly non-trivial. There are however some technicalities involved.

  1. In TGD framework it is not quite clear whether
    1. G still appears in the formula or
    2. whether G should be replaced with the square R2 of CP2 radius to give

      S= A/4π R2

      apart from numerical constant.

    For option a) one must include Planck constant explicitly to the formula to give S= A/4heffG: the entropy would decrease as heff=n× h increases. The condition heff=hgr= GM2/v0 would give S= v0/c<1. The entropy using b) would be by a factor of order 10-5 smaller and would not depend on the value of heff at all. It will be found that p-adic mass calculations lead to entropy allowing to circumvent these problems.
  2. There is also the question about the identification of the area A. For blackhole A would be determined by Schwartschild radius rS= 2GM depending on mass only. In TGD framework one has several candidates.
    1. The area of partonic 2-surface is an obvious first guess. One cannot however expect that the area of partonic 2-surface is constant. Could conformal gauge fixing fixes the 3-surfaces highly uniquely. Ordinary conformal invariance for partonic 2-surface does not however seem to be consistent with the fixing of the area of partonic 2-surface since conformal transformations do not preserve area.
    2. Could the area of partonic 2-surface be replaced with the area of the boundary of space-time sheet at which particle is topologically condensed and has size scale of order Compton length? This option looks the most feasible one on basis of p-adic mass calculations as will be found.

p-Adic variant of Bekenstein-Hawking law

When the 3-surface corresponds to elementary particle, a direct connection with p-adic thermodynamics suggests itself and allows to answer the questions above. p-Adic thermodynamics could be interpreted as a description of the entanglement with environment. In ZEO the entanglement could also correspond to time-like entanglement between the 3-D ends of the space-time surface at opposite light-like boundaries of CD. M-matrix, which can be seen as the analog of thermal S-matrix, decomposes to a product of hermitian square root of density matrix and unitary S-matrix and this hermitian matrix could also define p-adic thermodynamics.

  1. p-Adic thermodynamics would not be for energy but for mass squared (or scaling generator L0) would describe the entanglement of the particle with environment defined by the larger space-time sheet. Conformal weights would comes as positive powers of integers (pL0 would replace exp(-H/T) to guarantee the number theoretical existence and convergence of the Boltzmann weight: note that conformal invariance that is integer spectrum of L0 is also essential).
  2. The interactions with environment would excite very massive CP2 mass scale excitations (mass scale is about 10-4 times Planck mass) of the particle and give it thermal mass squared identifiable as the observed mass squared. The Boltzmann weights would be extremely small having p-adic norm about 1/pn, p the p-adic prime: M127=2127-1 for electron.
  3. I have proposed earlier p-adic entropy as a p-adic counterpart of Bekenstein-Hawking entropy. S= (R2/hbar2)× M2 holds true identically apart from numerical constant. Note that one could interpret R2M/hbar as the counterpart of Schwartschild radius. Note that this radius is proportional to 1/p1/2 so that the area A would correspond to the area defined by Compton length. This is in accordance with the third option.

What is the space-time correlate for negentropic entanglement?

The new element brought in by TGD framework is that number theoretic entanglement entropy is negative for negentropic entanglement assignable to unitary entanglement and NMP states that this negentropy increases. Since entropy is essentially number of energy degenerate states, a good guess is that the number n=heff/h of space-time sheets associated with heff defines the negentropy. An attractive space-time correlate for the negentropic entanglement is braiding. Braiding defines unitary S-matrix between the states at the ends of braid and this entanglement is negentropic. This entanglement gives also rise to topological quantum computation.

See the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article Surface area as geometric representation of entanglement entropy?.

The vanishing of conformal charges as a gauge conditions selecting preferred extremals of Kähler action

Classical TGD involves several key questions waiting for clearcut answers.

  1. The notion of preferred extremal emerges naturally in positive energy ontology, where Kähler metric assigns a unique (apart from gauge symmetries) preferred extremal to given 3-surface at M4 time= constant section of imbedding space H=M4× CP2. This would quantize the initial values of the time derivatives of imbedding coordinates and this could correspond to the Bohr orbitology in quantum mechanics.
  2. In zero energy ontology (ZEO) initial conditions are replaced by boundary conditions. One fixes only the 3-surfaces at the opposite boundaries of CD and in an ideal situation there would exist a unique space-time surface connecting them. One must however notice that the existence of light-like wormhole throat orbits at which the signature of the induced metric changes (det(g4)=0) its signature might change the situation. Does the attribute "preferred" become obsolete and does one lose the beautiful Bohr orbitology which looks intuitively compelling and would realize quantum classical correspondence?
  3. Intuitively it has become clear that the generalization of super-conformal symmetries by replacing 2-D manifold with metrically 2-D but topologically 3-D light-like boundary of causal diamond makes sense. Generalized super-conformal symmetries should apply also to the wormhole throat orbits which are also metrically 2-D and for which conformal symmetries respect detg(g4)=0 condition. Quantum classical correspondence demands that the generalized super-confornal invariance has classical counterpart. How could this classical counterpart be realized?
  4. Holography is one key aspect of TGD and mean that 3-surfaces dictate everything. In positive energy ontology the content w of this statement would be rather obvious and reduce to Bohr orbitology but in ZEO situation is different. On the other hand, TGD strongly suggests strong form of holography based stating that partonic 2-surfaces (the ends of wormhole throat orbits at boundaries of CD) and tangent space data at them code for quantum physics of TGD. General coordinate invariance would be realied in strong sense: one could formulate the theory either in terms of space-like 3-surfaces at the ends of CD or in terms of light-like wormhole throat orbits. This would realize Bohr orbitology also in ZEO by reducing the boundary conditions to those at partonic 2-surfaces. How to realize this explicitly at the level of field equations? This has been the challenge.
Answering questions is extremely useful activity. During last years Hamed has posed continually questions related to the basic TGD. At this time Hamed asked about the derivation of field equations of TGD. In "simple" field theories involving some polynomial non-linearities the deduction of field equations is of course totally trivial process but in the extremely non-linear geometric framework of TGD situation is quite different.

While answering the questions I made what I immediately dare to call a breakthrough discovery in the mathematical understanding of TGD. To put it concisely: one can assume that the variations at the light-like boundaries of CD vanish for all conformal variations which are not isometries. For isometries the contributions from the ends of CD cancel each other so that the corresponding variations need not vanish separately at boundaries of CD! This is extremely simple and profound fact. This would be nothing but the realisation of the analogs of conformal symmetries classically and give precise content for the notion of preferred external, Bohr orbitology, and strong form of holography. And the condition makes sense only in ZEO!

I attach below the answers to the questions of Hamed almost as such apart from slight editing and little additions, re-organization, and correction of typos.

The physical interpretation of the canonical momentum current

Hamed asked about the physical meaning of Tnk== ∂ L/∂(∂n hk) - normal components of canonical momentum labelled by the label k of imbedding space coordinates - it is good to start from the physical meaning of a more general vector field

Tαk == ∂ L/∂(∂α hk)

with both imbedding space indices k and space-time indices α - canonical momentum currents. L refers to Kähler action.

  1. One can start from the analogy with Newton's equations derived from action principle (Lagrangian). Now the analogs are the partial derivatives ∂ L/∂(dxk/dt). For a particle in potential one obtains just the momentum. Therefore the term canonical momentum current/density: one has kind of momentum current for each imbedding space coordinate.
  2. By contracting with generators of imbedding space isometries (Poincare and color) one indeed obtains conserved currents associated with isometries by Noether's theorem:

    jA α= TαkjAk .

    By field equations the divergences of these currents vanish and one obtains conserved charged- classical four-momentum and color charges:

    Dα TA α=0 .

  3. The normal component of conserved current must vanish at space-like boundaries if one has such


    if one has boundaries with Minkowskian signature of induced metric. Now one has wormhole throat orbits which are not genuine boundaries albeit analogous to them and one must be very careful. The quantity Tnk determines the values of normal components of currents and must vanish at possible space-like boundaries.

Note that in TGD field equations reduce to the conservation of isometry currents as in hydrodynamics where basic equations are just conservation laws.

The basic steps in the derivation of field equations

First a general recipe for deriving field equations from Kähler action - or any action as a matter of fact.

  1. At the first step one writes an expression of the variation of the Kähler action as sum of variations with respect to the induced metric g and induced Kähler form J. The partial derivatives in question are energy momentum tensor and contravariant Kähler form.
  2. After this the variations of g and J are expressed in terms of variations of imbedding space coordinates, which are the primary dynamical variables.
  3. The integral defining the variation can be decomposed to a total divergence plus a term vanishing for extremals for all variations: this gives the field equations. Total divergence term gives a boundary term and it vanishes by boundary conditions if the boundaries in question have time-like direction.

    If the boundary is space-like, the situation is more delicate in TGD framework: this will be considered in the sequel. In TGD situation is also delicate also because the light-like 3-surfaces which are common boundaries of regions with Minkowskian or Euclidian signature of the induced metric are not ordinary topological boundaries. Therefore a careful treatment of both cases is required in order to not to miss important physics.

Expressing this summary more explicitly, the variation of the Kahler action with respect to the gradients of the imbedding space coordinates reduces to an integral of

Tαkαδ hk + (∂ L/∂ hk) δ hk .

The latter term comes only from the dependence of the imbedding space metric and Kähler form on imbedding space coordinates. One can use a simple trick. Assume that they do not depend at all on imbedding space coordinates, derive field equations, and replaced partial derivatives by covariant derivatives at the end. Covariant derivative means covariance with respect to both space-time and imbedding space vector indices for the tensorial quantities involved. The trick works because imbedding space metric and Kähler form are covariantly constant quantities.

The integral of Tαkαδ hk decomposes to two parts.

  1. The first term, whose vanishing gives rise to field equations, is integral of

    Dα Tαk δ hk .

  2. The second term is integral of

    α (Tαk δ hk) .

    This term reduces as a total divergence to a 3-D surface integral over the boundary of the region of fixed signature of the induced metric consisting of the ends of CD and wormhole throat orbits (boundary of region with fixed signature of induced metric). This term vanishes if the normal components Tnk of canonical momentum currents vanishes at the boundary like region.

In the sequel the boundary terms are discussed explicitly and it will be found that their treatment indeed involves highly non-trivial physics.

Boundary conditions at boundaries of CD

In positive energy ontology one would formulate boundary conditions as initial conditions by fixing both the 3-surface and associated canonical momentum densities at either end of CD (positions and momenta of particles in mechanics). This would bring asymmetry between boundaries of CD.

In TGD framework one must carefully consider the boundary conditions at the boundaries of CDs. What is clear that the time-like boundary contributions from the boundaries of CD to the variation must vanish.

  1. This is true if the variations are assumed to vanish at the ends of CD. This might be however too strong a condition.
  2. One cannot demand vanishing of Ttk (t refers to time coordinate as normal coordinate) since this would give only vacuum extremals. One could however require quantum classical correspondence for any Cartan sub-algebra of isometries, whose elements define maximal set of isometry generators. The eigenvalues of quantal variants of isometry charge assignable to second quantized induced spinors at the ends of space-time surface are equal to the classical charges. Is this actually formulation of Equivalence Principle, is not quite clear to me.
While writing this a completely new idea popped to my mind. What if one poses the vanishing of the boundary terms at boundaries of CDs as additional boundary conditions for all variations except isometries? Or perhaps for all conformal variations (conformal in TGD sense)? This would not imply vanishing of isometry charges since the variations coming from the opposite ends of CD cancel each other! It soon became clear that this would allow to meet all the challenges listed in the beginning!
  1. These conditions would realize Bohr orbitology also to ZEO approach and define what "preferred extremal" means.
  2. The conditions would be very much like super-Virasoro conditions stating that super conformal generators with non-vanishing conformal weight annihilate states or create zero norm states but no conditions are posed on generators with vanishing conformal weight (now isometries). One could indeed assume only deformations, which are local isometries assignable to the generalised conformal algebra of the δ M4+/-× CP2. For arbitrary variations one would not require the vanishing. This could be the long sought for precise formulation of super-conformal invariance at the level of classical field equations!

    It is enough co consider the weaker conditions that the conformal charges defined as integrals of corresponding Noether currents vanish. These conditions would be direct equivalents of quantal conditions.

  3. The natural interpretation would be as a fixing of conformal gauge. This fixing would be motivated by the fact that WCW Kähler metric must possess isometries associated with the conformal algebra and can depend only on the tangent data at partonic 2-surfaces as became clear already for more than two decades ago. An alternative, non-practical option would be to allow all 3-surfaces at the ends of CD: this would lead to the problem of eliminating the analog of the volume of gauge group from the functional integral.
  4. The conditions would also define precisely the notion of holography and its reduction to strong form of holography in which partonic 2-surfaces and their tangent space data code for the dynamics.

Needless to say, the modification of this approach could make sense also at partonic orbits.

Isometry charges are complex

One must be careful also at the light-like 3-surfaces (orbits of wormhole throats) at which the induced metric changes its signature.

  1. Should one assume that det(g4)1/2 is imaginary in Minkowskian and real in Euclidian region? For Kähler action this is sensible and Euclidian region would give a real negative contribution giving rise to exponent of Kähler function of WCW ("world of classical worlds") making the functional integral convergent. Minkowskian regions would give imaginary contribution to the exponent causing interference effects absolutely essential in quantum field theory. This contribution would correspond to Morse function for WCW.

    The implication would be that the classical four-momenta in Euclidian/Minkowskian regions are imaginary/real. What could the interpretation be? Should one accept as a fact that four-momenta are complex.

  2. Twistor approach to TGD is now in quite good shape. M4× CP2 is the unique choice is one requires that the Cartesian factors allow twistor space with Kähler structure and classical TGD allows twistor formulation.

    In the recent formulation the fundamental fermions are assumed to propagate with light-like momenta along wormhole throats. At gauge theory limit particles must have massless or massive four-momenta. One can however also consider the possibility of complex massless momenta and in the standard twistor approach on mass shell massless particles appearing in graphs indeed have complex momenta. These complex momenta should by quantum classical correspondence correspond directly to classical complex momenta.

  3. A funny question popping in mind is whether the massivation of particles could be such that the momenta remain massless in complex sense! The complex variant of light-likeness condition would be

    p2Re= p2Im , pRe• pIm=0 .

    Could one interpret p2Im as the mass squared of the particle? Or could p2Im code for the decay width of an unstable particle?

Boundary conditions at the wormhole throat orbits and connection with quantum criticality and hierarchy of Planck constants defining dark matter hierarchy

The contributions from the orbits of wormhole throats are singular since the contravariant form of the induced metric develops components which are infinite (det(g4)=0). The contributions are real at Euclidian side of throat orbit and imaginary at the Minkowskian side so that they must be treated as independently.

  1. One can consider the possibility that under rather general conditions the normal components Tnkdet(g4) 1/2 approach to zero at partonic orbits since det(g4) is vanishing. Note however the appearance of contravariant appearing twice as index raising operator in Kähler action. If so, the vanishing of Tnkdet(g4) 1/2 need not fix completely the "boundary" conditions. In fact, I assign to the wormhole throat orbits conformal gauge symmetries so that just this is expected on physical grounds.
  2. Generalized conformal invariance would suggest that the variations defined as integrals of Tnkdet(g4) 1/2δ hk vanish in a non-trivial manner for the conformal algebra associated with the light-like wormhole throats with deformations respecting det(g4)=0 condition. Also the variations defined by infinitesimal isometries (zero conformal weight sector) should vanish since otherwise one would lose the conservation laws for isometry charges. The conditions for isometries might reduce to Tnkdet(g4) 1/2→ 0 at partonic orbits. Also now the interpretaton would be in terms of fixing of conformal gauge.
  3. Even Tnkdet(g4) 1/2=0 condition need not fix the partonic orbit completely. The Gribov ambiguity meaning that gauge conditions do not fix uniquely the gauge potential could have counterpart in TGD framework. It could be that there are several conformally non-equivalent space-time surfaces connecting 3-surfaces at the opposite ends of CD.

    If so, the boundary values at wormhole throats orbits could matter to some degree: very natural in boundary value problem thinking but new in initial value thinking. This would conform with the non-determinism of Kähler action implying criticality and the possibility that the 3-surfaces at the ends of CD are connected by several space-time surfaces which are physically non-equivalent.

    The hierarchy of Planck constants assigned to dark matter, quantum criticality and even criticality indeed relies on the assumption that heff=n× h corresponds to n-fold coverings having n space-time sheets which coincide at the ends of CD and that conformal symmetries act on the sheets as gauge symmetries. One would have as Gribov copies n conformal equivalence classes of wormhole throat orbits and corresponding space-time surfaces. Depending on whether one fixes the conformal gauge one has n equivalence classes of space-time surfaces or just one representative from each conformal equivalent class.

  4. There is also the question about the correspondence with the weak form of electric magnetic duality. This duality plus the condition that jαAα=0 in the interior of space-time surface imply the reduction of Kähler action to Chern-Simons terms. This would suggest that the boundary variation of the Kähler action reduces to that for Chern-Simons action which is indeed well-defined for light-like 3-surfaces.

    If so, the gauge fixing would reduce to variational equations for Chern-Simons action! A weaker condition is that classical conformal charges vanish. This would give a nice connection to the vision about TGD as almost topological QFT. In TGD framework these conditions do not imply the vanishing of Kähler form at boundaries. The conditions are satisfied if the CP2 projection of the partonic orbit is 2-D: the reason is that Chern-Simons term vanishes identically in this case.

  5. A further intuitively natural hypothesis is that there is a breaking of conformal symmetry: only the generators of conformal sub-algebra with conformal weight multiple of n act as gauge symmetries. This would give infinite hierarchies of breakings of conformal symmetry interpreted in terms of criticality: in the hierarchy the integers ni would satisfy ni divides ni+1.

    Similar degeneracy would be associated with the space-like ends at CD boundaries and I have considered the possibility that the integer n appearing in heff has decomposition n=n1n2 corresponding to the degeneracies associated with the two kinds of boundaries. Alternatively, one could have just n=n1=n2 from the condition that the two conformal symmetries are 3-dimensional manifestations of single 4-D analog of conformal symmetry.

As should have become clear, the derivation of field equations in TGD framework is not just an application of a formal recipe as in field theories and a lot of non-trivial physics is involved!

See the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article The vanishing of conformal charges as a gauge conditions selecting preferred extremals of Kähler action.

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