How unique is the choice of the action defining WCW Kähler metric? The problem is that twistor lift strongly suggests the identification of the preferred extremals as 4-D surfaces having 4-D generalization of complex structure and that a large number of general coordinate invariant actions constructible in terms of the induced geometry have the same preferred extremals.
1. Could twistor lift fix the choice of the action uniquely?
The twistor lift of TGD (see this, this , this , and this ) generalizes the notion of induction to the level of twistor fields and leads to a proposal that the action is obtained by dimensional reduction of the action having as its preferred extremals the counterpart of twistor space of the space-time surface identified as 6-D surface in the product T(M4)× T(CP2) twistor spaces of T(M4) and T(CP2) of M4 and CP2. Only M4 and CP2 allow a twistor space with Kähler structure (see this) so that TGD would be unique. Dimensional reduction is forced by the condition that the 6-surface has S2-bundle structure characterizing twistor spaces and the base space would be the space-time surface.
- Dimensional reduction of 6-D Kähler action implies that at the space-time level the fundamental action can be identified as the sum of Kähler action and volume term (cosmological constant). Other choices of the action do not look natural in this picture although they would have the same preferred extremals.
- Preferred extremals are proposed to correspond to minimal surfaces with singularities such that they are also extremals of 4-D Kähler action outside the singularities. The physical analogue are soap films spanned by frames and one can localize the violation of the strict determinism and of strict holography to the frames.
- The preferred extremal property is realized as the holomorphicity characterizing string world sheets, which generalizes to the 4-D situation. This in turn implies that the preferred extremals are the same for any general coordinate invariant action defined on the induced gauge fields and induced metric apart from possible extremals with vanishing CP2 Kähler action.
For instance, 4-D Kähler action and Weyl action as the sum of the tensor squares of the components of the Weyl tensor of CP2 representing quaternionic imaginary units constructed from the Weyl tensor of CP2 as an analog of gauge field would have the same preferred extremals and only the definition of Kähler function and therefore Kähler metric of WCW would change. One can even consider the possibility that the volume term in the 4-D action could be assigned to the tensor square of the induced metric representing a quaternionic or octonionic real unit.
Unique action is not the only way to achieve this. One cannot exclude the possibility that the Kähler gauge potential of WCW in the complex coordinates of WCW differs only by a complex gradient of a holomorphic function for different actions so that they would give the same Kähler form for WCW. This gradient is induced by a symplectic transformation of WCW inducing a U(1) gauge transformation. The Kähler metric is the same if the symplectic transformation is an isometry.
Symplectic transformations of WCW could give rise to inequivalent representations of the theory in terms of action at space-time level. Maybe the length scale dependent coupling parameters of an effective action could be interpreted in terms of a choice of WCW Kähler function, which maximally simplifies the computations at a given scale.
- The 6-D analogues of electroweak action and color action reducing to Kähler action in 4-D case exist. The 6-D analog of Weyl action based on the tensor representation of quaternionic imaginary units does not however exist. One could however consider the possibility that only the base space of twistor space T(M4) and T(CP2) have quaternionic structure.
- Kähler action has a huge vacuum degeneracy, which clearly distinguishes it from other actions. The presence of the volume term removes this degeneracy. However, for minimal surfaces having CP2 projections, which are Lagrangian manifolds and therefore have a vanishing induced Kähler form, would be preferred extremals according to the proposed definition. For these 4-surfaces, the existence of the generalized complex structure is dubious.
For the electroweak action, the terms corresponding to charged weak bosons eliminate these extremals and one could argue that electroweak action or its sum with the analogue of color action, also proportional Kähler action, defines the more plausible choice. Interestingly, also the neutral part of electroweak action is proportional to Kähler action.
The minimal possibility is that the M4 Kähler form vanishes: one can have a different representation of the Kähler gauge potential for it obtained as generalization of symplectic transformations acting non-trivially in M4. The recent picture about the second quantization of spinors of M4× CP2 assumes however non-trivial Kähler structure in M4.
2. Two paradoxes
TGD view leads to two apparent paradoxes.
- If the preferred extremals satisfy 4-D generalization of holomorphicity, a very large set of actions gives rise to the same preferred extremals unless there are some additional conditions restricting the number of preferred extremals for a given action.
- WCW metric has an infinite number of zero modes, which appear as parameters of the metric but do not contribute to the line element. The induced Kähler form depends on these degrees of freedom. The existence of the Kähler metric requires maximal isometries, which suggests that the Kähler metric is uniquely fixed apart from a conformal scaling factor \Omega depending on zero modes. This cannot be true: galaxy and elementary particle cannot correspond to the same Kähler metric.
2.1 The hierarchy subalgebras of supersymplectic algebra implies the decomposition of WCW into sectors with different actions
Supersymplectic algebra of δ M4+× CP2 is assumed to act as isometries of WCW (see this). There are also other important algebras but these will not be discussed now.
- The symplectic algebra A of δ M4+× CP2 has the structure of a conformal algebra in the sense that the radial conformal weights with non-negative real part, which is half integer, label the elements of the algebra have an interpretation as conformal weights.
The super symplectic algebra A has an infinite hierarchy of sub-algebras (see this) such that the conformal weights of sub-algebras An(SS) are integer multiples of the conformal weights of the entire algebra. The superconformal gauge conditions are weakened. Only the subalgebra An(SS) and the commutator [An(SS),A] annihilate the physical states. Also the corresponding classical Noether charges vanish for allowed space-time surfaces.
This weakening makes sense also for ordinary superconformal algebras and associated Kac-Moody algebras. This hierarchy can be interpreted as a hierarchy symmetry breakings, meaning that sub-algebra An(SS) acts as genuine dynamical symmetries rather than mere gauge symmetries. It is natural to assume that the super-symplectic algebra A does not affect the coupling parameters of the action.
- The generators of A correspond to the dynamical quantum degrees of freedom and leave the induced Kähler form invariant. They affect the induced space-time metric but this effect is gravitational and very small for Einsteinian space-time surfaces with 4-D M4 projection.
The number of dynamical degrees of freedom increases with n(SS). Therefore WCW decomposes into sectors labelled by n(SS) with different numbers of dynamical degrees of freedom so that their Kähler metrics cannot be equivalent and cannot be related by a symplectic isometry. They can correspond to different actions.
The number theoretical vision leads to the same conclusion as the hierarchy of HFFs. The number theoretic vision of TGD based on M8-H duality (see this) predicts a hierarchy with levels labelled by the degrees n(P) of rational polynomials P and corresponding extensions of rationals characterized by Galois groups and by ramified primes defining p-adic length scales.
These sequences allow us to imagine several discrete coupling constant evolutions realized at the level H in terms of action whose coupling parameters depend on the number theoretic parameters.
2.2.1 Coupling constant evolution with respect to n(P)
The first coupling constant evolution would be with respect to n(P).
- The coupling constants characterizing action could depend on the degree n(P) of the polynomial defining the space-time region by M8-H duality. The complexity of the space-time surface would increase with n(P) and new degrees of freedom would emerge as the number of the rational coefficients of P.
- This coupling constant evolution could naturally correspond to that assignable to the inclusion hierarchy of hyperfinite factors of type II1 (HFFs). I have indeed proposed (see this) that the degree n(P) equals to the number n(braid) of braids assignable to HFF for which super symplectic algebra subalgebra An(SS) with radial conformal weights coming as n(SS)-multiples of those of entire algebra A. One would have n(P)= n(braid)=n(SS). The number of dynamical degrees of freedom increases with n which just as it increases with n(P) and n(SS).
- The actions related to different values of n(P)=n(braid)=n(SS) cannot define the same Kähler metric since the number of allowed space-time surfaces depends on n(SS).
WCW could decompose to sub-WCWs corresponding to different actions, a kind of theory space. These theories would not be equivalent. A possible interpretation would be as a hierarchy of effective field theories.
- Hierarchies of composite polynomials define sequences of polynomials with increasing values of n(P) such that the order of a polynomial at a given level is divided by those at the lower levels. The proposal is that the inclusion sequences of extensions are realized at quantum level as inclusion hierarchies of hyperfinite factors of type II1.
A given inclusion hierarchy corresponds to a sequence n(SS)i such that n(SS)i divides n(SS)i+1. Therefore the degree of the composite polynomials increases very rapidly. The values of n(SS)i can be chosen to be primes and these primes correspond to the degrees of so called prime polynomials (see this) so that the decompositions correspond to prime factorizations of integers. The "densest" sequence of this kind would come in powers of 2 as n(SS)i= 2i. The corresponding p-adic length scales (assignable to maximal ramified primes for given n(SS)i) are expected to increase roughly exponentially, say as 2r2i. r=1/2 would give a subset of scales 2r/2 allowed by the p-adic length scale hypothesis. These transitions would be very rare.
A theory corresponding to a given composite polynomial would contain as sub-theories the theories corresponding to lower polynomial composites. The evolution with respect to n(SS) would correspond to a sequence of phase transitions in which the action genuinely changes. For instance, color confinement could be seen as an example of this phase transition.
- A subset of p-adic primes allowed by the p-adic length scale hypothesis p≈ 2k defining the proposed p-adic length scale hierarchy could relate to nS changing phase transition. TGD suggests a hierarchy of hadron physics corresponding to a scale hierarchy defined by Mersenne primes and their Gaussian counterparts (see this and this). Each of them would be characterized by a confinement phase transition in which nS and therefore also the action changes.
For a given value of n(P), one could have coupling constant sub-evolutions with respect to the set of ramified primes of P and dimensions n=heff/h0 of algebraic extensions. The action would only change by U(1) gauge transformation induced by a symplectic isometry of WCW. Coupling parameters could change but the actions would be equivalent.
The choice of the action in an optimal manner in a given scale could be seen as a choice of the most appropriate effective field theory in which radiative corrections would be taken into account. One can interpret the possibility to use a single choice of coupling parameters in terms of quantum criticality.
The range of the p-adic length scales labelled by ramified primes and effective Planck constants heff/h0 is finite for a given value of n(SS).
The first coupling constant evolution of this kind corresponds to ramified primes defining p-adic length scales for given n(SS).
- Ramified primes are factors of the discriminant D(P) of P, which is expressible as a product of non-vanishing root differents and reduces to a polynomial of the n coefficients of P. Ramified primes define p-adic length scales assignable to the particles in the amplitudes scattering amplitudes defined by zero energy states.
P would represent the space-time surface defining an interaction region in N--particle scattering. The N ramified primes dividing D(P) would characterize the p-adic length scales assignable to these particles. If D(P) reduces to a single ramified prime, one has elementary particle this), and the forward scattering amplitude corresponds to the propagator.
This would give rise to a multi-scale p-adic length scale evolution of the amplitudes analogous to the ordinary continuous coupling constant evolution of n-point scattering amplitudes with respect to momentum scales of the particles. This kind of evolutions extend also to evolutions with respect to n(SS).
- physical constraints require that n(P) and the maximum size of the ramified prime of P correlate (see this).
A given rational polynomial of degree n(P) can be always transformed to a polynomial with integer coefficients. If the integer coefficients are smaller than n(P), there is an upper bound for the ramified primes. This assumption also implies that finite fields become fundamental number fields in number theoretical vision (see this).
- p-Adic length scale hypothesis (see this) in its basic form states that there exist preferred primes p≈ 2k near some powers of 2. A more general hypothesis states that also primes near some powers of 3 possibly also other small primes are preferred physically. The challenge is to understand the origin of these preferred scales.
For polynomials P with a given degree n(P) for which discriminant D(P) is prime, there exists a maximal ramified prime. Numerical calculations suggest that the upper bound depends exponentially on n(P).
Could these maximal ramified primes satisfy the p-adic length scale hypothesis or its generalization? The maximal prime defines a fixed point of coupling constant evolution in accordance with the earlier proposal. For instance, could one think that one has p≈ 2k, k= n(SS)? Each p-adic prime would correspond to a p-adic coupling constant sub-evolution representable in terms of symplectic isometries.
2.3 Number theoretic discretization of WCW and maxima of WCW Kähler function
Number theoretic approach involves a unique discretization of space-time surface and also of WCW. The question is how the points of the discretized WCW correspond to the preferred extremals.
- The exponents of Kähler function for the maxima of Kähler function, which correspond to the universal preferred extremals, appear in the scattering amplitudes. The number theoretical approach involves a unique discretization of space-time surfaces defining the WCW coordinates of the space-time surface regarded as a point of WCW.
In (see this ) it is assumed that these WCW points appearing in the number theoretical discretization correspond to the maxima of the Kähler function. The maxima would depend on the action and would differ for ghd maxima associated with different actions unless they are not related by symplectic WCW isometry.
- The symplectic transformations of WCW acting as isometries are assumed to be induced by the symplectic transformations of δ M4+× CP2 (see this and this). As isometries they would naturally permute the maxima with each other.
For a summary of earlier postings see Latest progress in TGD.