The most recent vision about zero energy ontology and p-adicization
The generalization of the number concept obtained by fusing real and p-adics along rationals and common algbraics is the basic philosophy behind p-adicization. This however requires that it is possible to speak about rational points of the imbedding space and the basic objection against the notion of rational points of imbedding space common to real and various p-adic variants of the imbedding space is the necessity to fix some special coordinates in turn implying the loss of a manifest general coordinate invariance. The isometries of the imbedding space could save the situation provided one can identify some special coordinate system in which isometry group reduces to its discrete subgroup. The loss of the full isometry group could be compensated by assuming that WCW is union over sub-WCW:s obtained by applying isometries on basic sub-WCW with discrete subgroup of isometries.
The combination of zero energy ontology realized in terms of a hierarchy causal diamonds and hierarchy of Planck constants providing a description of dark matter and leading to a generalization of the notion of imbedding space suggests that it is possible to realize this dream. The article TGD: What Might be the First Principles? provides a brief summary about recent state of quantum TGD helping to understand the big picture behind the following considerations.
1. Zero energy ontology briefly
- The basic construct in the zero energy ontology is the space CD×CP2, where the causal diamond CD is defined as an intersection of future and past directed light-cones with time-like separation between their tips regarded as points of the underlying universal Minkowski space M4. In zero energy ontology physical states correspond to pairs of positive and negative energy states located at the boundaries of the future and past directed light-cones of a particular CD. CD:s form a fractal hierarchy and one can glue smaller CD:s within larger CD along the upper light-cone boundary along a radial light-like ray: this construction recipe allows to understand the asymmetry between positive and negative energies and why the arrow of experienced time corresponds to the arrow of geometric time and also why the contents of sensory experience is located to so narrow interval of geometric time. One can imagine evolution to occur as quantum leaps in which the size of the largest CD in the hierarchy of personal CD:s increases in such a manner that it becomes sub-CD of a larger CD. p-Adic length scale hypothesis follows if the values of temporal distance T between tips of CD come in powers of 2n. All conserved quantum numbers for zero energy states have vanishing net values. The interpretation of zero energy states in the framework of positive energy ontology is as physical events, say scattering events with positive and negative energy parts of the state interpreted as initial and final states of the event.
- In the realization of the hierarchy of Planck constants CD×CP2 is replaced with a Cartesian product of book like structures formed by almost copies of CD:s and CP2:s defined by singular coverings and factors spaces of CD and CP2 with singularities corresponding to intersection M2�CD and homologically trivial geodesic sphere S2 of CP2 for which the induced Kähler form vanishes. The coverings and factor spaces of CD:s are glued together along common M2�CD. The coverings and factors spaces of CP2 are glued together along common homologically non-trivial geodesic sphere S2. The choice of preferred M2 as subspace of tangent space of X4 at all its points and having interpretation as space of non-physical polarizations, brings M2 into the theory also in different manner. S2 in turn defines a subspace of the much larger space of vacuum extremals as surfaces inside M4×S2.
- Configuration space (the world of classical worlds, WCW) decomposes into a union of sub-WCW:s corresponding to different choices of M2 and S2 and also to different choices of the quantization axes of spin and energy and and color isospin and hyper-charge for each choice of this kind. This means breaking down of the isometries to a subgroup. This can be compensated by the fact that the union can be taken over the different choices of this subgroup.
- p-Adicization requires a further breakdown to discrete subgroups of the resulting sub-groups of the isometry groups but again a union over sub-WCW:s corresponding to different choices of the discrete subgroup can be assumed. Discretization relates also naturally to the notion of number theoretic braid.
Consider now the critical questions.
- Very naively one could think that center of mass wave functions in the union of sectors could give rise to representations of Poincare group. This does not conform with zero energy ontology, where energy-momentum should be assignable to say positive energy part of the state and where these degrees of freedom are expected to be pure gauge degrees of freedom. If zero energy ontology makes sense, then the states in the union over the various copies corresponding to different choices of M2 and S2 would give rise to wave functions having no dynamical meaning. This would bring in nothing new so that one could fix the gauge by choosing preferred M2 and S2 without losing anything. This picture is favored by the interpretation of M2 as the space of longitudinal polarizations.
- The crucial question is whether it is really possible to speak about zero energy states for a given sector defined by generalized imbedding space with fixed M2 and S2. Classically this is possible and conserved quantities are well defined. In quantal situation the presence of the lightcone boundaries breaks full Poincare invariance although the infinitesimal version of this invariance is preserved. Note that the basic dynamical objects are 3-D light-like "legs" of the generalized Feynman diagrams.
2. Definition of energy inzero energy ontology
Can one then define the notion of energy for positive and negative energy parts of the state? There are two alternative approaches depending on whether one allows or does not allow wave-functions for the positions of tips of light-cones.
Consider first the naive option for which four momenta are assigned to the wave functions assigned to the tips of CD:s.
- The condition that the tips are at time-like distance does not allow separation to a product but only following kind of wave functions
Ψ = exp(ip�m)Θ(m2) Θ(m0)× Φ(p) , m=m+-m-.
Here m+ and m- denote the positions of the light-cones and Q denotes step function. F denotes configuration space spinor field in internal degrees of freedom of 3-surface. One can introduce also the decomposition into particles by introducing sub-CD:s glued to the upper light-cone boundary of CD.
- The first criticism is that only a local eigen state of 4-momentum operators p� = (h/2p) �/i is in question everywhere except at boundaries and at the tips of the CD with exact translational invariance broken by the two step functions having a natural classical interpretation. The second criticism is that the quantization of the temporal distance between the tips to T = 2kT0 is in conflict with translational invariance and reduces it to a discrete scaling invariance.
The less naive approach relies of super conformal structures of quantum TGD assumes fixed value of T and therefore allows the crucial quantization condition T=2kT0.
- Since light-like 3-surfaces assignable to incoming and outgoing legs of the generalized Feynman diagrams are the basic objects, can hope of having enough translational invariance to define the notion of energy. If translations are restricted to time-like translations acting in the direction of the future (past) then one has local translation invariance of dynamics for classical field equations inside dM4� as a kind of semigroup. Also the M4 translations leading to interior of X4 from the light-like 2-surfaces surfaces act as translations. Classically these restrictions correspond to non-tachyonic momenta defining the allowed directions of translations realizable as particle motions. These two kinds of translations have been assigned to super-canonical conformal symmetries at dM4�×CP2 and and super Kac-Moody type conformal symmetries at light-like 3-surfaces. Equivalence Principle in TGD framework states that these two conformal symmetries define a structure completely analogous to a coset representation of conformal algebras so that the four-momenta associated with the two representations are identical .
- The condition selecting preferred extremals of Kähler action is induced by a global selection of M2 as a plane belonging to the tangent space of X4 at all its points . The M4 translations of X4 as a whole in general respect the form of this condition in the interior. Furthermore, if M4 translations are restricted to M2, also the condition itself - rather than only its general form - is respected. This observation, the earlier experience with the p-adic mass calculations, and also the treatment of quarks and gluons in QCD encourage to consider the possibility that translational invariance should be restricted to M2 translations so that mass squared, longitudinal momentum and transversal mass squared would be well defined quantum numbers. This would be enough to realize zero energy ontology. Encouragingly, M2 appears also in the generalization of the causal diamond to a book-like structure forced by the realization of the hierarchy of Planck constant at the level of the imbedding space.
- That the cm degrees of freedom for CD would be gauge like degrees of freedom sounds strange. The paradoxical feeling disappears as one realizes that this is not the case for sub-CDs, which indeed can have non-trivial correlation functions with either upper or lower tip of the CD playing a role analogous to that of an argument of n-point function in QFT description. One can also say that largest CD in the hierarchy defines infrared cutoff.
3. p-Adic variants of the imbedding space
Consider now the construction of p-adic variants of the imbedding space.
- Rational values of p-adic coordinates are non-negative so that light-cone proper time a4,+=�(t2-z2-x2-y2) is the unique Lorentz invariant choice for the p-adic time coordinate near the lower tip of CD. For the upper tip the identification of a4 would be a4,-=�((t-T)2-z2-x2-y2). In the p-adic context the simultaneous existence of both square roots would pose additional conditions on T. For 2-adic numbers T=2nT0, n � 0 (or more generally T=�k � n0bk 2k), would allow to satisfy these conditions and this would be one additional reason for T=2nT0 implying p-adic length scale hypothesis. The remaining coordinates of CD are naturally hyperbolic cosines and sines of the hyperbolic angle h�,4 and cosines and sines of the spherical coordinates q and f.
- The existence of the preferred plane M2 of un-physical polarizations would suggest that the 2-D light-cone proper times a2,+ = �(t2-z2) a2,- = �((t-T)2-z2) can be also considered. The remaining coordinates would be naturally h�,2 and cylindrical coordinates (r,f).
- The transcendental values of a4 and a2 are literally infinite as real numbers and could be visualized as points in infinitely distant geometric future so that the arrow of time might be said to emerge number theoretically. For M2 option p-adic transcendental values of r are infinite as real numbers so that also spatial infinity could be said to emerge p-adically.
- The selection of the preferred quantization axes of energy and angular momentum unique apart from a Lorentz transformation of M2 would have purely number theoretic meaning in both cases. One must allow a union over sub-WCWs labeled by points of SO(1,1). This suggests a deep connection between number theory, quantum theory, quantum measurement theory, and even quantum theory of mathematical consciousness.
- In the case of CP2 there are three real coordinate patches involved . The compactness of CP2 allows to use cosines and sines of the preferred angle variable for a given coordinate patch.
ξ1= tan(u)× cos(Θ/2)× exp(i(Ψ+Φ)/2) ,
ξ2= tan(u)× sin(Θ/2)× exp(i(Ψ-Φ)/2).
The ranges of the variables u,Q, F,Y are [0,p/2],[0,p],[0,4p],[0,2p] respectively. Note that u has naturally only the positive values in the allowed range. S2 corresponds to the values F = Y = 0 of the angle coordinates.
- The rational values of the (hyperbolic) cosine and sine correspond to Pythagorean triangles having sides of integer length and thus satisfying m2 = n2+r2 (m2=n2-r2). These conditions are equivalent and allow the well-known explicit solution . One can construct a p-adic completion for the set of Pythagorean triangles by allowing p-adic integers which are infinite as real integers as solutions of the conditions m2=r2�s2. These angles correspond to genuinely p-adic directions having no real counterpart. Hence one obtains p-adic continuum also in the angle degrees of freedom. Algebraic extensions of the p-adic numbers bringing in cosines and sines of the angles p/n lead to a hierarchy increasingly refined algebraic extensions of the generalized imbedding space. Since the different sectors of WCW directly correspond to correlates of selves this means direct correlation with the evolution of the mathematical consciousness. Trigonometric identities allow to construct points which in the real context correspond to sums and differences of angles.
- Negative rational values of the cosines and sines correspond as p-adic integers to infinite real numbers and it seems that one use several coordinate patches obtained as copies of the octant (x � 0,y � 0,z � 0,). An analogous picture applies in CP2 degrees of freedom.
- The expression of the metric tensor and spinor connection of the imbedding in the proposed coordinates makes sense as a p-adic numbers in the algebraic extension considered. The induction of the metric and spinor connection and curvature makes sense provided that the gradients of coordinates with respect to the internal coordinates of the space-time surface belong to the extensions. The most natural choice of the space-time coordinates is as subset of imbedding space-coordinates in a given coordinate patch. If the remaining imbedding space coordinates can be chosen to be rational functions of these preferred coordinates with coefficients in the algebraic extension of p-adic numbers considered for the preferred extremals of Kähler action, then also the gradients satisfy this condition. This is highly non-trivial condition on the extremals and if it works might fix completely the space of exact solutions of field equations. Space-time surfaces are also conjectured to be hyper-quaternionic , this condition might relate to the simultaneous hyper-quaternionicity and Kähler extremal property. Note also that this picture would provide a partial explanation for the decomposition of the imbedding space to sectors dictated also by quantum measurement theory and hierarchy of Planck constants.
4. p-Adic variants for the sectors of WCW
One can also wonder about the most general definition of the p-adic variants of the sectors of the world of classical worlds.
- The restriction of the surfaces in question to be expressible in terms of rational functions with coefficients which are rational numbers of belong to algebraic extension of rationals means that the world of classical worlds can be regarded as a a discrete set and there would be no difference between real and p-adic worlds of classical worlds: a rather unexpected conclusion.
- One can of course whether one should perform completion also for WCWs. In real context this would mean completion of the rational number valued coefficients of a rational function to arbitrary real coefficients and perhaps also allowance of Taylor and Laurent series as limits of rational functions. In the p-adic case the integers defining rational could be allowed to become p-adic transcendentals infinite as real numbers. Also now also Laurent series could be considered.
- In this picture there would be close analogy between the structure of generalized imbedding space and WCW. Different WCW:s could be said to intersect in the space formed by rational functions with coefficients in algebraic extension of rationals just real and p-adic variants of the imbedding space intersect along rational points. In the spirit of algebraic completion one might hope that the expressions for the various physical quantities, say the value of Kähler action, Kähler function, or at least the exponent of Kähler function (at least for the maxima of Kähler function) could be defined by analytic continuation of their values from these sub-WCW to various number fields. The matrix elements for p-adic-to-real phase transitions of zero energy states interpreted as intentional actions could be calculated in the intersection of real and p-adic WCW:s by interpreting everything as real.
For details see chapters TGD as a Generalized Number Theory I: p-Adicization Program.