1. Non-conserved gravitational four-momentum and conserved inertial momentum at 4-D space-time level
Consider first the situation at the level of classical theory identifiable in terms of classical dynamics for space-time surfaces.
- In TGD framework one must distinguish between non-conserved gravitational four-momentum and conserved inertial four-momentum identified as conserved Poincare four-momentum at the level of 4-D space-time dynamics and associated with the preferred extremals of Kähler action defining the analogs of Bohr orbits (no path integral over all possible space-time surfaces but functional integral over light-like partonic 3-surfaces). A collection of conserved vector currents rather than tensor results and this resolves the problems due to ill-definedness of four-momentum in General Relativity which served as the primary motivation for the identification of space-times as 4-surfaces of H=M4×CP2.
- Non-conserved gravitational four-momentum densities can be identified as a linear combination of Einstein tensor and metric tensor (cosmological constant) by contracting them with the Killing vectors of M4 translations. Collection of, in general non-conserved, 4-currents result but gravitational four-momentum is well-defined quite generally unlike in General Relativity. Only for the asymptotic stationary cosmologies corresponding to extremals of the curvature scalar plus constant for the induced metric gravitational four-momentum is conserved.
The first question is how non-conserved gravitational and conserved inertial four-momentum relate to each other. Certainly Equivalence Principle in a strong form cannot hold true.
- In zero energy ontology the total quantum numbers of states vanish and positive and negative energy parts of states have interpretation as initial and final states of particle reaction at elementary particle level where geometro-temporal distance between them is short (TGD inspired theory of consciousness forces to distinguish between geometric time and subjective time). Positive energy ontology emerges as an effective ontology at observational level when the temporal distance between positive and negative energy parts of the state is long as compared to the time scale of conscious observer. The recent understanding about bosons as wormhole contacts between space-time sheets with positive and negative time orientation suggests that the two space-time sheets in question correspond to positive and negative energy parts of the state. This brings in mind the picture of Connes about Higgs mechanism involving two copies of Minkowski space.
- The intuitive idea is that the conserved inertial four-momentum assignable to the positive energy part of the state is the average of the non-conserved gravitational four momentum and depends on the p-adic length scale characterizing the pair of space-time sheets connecting positive and negative energy states. The average is over a p-adic time scale characterizing the temporal span of the space-time sheet. This average is coded by the classical dynamics for the preferred extremal of Kähler action defining the generalized Bohr orbit.
A deeper level description of the situation is achieved at parton level. For light-like partonic 3-surfaces the dynamics is defined by almost topological QFT defined by Chern-Simons action for the induced Kähler form. The extrema have 2-D CP2 projection. Light-likeness implies the replacement of "topological" with "almost topological" by bringing in the notions of metric and four-momentum.
- The world of classical worlds (WCW) decomposes into a union of sub-WCW:s associated with preferred points of imbedding space H= M4+/-× CP2. The selection of preferred point of H means means a selection of tip of future/past directed light-cone in the case of M4+/- and selection of U(2) subgroup of SU(3) in the case of CP2. There is a further selection fixing rest system and angular momentum quantization axis (preferred plane in M4 defining non-physical polarizations for massless bosons) and quantization axis of color isospin and hyper-charge. That configuration space geometry reflects these choices conforms with quantum-classical correspondence requiring that everything quantal has a geometric correlate.
- At the level of S-matrix the preferred points of H defining past/future directed light-cones correspond to the arguments of n-point function. In the construction of S-matrix one integrates over the tips of the light-cones parameterizing sub-WCW:s consisting of partonic 3-surfaces residing inside these light-cones (×CP2). Hence a full Poincare invariance results meaning the emergence of conserved four-momentum identifiable as inertial four-momentum assignable to the preferred extremals of Kähler action defining Bohr orbits. These light-cones give rise to Russian doll cosmology with cosmologies within cosmologies such that elementary particles formally correspond to the lowest level in the hierarchy.
- Parton dynamics is associated with a given future/past light-cone. At parton level one has Lorentz invariance and only the mass squared is conserved for the partonic time evolution dictated by random light-likeness. There is a very delicate point involved here. Partonic four-momentum is non-vanishing only if CP2 Kähler gauge potential has also M4+/- component which is pure gauge. Mass squared is conserved (Lorentz invariance) if this component is in the direction of proper time coordinate a of the light-cone and if its magnitude is constant. From the point of view of spinor structure M4+/- and CP2 are not totally decoupled. This does not break gauge invariance since Kähler gauge potential does not give rise to U(1) gauge degeneracy but only to 4-D spin glass degeneracy.
- The natural identification of the conserved classical partonic four-momentum is as the non-conserved gravitational four-momentum defined for a space-time sheet characterized by a p-adic time scale. In accordance with zero energy ontology, a length scale dependent notion is in question. At single parton level Equivalence Principle would state that the conserved gravitational mass is equal to inertial mass but would not require equivalence of four-momenta.
- The natural hypothesis is that inertial four-momentum at partonic level is the temporal average of non-conserved gravitational four-momentum. This implies particle massivation in general since the motion of light-like parton is in general random zitterbewegung so that only mass squared is conserved. The average is defined always in some time scale identifiable as the p-adic time scale defining the mass scale via Uncertainty Principle. There is actually hierarchy of p-adic time scales coming as powers of p. Inertial mass vanishes only if the motion is non-random in the time scale considered and this never occurs exactly for even photon and graviton.
- The quantitative formulation of the averaging relies on p-adic thermodynamics for the integer valued conformal weight characterizing the particle. By number theoretic universality this description must be equivalent to real thermodynamics with quantized temperature. Quantization of the mass scale is purely number theoretical: p-adic thermodynamics based on standard Boltzman weight eL0/T does not make sense since exp(x) has always unit p-adic norm so that partition sum does not converge. One can however replace this Boltzman weight with pL0/Tp, which exists for Tp=1/n, n=1,2,..., if L0 is a generator of conformal scaling having non-negative integer spectrum. This predicts a discrete spectrum of p-adic mass scales and real thermodynamics is obtained by reversing the sign of exponent. Assuming a reasonable cutoff on conformal weight (only two lowest terms give non-negligible contributions to thermal average) and a prescription for the mapping of p-adic mass squared to its real counterpart the two descriptions are equivalent. Note that mass squared is the average of conformal weight rather than the average of four-momentum squared so that Lorentz invariance is not lost. Note also that in the construction of S-matrix four-momenta emerge only via the Fourier transform of n-point function and do not appear at fundamental vertices.
- Also the coupling to Higgs gives a contribution to the mass. Higgs corresponds to a wormhole contact with wormhole throats carrying fermion and antifermion quantum numbers as do all gauge bosons. Higgs expectation should have space-time correlate appearing in the modified Dirac operator. A good candidate is p-adic thermal average for the generalized eigenvalue of the modified Dirac operator vanishing for the zero modes. Thermal mass squared as opposed to Higgs contribution would correspond to the average of integer valued conformal weight. For bosons (in particular Higgs boson!) it is simply the sum of expectations for the two wormhole throats.
- Both contributions are basically thermal which raises the question whether the interpretation in terms of coherent state of Higgs field (and essentially quantal notion) is really appropriate unless also thermal states can be regarded as genuine quantum states. The matrix characterizing time-like entanglement for the zero energy quantum state can be also thermal S-matrix with respect to the incoming and outgoing partons (hyper-finite factors of type III allow the analog of thermal QFT at the level of quantum states). This allows also a first principle description of p-adic thermodynamics.
TGD allows several interpretations of Machian Principle and leads also to a generalization of the Principle.
- Machian Principle is true in the sense that the notion of completely free particle is non-sensible. Free CP2 type extremal (having random light-like curve as M4projection) is a pure vacuum extremal and only its topological condensation creates a wormhole throat (two of them) in the case of fermion (boson). Topological condensation to space-time sheet(s) generates all quantum numbers, not only mass. Both thermal massivation and massivation via the generation of coherent state of Higgs type wormhole contacts are due to topological condensation.
- Machian Principle has also interpretation in terms of p-adic physics. Most points of p-adic space-time sheets have infinite distance from the tip light-cone in the real sense. The discrete algebraic intersection of the p-adic space-time sheet with the real space-time sheet gives rise to effective p-adicity of the topology of the real space-time sheet if the number of these points is large enough. Hence p-adic thermodynamics with given p also assigned to the partonic 3-surface by the modified Dirac operator makes sense. The continuity and smoothness of the dynamics corresponds to the p-adic fractality and long range correlations for the real dynamics and allows to apply p-adic thermodynamics in the real context. p-Adic variant of Machian Principle says that p-adic dynamics of cognition and intentionality in literally infinite scale in the real sense dictates the values of masses among other things.
- A further interpretation of Machian Principle is in terms of number theoretic Brahman=Atman identity or equivalently, Algebraic Holography. This principle states that the number theoretic structure of the space-time point is so rich due to the presence of infinite hierarchy of real units obtained as ratios of infinite integers that single space-time point can represent the entire world of classical worlds. This could be generalized also to a criterion for a good mathematics: only those mathematical structures which are representable in the set of real units associated with the coordinates of single space-time point are really fundamental.
P.S. Anyone can some day wake up to the realization of having a scientific Archnemesis. Even the God of Old Testament can forgive but not your Archnemesis. The curse of Archnemesis follows you even from grave and to grave. What you can do is to cross your fingers and pray the help of mightier gods. For more about the topic here.