Machian Principle and TGD
Machian Principle has not played any role in the
development of TGD. Hence it is somewhat surprising
that this principle allows several interpretations
in TGD framework.
1. Nonconserved gravitational fourmomentum and conserved inertial momentum at 4D spacetime level
Consider first the situation at the level of classical theory identifiable in terms of classical dynamics for spacetime surfaces.
 In TGD framework one must distinguish between nonconserved gravitational fourmomentum and conserved inertial fourmomentum identified as conserved Poincare fourmomentum at the level of 4D spacetime dynamics and associated with the preferred extremals of Kähler action defining the analogs of Bohr orbits (no path integral over all possible spacetime surfaces but functional integral over lightlike partonic 3surfaces). A collection of conserved vector currents rather than tensor results and this resolves the problems due to illdefinedness of fourmomentum in General Relativity which served as the primary motivation for the identification of spacetimes as 4surfaces of H=M^{4}×CP_{2}.
 Nonconserved gravitational fourmomentum densities can be identified as a linear combination of Einstein tensor and metric tensor (cosmological constant) by contracting them with the Killing vectors of M^{4} translations. Collection of, in general nonconserved, 4currents result but gravitational fourmomentum is welldefined 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 fourmomentum is conserved.
2. Inertial fourmomentum as the average of gravitational fourmomentum
The first question is how nonconserved gravitational and conserved inertial fourmomentum 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 geometrotemporal 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 spacetime sheets with positive and negative time orientation suggests that the two spacetime 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 fourmomentum assignable to the positive energy part of the state is the average of the nonconserved gravitational four momentum and depends on the padic length scale characterizing the pair of spacetime sheets connecting positive and negative energy states. The average is over a padic time scale characterizing the temporal span of the spacetime sheet. This average is coded by the classical dynamics for the preferred extremal of Kähler action defining the generalized Bohr orbit.
3. Nonconserved gravitational fourmomentum and conserved inertial momentum at parton level
A deeper level description of the situation is achieved at parton level. For lightlike partonic 3surfaces the dynamics is defined by almost topological QFT defined by ChernSimons action for the induced Kähler form. The extrema have 2D CP_{2} projection. Lightlikeness implies the replacement of "topological" with "almost topological" by bringing in the notions of metric and fourmomentum.
 The world of classical worlds (WCW) decomposes into a union of subWCW:s associated with preferred points of imbedding space H= M^{4}_{+/}× CP_{2}. The selection of preferred point of H means means a selection of tip of future/past directed lightcone in the case of M^{4}_{+/} and selection of U(2) subgroup of SU(3) in the case of CP_{2}. There is a further selection fixing rest system and angular momentum quantization axis (preferred plane in M^{4} defining nonphysical polarizations for massless bosons) and quantization axis of color isospin and hypercharge. That configuration space geometry reflects these choices conforms with quantumclassical correspondence requiring that everything quantal has a geometric correlate.
 At the level of Smatrix the preferred points of H defining past/future directed lightcones correspond to the arguments of npoint function. In the construction of Smatrix one integrates over the tips of the lightcones parameterizing subWCW:s consisting of partonic 3surfaces residing inside these lightcones (×CP_{2}). Hence a full Poincare invariance results meaning the emergence of conserved fourmomentum identifiable as inertial fourmomentum assignable to the preferred extremals of Kähler action defining Bohr orbits. These lightcones give rise to Russian doll cosmology with cosmologies within cosmologies such that elementary p"/public_html/articles/ formally correspond to the lowest level in the hierarchy.
 Parton dynamics is associated with a given future/past lightcone. At parton level one has Lorentz invariance and only the mass squared is conserved for the partonic time evolution dictated by random lightlikeness. There is a very delicate point involved here. Partonic fourmomentum is nonvanishing only if CP_{2} Kähler gauge potential has also M^{4}+/ 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 lightcone and if its magnitude is constant. From the point of view of spinor structure M^{4}_{+/} and CP_{2} 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 4D spin glass degeneracy.
 The natural identification of the conserved classical partonic fourmomentum is as the nonconserved gravitational fourmomentum defined for a spacetime sheet characterized by a padic 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 fourmomenta.
4. Inertial fourmomentum as average of partonic fourmomentum and padic thermodynamics
 The natural hypothesis is that inertial fourmomentum at partonic level is the temporal average of nonconserved gravitational fourmomentum. This implies particle massivation in general since the motion of lightlike 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 padic time scale defining the mass scale via Uncertainty Principle. There is actually hierarchy of padic time scales coming as powers of p. Inertial mass vanishes only if the motion is nonrandom in the time scale considered and this never occurs exactly for even photon and graviton.
 The quantitative formulation of the averaging relies on padic 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: padic thermodynamics based on standard Boltzman weight e^{L0/T} does not make sense since exp(x) has always unit padic norm so that partition sum does not converge. One can however replace this Boltzman weight with p^{L0/Tp}, which exists for T_{p}=1/n, n=1,2,..., if L_{0} is a generator of conformal scaling having nonnegative integer spectrum. This predicts a discrete spectrum of padic 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 nonnegligible contributions to thermal average) and a prescription for the mapping of padic 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 fourmomentum squared so that Lorentz invariance is not lost. Note also that in the construction of Smatrix fourmomenta emerge only via the Fourier transform of npoint 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 spacetime correlate appearing in the modified Dirac operator. A good candidate is padic 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 timelike entanglement for the zero energy quantum state can be also thermal Smatrix with respect to the incoming and outgoing partons (hyperfinite factors of type III allow the analog of thermal QFT at the level of quantum states). This allows also a first principle description of padic thermodynamics.
5. Various interpretations of Machian Principle
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 nonsensible. Free CP_{2} type extremal (having random lightlike curve as M^{4}projection) 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 spacetime 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 padic physics. Most points of padic spacetime sheets have infinite distance from the tip lightcone in the real sense. The discrete algebraic intersection of the padic spacetime sheet with the real spacetime sheet gives rise to effective padicity of the topology of the real spacetime sheet if the number of these points is large enough. Hence padic thermodynamics with given p also assigned to the partonic 3surface by the modified Dirac operator makes sense. The continuity and smoothness of the dynamics corresponds to the padic fractality and long range correlations for the real dynamics and allows to apply padic thermodynamics in the real context. pAdic variant of Machian Principle says that padic 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 spacetime point is so rich due to the presence of infinite hierarchy of real units obtained as ratios of infinite integers that single spacetime 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 spacetime point are really fundamental.
For more details see the end of the chapter The Relationship Between TGD and GRT of "Classical Physics in ManySheeted SpaceTime".
