Wednesday, May 07, 2008

From Equivalence Principle to Zero Energy Ontology

The tension between analytic and constructive approaches is present even in Einstein's own theory and was the basic stimulus leading to TGD. Equivalence Principle states that gravitational and inertial masses are identical. This statement is however rather problematic as Einstein himself was first to admit. Einstein's equations express the identity for gravitational and inertial energy momentum densities as a consequence of a variational principle. There is however no global version of this statement because one cannot define the notions of inertial and gravitational four-momenta without adherence to perturbative approach.

The hypothesis that space-times are 4-D surfaces of a higher-dimensional space-time of form H=M4×S resolves the problem: since Poincare symmetries become symmetries of H rather than space-time itself. Inertial four-momentum can be defined as a conserved Noether charge and also gravitational four-momentum can be regarded as a Noether charge albeit non-conserved. Equivalence Principle can hold true only under some additional conditions. For instance, for the imbeddings of Robertson-Walker cosmologies inertial four-momentum density vanishes unlike gravitational four-momentum density, which for a long time remained quite a mystery. The real understanding of the situation became possible only after the introduction of what I call zero energy ontology.

In zero energy ontology one replaces positive energy states with zero energy states with positive and negative energy parts of the state at the boundaries of future and past direct light-cones forming a causal diamond. All conserved quantum numbers of the positive and negative energy states are of opposite sign so that these states can be created from vacuum. Äny physical state is creatable from vacuum" becomes thus a basic principle of quantum TGD and together with the notion of quantum jump resolves several philosophical problems (What was the initial state of universe?, What are the values of conserved quantities for Universe, Is theory building completely useless if only single solution of field equations is realized?).

At the level of elementary particle physics positive and negative energy parts of zero energy state are interpreted as initial and final states of a particle reaction so that quantum states become physical events. Equivalence Principle would hold true in the sense that the classical gravitational four-momentum of the vacuum extremal whose small deformations appear as the argument of configuration space spinor field is equal to the positive energy of the positive energy part of the zero energy quantum state. Equivalence Principle is expected to hold true for elementary particles and their composites but not for the quantum states defined around non-vacuum extremals.

More precisely, the inertial four-momentum assignable to the 3-D Chern-Simons action is non-vanishing only if one adds to the CP2 Kähler form a pure gauge part Aa=constant, where a denotes light cone proper time . A breaking of Poincare invariance is implied which is however compensated by the fact that configuration space corresponds to the union of configuration spaces associated with future and past directed light-cones. If the vacuum extremal is also an extremal of the curvature scalar, gravitational four-momentum is conserved.

In the case of CP2 type vacuum extremal gravitational stationarity transforms the M4 projection of the extremal from a random light-like curve to a light-like geodesic allowing an interpretation as incoming or outgoing on mass shell particle. General vacuum extremal corresponds to a virtual particle. At the classical level Equivalence Principle requires that the light-like gravitational four-momentum of CP2 vacuum extremal co-incides with the light-like inertial four-momentum associated with Chern-Simons action in this situation. This condition relates the value of Aa to gravitational constant G and CP2 radius R. G would thus appear as a fundamental constant and quantum criticality should dictate the ratio G/R2. The topologically condensed CP2 type vacuum extremal necessarily creates a non-vacuum region around it and the resulting inertial four-momentum would correspond to the gravitational four-momentum.

The strong form of Equivalence Principle would require that the classical 4-momentum associated with Kähler action of allowed small deformations co-incides with the conserved gravitational four-momentum of the vacuum extremal extremizing curvature scalar: this might have a natural interpretation in terms of Bohr orbitology but is not be consistent with zero energy ontology inspired picture unless one has double sheeted structure with sheets possessing opposite energies such that double sheeted structure is approximated by single sheet with Robertson-Walker cosmology in GRT framework. The identification of gauge bosons as wormhole contacts and gravitons as pairs of wormhole contacts supports double sheeted structure with sheets possessing opposite arrows of geometric time. Near the vicinity of wormhole contacts (pieces of CP2 type vacuum extremals) the sheets which are originally vacuum extremals are deformed to non-vacuum extremals and carry inertial four-momentum which should be equal to the gravitational four momentum of the vacuum extremal.

For background see the chapter TGD and GRT of "Classical Physics in Many-Sheeted Space-time". See also the article "Topological Geometrodynamics: an Overall View".

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