Zero energy ontology and quantum version of Robertson-Walker cosmology

Zero energy ontology has meant a real quantum leap in the understanding of the exact structure of the world of classical worlds (WCW). There are however still open questions and interpretational problems. The following comments are about a quantal interpretation of Robertson-Walker cosmology provided by zero energy ontology.

1. The light-like 3-surfaces -or equivalently corresponding space-time sheets- inside a particular causal diamond (CD) is the basic structural unit of world of classical worlds (WCW)). CD (or strictly speaking CD×CP₂) is characterized by the positions of the tips for the intersection of the future and past directed light-cones defining it. The Lorentz invariant temporal distance a between the tips allows to characterize the CDs related by Lorentz boosts and SO(3) acts as the isotropy group of a given CD. CDs with a given value of a are parameterized by Lobatchevski space -call it L(a)- identifiable as a²=constant hyperboloid of the future light-cone and having interpretation as a constant time slice in TGD inspired cosmology.

2. The moduli space for CDs characterized by a given value of a is M⁴×L(a). If one poses no restrictions on the values of a, the union of all CDs corresponds to M⁴×M⁴₊, where M⁴₊ corresponds to the interior of future light-cone. F-theorist might get excited about dimension 12 for M⁴×M⁴₊×CP₂: this is of course just a numerical co-incidence.

3. p-Adic length scale hypothesis follows if it is assumed that a comes as octaves of CP₂ time scale: a_n = 2ⁿT_CP2. For this option the moduli space would be discrete union of spaces M⁴×L(a_n). A weaker condition would be that a comes as prime multiples of T_CP2. In this case the preferred p-adic primes p ≈ 2ⁿ correspond to a=a_n and would be natural winners in fight for survival. If continuum is allowed, p-adic length scale hypothesis must be be a result of dynamics alone. Algebraic physics favors quantization at the level of moduli spaces.

4. Also unions of CDs are possible. The proposal has been that CDs form a fractal hierarchy in the sense that there are CDs within CDs but that CDs to not intersect. A more general option would allow also intersecting CDs.

Consider now the possible cosmological implications of this picture. In TGD framework Robertson-Walker cosmologies correspond to Lorentz invariant space-time surfaces in M⁴₊ and the parameter a corresponds to cosmic time.

1. First some questions. Could Robertson Walker coordinates label CDs rather than points of space-time surface at deeper level? Does the parameter a labeling CDs really correspond to cosmic time? Do astrophysical objects correspond to sub-CDs?

2. An affirmative answer to these questions is consistent with classical causality since the observer identified as -say- upper boundary of CD receives classical positive/negative energy signals from sub-CDs arriving with a velocity not exceeding light-velocity. M⁴×L(a) decomposition provides also a more precise articulation of the answer to the question how the non-conservation of energy in cosmological scales can be consistent with Poincare invariance. Note also that the empirically favored sub-critical Robertson-Walker cosmologies are unavoidable in this framework whereas the understanding of sub-criticality is one of the fundamental open problems in General Relativity inspired cosmology.

3. What objections against this interpretation can one imagine?
   1. Robertson-Walker cosmology reduces to future light-cone only at the limit of vanishing density of gravitational mass. One could however argue that the scaling factor of the metric of L(a) need not be a² corresponding to M⁴₊ but can be more general function of a. This would allow all Robertson-Walker cosmologies with sub-critical mass density. This argument makes sense also for a = a_n option.

   2. Lorentz invariant space-time surfaces in CD provide an elegant and highly predictive model for cosmology. Should one give up this model in favor of the proposed model? This need not to be the case. Quantum classical correspondence requires that also the quantum cosmology has a representation at space-time level.

4. What is then the physical interpretation for the density of gravitational mass in Robertson- Walker cosmology in the new framework? A given CD characterized by a point of M⁴×L(a), has certainly a finite gravitational mass identified as the mass assignable to positive/negative energy state at either upper or lower light-like boundary or CD. In zero energy ontology this mass is actually an average over a superposition of pairs of positive and negative energy states with varying energies. Since quantum TGD can be seen as square root of thermodynamics the resulting mass has only statistical meaning. One can assign a probability amplitude to CD as a wave function in M⁴×L(a) as a function of various quantum numbers. The cosmological density of gravitational mass would correspond to the quantum average of the mass density determined by this amplitude. Hence the quantum view about cosmology would be statistical as is also the view provided by standard cosmology.

5. Could cosmological time be really quantized as a=a_n = 2ⁿT(CP₂)? Note that other values of a are possible at the pages of the book like structure representing the generalized imbedding space since a scales as r=hbar/hbar)₀ at these pages. All rational multiples of a_n are possible for the most general option. The quantization of a does not lead to any obvious contradiction since M⁴ time would correspond to the time measured in laboratory and there is no clock keeping count about the flow of a and telling whether it is really discrete or continuous. It might be however possible to deduce experimental tests for this prediction since it holds true in all scales. Even for elementary particles the time scale a is macroscopic. For electron it is .1 seconds, which defines the fundamental bio-rhythm.

6. The quantization for a encourages also to consider the quantization for the space of Lorentz boosts characterized by L(a) obtained by restricting the boosts to a subgroup of Lorentz group. A more concrete picture is obtained from the representation of SL(2,C) as Möbius transformations of plane.
   1. The restriction to a discrete subgroup of Lorentz group SL(2,C) is possible. This would allow an extremely rich structure. The most general discrete subgroup would be subgroup of SL(2,Q_C), where Q_C could be any algebraic extension of complex rational numbers. In particular, discrete subgroups of rotation group and powers Lⁿ of a basic Lorentz boost L=exp(η) to a motion with a fixed velocity v₀ = tanh(η) define lattice like structures in L(a). This would effectively mean a cosmology in 4-D lattice. Note that everything is fully consistent with the basic symmetries.

   2. The alternative possibility is that all points of L(a) are possible but that the probability amplitude is invariant under some discrete subgroup of SL(2,Q_C). The first option could be seen as a special case of this.

   3. One can consider also the restriction to a discrete subgroup of SL(2,R) known as Fuschian groups. This would mean a spontaneous breaking of Lorentz symmetry since only boosts in one particular direction would be allowed. The modular group SL(2,Z) and its subgroups known as congruence subgroups define an especially interesting hierarchy of groups if this kind: the tesselations of hyperbolic plane provide a concrete representation for the resulting hyperbolic geometries.

   4. Is there any experimental support for these ideas. There are indeed claims for the quantization of cosmic recession velocities of quasars (See Fang, L., Z. and Sato, H. (1985): Is the Periodicity in the Distribution of Quasar Red Shifts an Evidence of Multiple Connectedness of the Universe?, Gen. Rel. and Grav. Vol 17 , No 11.). For non-relativistic velocities this means that in a given direction there are objects for which corresponding Lorentz boosts are powers of a basic boost exp(η). The effect could be due to a restriction of allowed Lorentz boosts to a discrete subgroup or to the invariance of the cosmic wave function under this kind of subgroup. These effects should take place in all scales: in particle physics they could manifest themselves as a periodicity of production rates as a function of η closely related to the so called rapidity variable y.

7. The possibility of CDs would mean violent collisions of sub-cosmologies. One could consider a generalized form of Pauli exclusion principle denying the intersections.

For a background see the chapter TGD and Cosmology of the book "Physics in Many-Sheeted Space-time".