Quantum dynamics for the moduli associated of CDs and the arrow of geometric time

How the arrow of geometric time at the level of space-time and imbedding space is induced from the arrow of subjective time identified in terms of sequence of quantum jumps forming a fractal hierarchy of quantum jumps within quantum jumps? This is one of the long lasting puzzles of TGD and TGD inspired theory of consciousness. I have been pondering this question quite intensively during last years. The latest blog posting about the problem has title Mystery of time again.

In zero energy ontology (ZEO) the geometry of CD (I often use the sloppy notation CD== CD× CP₂, where the latter CD is defined as the intersection of future and past directed light-cones) is that of double light-cone (double pyramid) and this must relate closely to the problem at hand. An easy manner to obtain absolute arrow of geometric time at least statistically is to assume that imbedding space is M⁴+× CP₂ - that is product of future like cone with CP₂. The problem is however that of finding a convincing quantal mechanism generating the arrow of time, and also explaining why the geometric arrow of time sometimes changes from the standard one (say for phase conjugate laser beams).

The latest vision about the generation of the arrow of geometric time the level of imbedding space and space-time involves rather radical features but is consistent with the second law if generalized so that the geometric arrow of time at the level of imbedding level alternates as state function reduction takes place alternately at opposite light-like boundaries of a fixed CD. If the partially non-deterministic dynamics at space-time level defines a correlate for the dissipative dynamics of quantum jumps, the arrow of geometric time level at space-time level is constant (space-time surface can assignable to the state function reductions can be seen as folded surface spanned between boundaries of CD) and entropy defines monotonically increasing time coordinate. This is rather radical revision of the standard view but makes definite predictions: in particular syntropic aspects of the physics of living matter could be assigned with the non-standard direction of geometric time at the space-time level.

This approach hower still suffers from a defect. CDs are regarded as completely non-dynamical: once CD is created it remains the same from quantum jump to quantum jump and thus serves as a fixed arena of dynamics. This cannot be the case.

Some questions about CDs and their quantum dynamics

One can raise several questions relating to CDs.

1. CDs are assumed to form a fractal hierarchy of CDs within CDs. The size scale of CD has been argued to come as an integer multiple of CP₂ size scale on basis of number theoretic arguments. One can ask whether CDs can overlap and interact and what interaction means.
2. What is the proper interpretation of CD? Could CD correspond to a spotlight of consciousness directed to a particular region of space-time surface, so that space-time surface need not end at the boundaries of CD as also generalized Feynman diagrammatics mildly suggests? Or do the space-time surfaces end at the boundaries of CD so that CD defines a sub-Universe?
3. Should one assign CD to every subsystem - even elementary particles and fermion serving as their building bricks? Can one identify CD as a carrier of topologically quantized classical fields associated with a particle?

1. In ZEO one can in principle imagine a creation of CD from and its disappearance to vacuum. It is still unclear whether the space-time sheets associated with CD restricted to the interior of CD or whether they can continue outside CD.

   For the first option appearance of CD would be a creation of sub-Universe contained by CD. CD could be assigned with any sub-system. For the latter option the appearance of CD would be a generation of spotlight of consciousness directing attention to a particular region of imbedding space and thus to the portions of space-time surfaces inside it. Quantum superposition of space-time surfaces is actually in question and should be determined before the presence of CD by vacuum functional. How to describe possible creation and disappearance of CDs quantally, is not clear. For instance, what is the amplitude for the appearance of a new CD from vacuum in given quantum jump?

2. CDs have various moduli and one could assign to them quantum dynamics. The position of cm or either tip of CD in M⁴ defines moduli as does also the point of CP₂ defining the origin of complex Eguchi-Hanson coordinates in which U(2)⊂ SU(3) acts linearly: these points are in general assumed to be different at the two ends of CD. If either tip of CD is fixed the Lorentz boost leaving the tip fixed, moves the other along constant proper time hyperboloid H³ and the tesselations defined by the factor space H³/Γ, where Γ is discrete subgroup of SL(2,C), are favored for number theoretical reasons.

   Quantum classical correspondence inspires the question whether the boost is determined completely by the four-momentum assignable to the positive/negative energy part of zero energy states and corresponds to the four-velocity β defined by the ratio P/M of total four-momentum and mass for the CD in question. It seems that this kind of assumption can be justified only in semiclassical approximation.

3. In ZEO cm degrees of freedom of CD cannot carry Poincare charges. One can however assign the Poincare charges of the positive energy part of zero energy state to a wave function depending on the coordinate differences m₁₂ defining the relative coordinate for the tips of the CD.

   The most general option is that the size scale of CD is continuous. This would allow to realize momentum eigen state as the analogs of plane waves as a function of the position m₁₂ of the (say) upper tip of CD relative to the lower tip.

   The size scale of CD has been however assumed to be quantized. That is, the temporal distance T between the tips comes as an integer multiple of CP₂ time T_CP2: this scale is about 10⁴ Planck lengths so that this discretization has not practical consequences. Discretization is suggested both by the number theoretical vision, the finite measurement resolution, and by the general features of the U-matrix expressible as collection of M-matrices. Indeed in ZEO, one naturally obtains an infinite collection of U-matrices labelled by an integer, which would correspond to the Lorentz invariant temporal distance T_n=nT_CP2 between the tips. The scaling up of the temporal distance would represent scaling of CD in the rest system defined by the fixed tip thus translating the second tip with integer multiple of T_CP2 from T_n1 to T_n2.

   A further quantization would relate to the tesselations defined by the subgroups Γ. The counterparts of plane waves for the momentum eigenstates would be defined in a discretized version of Minkowski space obtained by dividing it to a sequence of discretized hyperboloids with proper time distance a=nT_CP2 from the lower tip of CD.

4. There is evidence that one can assign a CDs with a fixed size scale to a given particle as secondary p-adic length scale: for electron this size scale would correspond to Mersenne prime M₁₂₇ and frequency 10 Hz defining a fundamental biorhythm. This would give a deep connection between elementary particle physics and physics in macroscopic length scales. The integer multiples of the secondary p-adic length size scale would correspond to integer values of the effective Planck constant.

   A natural interpretation of this scale would be as infrared cutoff so that the wave functions approximating momentum eigenstates and depending on the relative coordinate m₁₂ would be restricted in the region between light-cone boundary and hyperboloid a=M₁₂₇T₀. Similar restriction would take place for all elementary particles. For particle with effective Planck constant hbar_eff=nhbar₀ the IR cutoff would be n-multiple of that defined by the secondary p-adic time scale.

Could CDs allow to understand the simultaneous wave-particle nature of quantum states?

One of the paradoxical features of quantum theory is that we observe always particles - even with well-defined momentum - to have rather well-defined spatial orbits. As if spatial localization would occur in quantum measurements always and would be a key element of perception and state function reduction process. This raises a heretic question: could it be possible that the localized particles in some sense have a well-defined momentum. In standard quantum theory this is definitely not possible. The assignment of CD with particle - or any physical system - however suggests that that this paradoxical looking assignment is possible. Particle would be localized with respect to (say) the lower tip of CD and delocalized with respect to (say) the upper tip and localization of the the lower tip would imply delocalization of the upper tip.

It is indeed natural to assume that either tip of CD - say lower one - is localized in M⁴ in state function reduction. Unless one is willing to make additional assumptions, this implies not only the non-prepared character of the state at the upper tip, but also a delocalization of the upper tip itself by non-triviality of M-matrix: one has quantum superpositions of worlds characterized CDs with fixed lower tip. The localization at the lower tip would correspond to the fact that we experience the world as classical. Each zero energy state would be prepared at the either (say lower) end of CD so that its lower tip would have a fixed position in M⁴. The unprepared upper tip could have a wave function in the space of all possible CDs with a fixed lower tip.

One could also assign the spinor harmonics of M⁴× CP₂ to the relative coordinates m₁₂ and their analogs in CP₂ degrees of freedom. The notion of CD would therefore make possible to realize simultaneously the paricle lbehavior in position space (localization of the lower tip of CD) and wave like nature of the state (superposition of momentum eigenstates for the upper tip relative to the lower tip).

This vision is only a heuristic guess. One should demonstrate that the average dynamical behavior for coordinate differences m₁₂ corresponds to that for a free particle with given four-momentum for a given CD and fixed quantum numbers for the positive energy part of the state.

The arrow of geometric time at the level of imbedding space and CDs

In the earlier argument the arrow of geometric time at imbedding space level was argued to relate to the fact that zero energy states are prepared only at the either end of CD but not both. This is certainly part of the story but something more concrete would be needed. In any case, the experienced flow of time should relate to what happens CDs but in the proposed model CDs are not affected in the quantum jump. Th is would leave only the drifting of sub-CDs as a mechanism generating the arrow of geometric time at imbedding space level. It is however difficult to concretize this option.

Could one understand the arrow of geometric time at imbedding space level as an increase of the size of the size of CDs appearing in zero energy state? The moduli space of CDs with a fixed upper/lower tip is without discretization future/past light-cone. Therefore there is more room in the future than in past for a particular CD and the situation is like diffusion in future light-cone meaning that the temporal distance from the tip is bound to increase in statistical sense. This means gradual scaling up of the size of the CD. A natural interpretation would be in terms of cosmological expansion.

There are two options to consider depending on whether the imbedding space is M⁴× CP₂ or M⁴₊× CP₂. The latter option allows local Poincare symmetry and is consistent with standard Robertson-Walker cosmology so that it cannot be excluded. The first option leads to Russian doll cosmology containing cosmologies within cosmologies in ZEO and is aesthetically more pleasing.

1. Consider first the M⁴× CP₂ option. At each tip of CD one has arrow of geometric time at the level of imbedding space and these arrows are opposite. What does this mean? Do the tips correspond to separate conscious entities becoming conscious alternately in state function reductions? Or do they correspond to a single conscious entity with memories?

   Could sleep awake cycle correspond to a sequence of state function reductions at opposite ends of personal CD? It would seem that we are conscious (in the sense we understand consciousness) only after state function reduction. Could we be conscious and have sensory percepts about the other end of CD during sleep state but have no memories about this period so that we would be living double life without knowing it? Does the unprepared and delocalized part (with respect to m₁₂) of zero energy state contribute to the conscious experience accompanying state function reduction? Holography would suggest that this is not the case.

   If CD corresponds to a spotlight of consciousness, the time span of conscious experience could increase in both time directions for the latter option. The span of human collective consciousness has been increasing in both direction all the time: we are already becoming conscious what has probably happened immediately after the Big Bang. Could this evolution be completely universal and coded to the fundamental physics?

2. If the imbedding space is assumed to be M⁴₊× CP₂, one obtains only one arrow of time in the long run. The reason is that the lower tip of any CD sooner or later reaches δ M⁴₊× CP₂ and further expansion in this direction becomes impossible so that only the expansion of CD to the future direction becomes possible.

Summary

The proposed vision for the dynamics of the moduli of CDs is rather general and allows a concrete understanding of the arrow of geometric time at imbedding space level and binds it directly to expansion of CDs as analog of cosmic expansion. The previous vision about how the arrow of geometric time could emerge at the level of space-time level remains essentially un-changed and allows the increase of syntropy to be understood as the increase of entropy but for a non-standard correspondence between the arrows of subjective time and the arrow of imbedding space time.

Imbedding space spinor harmonics characterizing the ground states of the representations of symplectic group of δ M⁴_+/-× CP₂ define the counterparts of single particle wave functions assignable to the relative coordinates of the second tip of CD with respect to the one fixed in state function reduction. The surprising outcome is the possibility to understand the paradoxical aspects of wave-particle duality in terms of bi-local character of CD: localization of given tip implies delocalization of the other tip.

For backbground see the chapter About the Nature of Time.