Consider first what was said in the post "Undecidability, Uncomputability and the Unity of Physics. Part 1"
by Tim Palmer.
- I understood (perhaps mis-) that the idea is to reduce quantum randomness to classical chaos. If this is taken to mean that quantum theory reduces to chaos theory, I will not follow. The precise rules of quantum measurement having interpretation as measurements performed for the observables - typically generators of symmetries - are very restrictive and it is extremely difficult to image that classical physics could explain them. Quantum theory is much more than probability theory. Probabilities are essentially moduli squared for probability amplitudes and this gives rise to interference and entanglement. Therefore the idea of reducing state function reduction (SFR) and quantum randomness to classical chaos does not look promising. One could however consider the possibility classical chaos is in some sense as a correlate for quantum randomness or associated with state function reductions.
- The difficulty to combine general relativity (GRT) to quantum gravity was mentioned. The difficulty is basically due to the loss of Poincare symmetries in curved space-time. Already string models solve the problem by assuming that strings live in M10 or its spontaneous compactification. Strings are however 2-D, not 4-D, and this leads to a catastrophe. In TGD H= M4× CP2 allows to have Poincare invariance and conservation laws are not lost. In QFT picture this means that the existence of energy guarantees existence of Hamiltonian defining time evolution operator and S-matrix.
- It was noticed that chaos in quantum theory cannot be assigned to Schrödinger equation. This is true and applies quite generally to unitary time evolution generated by unitary S-matrix acting linearly. It as also noticed that in statistical mechanism Liouville operator defines a linear equation for phase space probability distribution analogous to Schrödinger equation. Liouville equation allows the classical system to be non-linear and chaotic. Could Schrödinger equation in some sense replace Liouville equation in in quantum theory since phase space ceases to make sense by Uncertainty Principle.
Could Schrödinger equation allow in some sense non-linear chaotic classical systems? In Copenhagen interpretation no classical system exists except at macroscopic limit as an approximation. One has only wave function coding for the knowledge about physical system changing in quantum measurement. There is no classical reality and there are no classical orbits of particle since one gives up the notion of Bohr orbit. Could Bohr orbit be more than approximation?
- Chaos is difficult to define in GRT. The replacement time coordinate with its logarithm exponentially growing difference becomes linearly growing and one does not have chaos. By general coordinate invariance this definition of chaos does not therefore make sense.
- Strange attractors are typical asymptotic situations in chaotic systems and can make sense also in general relativity (GRT). They represent lower dimensional manifolds to which the dynamics of the system is restricted asymptotically. It is not possible to predict to which strange attractor the chaotic dynamical system ends up. This definition of chaos makes sense also in GRT.
Could TGD allow realization of Palmer's idea in some form?
It came as a surprise to me that these to notions could a have deep relationship in TGD framework.
- Strong form of Palmer's idea stating that quantum randomness reduces to classical chaos certainly fails but one can consider weaker forms of the idea. Even these variants fail in Copenhagen interpretation since strictly speaking there is no classical reality, only wave function coding for the knowledge about the system. Bohr orbits should be more than approximation and in TGD framework space-time surface as preferred extremal of action is analogous to Bohr orbit and classical physics defined by Bohr orbits is an exact part of quantum theory.
- In the zero energy ontology (ZEO) of TGD the idea works in weaker form and has very strong implications for the more detailed understanding of ZEO and M8-M4× CP2 duality. Ordinary ("big") state functions (BSFRs) meaning the death of the system in a universal sense and re-incarnation with opposite arrow of time would involve quantum criticality accompanied by classical chaos assignable to the correspondence between geometric time and subjective time identified as sequence of "small" state function reductions (SSFRs) as analogs of weak measurements. The findings of Minev et al give strong support for this view and Libet's findings about active aspects of consciousness can be understood if the act of free will corresponds to BSFR.
Besides 4-D space-time surfaces there are also brane-like 6-surfaces corresponding to roots ri,k of Pi(r) and defining "special moments in the life of self having ti=ri,k ball as M4+ projection. The number of roots and their density increases rapidly in the sequence of SSFRs. The condition that the largest root belongs to CD gives a lower bound to it size L as largest root. Note that L increases.
Concerning the approach to chaos, one can consider three options.
Option I: The sequence of steps consisting of unitary evolutions followed by SSFR corresponds to a functional factorization at the level of polynomials as sequence P2=Q1∘ Q2∘ .. Qn. The size L of CD increases if it corresponds to the largest root, also the tip of active boundary of CD must shift so that the argument of P2 L-r is replaced in each iteration step to with updated argument with larger value of L identifiable as the largest root of P2.
Option II: A completely unexpected connection with the iteration of analytic functions and Julia sets, which are fractals assigned also with chaos interpreted as complexity emerges. In a reasonable approximation quantum time evolution by SSFRs could be induced by an iteration of a polynomial or even an analytic function: P2 =P2→ P2∘ 2 →.... For P2(0)=0, the roots of P2° N consist of inverse images of roots of P2 by P2° -k for k=0,...,N-1.
Suppose that M8 and X4 are complexified and thus also t=r and "real" X4 is the projection of X4c to real M8. Complexify also the coefficients of polynomials P. If so, the Mandelbrot and Julia sets (see this and this) characterizing fractals would have a physical interpretation in ZEO.
Chaos is approached in the sense that the inverse images of the roots of P2 assumed to belong to filled Julia set approaching to points of Julia set of P2 as the number N of iterations increases in statistical sense. The size L as largest root of P2° N would increase with N if CD is assumed to contain all roots. The density of the roots in Julia set increases near L since the size of CD is bounded by the size Julia set. One could perhaps say that near the t= L in the middle of CD the life of self when the size of CD has become almost stationary, is the most intensive.
Option III: A conservative option is to consider only real polynomials P2(r) with real argument r. Only non-negative real roots rn are of interest whereas in the general case one considers all values of r. For a large N the inverse iterates of the roots of P2 would approach to the real Julia set obtained as a real projection of Julia set for complex iteration.
How the size L of CD is determined and when can BSFR occur?
Option I: If L is minimal and thus given by the largest root of P2° N in Julia set, it is bound to increase in the iteration (this option is perhaps too deterministic). Should L be smaller than the sizes of Julia sets of both A and B if the iteration gives no roots outside Julia set.
Could BSFR become probable when L as the largest allowed root for P2° N is larger than the size of Julia set of A? There would be no more new "special moments in the life of self and this would make death and re-incarnation with opposite arrow of time probable. The size of CD could decrease dramatically in the first iteration for P1 if it is determined as the largest allowed root of P1: the re-incarnated self would have childhood.
Option II: The size of CD could be determined in SSFR statistically as an allowed root of P2. Since the density of roots increases, one would have a lot of choices and quantum criticality and fluctuations of the order of clock time τ=2L: the order of subjective time would not anymore correspond to that for clock time. BSFR would occur for the same reason as for the first option.
The fact that fractals quite generally assignable to iteration (see this) appear everywhere gives direct support for the ZEO based view about consciousness and self-organization and would give a completely new meaning for "self" in "self-organization". Fractals, quantum measurement theory, theory of self-organization, and theory of consciousness would be closely related.
See the article Could quantum randomness have something to do with classical chaos? or the chapter with the same title.
For a summary of earlier postings see Latest progress in TGD.