About number theoretic aspects of NMP
There is something in NMP that I still do not understand: every time I begin to explain what NMP is I have this unpleasant gut feeling. I have the habit of making a fresh start everytime rather than pretending that everything is crystal clear. I have indeed considered very many variants of NMP. In the following I will consider two variants of NMP. Second variant reduces to a pure number theory in adelic framework inspired by number theoretic vision. It is certainly the simplest one since it says nothing explicit about negentropy. Second variant says essentially the as "strong form of NMP", when the reduction occurs to an eigen-space of density matrix.
I will not consider zero energy ontology (ZEO) related aspects and the aspects related to the hierarchy of subsystems and selves since I dare regard these as "engineering" aspects.
What NMP should say?
What NMP should state?
- NMP takes in some sense the role of God and the basic question is whether we live in the best possible world or not. Theologists asks why God allows sin. I ask whether NMP demand increase of negentropy always or does it allow also reduction of negentropy? Why? Could NMP lead to increase of negentropy only in statistical sense - evolution? Could it only give potential for gaining a larger negentropy?
These questions have turned to be highly non-trivial. My personal experience is that we do not live in the best possible world and this experience plus simplicity motivates the proposal to be discussed.
- Is NMP a separate principle or could NMP be reduced to mere number theory? For the latter option state function would occur to an eigenstate/eigenspace of density matrix only if the corresponding eigenvalue and eigenstate/eigenspace are expressible using numberes in the extension of rationals defining the adele considered. A phase transition to an extension of an extension containing these coefficients would be required to make possible reduction. A step in number theoretic evolution would occur. Also an entanglement of measured state pairs with those of measuring system in containing the extension of extension would make possible the reduction. Negentropy would be reduced but higher-D extension would provide potential for more negentropic entanglement. I will consider this option in the following.
- If one has higher-D eigenspace of density matrix, p-adic negentropy is largest for the entire subspace and the sum of real and p-adic negentropies vanishes for all of them. For negentropy identified as total p-adic negentropy strong from of NMP would select the entire sub-space and NMP would indeed say something explicit about negentropy.
The notion of entanglement negentropy
- Number theoretic universality demands that density matrix and entanglement coefficients are numbers in an algebraic extension of rationals extended by adding root of e. The induced p-adic extensions are finite-D and one obtains adele assigned to the extension of rationals. Real physics is replaced by adelic physics.
- The same entanglement in coefficients in extension of rationals can be seen as numbers is both real and various p-adic sectors. In real sector one can define real entropy and in various p-adic sectors p-adic negentropies (real valued).
- Question: should one define total entanglement negentropy as
Both options can be considered.
- sum of p-adic negentropies or
- as difference for the sum of p-adic negentropies and real etropy. For rational entanglement probabilities real entropy equals to the sum of p-adic negentropies and total negentropy would vanish. For extensions this negentropy would be positive under natural additional conditions as shown earlier.
State function reduction as universal measurement interaction between any two systems
- The basic vision is that state function reductions occur all the for all kinds of matter and involves a measurement of density matrix ρ characterizing entanglement of the system with environment leading to a sub-space for which states have same eigenvalue of density matrix. What this measurement really is is not at all clear.
- The measurement of the density matrix means diagonalization of the density matrix and selection of an eigenstate or eigenspace. Diagonalization is possible without going outside the extension only if the entanglement probabilities and the coefficients of states belong to the original extension defining the adele. This need not be the case!
More precisely, the eigenvalues of the density matrix as roots of N:th order polynomial with coefficients in extension in general belong to N-D extension of extension. Same about the coefficients of eigenstates in the original basis. Consider as example the eigen values and eigenstates of rational valued N× N entanglement matrix, which are roots of a polynomial of degree N and in general algebraic number.
Question: Is state function reduction number theoretically forbidden in the generic case? Could entanglement be stable purely number theoretically? Could NMP reduce to just this number theoretic principle saying nothing explicit about negentropy? Could phase transition increasing the dimension of extension but keeping the entanglement coefficients unaffected make reduction possible. Could entanglement with an external system in higher-D extension -intelligent observer - make reduction possible?
- There is a further delicacy involved. The eigen-space of density matrix can be N-dimensional if the density matrix has N-fold degenerate eigenvalue with all N entanglement probabilities identical. For unitary entanglement matrix the density matrix is indeed N×N unit matrix. This kind of NE is stable also algebraically if the coefficients of eigenstates do not belong to the extension. If they do not belong to it then the question is whether NMP allows a reduction to subspace of and eigen space or whether only entire subspace is allowed.
For total negentropy identified as the sum of real and p-adic negentropies for any eigenspace would vanish and would not distinguish between sub-spaces. Identification of negentropy as as p-adic negentropy would distinguish between sub-spaces and´NMP in strong form would not allow reduction to sub-spaces. Number theoretic NMP would thus also say something about negentropy.
I have also consider the possibility of weak NMP. Any subspace could be selected and negentropy would be reduced. The worst thing to do in this case would be a selection of 1-D subspace: entanglement would be totally lost and system would be totally isolated from the rest of the world. I have proposed that this possibility corresponds to the fact that we do not seem to live in the best possible world.
NMP as a purely number theoretic constraint?
Let us consider the possibility that NMP reduces to the number theoretic condition tending to stabilize
- Density matrix characterizing entanglement with the environment is a universal observable. Reduction can occur to an eigenspace of the density matrix. For rational entanglement probabilities the total negentropy would vanish so that NMP formulated in terms of negentropy cannot say anything about the situation. This suggests that NMP quite generally does not directly refer to negentropy.
- The condition that eigenstates and eigenvalues are in the extension of rationals defining the adelic physics poses a restriction. The reduction could occur only if these numbers are in the original extension. Also rational entanglement would be stable in the generic case and a phase transition to higher algebraic extension is required for state function reduction to occur. Standard quantum measurement theory would be obtained when the coefficients of eigenstates and entanglement probabilities are in the original extension.
- If this is not the case, a phase transition to an extension of extension containing the N-D extension of it could save the situation. This would be a step in number theoretic evolution. Reduction would lead to a reduction of negentropy but would give potential for gaining a larger entanglement negentropy. Evolution would proceed through catastrophes giving potential for more negentropic entanglement! This seems to be the case!
Alternatively, the state pairs of the system + complement could be entangled with observer in an extension of rationals containg the needed N-D extension of extension and state function possible for observer would induce reduction in the original system. This would mean fusion with a self at higher level of evolutionary hierarchy - kind of enlightment. This would give an active role to the intelligent observer (intelligence characterized by the dimension of extension of rationals). Intelligent observer would reduce the negentropy and thus NMP would not hold true universally.
Since higher-D extension allows higher negentropy and in the generic case NE is stable, one might hope that NMP holds true statistically (for rationals total negentropy as sum or real and total p-adic negentropies vanishes).
The Universe would evolve rather than being a paradize: the number theoretic NMP would allow temporary reduction of negentropy but provide a potential for larger negentropy and the increase of negentropy in statistical sense is highly suggestive. To me this option looks like simplest and most realistic one.
- If negentropy is identified as total p-adic negentropy rather than sum of real and p-adic negentropies, strong form of NMP says something explicit about negentropy: the reduction would take place to the entire subspace having the largest p-adic negentropy.
For background see the chapter Negentropy Maximization Principle.
or the article About number theoretic aspects of NMP.