### Details related to adelic NMP

What happens in state function reduction and what NMP really says is still far from being completely clear. The basic condition is that standard measurement theory emerges as a special case and is forced by NMP. This does not however fix the NMP completely.

1. Adelic NMP as the only reasonable option

During years I have considered two options for NMP.

1. In the original approach to NMP it was assumed that both generic entanglement with real entanglement probabilities and entanglement with algebraic entanglement probabilities are possible. Real entanglement is entropic and demands standard measurement theory leading to a 1-D eigen-space of the density matrix. Algebraic entanglement can be negentropic in number theoretic sense for some p-adic primes, and in this case state function reduction occurs only if it increases negentropy. It takes place to N-dimensional eigen-space of the density matrix. The basic objection is that real entanglement is transcendental in the generic case reducing to algebraic entanglement only as a special case. Algebraic entanglement is also extremely rare without additional physical assumptions.
2. In the adelic approach entanglement coefficients and therefore also entanglement probabilities are always algebraic numbers from the condition that the notion of p-adic Hilbert space makes sense. Also extensions of rationals defining finite-dimensional extension of p-adic numbers (roots of e can appear in extension) must be allowed. Same entanglement can be seen from both real (sensory) and p-adic perspectives (cognitive). The entanglement is always entropic in the real sector but can be negentropic in some p-adic sectors. It is now clear that the adelic option is the only sensible one.

2. Variants of the adelic NMP

The adelic option allows to consider several variants.

1. Negentropy could correspond a) to the sum N= NR+∑p Np of real and various p-adic negentropies or b) to the sum N=∑ Np of only p-adic negentropies. Np is non-vanishing for a finite number of p-adic primes only as is easy to find. In both cases ∑p Np could be interpreted as negentropy assignable to cognition. NR might have interpretation as a measure of ignorance of one of the entangled systems about the state of other.
2. NMP implies that state function reduction (measurement of density matrix leading to its eigen-space) occurs if negentropy 1) is not reduced or 2) increases. This means that negentropic entanglement is stable against NMP.
Can one select between these options?
1. For option a) NMP becomes trivial for rational entanglement probabilities as is easy to find: one has N= NR+∑p Np=0. NMP does not force state function reduction to occur but it could occur and imply ordinary state function reduction as a special case for option 1) (when eigen-spaces are 1-dimensional). Therefore one would have option 1a).
2. If option 1a) is unrealistic, only the options 1b) and 2b) with N= ∑p Np are left. For option 2b) state function necessarily occurs for N=∑p Np<0 but not for N=0 - not even in rational case. For option 2b) the state function reduction could occur also for N=0. However, since Np is proportional to log(p) and the numbers log(p) are algebraically independent, N=0 is not actually possible so that 1b) and 2b) are equivalent. Therefore NMP states that N=∑p Np must increase for N<0: this forces state function reduction to an eigen-space of density matrix.

But is it really possible to have ∑ Np<0 making possible ordinary state function reduction? For rational entanglement probabilities this is not possible by SR= ∑p Np and one might even speculate that for algebraic extensions one as ∑p Np≥ SR. Mathematician could probably check the situation. ∑p Np≥ SR holds true, entanglement is stable against NMP and ordinary state function reduction is not possible. This would leave only the option 1a) and negentropic entanglement with N>0 would be stable also now. N=0 entanglement (possibly rational always) would allow ordinary state function reduction.

This leaves still two options. Negentropy gain is A) maximal or B) non-negative but not necessarily maximal: I have considered the latter option earlier. For option 1a) reduction is possible only for N=0 and in this case negentropy gain is zero for all possible eigen-spaces of density matrix and maximality condition does not say anything.
1. For option 1a) reduction is possible only for N=0 and in this case negentropy gain is zero for all possible eigen-spaces of density matrix and A) and B) are equivalent. One obtains ordinary state function reductions.
2. Consider next the equivalent options 1b) and 2b) making sense if ∑p Np<0 is possible. For option A) negentropy gain is maximal and the reduction occurs to an eigen-space with maximum dimension N=Nmax. There can be several eigen-spaces with the same maximal dimension. As a special case one obtains ordinary state function reduction. The reduction probability is same as in standard quantum measurement theory.

For option B) the reduction could occur also to any N-dimensional eigen-space or its sub-space. The idea would be that NMP allows something analogous to a choice between good and evil: the negentropy gain could in this case be also smaller than the maximal one corresponding to log(Nmax). This would conform with the intuition that we do not seem to live in best possible world. On the other hand, negentropy transfer between systems could be also seen as stealing in some situations and metabolism identified as negentropy transfer could be seen as the fundamental "crime" to which all other forms of reduce.

3. Could quantum measurement involve also adelic localization?

For option B) there is still one possible refinement involved. p-Adic mass calculations lead to the conclusion that elementary particles are characterized by p-adic primes and that p-adic length scale hypothesis p≈ 2k holds true: a more general form of hypothesis allows also to consider primes near powers qn of some small prime such as q=3.

Could state function reduction imply also adelic/cognitive localization in the sense that the negentropy is nonzero and positive for only single p-adic prime in the final state? The reduction would occur to pk-dimensional eigen-space with pk dividing N: any divisor would be allowed. Note that Hilbert spaces with prime dimension are prime with respect to the decomposition to tensor product so that reduction would select prime power factor of the eigen-space.

The information theoretic meaning would be that prime-dimensional Hilbert spaces are stable against decomposition to tensor products so that the notion of entanglement would not make sense and therefore also the change of the state by the reduction of entanglement would be impossible. I have considered the possibility that prime-dimensional state spaces could make possible stable storage of quantum information. The prime-dimensional state when imbedded to higher-dimensional space could be seen as an entangled state.

This hypothesis would provide considerable insights to the origin of p-adic length scale hypothesis. To get a contact with physics consider electron as an example.

1. In the case of electron one would have p=M127=2127-1∼ 1038. Could electron decompose to two entangled subsystems with density matrix equal to p× p identity matrix? The dimension of eigen-space would be huge and electron would carry negentropy of 127 bits: also p-adic mass calculations combined with a generalization of Hawking-Bekenstein formula suggest that electron carries entropy of 127 bits: in adelic picture these views are mutually consistent.

The recent view indeed is that all elementary particles correspond to closed monopole magnetic flux tubes with a shape of highly flattened rectangles with short sides identifiable as extremely short wormhole contacts (CP2 size) and long sides with length of order Compton length. Magnetic monopole flux traverses along first space-time sheet between wormhole throats, goes through wormhole contact, and returns back along second space-time sheet. Many-fermion states are assigned with the throats and are located at the ends of strings traversing along the flux tubes.

Could this structure be in the case of electron a 127-sheeted structure such that the two wormhole contacts carry a superposition of pairs formed by states containing n ∈{1,...,127} fermions at second contact and n antifermions with opposite charges at second contact so that 2127-1 dimensional eigen-space would be obtained for a fermion with given spin and isospin. For instance, n=0 state with no fermion-pairs could be excluded.

2. Right-handed neutrinos and antineutrinos are candidates for the generators of N=2 supersymmetry in TGD framework. It however seems that SUSY is not manifested at LHC energies, and one can wonder whether right-handed neutrinos might be realized in some other manner. Also the mathematics involved remains still somewhat unclear. For right-handed neutrinos, which are not covariantly constant transformation to left-handed neutrinos is possible and leads to the mixing and massivation of neutrinos. For covariantly constant right handed neutrino spinors this does not happen but they can included into the spectrum only if they have non-vanishing norm.

This might be the case with a proper definition of norm with Ψbar pkγkΨ replaced by Ψbar; nkγkΨ: here nk defines normal of the light-like boundary of CD. Covariantly constant right-handed neutrinos have neither electro-weak, color, nor gravitational interactions so that their negentropic entanglement would be highly stable. Unfortunately, the situation is still unclear and this leaves open the idea that right-handed neutrinos might play fundamental role in cognition and negentropy storage. Amusingly, I proposed the notion of cognitive neutrino long time ago but based on arguments which turned out to be wrong.

One could indeed consider the possibility that each sheet of the 127-sheeted structure contains at most one νR at the neutrino end of the flux tube accompanied by νbarR at anti-neutrino end. One would have a superposition p=2127-1 states formed by many-neutrino states and their CP conjugates at opposite "ends" of the flux tube. It is also possible that νbarRR pairs are spin singlets so that one has superposition over many-particle states formed from these analogous to coherent state.

This is not the only possibility. The proposal for how the finite range of weak interactions emerges suggests a possible realization for how the number of states in superposition reduces from 2127 to 2127-1. The left weak isospin of fermion at wormhole throat is compensated by the opposite weak isospin of neutrino/antineutrino plus νbarRR or cancelling its fermion number: therefore weak charges vanish in scales longer than the flux tube length of order of the Compton length. The physical picture is that massless weak boson exchanges occur inside the flux tube which therefore defines the range of weak interactions. Same mechanism could be at work for both wormhole throat pairs and therefore also for fermion and anti-fermion at opposite wormhole throats defining building bricks of bosons. The state νbarRR would be excluded from the superposition of pairs of many-particle states and superposition would contain p=2127-1 states.
3. Could this relate to heff=n× h hypothesis? It has been assumed that heff/h=n corresponds to space-time surfaces representable as n-fold singular coverings, whose sheets co-incide at the 3-D ends of the space-time surface at opposite boundaries of CD. There is of course no need to assume that the covering considered above corresponds to singular covering and the vision that only particles with same value of n appear in same vertices suggests that n=1 holds true for visible matter.

One can still ask whether the elementary particle characterized by p ≈ 2k could corresponds to k-fold singular covering and to heff/h= k? This would require that phase transitions changing the value of k take place at the lines of scattering diagrams to guarantee that all particles have the same value of k in given vertex. These phase transitions are a key element of TGD inspired quantum biology.

In the first order of perturbation theory this would not mean any deviations from standard quantum theory for given k and the general vision that loop corrections from the functional integration over WCW vanish suggests that there are no effects in perturbation theory for given k. p-Adic coupling constant evolution would be discrete and make itself visible by the phase transitions at the lines of scattering diagrams (not identifiable as Feynman diagrams). The different values of heff/h=n be also seen through non-perturbative effects assignable to the bound states and also via the proportionality of p-adic mass scales to p-1/2≈ 2-k/2 predicted by p-adic mass calculations.

See the new chapter Non-locality in quantum theory, in biology and neuroscience, and in remote mental interactions: TGD perspective or article with the same title.