- The natural assumption is that entanglement is a number-theoretically universal concept and therefore makes sense in both real and various p-adic senses. This is guaranteed if the entanglement coefficients are in an extension E of rationals associated with the polynomial Q defining the space-time surface in M
^{8}and having rational coefficients.In the general case, the diagonalized density matrix ρ produced in a state function reduction (SFR) has eigenvalues in an extension E

_{1}of E. E_{1}is defined by the characteristic polynomial P of ρ. - Is the selection of one of the eigenstates in SFR possible if E
_{1}is non-trivial? If not, then one would have a number-theoretic entanglement protection. - On the other hand, if the SFR can occur, does it require a phase transition replacing E with its extension by E
_{1}required by the diagonalization?

- Suppose that the observer and measured system correspond to 4-surfaces defined by the polynomials O and S somehow composed to define the composite system and reflecting the asymmetric relationship between O and S. The simplest option is Q=O∘ S but one can also consider as representations of the measurement action deformations of the polynomial O× P making it irreducible. Composition conforms with the properties of tensor product since the dimension of extension of rationals for the composite is a product of dimensions for factors.
- The loss of correlations would suggest that a classical correlate for the outcome is a union of uncorrelated surfaces defined by O and S or equivalently by the reducible polynomial defined by the O× S (see this). Information would be lost and the dimension for the resulting extension is the sum of dimensions for the composites. O however gains information and quantum classical correspondence (QCC) suggests that the polynomial O is replaced with a new one to realize this.
- QCC suggests the replacement of the polynomial O the polynomial P∘ O, where P is the characteristic polynomial associated with the diagonalization of the density matrix ρ. The final state would be a union of surfaces represented by P∘ O and S: the information about the measured observable would correspond to the increase of complexity of the space-time surface associated with the observer. Information would be transferred from entangled Galois degrees of freedom including also fermionic ones to the geometric degrees of freedom P∘ O. The information about the outcome of the measurement would in turn be coded by the Galois groups and fermionic state.
- This would give a direct quantum classical correspondence between entanglement matrices and polynomials defining space-time surfaces in M
^{8}. The space-time surface of O would store the measurement history as kinds of Akashic records. If the density matrix corresponds to a polynomial P which is a composite of polynomials, the measurement can add several new layers to the Galois hierarchy and gradually increase its height.The sequence of SFRs could correspond to a sequence of extensions of extensions of..... This would lead to the space-time analog of chaos as the outcome of iteration if the density matrices associated with entanglement coefficients correspond to a hierarchy of powers P

^{k}.

- Consider an extension, which is a sequence of extensions E
_{1}→ ..E_{k}→ E_{k+1}..→ E_{n}defined by the composite polynomial P_{n}∘ ....∘ P_{1}. The lowest level corresponds to a simple Galois group having no non-trivial normal subgroups. - The state in the group algebra of Galois group G= G
_{n}having G_{n-1}as a normal subgroup can be expressed as an entangled state associated with the factor groups G_{n}/G_{n-1}and subgroup G_{n-1}and the first cognitive measurement in the cascade would reduce this entanglement. After that the process could but need not to continue down to G_{1}. Cognitive measurements considerably generalize the usual view about the pair formed by the observer and measured system and it is not clear whether O-S pair can be always represented in this manner as assumed above: also small deformations of the polynomial O× S can be considered.These considerations inspire the proposal the space-time surface assigned to the outcome of cognitive measurement G

_{k},G_{k-1}corresponds to polynomial the Q_{k,k-1}∘ P_{n}, where Q_{k,k-1}is the characteristic polynomial of the entanglement matrix in question.

For a summary of earlier postings see Latest progress in TGD.

## No comments:

Post a Comment