- 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 M8 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 E1 of E. E1 is defined by the characteristic polynomial P of ρ.
- Is the selection of one of the eigenstates in SFR possible if E1 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 E1 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 M8. 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 Pk.
- Consider an extension, which is a sequence of extensions E1→ ..Ek → Ek+1..→ En defined by the composite polynomial Pn∘ ....∘ P1. 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= Gn having Gn-1 as a normal subgroup can be expressed as an entangled state associated with the factor groups Gn/Gn-1 and subgroup Gn-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 G1. 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 Gk,Gk-1 corresponds to polynomial the Qk,k-1∘ Pn, where Qk,k-1 is the characteristic polynomial of the entanglement matrix in question.
For a summary of earlier postings see Latest progress in TGD.