Is the sum of p-adic negentropies equal to real entropy?
I ended almost by accident to a fascinating and almost trivial theorem. Adelic theorem for information would state that conscious information represented as sum of p-adic negentropies (entropies, which are negative) is equal to real entropy. The more conscious information, the larger the chaos in the environment as everyone can verify by just looking around;-)
This looks bad! Luckily, it turned out that this statement is true for rational probabilities only. For algebraic extensions it cannot be true as is easy to see. That negentropic entanglement is possible only for algebraic extensions of rationals conforms with the vision that algebraic extensions of rationals characterize evolutionary hierarchy. The rationals represent the lowest level at which there either conscious information vanishes or if equal to p-adic contribution to negentropy is companied by equally large real entropy.
It is not completely obvious that the notion of p-adic negentropy indeed makes sense for algebraic extensions of rationals. A possible problem is caused by the fact that the decomposition of algebraic integer to primes is not unique. Simple argument however strongly suggests that the various p-adic norms of the factors do not depend on the factorization. Also a formula for the difference of the total p-adic negentropy and real entropy is deduced.
p-Adic contribution to negentropy equals to real entropy for rational probabilities but not for
The following argument shows that p-adic negentropy equals to real entropy for rational probabilities.
- The fusion of real physics and various p-adic physics (identified as correlates for cognition, imagination, and intentionality) to single coherent whole leads to what I call adelic physics. Adeles associated with given extension of rationals are Cartesian product of real number field with all p-adic number fields extended by the extension of rationals. Besides algebraic extensions also the extension by any root of e is possible since it induces finite-dimensional p-adic extension. One obtains hierarchy of adeles and of corresponding adelic physics interpreted as an evolutionary hierarchy.
An important point is that p-adic Hilbert spaces exist only if one restricts the p-adic numbers to an algebraic extension of rationals having interpretation as numbers in any number field. This is due to the fact that sum of the p-adic valued probabilities can vanish for general p-adic numbers so that the norm of state can vanish. One can say that the Hilbert space of states is universal and is in the algebraic intersection of reality and various p-adicities.
- Negentropy Maximization Principle (NMP) is the variational principle of consciousness in TGD framework reducing to quantum measurement theory in Zero Energy Ontology assuming adelic physics. One can define the p-adic counterparts of Shannon entropy for all finite-dimensional extensions of p-adic numbers, and the amazing fact is that these entropies can be negative and thus serve as measures for information rather than for lack of it. Furthermore, all non-vanishing p-adic negentropies are positive and the number of primes contributing to negentropy is finite since any algebraic number can be expressed using a generalization of prime number decomposition of rational number. These p-adic primes characterize given systen, say elementary particle.
NMP states that the negentropy gain is maximal in the quantum jump defining state function reduction. How does one define the negentropy? As the sum of p-adic negentropies or as the sum of real negative negentropy plus the sum of p-adic negentropies? The latter option I proposed for some time ago without checking what one obtains.
- The adelic theorem says that the norm of rational number is equal to the product of the inverses of its p-adic norms. The statement that the sum of real and p-adic negentropies is zero follows more or less as a statement that the logarithms of real norm and the product of p-adic norms for prime factors of rational sum up to zero.
The core formula is adelic formula stating that the real norm of rational number is product of its p-adic norms. This implies that the logarithm of the rational number is sum over the logarithms of its p-adic norms. Since in p-adic entropy assigned to prime p logarithms of probabilities are replaced by their p-adic norms, this implies that for rational probabilities the real entropy equals to p-adic negentropy. If real entropy is counted as conscious information, the negentropy vanishes identically for rational probabilities.
It would seem that the negentropy appearing in the definition of NMP must be the sum of p-adic negentropies and real entropy should have interpretation as a measure for ignorance about the state of either entangled system. The sum of p-adic negentropies would serve as a measure for the information carried by a rule with superposed state pairs representingt the instances of the rule. The information would be conscious information and carried by the negentropically entangled system.
- What about probabilities in algebraic extensions? The probabilities are now algebraic numbers. Below
an argument is develoed that the p-adic norms of of probabilities are uniquely defined and are always powers of primes so that the adelic formula cannot be true since on the real side one has logarithms of algebraic numbers and on the p-adic side only logarithms of primes.
What could be the interpretation?
One however expects that the value of real entropy correlates strongly with the value of negentropy. One expects that the value of real entropy correlates strongly with the value ofp-adic total negentropy. This would conform with the observation that large entropy seems to be a prerequisite for life by providing large number of states with degenerate energies providing large representative capacity. For instance, Jeremy England has made this proposal: I have commented this proposal from TGD point of view.
- If conscious information corresponds to N-P, it accompanies the emergence of algebraic extensions of rationals at the level of Hilbert space.
- If N corresponds to conscious information, then at the lowest level conscious information is necessary accompanied by entropy but for algebraic extensions N-P could be positive since N is maximized. This option looks more plausible.
Formula for the difference of total p-adic negentropy and real entanglement entropy
In the following some non-trivial details related to the definition of p-adic norms for the rationals in the algebraic extension of rationals are discussed.
The induced p-adic norm Np(x) for n-dimensional extension of Q is defined as the determinant det(x) of the linear map defined by multiplication with x. det(x) is rational number. The corresponding p-adic norm is defined as the n:th root Np(det(x))1/n of the ordinary p-adic norm. Root guarantees that the norm co-incides with the ordinary p-adic norm for ordinary p-adic integers. One must perform now a factorization to algebraic primes. Below an argument is given that although the factorization to primes is not always unique, the product of p-adic norms for given algebraic rational defined as ratio of algebraic integers is unique.
Can one write an explicit formula the difference of total p-adic entanglement negentropy (positive) and real entanglement entropy using prime factorization in finite dimensional algebraic extension (note that for algebraic numbers defining infinite-dimensional extension of rationals factorization does not even exist since one can write a=a1/2a1/2=...)? This requires that total p-adic entropy is uniquely defined. There is a possible problem due to the non-uniqueness of the prime factorization.
For background see the chapter Negentropy Maximization Principle.
- For Dedekind rings, in particular rings of integers, there exists by definition a unique factorization of proper ideals to prime ideals (see this). In contrast, the prime factorization in the extensions of Q is not always unique. Already for Q((-5)1/2) one has 6=2× 3= (1+(-5)1/2)(1-(-5)1/2) and the primes involved are not related by multiplication with units.
Various factorizations are characterized by so called class group and class field theory is the branch of number theory studying factorizations in algebraic extensions of integer rings. Factorization is by definition unique for Euclidian domains. Euclidian domains allow by definition so called Euclidian function f(x) having values in R+ with the property that for any a and b one has either a=qb or a= qb+r with f(r)<f(b). It seems that one cannot restrict to Euclidian domains in the recent situation.
- Even when the factorization in the extension is not unique, one can hope that the product of various p-adic norms for the factors is same for all factorizations. Since the p-adic norm for the extensions of primes is induced by ordinary p-adic number this requires that the p-adic prime for which the induced p-adic norm differs from unity are same for all factorizations and that the products of p-adic norms differing from unity are same. This independence on the representative for factorization would be analogous to gauge invariance in physicist's conceptualization.
The probabilities Pk belongs to a unique product of ideals labelled by primes of extension. The ideals are characterized by norms and if this norm is product of p-adic norms for any prime factorization as looks natural then the independence on the factorization follows. Number theorist can certainly immediately tell whether this is true. What is encouraging that for Q((-5)1/2) z=x+(-5)1/2y has determinant det(z)=x2+5y2 and for z==1+/- (-5)1/2 one has has det(z)=6 so that for the products of p-adic norms for the factorizations 6=2× 3 and (1+(-5)1/2)(1-(-5)1/2) are equal.
- If this guess is true, one can write the the difference of total p-adic negentropy N and real entanglement entropy S as
N-S= ∑ Pk log(Pk/∏p Np(Pk)) .
Here ∏p Np(Pk) would not depend on particular factorization. The condition ∑ Pk=1 poses an additional condition. It would be nice to understand whether N-S≥ 0 holds true generally and if not, what are the conditions guaranteeing this. The p-adic numbers of numerators of rationals involved give positive contributions to N-S as the example Pk=1/N in rational case shows.