What's new inPhysics as a Generalized Number TheoryNote: Newest contributions are at the top! 
Year 2012 
Does the square root of padic thermodynamics make sense?In zero energy ontology Mmatrix is in a welldefined sense "complex" square root of density matrix reducing to a product of Hermitian square root of density matrix multiplied by unitary Smatrix. A natural guess is that padic thermodynamics possesses this kind of square root or better to say: is modulus squared for it. For fermions the value of padic temperature is however T=1 and thus minimal. It is not possible to construct real square root by simply taking the square root of thermodynamical probabilities for various conformal weights. One manner to solve the problem is to assume that one has quadratic algebraic extension of padic numbers in which the padic prime splits as p= ππ*, π= m+(k)^{1/2}n. For k=1 primes p mod 4=1 allow a representation as product of Gaussian prime and its conjugate. For primes p mod 4=3 Gaussian primes do not help. Mersenne primes rerpesent an important examples of these primes. Eisenstein primes provide the simplest extension of rationals splitting Mersenne primes. For Eisenstein primes one has k=3 and all ordinary primes satisfying either p=3 or p mod 3=1 (true for Mersenne primes) allows this splitting. For the square root of padic thermodynamics the complex square roots of probabilities would be given by π^{(L0/T)}/Z^{1/2}, and the moduli squared would give thermodynamical probabilities as p^{(L0/T)}/Z. Here Z is the partition function. An interesting question is whether T=1 for fermions means that complex square of thermodynamics is indeed complex and whether T=2 for bosons means that the square root is actually real. For background see the chapter Physics as Generalized Number Theory: pAdicization Program. 
Quantum Mechanics and Quantum MathematicsQuantum Mathematics (QM) suggests that the basic structures of Quantum Mechanics (QM) might reduce to fundamental mathematical and metamathematical structures, and that one even consider the possibility that Quantum Mechanics reduces to Quantum Mathematics with mathematician included or expressing it in a concice manner: QM=QM! The notes below were stimulated by an observation raising a question about a possible connection between multiverse interpretation of quantum mechanics and quantum mathematics. The heuristic idea of multiverse interpretation is that quantum state repeatedly branches to quantum states which in turn branch again. The possible outcomes of the state function reduction would correspond to different branches of the multiverse so that one could save keep quantum mechanics deterministic if one can give a welldefined mathematical meaning to the branching. Could quantum mathematics allow to somehow realize the idea about repeated branching of the quantum universe? Or at least to identify some analog for it? The second question concerns the identification of the preferred state basis in which the branching occurs. Quantum Mathematics briefly Quantum Mathematics replaces numbers with Hilbert spaces and arithmetic operations + and × with direct sum ⊕ and tensor product ⊗.
The minimal view about unitary process and state function reduction is provided by ZEO.
Note that direct sum, tensor product, and the counterpart of second quantization for Hilbert spaces in the proposed sense would be quantum mathematics counterpart for set theoretic operations, Cartesian product and formation of the power set in set theory. ZEO, state function reduction, unitary process, and quantum mathematics State function reduction acts in a tensor product of Hilbert spaces. In the padic context to be discussed n the following x_{n}⊗ p^{n} is the natural candidate for this tensor product. One can assign a density matrix to a given entangled state of this system and calculate the Shannon entropy. One can also assign to it a number theoretical entropy if entanglement coefficients are rationals or even algebraic numbers, and this entropy can be negative. One can apply Negentropy Maximization Principle to identify the preferred states basis as eigenstates of the density matrix. For negentropic entanglement the quantum jump does not destroy the entanglement. Could the state function reduction take place separately for each subspace x_{n}⊗ p^{n} in the direct sum ⊕_{n} x_{n}⊗ p^{n} so that one would have quantum parallel state function reductions? This is an old proposal motivated by the manysheeted spacetime. The direct summands in this case would correspond to the contributions to the states localizable at various spacetime sheets assigned to different powers of p defing a scale hierarhcy. The powers p^{n} would be associated with zero modes by the previous argument so that the assumption about independent reduction would reflect the superselection rule for zero modes. Also different values of padic prime are present and tensor product between them is possible if the entanglement coefficients are rationals or even algebraics. In the formulation using adeles the needed generalization could be formulated in a straightforward manner. How can one select the entangled states in the summands x_{n}⊗ p^{n}? Is there some unique choice? How do unitary process and state function reduction relate to this choice? Could the dynamics of Quantum Mathematics be a structural analog for a sequence of state function reductions taking place at the opposite ends of CD with unitary matrix U relating the state basis for which single particle states have well defined quantum numbers either at the upper or lower end of CD? Could the unitary process and state function reduction be identified solely from the requirement that zero energy states correspond to tensor products Hilbert spaces, which correspond to inverses of each other as numbers? Could the extension of arithmetics to include coarithmetics make the dynamics in question unique? What multiverse branching could mean? Could QM allow to identify a mathematical counterpart for the branching of quantum states to quantum states corresponding to preferred basis? Could one can imagine that a superposition of states ∑ c_{n}Ψ_{n} in a direct summand x_{n}⊗ p^{n} is replaced by a state for which Ψ_{n} belong to different direct summands and that branching to nonintefering subuniverses is induced by the proposed superselection rule or perhaps even induces state function reduction? These two options seem to be equivalent experimentally. Could this decoherence process perhaps correspond to the replacement of the original Hilbert space characterized by number x with a new Hilbert space corresponding to number y inducing the splitting of x_{n}⊗ p^{n}? Could the interpretation of finite integers x_{n} and p^{n} as padic numbers p_{1}≠ p induce the decoherence? This kind of situation is encountered also in symmetry breaking. The irreducible representation of a symmetry group reduces to a direct sum of representations of a subgroup and one has in practice superselection rule: one does not talk about superpositions of photon and Z^{0}. In quantum measurement the classical external fields indeed induce symmetry breaking by giving different energies for the components of the state. In the case of the factor x_{n}⊗ p^{n} the entanglement coefficients define the density matrix characterizing the preferred state basis. It would seem that the process of branching decomposes this state space to a direct sum 1D state spaces associated with the eigenstates of the density matrix. In symmetry breaking superposition principle holds true and instead of quantum superposition for different orientations of "Higgs field" or magnetic field a localization selecting single orientation of the "Higgs field" takes place. Could state function reduction be analogous process? Could nonquantum fluctuating zero modes of WCW metric apper as analogs of "Higgs fields". In this picture quantum superposition of states with different values of zero modes would not be possible, and state function reduction might take place only for entanglement between zero modes and nonzero modes. The replacement of a point of Hilbert space with Hilbert space as a second quantization The fractal character of the Quantum Mathematics is what makes it a good candidate for understanding the selfreferentiality of consciousness. The replacement of the Hilbert space with the direct sum of Hilbert spaces defined by its points would be the basic step and could be repeated endlessly corresponding to a hierarchy of statements about statements or hierarchy of n^{th} order logics. The construction of infinite primes leads to a similar structure. What about the step leading to a deeper level in hierarchy and involving the replacement of each point of Hilbert space with Hilbert space characterizing it number theoretically? What could it correspond at the level of states?
For background see the chapter Quantum Adeles. 
Riemann Hypothesis and Zero Energy Ontology
Ulla mentioned in the comment section of the earlier posting an intervew of Matthew Watkins. The pages of Matthew Watkins about all imaginable topics related to Riemann zeta are excellent and I can only warmly recommend. I was actually in contact with him for years ago and there might be also TGD inspired proposal for strategy proving Riemann hypothesis at the pages of Matthew Watkins. The interview was very inspiring reading. MW has very profound vision about what mathematics is and he is able to express it in understandable manner. MW tells also about the recent work of Connes applying padics and adeles(!) to the problem. I would guess that these are old ideas and I have myself speculated about the connection with padics for long time ago. MW tells in the interview about the thermodynamical interpretation of zeta function. Zeta reduces to a product ζ(s)= ∏_{p}Z_{p}(s) of partition functions Z_{p}(s)=1/[1p^{s}] over particles labelled by primes p. This relates very closely also to infinite primes and one can talk about Riemann gas with particle momenta/energies given by log(p). s is in general complex number and for the zeros of the zeta one has s=1/2+iy. The imaginary part y is nonrational number. At s=1 zeta diverges and for Re(s)≤1 the definition of zeta as product fails. Physicist would interpret this as a phase transition taking place at the critical line s=1 so that one cannot anymore talk about Riemann gas. Should one talk about Riemann liquid? Or  anticipating what follows about quantum liquid? What the vanishing of zeta could mean physically? Certainly the thermodynamical interpretation as sum of something interpretable as thermodynamical probabilities apart from normalization fails. The basic problem with this interpretation is that it is only formal since the temperature parameter is complex. How could one overcome this problem? A possible answer emerged as I read the interview.
What the vanishing of the zeta could mean if one accepts the interpretation quantum theory as a square root of thermodynamics?
For background see the chapter Riemann Hypothesis and Physics. 
pAdic homology and finite measurement resolutionDiscretization in dimension D in terms of pinary cutoff means division of the manifold to cubelike objects. What suggests itself is homology theory defined by the measurement resolution and by the fluxes assigned to the induced Kähler form.
The simplest realization of this homology theory in padic context could be induced by canonical identification from real homology. The homology of padic object would the homology of its canonical image.

Hilbert padics, hierarchy of Planck constants, and finite measurement resolutionThe hierarchy of Planck constants assigns to the Nfold coverings of the imbedding space points Ndimensional Hilbert spaces. The natural identification of these Hilbert spaces would be as Hilbert spaces assignable to spacetime points or with points of partonic 2surfaces. There is however an objection against this identification.
One of the basic challenges of quantum TGD is to find an elegant realization for the notion of finite measurement resolution. The notion of resolution involves observer in an essential manner and this suggests that cognition is involved. If padic physics is indeed physics of cognition, the natural guess is that padic physics should provide the primary realization of this notion. The simplest realization of finite measurement resolution would be just what one would expect it to be except that this realization is most natural in the padic context. One can however define this notion also in real context by using canonical identification to map padic geometric objets to real ones. Does discretization define an analog of homology theory? Discretization in dimension D in terms of pinary cutoff means division of the manifold to cubelike objects. What suggests itself is homology theory defined by the measurement resolution and by the fluxes assigned to the induced Kähler form.
Does the notion of manifold in finite measurement resolution make sense? A modification of the notion of manifold taking into account finite measurement resolution might be useful for the purposes of TGD.
Hierachy of finite measurement resolutions and hierarchy of padic normal Lie groups The formulation of quantum TGD is almost completely in terms of various symmetry group and it would be highly desirable to formulate the notion of finite measurement resolution in terms of symmetries.
For background see the chapter Quantum Adeles. 
Quantum MathematicsThe comment of Pesla to previous posting contained something relating to the selfreferentiality of consciousness and inspired a comment which to my opinion deserves a status of posting. The comment summarizes the recent work to which I have associated the phrase "quantum adeles" but to which I would now prefer to assign the phrase "quantum mathematics". To my view the self referentiality of consciousness is the real "hard problem". The "hard problem" as it is usually understood is only a problem of dualistic approach. My hunch is that the understanding of selfreferentiality requires completely new mathematics with explicitly builtin selfreferentiality. During last weeks I have been writing and rewriting chapter about quantum adeles and end up to propose what this new mathematics might be. The latest draft is here . 1. Replace of numbers with Hilbert spaces and + and × with direct sum and tensor product The idea is to start from arithemetics : + and × for natural numbers and generalize it .
Replace also the coordinates of points of Hilbert spaces with Hilbert spaces again and again! The second key observation is that one can do all this again but at new level. Replace the numbers defining vectors of the Hilbert spaces (number sequences) assigned to numbers with Hilbert spaces! Continue ad infinitum by replacing points with Hilbert spaces again and again. You get sequence of abstractions, which would be analogous to a hierarchy of n:th order logics. At lowest levels would be just predicate calculus: statements like 4=2^{2}. At second level abstractions like y=x^{2}. At next level collections of algebraic equations, etc.... Connection with infinite primes and endless second quantization This construction is structurally very similar to  if not equivalent with  the construction of infinite primes which corresponds to repeated second quantization in quantum physics. There is also a close relationship to  maybe equivalence with  what I have called algebraic holography or number theoretic Brahman=Atman identity. Numbers have infinitely complex anatomy not visible for physicist but necessary for understanding the self referentiality of consciousness and allowing mathematical objects to be holograms coding for mathematics. Hilbert spaces would be the DNA of mathematics from which all mathematical structures would be built! Generalized Feynman diagrams as mathematical formulas? I did not mention that one can assign to direct sum and tensor product their cooperations and sequences of mathematical operations are very much like generalized Feynman diagrams. Coproduct for instance would assign to integer m all its factorizations to a product of two integers with some amplitude for each factorization. Same for cosum. Operation and cooperation would together give meaning to 3particle vertex. The amplitudes for the different factorizations must satisfy consistency conditions: associativity and distributivity might give constraints to the couplings to different channels as particle physicist might express it. The proposal is that quantum TGD is indeed quantum arithmetics with product and sum and their cooperations. Perhaps even something more general since also quantum logics and quantum set theory could be included! Generalized Feynman diagrams would correspond to formulas and sequences of mathematical operations with stringy 3vertex as fusion of 3 surfaces corresponding to ⊕ and Feynmannian 3vertex as gluing of 3surfaces along their ends, which is partonic 2surface, corresponding to ⊗! One implication is that all generalized Feynman diagrams would reduce to a canonical form without loops and incoming/outgoing legs could be permuted. This is actually a generalization of old fashioned string model duality symmetry that I proposed years ago but gave it up as too "romantic": see this. For details see the new chapter Quantum Adeles. 
Updated view about quantum adelesI have been working last weeks with quantum adeles. This has involved several wrong tracks and about five days ago a catastrophe splitting the chapter "Quantum Adeles" to two pieces entitled "Quantum Adeles" and "About Absolute Galois Group" took place, and simplified dramatically the view about what adeles are and led to the notion of quantum mathematics. At least now the situation seems to be settled down and I see no signs about possible new catastrophes. I glue the abstract of the reincarnated "Quantum Adeles" below. Quantum arithmetics provides a possible resolution of a longlasting challenge of finding a mathematical justification for the canonical identification mapping padics to reals playing a key role in TGD  in particular in padic mass calculations. pAdic numbers have padic pinary expansions ∑ a_{n}p^{n} satisfying a_{n}<p. of powers p^{n} to be products of primes p_{1}<p satisfying a_{n}<p for ordinary padic numbers. One could map this expansion to its quantum counterpart by replacing a_{n} with their counterpart and by canonical identification map p→ 1/p the expansion to real number. This definition might be criticized as being essentially equivalent with ordinary padic numbers since one can argue that the map of coefficients a_{n} to their quantum counterparts takes place only in the canonical identification map to reals. One could however modify this recipe. Represent integer n as a product of primes l and allow for l all expansions for which the coefficients a_{n} consist of primes p_{1}<p but give up the condition a_{n}<p. This would give 1tomany correspondence between ordinary padic numbers and their quantum counterparts. It took time to realize that l<p condition might be necessary in which case the quantization in this sense  if present at all  could be associated with the canonical identification map to reals. It would correspond only to the process taking into account finite measurement resolution rather than replacement of padic number field with something new, hopefully a field. At this step one might perhaps allow l>p so that one would obtain several real images under canonical identification. This did not however mean giving up the notion of the idea of generalizing number concept. One can replace integer n with ndimensional Hilbert space and sum + and product × with direct sum ⊕ and tensor product ⊗ and introduce their cooperations, the definition of which is highly nontrivial. This procedure yields also Hilbert space variants of rationals, algebraic numbers, padic number fields, and even complex, quaternionic and octonionic algebraics. Also adeles can be replaced with their Hilbert space counterparts. Even more, one can replace the points of Hilbert spaces with Hilbert spaces and repeat this process, which is very similar to the construction of infinite primes having interpretation in terms of repeated second quantization. This process could be the counterpart for construction of n^{th} order logics and one might speak of Hilbert or quantum mathematics. The construction would also generalize the notion of algebraic holography and provide selfreferential cognitive representation of mathematics. This vision emerged from the connections with generalized Feynman diagrams, braids, and with the hierarchy of Planck constants realized in terms of coverings of the imbedding space. Hilbert space generalization of number concept seems to be extremely well suited for the purposes of TGD. For instance, generalized Feynman diagrams could be identifiable as arithmetic Feynman diagrams describing sequences of arithmetic operations and their cooperations. One could interpret ×_{q} and +_{q} and their coalgebra operations as 3vertices for number theoretical Feynman diagrams describing algebraic identities X=Y having natural interpretation in zero energy ontology. The two vertices have direct counterparts as two kinds of basic topological vertices in quantum TGD (stringy vertices and vertices of Feynman diagrams). The definition of cooperations would characterize quantum dynamics. Physical states would correspond to the Hilbert space states assignable to numbers. One prediction is that all loops can be eliminated from generalized Feynman diagrams and diagrams are in projective sense invariant under permutations of incoming (outgoing legs). I glue also the abstract for the second chapter "About Absolute Galois" group which came out from the catastrophe. The reason for the splitting out was that the question whether Absolute Galois group might be isomorphic with the analog of Galois group assigned to quantum padics ceased to make sense. Absolute Galois Group defined as Galois group of algebraic numbers regarded as extension of rationals is very difficult concept to define. The goal of classical Langlands program is to understand the Galois group of algebraic numbers as algebraic extension of rationals  Absolute Galois Group (AGG)  through its representations. Invertible adeles ideles  define Gl_{1} which can be shown to be isomorphic with the Galois group of maximal Abelian extension of rationals (MAGG) and the Langlands conjecture is that the representations for algebraic groups with matrix elements replaced with adeles provide information about AGG and algebraic geometry. I have asked already earlier whether AGG could act is symmetries of quantum TGD. The basis idea was that AGG could be identified as a permutation group for a braid having infinite number of strands. The notion of quantum adele leads to the interpretation of the analog of Galois group for quantum adeles in terms of permutation groups assignable to finite l braids. One can also assign to infinite primes braid structures and Galois groups have lift to braid groups (see this). Objects known as dessins d'enfant provide a geometric representation for AGG in terms of action on algebraic Riemann surfaces allowing interpretation also as algebraic surfaces in finite fields. This representation would make sense for algebraic partonic 2surfaces, and could be important in the intersection of real and padic worlds assigned with living matter in TGD inspired quantum biology, and would allow to regard the quantum states of living matter as representations of AGG. Adeles would make these representations very concrete by bringing in cognition represented in terms of padics and there is also a generalization to Hilbert adeles. For details see the new chapters Quantum Adeles and About Absolute Galois Group. 
Two little observations about quantum padicsThe two little observations to be made require some background about quantum padics.
Consider now the two little observations.
For details see the new chapter Quantum Adeles of "Physics as Generalized Number Theory". 
Progress in number theoretic vision about TGDDuring last weeks I have been writing a new chapter Quantum Adeles. The key idea is the generalization of padic number fields to their quantum counterparts and they key problem is what quantum padics and quantum adeles mean. Second key question is how these notions relate to various key ideas of quantum TGD proper. The new chapter gives the details: here I just list the basic ideas and results. What quantum padics and quantum adeles really are? What quantum padics are? The first guess is that one obtains quantum padics from padic integers by decomposing them to products of primes l first and after then expressing the primes l in all possible manners as power series of p by allowing the coefficients to be also larger than p but containing only prime factors p_{1}<p. In the decomposition of coefficients to primes p_{1}<p these primes are replaced with quantum primes assignable to p. One could pose the additional condition that coefficients are smaller than p^{N} and decompose to products of primes l<p^{N} mapped to quantum primes assigned with q= exp(i2π/p^{N}). The interpretation would be in terms of pinary cutoff. For N=1 one would obtain the counterpart of padic numbers. For N>1 this correspondence assigns to ordinary padic integer larger number of quantum padic integers and one can define a natural projection to the ordinary padic integer and its direct quantum counterpart with coefficients a_{k}<p in pinary expansion so that a covering space of padics results. One expects also that it is possible to assign what one could call quantum Galois group to this covering and the crazy guess is that it is isomorphich with the Absolute Galois Group defined as Galois group for algebraic numbers as extension of rationals. One must admit that the details are not fully clear yet here. For instance, one can consider quantum padics defined in power series of p^{N} with coefficients a_{n}<p^{N} and expressed as products of quantum primes l<p^{N}. Even in the case that only N=1 option works the work has left to surprisingly detailed understanding of the relationship between different pieces of TGD. This step is however not enough for quantum padics.
A beautiful physical interpretation for the number theoretic Feynman diagrams emerges.
The connection with infinite primes A beautiful connection with the hierarchy of infinite primes emerges.
The interpretation of integers representing particles a Hilbert space dimensions In number theoretic dynamics particles are labeled by integers decomposing to primes interpreted as labels for braid strands. Both timelike and spacelike braids appear. The interpretation of sum and product in terms of direct sum and tensor product implies that these integers must correspond to Hilbert space dimensions. Hilbert spaces indeed decompose to tensor product of primedimensional Hilbert spaces stable against further decomposition. Second natural decomposition appearing in representation theory is into direct sums. This decomposition would take place for primedimensional Hilbert spaces with dimension l with dimensions a_{n}p^{n} in the padic expansion. The replacement of a_{n} with quantum integer would mean decomposition of the summand to a tensor product of quantum Hilbert spaces with dimensions which are quantum primes and of p^{n}dimensional ordinary Hilbert space. This should relate to the finite measurement resolution. ×_{q} vertex would correspond to tensor product and +_{q} to direct sum with this interpretation. Tensor product automatically conserves the number theoretic multiplicative momentum defined by n in the sense that the outgoing Hilbert space is tensor product of incoming Hilbert spaces. For +_{q} this conservation law is broken. Connection with the hierarchy of Planck constants, dark matter hierarchy, and living matter The obvious question concerns the interpretation of the Hilbert spaces assignable to braid strands. The hierarchy of Planck constants interpreted in terms of a hierarchy of phases behaving like dark matter suggests the answer here.
Summary The work with quantum padics and quantum adeles and generalization of number field concept to quantum number field in the framework of zero energy ontology has led to amazingly deep connections between padic physics as physics of cognition, infinite primes, hierarchy of Planck constants, vacuum degeneracy of Kähler action, generalized Feynman diagrams, and braids. The physics of life would rely crucially on padic physics of cognition. The optimistic inside me even insists that the basic mathematical structures of TGD are now rather wellunderstood. This fellow even uses the word "breakthrough" without blushing. I have of course continually admonished him for his reckless exaggerations but in vain. The skeptic inside me continues to ask how this construction could fail. A possible Achilles heel relates to the detailed definition of the notion of quantum padics. For N=1 it reduces essentially to ordinary padic number field mapped to reals by quantum variant of canonical identification. Therefore most of the general picture survives even for N=1. What would be lost are wave functions in the space of quantum variants of a given prime and also the crazy conjecture that quantum Galois group is isomorphic to Absolute Galois Group. For detais see the new chapter Quantum Adeles. 
Progress in understanding of quantum padicsQuantum arithmetics is a notion which emerged as a possible resolution of longlived challenge of finding mathematical justification for the canonical identification mapping padics to reals playing key role in padic mass calculations. The model for Shnoll effect was the bridge leading to the discovery of quantum arithmetics. I have been gradually developing the notion of quantum padics and during the weekend made quite a step of progress in understanding the concept and dare say that the notion now rests on a sound basis.
To sum up, the vision abut "Physics as generalized number theory" can be also transformed to "Number theory as quantum physics"! For detais see the new chapter Quantum Adeles. 
Quantum AdelesQuantum arithmetics is a notion which emerged as a possible resolution of longlived challenge of finding mathematical justification for the canonical identification mapping padics to reals playing key role in padic mass calculations. The model for Shnoll effect was the bridge leading to the discovery of quantum arithmetics.
The ring of adeles is essentially Cartesian product of different padic number fields and reals.
Ordinary adeles play a fundamental technical tool in Langlands correspondence. The goal of classical Langlands program is to understand the Galois group of algebraic numbers as algebraic extension of rationals  Absolute Galois Group (AGG)  through its representations. Invertible adeles define Gl_{1} which can be shown to be isomorphic with the Galois group of maximal Abelian extension of rationals (MAGG) and the Langlands conjecture is that the representations for algebraic groups with matrix elements replaced with adeles provide information about AGG and algebraic geometry. The crazy question is whether quantum adeles could be isomorphic with algebraic numbers and whether the Galois group of quantum adeles could be isomorphic with AGG or with its commutator group. If so, AGG would naturally act is symmetries of quantum TGD. The connection with infinite primes leads to a proposal what quantum padics and quantum adeles associated with algebraic extensions of rationals could be and provides support for the conjecture. The Galois group of quantum padic prime p would be isomorphic with the ordinary Galois group permuting the factors in the representation of this prime as product of primes of algebraic extension in which the prime splits. Objects known as dessins d'enfant provide a geometric representation for AGG in terms of action on algebraic Riemann surfaces allowing interpretation also as algebraic surfaces in finite fields. This representation would make sense for algebraic partonic 2surfaces, and could be important in the intersection of real and padic worlds assigned with living matter in TGD inspired quantum biology, and would allow to regard the quantum states of living matter as representations of AGG. Quantum Adeles would make these representations very concrete by bringing in cognition represented in terms of quantum padics. Quantum Adeles could allow to realize number theoretical universality in TGD framework and would be essential in the construction of generalized Feynman diagrams as amplitudes in the tensor product of state spaces assignable to real and padic number fields. Canonical identification would allow to map the amplitudes to reals and complex numbers. Quantum Adeles also provide a fresh view to conjectured M^{8}M^{4}×CP_{2} duality, and the two suggested realizations for the decomposition of spacetime surfaces to associative/quaternionic and coassociative/coquaternionic regions. For detais see the new chapter Quantum Adeles. 
Quantum padic deformations of spacetime surfaces as a representation of finite measurement resolution?A mathematically fascinating question is whether one could use quantum arithmetics as a tool to build quantum deformations of partonic 2surfaces or even of spacetime surfaces and how could one achieve this. These quantum spacetimes would be commutative and therefore not like noncommutative geometries assigned with quantum groups. Perhaps one could see them as commutative semiclassical counterparts of noncommutative quantum geometries just as the commutative quantum groups (see this) could be seen commutative counterparts of quantum groups. As one tries to develop a new mathematical notion and interpret it, one tends to forget the motivations for the notion. It is however extremely important to remember why the new notion is needed.
Consider now in more detail the identification of the quantum deformations of spacetime surfaces.
Consider now how the notion of finite measurement resolution allows to circumvent the objections against the construction.

Anatomy of quantum jump in zero energy ontologyConsider now the anatomy of quantum jump identified as a moment of consciousness in the framework of Zero energy ontology (ZEO).
The irreversibility is realized as a property of zero energy states (for ordinary positive energy ontology it is realized at the level of dynamics) and is necessary in order to obtain nontrivial Umatrix. State function reduction should involve several parts. First of all it should select the density matrix or rather its Hermitian square root. After this choice it should lead to a state which prepared either at the upper or lower boundary of CD but not both since this would be in conflict with the counterpart for the determinism of quantum time evolution. Generalization of Smatrix ZEO forces the generalization of Smatrix with a triplet formed by Umatrix, Mmatrix, and Smatrix. The basic vision is that quantum theory is at mathematical level a complex square roots of thermodynamics. What happens in quantum jump was already discussed.
Unitary process and choice of the density matrix Consider first unitary process followed by the choice of the density matrix.
State function preparation Consider next the counterpart of the ordinary state preparation process.
State function reduction process The process which is the analog of measuring the final state of the scattering process is also needed and would mean state function reduction at the upper end of CD  to state  n^{}> now.
Can the arrow of geometric time change? A highly interesting question is what happens if the first state preparation leading to a state  K^{+}> is followed by a Uprocess of type U^{} rather than by the state function reduction process K^{+}> → L^{}>. Does this mean that the arrow of geometric time changes? Could this change of the arrow of geometric time take place in living matter? Could processes like molecular self assembly be entropy producing processes but with nonstandard arrow of geometric time? Or are they processes in which negentropy increases by the fusion of negentropic parts to larger ones? Could the variability relate to sleepawake cycle and to the fact that during dreams we are often in our childhood and youth. Old people are often said to return to their childhood. Could this have more than a metaphoric meaning? Could biological death mean return to childhood at the level of conscious experience? I have explained the recent views about the arrow of time here . For background see chapter Negentropy Maximization Principle. 