What's new inPhysics as a Generalized Number TheoryNote: Newest contributions are at the top! 
Year 2011 
How quantum arithmetics affects basic TGD and TGD inspired view about life and consciousness?The vision about real and padic physics as completions of rational physics or physics associated with extensions of rational numbers is central element of number theoretical universality. The physics in the extensions of rationals are assigned with the interaction of real and padic worlds.
How does the replacement of rationals with quantum rationals modify quantum TGD and the TGD inspired vision about quantum biology and consciousness? What happens to padic mass calculations and quantum TGD? The basic assumption behind the padic mass calculations and all applications is that one can assign to a given partonic 2surface (or even lightlike 3surface) a preferred padic prime (or possibly several primes). The replacement of rationals with quantum rationals in padic mass calculations implies effects, which are extremely small since the difference between rationals and quantum rationals is extremely small due to the fact that the primes assignable to elementary particles are so large (M_{127}=2^{127}1 for electron). The predictions of padic mass calculations remains almost as such in excellent accuracy. The bonus is the uniqueness of the canonical identification making the theory unique. The problem of the original padic mass calculations is that the number of common rationals (plus possible algebraics in some extension of rationals) is same for all primes p. What is the additional criterion selecting the preferred prime assigned to the elementary particle? Could the preferred prime correspond to the maximization of number theoretic negentropy for a quantum state involved and therefore for the partonic 2surface by quantum classical correspondence? The solution ansatz for the modified Dirac equation indeed allows this assignment (see this): could this provide the first principle selecting the preferred padic prime? Here the replacement of rationals with quantum rationals improves the situation dramatically.
What happens to TGD inspired theory of consciousness and quantum biology? The vision about rationals as common to reals and padics is central for TGD inspired theory of consciousness and the applications of TGD in biology.
For details and background see the new chapter Quantum Arithmetics and the Relationship between Real and pAdic Physics.

Quantum Arithmetics and the Relationship between Real and pAdic PhysicspAdic physics involves two only partially understood questions.
A possible answer to these questions relies on the following ideas inspired by the model of Shnoll effect. The first piece of the puzzle is the notion of quantum arithmetics formulated in nonrigorous manner already in the model of Shnoll effect.
Quantum arithmetics inspires the notion of quantum matrix group as counterpart of quantum group for which matrix elements are ordinary numbers. Quantum classical correspondence and the notion of finite measurement resolution realized at classical level in terms of discretization suggest that these two views about quantum groups are closely related. The preferred prime p defining the quantum matrix group is identified as padic prime and canonical identification p→ 1/p is group homomorphism so that symmetries are respected.
The findings about quantum SO(3) suggest a possible explanation for padic length scale hypothesis and preferred padic primes.
For details and background see the new chapter Quantum Arithmetics and the Relationship between Real and pAdic Physics.

Could TGD be an integrable theory?During years evidence supporting the idea that TGD could be an integrable theory in some sense has accumulated. The challenge is to show that various ideas about what integrability means form pieces of a bigger coherent picture. Of course, some of the ideas are doomed to be only partially correct or simply wrong. Since it is not possible to know beforehand what ideas are wrong and what are right the situation is very much like in experimental physics and it is easy to claim (and has been and will be claimed) that all this argumentation is useless speculation. This is the price that must be paid for the luxury of genuine thinking. Integrable theories allow to solve nonlinear classical dynamics in terms of scattering data for a linear system. In TGD framework this translates to quantum classical correspondence. The solutions of modified Dirac equation define the scattering data. The conjecture is that octonionic realanalyticity with spacetime surfaces identified as surfaces for which the imaginary part of the biquaternion representing the octonion vanishes solves the field equations. This conjecture generalizes the conformal invariance to its octonionic analog. If this conjecture is correct, the scattering data should define a real analytic function whose octonionic extension defines the spacetime surface as a surface for which its imaginary part in the representation as biquaternion vanishes. There are excellent hopes about this thanks to the reduction of the modified Dirac equation to geometric optics. For details and background the reader can consult to the article An attempt to understand preferred extremals of Kähler action and to the chapter TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts.

pAdic fractality, canonical identification, and symmetriesThe original motivation for the canonical identification I: ∑ x_{n}p^{n} →∑ x_{n}p^{n}, and its variants  in particular the variant mapping real rationals with the defining integers below a pinary cutoff to padic rationals  was that it defines a continuous map from padics to reals and produces beautiful padic fractals as a map from reals to padics by canonical identification followed by a padically smooth map in turn followed by the inverse of the canonical identification. The first drawback was that the map does not commute with symmetries. Second drawback was that the standard canonical identification from reals to padics with finite pinary cutoff is twovalued for finite integers. The canonical real images of these transcendentals are also transcendentals. These are however countable whereas padic algebraics and transcendentals having by definition a nonperiodic pinary expansion are uncountable. Therefore the map from reals to padics is single valued for almost all padic numbers. On the other hand, padic rationals form a dense set of padic numbers and define "almost all" for the purposes of numerics! Which argument is heavier? The direct identification of reals and padics via common rationals commutes with symmetries in an approximation defined by the pinary cutoff an is used in the canonical identification with pinary cutoff mapping rationals to rationals. Symmetries are of extreme importance in physics. Is it possible to imagine the action of say Poincare transformations commuting with the canonical identification in the sets of padic and real transcendentals? This might be the case.
Note that the analog of Wick rotation could be used also to define padic integrals by mapping the padic integration region to real one by some variant of canonical identification continuously, performing the integral in the real context, and mapping the outcome of the integral to padic number by canonical identification. Again preferred coordinates are essential and in TGD framework such coordinates are provided by symmetries. This would allow a numerical treatment of the padic integral but the map of the resulting rational to padic number would be two valued. The difference between the images would be determined by the numerical accuracy when padic expansions are used. This method would be a numerical analog of the analytic definition of padic integrals by analytic continuation from the intersection of real and padic worlds defined by rational values of parameters appearing in the expressions of integrals. For details and background see the chapter pAdic numbers and generalization of number concept. 
Could octonion analyticity solve the field equations of quantum TGD?
There are pressing motivations for understanding the preferred extremals of Kähler action. For instance, the conformal invariance of string models naturally generalizes to 4D invariance defined by quantum Yangian of quantum affine algebra (KacMoody type algebra) characterized by two complex coordinates and therefore explaining naturally the effective 2dimensionality (see this). The problem is however how to assign a complex coordinate with the string world sheet having Minkowskian signature of metric. One can hope that the understanding of preferred extremals could allow to identify two preferred complex coordinates whose existence is also suggested by number theoretical vision giving preferred role for the rational points of partonic 2surfaces in preferred coordinates. The best one could hope is a general solution of field equations in accordance with the hints that TGD is integrable quantum theory. A lot is is known about properties of preferred extremals and just by trying to integrate all this understanding, one might gain new visions. The problem is that all these arguments are heuristic and rely heavily on physical intuition. The following considerations relate to the spacetime regions having Minkowskian signature of the induced metric. The attempt to generalize the construction also to Euclidian regions could be very rewarding. Only a humble attempt to combine various ideas to a more coherent picture is in question. The core observations and visions are following.
The considerations of this section lead to a revival of an old very ambitious and very romantic number theoretic idea, which I already thought to be dead.
For background see the chapter TGD as Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts and the article An attempt to understand preferred extremals of Kauml;hler action 
How to understand transcendental numbers in terms of infinite integers?Santeri Satama made in my blog a very interesting question about transcendental numbers. The reformulation of Santeri's question could be "How can one know that given number defined as a limit of rational number is genuinely algebraic or transcendental?". I answered to the question and since it inspired a long sequence of speculations during my morning walk on the sands of Tullinniemi I decided to expand my hasty answer to a blog posting. The basic outcome was the proposal that by bringing TGD based view about infinity based on infinite primes, integers, and rationals one could regard transcendental numbers as algebraic numbers by allowing genuinely infinite numbers in their definition.
1. How can one know that the real number is transcendental? The difficulty of telling whether given real number defined as a limit of algebraic number boils down to the fact that there is no numerical method for telling whether this kind of number is rational, algebraic, or transcendental. This limitation of numerics would be also a restriction of cognition if padic view about it is correct. One can ask several questions. What about infiniteP padic numbers: if they make sense could it be possible to cognize also transcendentally? What can we conclude from the very fact that we cognize transcendentals? Transcendentality can be proven for some transcendentals such as π. How this is possible? What distinguishes "knowably transcendentals" like π and e from those, which are able to hide their real number theoretic identity?
2. Exponentiation and formation of set of subsets as transcendence What is so special in exponentiation? Why it sends algebraic numbers to "knowably transcendentals". One could try to understand this in terms of exponentiation which for natural numbers has also an interpretation in terms of power set just as product has interpretation in terms of Cartesian product.
3. Infinite primes and transcendence TGD suggests also a different identification of transcendence not expressible as formation of a power set or its generalizations.
This idea inspires some questions.
4. Identification of real transcendentals as infinite algebraic numbers with finite value as real numbers The following observations suggests that it could be possible to reduce transcendentals to generalized algebraic numbers in the framework provided by infinite primes. This would mean that not only physics but also mathematics (or at least "physical mathematics") could be seen as generalized number theory.
One can consider some examples to illustrate the situation.
To conclude, the talk about infinite primes might sound weird in the ears of a layman but mathematicians do not lose their peace of mind when they here the word "infinity". The notion of infinity is relative. For instance, infinite integers are completely finite in padic sense. One can also imagine completely "realworldish" realizations for infinite integers (say as states of repeatedly second quantized arithmetic quantum field theory and this realization might provide completely new insights about how to undestand bound states in ordinary QFT).
For details and background see the chapter Physics as Generalized Number Theory: Infinite Primes or to the article How infinite primes relate to other views about mathematical infinity?. 
About the structure of the Yangian algebraThe attempt to understand Langlands conjecture in TGD framework (see this) led to a completely unexpected progress in the understanding of the Yangian symmetry expected to be the basic symmetry of quantum TGD (see this) and the following vision suggesting how conformal field theory could be generalized to fourdimensional context is a fruit of this work. The structure of the Yangian algebra is quite intricate and in order to minimize confusion easily caused by my own restricted mathematical skills it is best to try to build a physical interpretation for what Yangian really is and leave the details for the mathematicians.
Slicing of spacetime sheets to partonic 2surfaces and string world sheets The proposals is t the preferred extremals of Kähler action are involved in an essential manner the slicing of the spacetime sheets by partonic 2surfaces and string world sheets. Also an analogous slicing of Minkowski space is assumed and there are infinite number of this kind of slicings defining what I have called HamiltonJaboci coordinates (see this). What is really involved is far from clear. For instance, I do not really understand whether the slicings of the spacetime surfaces are purely dynamical or induced by special coordinatizations of the spacetime sheets using projections to special kind of submanifolds of the imbedding space, or are these two type of slicings equivalent by the very property of being a preferred extremal. Therefore I can represent only what I think I understand about the situation.
Physical interpretation of the Yangian of quantum affine algebra What the Yangian of quantum affine algebra or more generally, its super counterpart could mean in TGD framework? The key idea is that this algebra would define a generalization of super conformal algebras of super conformal field theories as well as the generalization of super Virasoro algebra. Optimist could hope that the constructions associated with conformal algebras generalize: this includes the representation theory of super conformal and super Virasoro algebras, coset construction, and vertex operator construction in terms of free fields. One could also hope that the classification of extended conformal theories defined in this manner might be possible.
How to construct the Yangian of quantum affine algebra? The next step is to try to understand the construction of the Yangian of quantum affine algebra.
These arguments are of course heuristic and do not satisfy any criteria of mathematical rigor and the details could of course change under closer scrutinity. The whole point is in the attempt to understand the situation physically in all its generality. The most important outcome is the conjecture that the incredibly powerful mathematical apparatus of 2dimensional conformal field theories might have a generalization to fourdimensional context based on Yangians of quantum affined algebras. This might explain the miracles of both twistor approach and string approach. How 4D generalization of conformal invariance relates to strong form of general coordinate invariance? The basic objections that one can rise to the extension of conformal field theory to 4D context come from the successes of padic mass calculations. pAdic thermodynamics relies heavily on the properties of partition functions for superconformal representations. What happens when one replaces affine algebra with (quantum) Yangian of affine algebra? Ordinary Yangian involves the original algebra and its dual and from these higher multilocal generators are constructed. In the recent case the obvious interpretation for this would be that one has KacMoody type algebra with expansion with respect to complex coordinate w for partonic 2surfaces and its dual algebra with expansion with respect to hypercomplex coordinate of string world sheet. pAdic mass calculations suggest that the use of either algebra is enough to construct single particle states. Or more precisely, local generators are enough. I have indeed proposed that the multilocal generators are relevant for the construction of bound states. Also the strong form of general coordinate invariance implying strong form of holography, effective 2dimensionality, electricmagnetic duality and Sduality suggest the same. If one could construct the states representing elementary particles solely in terms of either algebra, there would be no danger that the results of padic mass calculations are lost. Note that also the necessity to restrict the conformal weights of conformal representations to be nonnegative would have nice interpretation in terms of the duality. For details and background the reader can consult either to the chapter Physics as Generalized Number Theory: Infinite Primes or to the article Langlands conjectures in TGD framework. 
Is quantal Boolean reverse engineering possible?The quantal version of Boolean algebra means that the basic logical functions have quantum inverses. The inverse of C=A ∧ B represents the quantum superposition of all pairs A and B for which A∧ B=C hols true. Same is true for ∨. How could these additional quantum logical functions with no classical counterparts extend the capacities of logician? What comes in mind is logical reverse engineering. Consider the standard problem solving situation repeatedly encountered by my hero Hercule Poirot. Someone has been murdered. Who could have done it? Who did it? Actually scientists who want to explain instead of just applying the method to get additional items to the CVC, meet this kind of problem repeatedly. One has something which looks like an experimental anomaly and one has to explain it. Is this anomaly genuine or is it due to a systematic error in the information processing? Could the interpretation of data be somehow wrong? Is the model behind experiments based on existing theory really correct or has something very delicate been neglected? If a genuine anomaly is in question (someone has been really murdered this is always obvious in the tales about the deeds of Hercule Poirot since the mere presence of Hercule guarantees the murder unless it has been already done) one encounters what might be called Poirot problem in honor of my hero. As a matter fact, from the point of view of Boolean algebra, one has the same reverse Boolean engineering problem irrespective of whether it was a genuine anomaly or not. This brings in my mind the enormously simplified problem. The logical statement C is found to be true. Which pairs A,B could have implied C as C=A∧ B (or A∨ B). Of course, much more complex situations can be considered where C corresponds to some logical function C=f(A_{1},A_{2},...,A_{n}). Quantum Poirot could use quantum computer able to realize the cogates for gates AND and OR (essentially time reversals) and write a quantum computer program solving the problem by constructing the Boolean cofunction of Boolean function f. What would happen in TGD Universe obeying zero energy ontology is following.
To conclude, I am a Boolean dilettante and know practically nothing about what quantum computer theorists have done in particular I do not know whether they have considered quantum inverse gages. My feeling is that only the gates with bits replaced with qubits are considered: very natural when one thinks in terms of Boolean logic. If this is really the case, quantal coAND and coOR having no classical counterparts would bring a totally new aspect to quantum computation in solving problems in which one cannot do without (quantum) Poirot and his little gray (quantum) brain cells. For details and background the reader can consult either to the chapter Physics as Generalized Number Theory: Infinite Primes or to the article How infinite primes relate to other views about mathematical infinity?. 
TGD and Physical Mathematics
I discussed in What's New the TGD based vision about Langlands program. I have actually written several blog postings (and What's News) related to the relationship between TGD and physical mathematics during this year (see this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, and this). I try also to become (and remain!) conscious about possible sources of inconsistencies to see what might go wrong. I see the attempt to understand the relation between Langlands program and TGD as a part of a bigger project the goal of which is to relate TGD to physical mathematics. The basic motivations come from the mathematical challenges of TGD and from the almostbelief that the beautiful mathematical structures of the contemporary physical mathematics must be realized in Nature somehow. The notion of infinite prime is becoming more and more important concept of quantum TGD and also a common denominator. The infinitedimensional symplectic group acting as the isometry group of WCW geometry and symplectic flows seems to be another common denominator. Zero energy ontology together with the notion of causal diamond is also a central concept. A further common denominator seems to be the notion of finite measurement resolution allowing discretization. Strings and supersymmetry so beautiful notions that it is difficult to imagine physics without them although super string theory has turned out to be a disappointment in this respect. In the following I mention just some examples of problems that I have discussed during this year. Infinite primes are certainly something genuinely TGD inspired and it is reasonable to consider their possible role in physical mathematics.
The notion of finite measurement resolution realized at quantum level as inclusions of hyperfinite factors and at spacetime level in terms of braids replacing the orbits of partonic 2surfaces  is also a purely TGD inspired notion and gives good hopes about calculable theory.
TGD is a generalization of string models obtained by replacing strings with 3surfaces. Therefore it is not surprising that stringy structures should appear also in TGD Universe and the strong form of general coordinate invariance indeed implies this. As a matter fact, string like objects appear also in various applications of TGD: consider only the notions of cosmic string (see this) and nuclear string (see this). Magnetic flux tubes central in TGD inspired quantum biology making possible topological quantum computation (see this) represent a further example.
When I look what I have written about various topics during this year I find that symplectic invariance and symplectic flows appear repeatedly.
Number theoretical universality is one of the corner stones of the vision about physics as generalized number theory. One might perhaps say that a similar vision has guided Grothendieck and his followers.
Twistor approach has led to the emergence of motives to physics and twistor approach is also what gives hopes that some day quantum TGD could be formulated in terms of explicit Feynman rules or their twistorial generalization (see this and this).
For details and background the reader can consult either to the chapter Langlands Program and TGD of "Physics as Generalized Number Theory" or to the articles Langlands Conjectures in TGD Framework, How infinite primes relate to other views about mathematical infinity?, Motives and Infinite primes, and Could one generalize braid invariant defined by vacuum expectation of Wilson loop to an invariant of braid cobordisms and of 2knots?. Also the previous blog postings during this year give a view about the development of ideas. 
Langlands Conjectures in TGD Framework
During last years I have done work in attempt to relate TGD to the new developments in mathematics. The evolution of ideas has been especially fast during last year and I have reported about these developments in various postings. The latest articles are How infinite primes relate to other views about mathematical infinity? and Motives and Infinite Primes. What makes me happy is that TGD is not only receiving experimental support from LHC and other particle accelerators but also providing profound insights inspiring mathematical conjectures. What is also highly satisfying that the physically motivated visions such as the need for number theoretic universality are guiding the development of modern mathematics. The notion of motives introduced by Grothendieck is a good example about number theoretical universality and relates to the need to define integral at least in cohomological sense for all number fields: very concrete challenge also in TGD framework. Langlands program is one of the hot areas of what might be called physical mathematics. The above mentioned number theoretical universality is one of the guiding lines in this approach. The program relies on very general conjectures about a connection between number theory and harmonic analysis relating the representations of Galois groups with the representations of certain kinds of Lie groups to each other. Langlands conjecture has many forms and it is indeed a conjecture and many of them are inprecise since the notions involved are not sharply defined. Peter Woit noticed that Ed Frenkel had given a talk with rather interesting title "What do Fermat's Last Theorem and Electromagnetic Duality Have in Common?"? I listened the talk and found it very inspiring. The talk provides bird's eye of view about some basic aspects of Langlands program using the language understood by physicist. Also the ideas concerting the connection between Langlands duality and electricmagnetic duality generalized to Sduality in the context of nonAbelian gauge theories and string theory context and developed by Witten and Kapustin and followers are summarized. In this context D=4 and twisted version of N=4 SYM familiar from twistor program and defining a topological QFT appears. For some years ago I made my first attempt to understand what Langlands program is about and tried to relate it to TGD framework (see this). At that time I did not really understand the motivations for many of the mathematical structures introduced. In particular, I did not really understand the motivations for introducing the gigantic Galois group of algebraic numbers regarded as algebraic extension of rationals.
The talk of Frenkel inspired me to look again for Langlands program in TGD framework taking into the account of various developments that have occured in TGD during these years. I realized again that ideas develop unconsciously during the years and that many questions which remained unanswered for some years ago had found obvious answers. Instead of writing a 10 page posting I attach the abstract of pdf article "Langlands conjectures in TGD framework" at my homepage. The arguments of this article support the view that in TGD Universe number theoretic and geometric Langlands conjectures could be understood very naturally. The basic notions are following.
For details and background the reader can consult either to the chapter Langlands Program and TGD or to the article Langlands Conjectures in TGD Framework. 
Quantum Boolean algebra instead of Boolean algebra?
Mathematical logic relies on the notion of Boolean algebra, which has a wellknown representation as the algebra of sets which in turn has in algebraic geometry a representation in terms of algebraic varieties. This is not however attractive at spacetime level since the dimension of the algebraic variety is different for the intersection resp. union representing AND resp. OR so that only only finite number of ANDs can appear in the Boolean function. TGD inspired interpretation of the fermionic sector of the theory in terms of Boolean algebra inspires more concrete ideas about the the realization of Boolean algebra at both quantum level and classical spacetime level and also suggests a geometric realization of the basic logical functions respecting the dimension of the representative objects.
These observations suggest that generalized Feynman diagrams and their spacetime counterparts could have a precise interpretation in quantum Boolean algebra and that one should perhaps consider the extension of the mathematical logic to quantum logic. Alternatively, one could argue that quantum Boolean algebra is more like a model for what mathematical cognition could be in the real world. For details and background the reader can consult either to the chapter Physics as Generalized Number Theory: Infinite Primes or to the article How infinite primes relate to other views about mathematical infinity?. 
How infinite primes relate to other views about mathematical infinity?
Infinite primes is a purely TGD inspired notion. The notion of infinity is number theoretical and infinite primes have well defined divisibility properties. One can partially order them by the real norm. pAdic norms of infinite primes are well defined and finite. The construction of infinite primes is a hierarchical procedure structurally equivalent to a repeated second quantization of a supersymmetric arithmetic quantum field theory. At the lowest level bosons and fermions are labelled by ordinary primes. At the next level one obtains free Fock states plus states having interpretation as bound many particle states. The many particle states of a given level become the single particle states of the next level and one can repeat the construction ad infinitum. The analogy with quantum theory is intriguing and I have proposed that the quantum states in TGD Universe correspond to octonionic generalizations of infinite primes. It is interesting to compare infinite primes (and integers) to the Cantorian view about infinite ordinals and cardinals. The basic problems of Cantor's approach relate to the axiom of choice, continuum hypothesis, and Russell's antinomy: all these problems are due to the definition of ordinals as sets. In TGD framework infinite primes, integers, and rationals are defined purely algebraically so that these problems are avoided. It is not surprising that these approaches are not equivalent. For instance, sum and product for Cantorian ordinals are not commutative unlike for infinite integers defined in terms of infinite primes. Set theory defines the foundations of modern mathematics. Set theory relies strongly on classical physics, and the obvious question is whether one should reconsider the foundations of mathematics in light of quantum physics. Is set theory really the correct approach to axiomatization?
It is interesting to discuss the possible impact of these observations on the foundations of physical mathematics assuming that one accepts the TGD inspired view about infinity, about the notion of number, and the restrictions on the notion of set suggested by classical TGD. I do not bother to type all the text here. For details the reader can consult either to the chapter Physics as Generalized Number Theory: Infinite Primes or to the article How infinite primes relate to other views about mathematical infinity?. 
How could one calculate padic integrals numerically?
Riemann sum gives the simplest numerical approach to the calculation of real integrals. Also padic integrals should allow a numerical approach and very probably such approaches already exist and "motivic integration" presumably is the proper word to google. The attempts of an average physicist to dig out this kind of wisdom from the vastness of mathematical literature however lead to a depression and deep feeling of inferiority. The only manner to avoid the painful question "To whom should I blame for ever imagining that I could become a real mathematical physicist some day?" is a humble attempt to extrapolate real common sense to padic realm. One must believe that the almost trivial Riemann integral must have an almost trivial padic generalization although this looks far from obvious. 1. A proposal for padic numerical integration The physical picture provided by quantum TGD gives strong constraints on the notion of padic integral.
2. General coordinate invariance From the point of view of physics general coordinate invariance of the volume integral and more general integrals is of utmost importance.
A convenient iteration procedure is based on the representation of integrand f as sum ∑_{k}f_{k} of functions associated with different padic valued branches z_{k}=z_{k}(x) for the surface in the coordinates chosen and identified as a subset of preferred imbedding space coordinates. The number of branches z_{k} contributing is by padic continuity locally constant. The function f_{k}  call it g for simplicity  can in turn be decomposed into a sum of piecewise constant functions by introducing first the piecewise constant pinary cutoffs g_{n}(x) obtained in the approximation O(p^{n+1})=0. One can write g as g(x)= ∑ h_{n} (x) , h_{0}(x)=g_{0}(x) , h_{n}=g_{n}(x)g_{n1}(x) for n>0 . Note that h_{n}(x) is of form g_{n}(x)= a_{n}(x)p^{n}, a_{n}(x) ∈ {0,p1} so that the representation for integral as a sum of integrals for piecewise constant functions h_{n} converge rapidly. The technical problem is the determination of the boundaries of the regions inside which these functions contribute. The integral reduces to the calculation of the number of points for given value of h_{n}(x) and by the local constancy for the number of padic valued roots z_{k}(x) the number of points for N_{0} ∑_{k≥ 0} p^{k}= N_{0}/(1p), where N_{0} is the number of points x with the property that not all points y= x(1+O(p)) represent padic points z(x). Hence a finite number of calculational steps is enough to determine completely the contribution of given value to the integral and the only approximation comes from the cutoff in n for h_{n(x). }
4. Number theoretical universality This picture looks nice but it is far from clear whether the resulting integral is that what physicist wants. It is not clear whether the limit Vol(V,n), n→ ∞, exists or even should exist always.
The equivalence of the proposed numerical integral with the algebraic definition of padic integral motivated by the algebraic formula in real context expressed in terms of various parameters defining the variety and the integrand and continued to all number fields would be such a number theoretical miracle that it deserves italics around it: For algebraic surfaces the real volume of the variety equals apart from constant C to the number of padic points of the variety in the case that the volume is expressible as padic integer. The proportionality constant C can depend on padic number field , and the previous numerical argument suggests that the constant could be simply the factor 1/(1p) resulting from the sum of padic points in padic scales so short that the number of the padic branches z_{k}(x) is locally constant. This constant is indeed needed: without it the real integrals in the intersection of real and padic worlds giving integer valued result I=m would correspond to functions for which the number of padic valued points is finite. The statement generalizes to apply also to the integrals of rational and perhaps even more general functions. The equivalence should be considered in a weak form by allowing the transcendentals contained by the formulas have different meanings in real and padic number fields. Already the integrals of rational functions contain this kind of transcendentals. The basic objection that number of padic points without cannot give something proportional to real volume with an appropriate interpretation cannot hold true since real integral contains the determinant of the induced metric. As already noticed the preferred coordinates for the imbedding space are fixed by the isometries of the imbedding space and therefore the information about metric is actually present. For constant function the correspondence holds true and since the recipe for performing of the integral reduce to that for an infinite sum of constant functions, it might be that the miracle indeed happens. The proposal can be tested in a very simple manner. The simplest possible algebraic variety is unit circle defined by the condition x^{2}+y^{2}=1.
6. pAdic thermodynamics for measurement resolution? The proposed definition is rather attractive number theoretically since everything would reduce to the counting of padic points of algebraic varieties. The approach generalizes also to algebraic extensions of padic numbers. Mathematicians and also physicists love partition functions, and one can indeed assign to the volume integral a partition function as padic valued power series in powers Z(t)=∑ v_{n}t^{n} with the coefficients v_{n} giving the volume in O(p^{n})=0 cutoff. One can also define partition functions Z_{f}(t)= ∑ f_{n}t^{n}, with f_{n} giving the integral of f in the same approximation. Could this kind of partition functions have a physical interpretation as averages over physical measurements over different pinary cutoffs? pAdic temperature can be identified as t = p^{1/T}, T=1/k. For padically small temperatures the lowest terms corresponding to the worst measurement resolution dominate. At first this sounds counterintuitive since usually low temperatures are thought to make possible good measurement resolution. One can however argue that one must excite padic short range degrees of freedom to get information about them. These degrees of freedom correspond to the higher pinary digits by padic length scale hypothesis and high energies by Uncertainty Principle. Hence high padic temperatures are needed. Also measurement resolution would be subject to padic thermodynamics rather than being freely fixed by the experimentalist. For details see the new chapter Motives and Infinite Primes or the article with same title. 
What about counterparts of T, S, and U dualities in TGD framework?The natural question is what could be the TGD counterparts of S, T and Udualities. If one accepts the identification of Uduality as product U=ST and the proposed counterpart of T duality as a strong form of general coordinate invariance discussed in previous posting, it remains to understand the TGD counterpart of Sduality  in other words electricmagnetic duality  relating the theories with gauge couplings g and 1/g. Quantum criticality selects the preferred value of g_{K}: Kähler coupling strength is very near to fine structure constant at electron length scale and can be equal to it. Since there is no coupling constant evolution associated with α_{K}, it does not make sense to say that g_{K} becomes strong and is replaced with its inverse at some point. One should be able to formulate the counterpart of Sduality as an identity following from the weak form of electricmagnetic duality and the reduction of TGD to almost topological QFT. This seems to be the case. TGD based view about Sduality The following arguments suggests that in TGD framework S duality is realized for each preferred extremal of Kähler action separately whereas in standard view the duality would be realized only at the level of path integral defining the partition function.
The boundary condition J_{E}=J_{B} at the Euclidian side of the wormhole throat inspires the question whether all Euclidian regions could be selfdual so that the density of Kähler action would be just the instanton density. Selfduality follows if the deformation of the metric induced by the deformation of the canonically imbedded CP_{2} is such that in CP_{2} coordinates for the Euclidian region the tensor (g^{αβ}g^{μν} g^{αν}g^{μβ})/g^{1/2} remains invariant. This is certainly the case for CP_{2} type vacuum extremals since by the lightlikeness of M^{4} projection the metric remains invariant. Also conformal scalings of the induced metric would satisfy this condition. Conformal scaling is not consistent with the degeneracy of the 4metric at the wormhole throat. Selfduality is indeed an unnecessarily strong condition. Comparison with standard view about dualities One can compare the proposed realization of T, S and Uduality to the more general dualities defined by the modular group SL(2,Z), which in QFT framework can hold true for the path integral over all possible gauge field configurations. In the resent case the dualities hold true for every preferred extremal separately and the functional integral is only over the spacetime projections of fixed Kähler form of CP_{2}. Modular invariance for Maxwell action was discussed by E. Verlinde for Maxwell action with θ term for a general 4D compact manifold with Euclidian signature of metric. In this case one has path integral giving sum over infinite number of extrema characterized by the cohomological equivalence class of the Maxwell field the action exponential to a high degree. Modular invariance is broken for CP_{2}: one obtains invariance only for τ→ τ+2 whereas S induces a phase factor to the path integral.
CP breaking and ground state degeneracy Ground state degeneracy due to the possibility of having both signs for Minkowskian contribution to the exponent of vacuum functional provides a general view about the description of CP breaking in TGD framework.
For details see the new chapter Motives and Infinite Primes or the article with same title. 
Ktheory, branes, and TGDKtheory is an essential part of the motivic cohomology. Unfortunately, this theory is very abstract and the articles written by mathematicians are usually incomprehensible for a physicist. Hence the best manner to learn Ktheory is to learn about its physics applications. The most important applications are brane classification in super string models and Mtheory. The excellent lectures by Harah Evslin with title What doesn't Ktheory classify? make it possible to learn the basic motivations for the classification, what kind of classifications are possible, and what are the failures. Also the Wikipedia article gives a bird's eye of view about the problems. As a byproduct one learns something about the basic ideas of Ktheory. In the sequel I will discuss critically the basic assumptions of brane world scenario, sum up my understanding about the problems related to the topological classification of branes and also to the notion itself, ask what goes wrong with branes and demonstrate how the problems are avoided in TGD framework, and conclude with a proposal for a natural generalization of Ktheory to include also the division of bundles inspired by the generalization of Feynman diagrammatics in quantum TGD, by zero energy ontology, and by the notion of finite measurement resolution. Brane world scenario The brane world scenario looks attractive from the mathematical point of view ine one is able to get accustomed with the idea that basic geometric objects have varying dimensions. Even accepting the varying dimensions, the basic physical assumptions behind this scenario are vulnerable to criticism.
In this framework the brave brane world episode would have been a very useful Odysseia. The possibility to interpret various geometric objects physically has proved to be an extremely powerful tool for building provable conjectures and has produced lots of immensely beautiful mathematics. As a fundamental theory this kind of approach does not look convincing to me. The basic challenge: classify the conserved brane charges associated with branes One can of course forget these critical arguments and look whether this general picture works. The first thing that one can do is to classify the branes topologically. I made the same question about 32 years ago in TGD framework: I thought that cobordism for 3manifolds might give highly interesting topological conservation laws. I was disappointed. The results of Thom's classical article about manifold cobordism demonstrated that there is no hope for really interesting conservation laws. The assumption of Lorentz cobordism meaning the existence of global timelike vector field would make the situation more interesting but this condition looked too strong and I could not see a real justification for it. In generalized Feynman diagrammatics there is no need for this kind of condition. There are many alternative approaches to the classification problem. One can use homotopy, homology, cohomology and their relative and other variants, topological or algebraic Ktheory, twisted Ktheory, and variants of Ktheory not yet existing but to be proposed within next years. The list is probably endless unless something like motivic cohomology brings in enlightment.
The challenge is to find the mathematical classification which suits best the physical intuitions (, which might be fatally wrong as already proposed) but is universal at the same time. This challenge has turned out to be tough. The RamondRamond (RR) pform fields of type II superstring theory are rather delicate objects and a source of most of the problems. The difficulties emerge also by the presence of NeveuSchwartz 3form H =dB defining classical background field. Ktheory has emerged as a good candidate for the classification of branes. It leaves the confines of homology and uses bundle structures associated with branes and classifies these. There are many Ktheories. In topological Ktheory bundles form an algebraic structure with sum, difference, and multiplication. Sum is simply the direct sum for the fibers of the bundle with common base space. Product reduces to a tensor product for the fibers. The difference of bundles represents a more abstract notion. It is obtained by replacing bundles with pairs in much the same way as rationals can be thought of as pairs of integers with equivalence (m,n)= (km,kn), k integer. Pairs (n,1) representing integers and pairs (1,n) their inverses. In the recent case one replaces multiplication with sum and regards bundle pairs and (E,F) and (E+G,F+G) equivalent. Although the pair as such remains a formal notion, each pair must have also a real world representativs. Therefore the sign for the bundle must have meaning and corresponds to the sign of the charges assigned to the bundle. The charges are analogous to winding of the brane and one can call brane with negative winding antibrane. The interpretation in terms of orientation looks rather natural. Later a TGD inspired concrete interpretation for the bundle sum, difference, product and also division will be proposed. Problems related to the existence of spinor structure Many problems in the classification of brane charges relate to the existence of spinor structure. The existence of spinor structure is a problem already in general general relativity since ordinary spinor structure exists only if the second StiefelWhitney class of the manifold is nonvanishing: if the third StiefelWhitney class vanishes one can introduce so called spin^{c} structure. This kind of problems are encountered already in lattice QCD, where periodic boundary conditions imply nonuniqueness having interpretation in terms of 16 different spinor structures with no obvious physical interpretation. One the strengths of TGD is that the notion of induced spinor structure eliminates all problems of this kind completely. One can therefore find direct support for TGD based notion of spinor structure from the basic inconsistency of QCD lattice calculations!
Ashoke Sen has proposed a grand vision for understanding the brane classification in terms of tachyon condensation in absence of NSNS field H. The basic observation is that stacks of spacefilling D and anti Dbranes are unstable against process called tachyon condensation which however means fusion of p+1D brane orbits rather than pdimensional time slicse of branes. These branes are however accompanied by lowerdimensional branes and the decay process cannot destroy these. Therefore the idea arises that suitable stacks of D9 branes and antiD9branes could code for all lowerdimensional brane configurations as the end products of the decay process. This leads to a creation of lowerdimensional branes. All decay products of branes resulting in the decay cascade would be by definition equivalent. The basic step of the decay process is the fusion of Dbranes in stack to single brane. In bundle theoretic language one can say that the Dbranes and antiD branes in the stack fuse together to single brane with bundle fiber which is direct sum of the fibers on the stack. This fusion process for the branes of stack would correspond in topological Ktheory. The fusion of Dbranes and antiD branes would give rise to nothing since the fibers would have opposite sign. The classification would reduce to that for stacks of D9branes and anti D9branes. Problems with Hodge duality and Sduality The Ktheory classification is plagued by problems all of which need not be only technical.
The existence of nonrepresentable 7D homology classes for targent space dimension D>9 There is a further nasty problem which destroys the hopes that twisted Ktheory could provide a satisfactory classification. Even worse, something might be wrong with the superstring theory itself. The problem is that not all homology classes allow a representation as nonsingular manifolds. The first dimension in which this happens is D=10, the dimension of superstring models! Situation is of course the same in Mtheory. The existence of the nonrepresentables was demonstrated by Thom  the creator of catastrophe theory and of cobordism theory for manifolds for a long time ago. What happens is that there can exist 7D cycles which allow only singular imbeddings. A good example would be the imbedding of twistor space CP_{3}, whose orbit would have conical singularity for which CP_{3} would contract to a point at the "moment of big bang". Therefore homological classification not only allows but demands branes which are orbifolds. Should orbifolds be excluded as unphysical? If so then homology gives too many branes and the singular branes must be excluded by replacing the homology with something else. Could twisted Ktheory exclude nonrepresentable branes as unstable ones by having nonvanishing w_{3}+[H]? The answer to the question is negative: D6branes with w_{3}+[H]=0 exist for which Ktheory charges can be both vanishing or nonvanishing. One can argue that nonrepresentability is not a problem in superstring models (Mtheory) since spontaneous compactification leads to M× X_{6} (M× X_{7}). On the other hand, Cartesian product topology is an approximation which is expected to fail in high enough length scale resolution and near big bang so that one could encounter the problem. Most importantly, if Mtheory is theory of everything it cannot contain this kind of beauty spots. What could go wrong with super string theory and how TGD circumvents the problems? As a proponent of TGD I cannot avoid the temptation to suggest that at least two things could go wrong in the fundamental physical assumptions of superstrings and Mtheory.
Can one identify the counterparts of RR and NSNS fields in TGD? RR and NSNS 3forms are clearly in fundamental role in Mtheory. Since in TGD partonic 2surfaces define the analogs of fundamental M2branes, one can wonder whether these 3forms could have TGD counterparts.
Could one divide bundles? TGD differs from string models in one important aspects: stringy diagrams do not have interpretation as analogs of vertices of Feynman diagrams: the stringy decay of partonic 2surface to two pieces does not represent particle decay but a propagation along different paths for incoming particle. Particle reactions in turn are described by the vertices of generalized Feynman diagrams in which the ends of incoming and outgoing particles meet along partonic 2surface. This suggests a generalization of Ktheory for bundles assignable to the partonic 2surfaces. It is good to start with a guess for the concrete geometric realization of the sum and product of bundles in TGD framework.
Why not define also division of bundles in terms of the division for tensor product? In terms of the 3vertex for generalized Feynman diagrams A⊗ B=C representing tensor product B would be by definition C/A. Therefore TGD would extend the Ktheory algebra by introducing also division as a natural operation necessitated by the presence of the join along ends vertices not present in string theory. I would be surprised if some mathematician would not have published the idea in some exotic journal. Below I represent an argument that this notion could be also applied in the mathematical description of finite measurement resolution in TGD framework using inclusions of hyperfinite factor. Division could make possible a rigorous definition for for noncommutative quantum spaces. Tensor division could have also other natural applications in TGD framework.
For more details see the new chapter Infinite Primes and Motives or the article with same title. 
How detailed the quantum classical correspondence can be?Can the dynamics defined by preferred extremals of Kähler action be dissipative in some sense? The generation of the arrow of time has a nice realization in zero energy ontology as a choice of welldefined particle numbers and other quantum numbers at the "lower" end of CD. By quantum classical correspondence this should have a spacetime correlate. Gradient dynamics is a highly phenomenological realization of the dissipative dynamics and one must try to identify a microscopic variant of dissipation in terms of entropy growth of some kind. If the arrow of time and dissipation has spacetime correlate, there are hopes about the identification of this kind of correlate. Quantum classical correspondence has been perhaps the most useful guiding principle in the construction of quantum TGD. What is says that not only quantum numbers but also quantum jump sequences should have spacetime correlates: about this the failure of strict determinism of Kähler action gives good hopes. Even the quantum superposition  at least for certain situations  might have spacetime correlates.
The following discussion concentrates on possible spacetime correlates for the quantum superposition of WCW spinor fields and for the arrow of time.

Floer homology and TGDTGD can be seen as almost topological quantum field theory. This could have served as a motivation for spending most of last months to the attempt to learn some of the mathematics related to various kind of homologies and cohomologies. The decisive stimulus came from the attempt to understand the basic ideas of motivic cohomology. I am not a specialist and do not have any ambition or abilities to become such. My goals is to see whether these ideas could be applied in quantum TGD. Documentation is the best manner to develop ideas and the learning process has materialized as a new chapter entitled Infinite Primes and Motives of "Physics as Generalized Number Theory". It soon became clear that much of the mathematics needed by TGD has existed for decades and developing all the time. The difficult task is to understand the essentials of this mathematics and translate to the language that I talk and understand. Also generalization is unavoidable. Those who think that new physics can be done by taking math as such are wasting their time. Among another things I have been learning about various cohomologies and homologies  about quantum cohomology, about Floer homology and topological string theories, about GromovWitten invariants,... It would be very naive to think that these notions would work as such in TGD framework. It looks however very plausible that the their generalizations to TGD exist, and could be very useful in the more detailed formulation of quantum TGD. The crucially important notion is finite measurement resolution making everything almost topological and highly number theoretic. In this brainstormy spirit I have even become a proud father of my own pet homology, which I have christened as braided Galois homology. It is based on the correspondence between infinite primes and polynomials of several variables and is formulated in braided group algebras with braidings realized as symplectic flows and generalizing somewhat the usual notion of homology meaning that the square of boundary operation gives something in commutator group reducing to unit element of ordinary homology only in the factor group obtained by dividing with the commutator group. Floer homology in its original form replaces Morse function in symplectic manifold M in the loop space LM of M. The loops can be seen as homotopies of Hamiltonians and paths in loops space describe cylinders in M. With an appropriate choice of symplectic action these cylinderes can be regarded as (pseudo)holomorphic surface completely analogous to string orbits. By combining Floer's theory with Witten's discovery about the connection between the Morse theory and supersymmetry one ends up with topological QFTs as a manner to formulate Floer homology and various variants of this notion in particular topological QFTs characterizing topology of threemanifolds. This kind of learning periods are very useful as a rule since they allow to improve bird's eye of view about TGD and its problems. The understanding of both quantum TGD and its classical counterpart is still far from from comprehensive. For instance, the view about the physical and mathematical roles of Kähler actions for Euclidian and Minkowskian spacetime regions is far from clear. Do they provide dual descriptions as suggested or are both needed? Kähler action for preferred extremal in Euclidian regions defines naturally positive definite Kähler function. But can one regard the Kähler action in Minkowskian regions as equivalent definition for Kähler function or should one regard it as imaginary as the presence of square root of metric determinant would suggest? What could be the interpretation in this case? The basic ideas about Floer homology suggest and answer to these questions.
Should one assume that the reduction to ChernSimons terms occurs for the preferred extremals in both Minkowskian and Euclidian regions or only in Minkowskian regions?
Floer homology and GromovWitten invariants provide also other insights about quantum TGD. For more details see the new chapter Infinite Primes and Motives or the article with same title.

Twistors, hyperbolic 3manifolds, and zero energy ontologyWhile performing web searches for twistors and motives I have begun to realize that Russian mathematicians have been building the mathematics needed by quantum TGD for decades while realizing the great visions of Grothendieck. Maybe I am also beginning to vaguely grasp something about the connection of Grassmannian twistor approach to the motivic integrals. In the following I make comments about three articles that I found from web. The latest finding was the article Volumes of hyperbolic manifolds and mixed Tate motives by Goncharov one of the great Russian mathematicians involved with the drama. The article is about polylogarithms emerging in twistor calculations and their relationship to the volumes of hyperbolic nmanifolds. I do not of course understand anything about the jargon of the article: it is written by a specialist for specialists and I can only try to understand the general notions and the possible meaning of the results from TGD point of view. Hyperbolic nmanifolds are nmanifolds equipped with complete Riemann metric having constant sectional curvature equal to 1 (with a suitable choice of length unit) and therefore obeying Einstein's equations with cosmological constant. They are obtained as coset spaces on propertime constant hyperboloids of n+1dimensional Minkowski space by dividing by the action of discrete subgroup of SO(n,1), whose action defines a lattice like structure on the hyperboloid. What is remarkable is that the volumes of these closed spaces are homotopy invariants in a welldefined sense. What is even more remarkable that hyperbolic 3manifolds are completely exceptional in that there are very many of them. The complements of knots and links in 3sphere are often cusped hyperbolic 3manifolds (having therefore tori as boundaries). Also Haken manifolds are hyperbolic. Says Wikipedia: According to Thurston's geometrization conjecture, proved by Perelman, any closed, irreducible, atoroidal 3manifold with infinite fundamental group is hyperbolic. There is an analogous statement for 3manifolds with boundary. Therefore there are very many hyperbolic 3manifolds. The geometrization conjecture of Thurston allows to see hyperbolic 3manifolds in a wider framework. The theorem states that compact 3manifolds can be decomposed canonically into submanifolds that have geometric structures. It was Perelman who sketched the proof of the conjecture. The prime decomposition with respect to connected sum reduces the problem to the classification of prime 3manifolds and geometrization conjecture states that closed 3manifold can be cut along tori such that the interior of each piece has a geometric structure with finite volume serving as a topological invariant. There are 8 possible geometric structures in dimension three and they are characterized by the isometry group of the geometry and the isotropy group of point. Important is also the behavior under Ricci flow ∂_{t}g_{ij}= 2R_{ij}: here t is not spacetime coordinate but a parameter of homotopy. If I have understood correctly, Ricci flow is a dissipative flow gradually polishing the metric for a particular region of 3manifold to one of the 8 highly symmetric local metrics defining topological invariants. This conforms with the general vision about dissipation as source of maximal symmetries. For compact nmanifolds the normalized Ricci flow ∂_{t}g_{ij}= 2R_{ij} +(2/n)Rg_{ij} preserving the volume makes sense. Interestingly, for n=4 the right hand side is Einstein tensor so that the solutions of vacuum Einstein's equations in dimension four are fixed points of normalized Ricci flow. Ricci flow expands the negatively curved regions and contracts the positively curved regions of spacetime time. Hyperbolic geometries represent one these 8 geometries and for the Ricci flow is expanding. The outcome is amazingly simple and gives also support for the idea that the preferred extremals of Kähler action could represent maximally symmetries 4geometries defining topological invariants: the preferred extremals would be maximally symmetric representatives with a given topology or algebraic geometry. The volume spectrum for hyperbolic 3manifolds forms a countable set which is however not discrete: the statement that one can assign to them ordinal ω^{ω} does not have any obvious meaning for the man of the street;). What comes into my simple mind is that padic integers and more generally, profinite spaces with infinite number of points, might be something similar: one can enumerate them by infinitely long sequences of pinary digits so that they are countable (I do not know whether also infinite padic primes must be allowed and whether they could somehow correspond the hierarchy of infinite ordinals). They are totally disconnected in real sense but do not form a discrete set since since can connect any two points by a padically continuous curve. What makes twistor people excited is that the polylogarithms emerging from twistor integrals (see this and this) seem to be expressible in terms of the volumes of hyperbolic manifolds. What fascinates me is that the polylogarithms in question make sense also padically and that the moduli spaces for causal diamonds or rather, for the double lightcones associated with their M^{4} projections with second tip fixed  are naturally lattices of the 3dimensional hyperbolic space defined by all positions of the second tip and 3dimensional hyperbolic spaces are the most interesting ones! In the intersection of the real and padic worlds both algebraic universality and finite measurement resolution require number theoretic discretization so that the 3volume volume could be quantized in discrete manner. For n=3 the group defining the lattice is a discrete subgroup of the group of SO(3,1) which equals to PSL(2,C) obtained by identifying SL(2,C) matrices with opposite sign. The divisor group defining the lattice and hyperbolic spaces as its lattice cell is therefore a subgroup of PSL(2,Z_{c}), where Z_{c} denotes complex integers. Recall that PSL(2,Z_{c}) acts also in complex plane (and therefore on partonic 2surfaces) as discrete Möbius transformations whereas PSL(2,Z) correspond to 3braid group. Reader is perhaps familiar with fractal like orbits of points of plane under iterated Möbius transformations. The lattice cell of this lattice obtained by identifying symmetry related points defines hyperbolic 3manifolds. Therefore zero energy ontology realizes directly the hyperboliic manifolds whose volumes should somehow represent the polylogarithms. The volumes are topological invariants in the sense that homeomorphism does not affect the volume of the space in question if it is given hyperbolic metric. The spectrum of volumes is said to be highly transcendental. In the intersection of real and padic worlds only algebraic volumes are possible unless one allows extension by say finite number of roots of e (e^{p} is padic number). The padic existence of polylogarithms suggests that also padic variants of hyperbolic spaces make sense and that one can assign to them volume as topological invariant although the notion of ordinary volume integral is problematic. In fact, hyperbolic spaces are symmetric spaces and the general arguments that I have developed earlier allow to imagine what the padic variants of real symmetric spaces could be. Not surprisingly, also AdSCFT enthusiasts would like to have similar invariants for for AdS (Minkowskian analog of hyperbolic space) and even dS (Minkowskian analog of sphere). Mitchell Porter gives a link to the talk of Maldacena. The expected noncompactness of these spaces implies infinite volume and this problem should be circumvented somehow. Maybe the preferred role of hyperbolic spaces over AdS and dS might finally select between TGD and Mtheory like approach. This would simplify matters enormously since 10dimensional holography would reduce to 4dimensional one and would have a direct connection with physics as we have used to know it. For condensed matter physicists expected to say something interesting about this real world already the complexities of 3D world represent a tough enough challenge and the formulation of the problems in terms of 10dimensional blackholes migh be too much;). For more details see the new chapter Infinite Primes and Motives or the article with same title. 
Motives and twistors in TGDMotivic cohomology has turned out to pop up in the calculations of the twistorial amplitudes using Grassmannian approach (see this and this). The amplitudes reduce to multiple residue integrals over smooth projective subvarieties of projective spaces. Therefore they represent the simplest kind of algebraic geometry for which cohomology theory exists. Also in Grothendieck's vision about motivic cohomology projective spaces are fundamental as spaces to which more general spaces can be mapped in the construction of the cohomology groups (factorization). In the previous posting I gave an abstract of a chapter about motives and TGD explaining a proposal for a noncommutative variant of homology theory based on a hierarchy of Galois groups assigned with the zero locus of polynomial and its restrictions to lover dimension planes obtained by putting variables appearing in it to zero one by one: the basic idea is simple but I would have never discovered it without infinite primes. The basic problem is to define boundary homomorphism for the hierarchy of Galois groups G_{k} satisfying the nonabelian generalization of δ^{2}=0 stating that the image under δ^{2} belongs to the commutator subgroup of G_{k2} and therefore is mapped to zero in abelianization, which means division by commutator subgroup.
Summarizing, the infinite prime  irreducible polynomial  braid  quantum state connection suggests very deep connections between number theory, algebraic geometry, topological quantum field theories, and supersymmetric quantum field theories. The article Motives and Infinite Primes gives a more detailed discussion. Defining integration in padic context is one of the basic challenges of quantum TGD in which real and various padic physics ought to be unified to a larger theory by realizing what I have called number theoretical Universality. Grothendieck's motivic comology can be seen as a program for the realization of integration of forms making sense also in padic context. In the following I shall discuss some aspects of the problem in TGD framework. The discussion of course fails to satisfy all standards of mathematical rigor but it relies of extremely deep and general physical principles and my conviction is that good physics is the best guideline for developing good mathematics. Number theoretic universality, residue integrals, and symplectic symmetry A key challenge in the realization of the number theoretic universality is the definition of padic definite integral. In twistor approach integration reduces to the calculation of multiple residue integrals over closed varieties. These could exist also for padic number fields. Even more general integrals identifiable as integrals of forms can be defined in terms of motivic cohomology. Yangian symmetry (see this and this) is the symmetry behind the successes of twistor Grassmannian approach and has a very natural realization in zero energy ontology (see this). Also the basic prerequisites for twistorialization are satisfied. Even more, it is possible to have massive states as bound states of massless ones and one can circumvent the IR difficulties of massless gauge theories. Even UV divergences are tamed since virtual particles consist of massless wormhole throats without bound state condition on masses. Spacelike momentum exchanges correspond to pairs of throats with opposite sign of energy. Algebraic universality could be realized if the calculation of the scattering amplitudes reduces to multiple residue integrals just as in twistor Grassmannian approach. This is because also padic integrals could be defined as residue integrals. For rational functions with rational coefficients field the outcome would be an algebraic number apart from power of 2π, which in padic framework is a nuisance unless it is possible to get rid of it by a proper normalization or unless one can accepts the infinitedimensional transcendental extension defined by 2π. It could also happen that physical predictions do not contain the power of 2π. Motivic cohomology defines much more general approach allowing to calculate analogs of integrals of forms over closed varieties for arbitrary number fields. In motivic integration  to be discussed below  the basic idea is to replace integrals as real numbers with elements of so called scissor group whose elements are geometric objects. In the recent case one could consider the possibility that (2π)^{n} is interpreted as torus (S^{1})^{n} regarded as an element of scissor group which is free group formed by formal sums of varieties modulo certain natural relations meaning. Motivic cohomology allows to realize integrals of forms over cycles also in padic context. Symplectic transformations are transformation leaving areas invariant. Symplectic form and its exterior powers define natural volume measures as elements of cohomology and padic variant of integrals over closed and even surfaces with boundary might make sense. In TGD framework symplectic transformations indeed define a fundamental symmetry and quantum fluctuating degrees of freedom reduce to a symplectic group assignable to δ M^{4}_{+/}× CP_{2} in welldefined sense (see this). One might hope that they could allow to define scissor group with very simple canonical representatives perhaps even polygons so that integrals could be defined purely algebraically using elementary area (volume) formulas and allowing continuation to real and padic number fields. The basic argument could be that varieties with rational symplectic volumes form a dense set of all varieties involved. How to define the padic variant for the exponent of Kähler action? The exponent of Kähler function defined by the Kähler action (integral of Maxwell action for induced Kähler form) is central for quantum at least in the real sector of WCW. The question is whether this exponent could have padic counterpart and if so, how it should be defined. In the real context the replacement of the exponent with power of p changes nothing but in the padic context the interpretation is affected in a dramatic manner. Physical intuition provided by padic thermodynamics (see this) suggests that the exponent of Kähler function is analogous to Bolzmann weight replaced in the padic context with nonnegative power of p in order to achieve convergence of the series defining the partition function not possible for the exponent function in padic context.
If this picture inspired by padic thermodynamics holds true, padic integration at the level of WCW would give analog of partition function with Boltzman weight replaced by a power of p reducing a sum over contributions corresponding to different powers of p with WCW integra.l over spacetime sheets with this value of Kähler action defining the analog for the degeneracy of states with a given value of energy. The integral over spacetime sheets corresponding to fixed value of Kähler action should allow definition in terms of a symplectic form defined in the padic variant of WCW. In finitedimensional case one could worry about odd dimension of this submanifold but in infinitedimensional case this need not be a problem. Kähler function could defines one particular zero mode of WCW Kähler metric possessing an infinite number of zero modes. One should also give a meaning to the padic integral of Kähler action over spacetime surface assumed to be quantized as multiples of log(m/n).
Since padic objects do not possess boundaries, one could argue that only the integrals over closed varieties make sense. Hence the basic premise of cohomology would fail when one has padic integral over braid strand since it does not represent closed curve. The question is whether one could identify the end points of braid in some sense so that one would have a closed curve effectively or alternatively relative cohomology. Periodic boundary conditions is certainly one prerequisite for this kind of identification.
Motivic integration While doing web searches related to motivic cohomology I encountered also the notion of motivic measure proposed first by Kontsevich. Motivic integration is a purely algebraic procedure in the sense that assigns to the symbol defining the variety for which one wants to calculate measure. The measure is not real valued but takes values in so called scissor group, which is a free group with group operation defined by a formal sum of varieties subject to relations. Motivic measure is number theoretical universal in the sense that it is independent of number field but can be given a value in particular number field via a homomorphism of motivic group to the number field with respect to sum operation. Some examples are in order.
Infinite rationals and multiple residue integrals as Galois invariants and Galois groups as symmetry groups of quantum physics In TGD framework one could consider also another kind of cohomological interpretation. The basic structures are braids at lightlike 3surfaces and spacelike 3surfaces at the ends of spacetime surfaces. Braids intersects have common ends points at the partonic 2surfaces at the lightlike boundaries of a causal diamond. String world sheets define braid cobordism and in more general case 2knot (see this)). Strong form of holography with finite measurement resolution would suggest that physics is coded by the data associated with the discrete set of points at partonic 2surfaces. Cohomological interpretation would in turn would suggest that these points could be identified as intersections of string world sheets and partonic 2surface defining dual descriptions of physics and would represent intersection form for string world sheets and partonic 2surfaces. Infinite rationals define rational functions and one can assign to them residue integrals if the variables x_{n} are interpreted as complex variables. These rational functions could be replaced with a hierarchy of subvarieties defined by their poles of various dimensions. Just as the zeros allow realization as braids or braids also poles would allow a realization as braids of braids. Hence the nfold residue integral could have a representation in terms of braids. Given level of the braid hierarchy with n levels would correspond to a level in the hierarchy of complex varieties with decreasing complex dimension. One can assign also to the poles (zeros of polynomial in the denominator of rational function) Galois group and obtains a hierarchy of Galois groups in this manner. Also the braid representation would exists for these Galois groups and define even cohomology and homology if they do so for the zeros. The intersections of braids with of the partonic 2surfaces would represent the poles in the preferred coordinates and various residue integrals would have representation in terms of products of complex points of partonic 2surface in preferred coordinates. The interpretation would be in terms of quantum classical correspondence. Galois groups transform the poles to each other and one can ask how much information they give about the residue integral. One would expect that the nfold residue integral as a sum over residues expressible in terms of the poles is invariant under Galois group. This is the case for the simplest integrals in plane with n poles and probably quite generally. Physically the invariance under the hierarchy of Galois group would mean that Galois groups act as the symmetry group of quantum physics. This conforms with the number theoretic vision and one could justify the formula for the residue integral also as a definition motivated by the condition of Galois invariance. Of course, all symmetric functions of roots would be Galois invariants and would be expected to appear in the expressions for scattering amplitudes. The Galois groups associated with zeros and poles of the infinite rational seem to have a clear physical significance. This can be understood in zero energy ontology if positive (negative) physical states are indeed identifiable as infinite integers and if zero energy states can be mapped to infinite rationals which as real numbers reduce to real units. The positive/negative energy part of the zero energy state would correspond to zeros/poles in this correspondence. An interesting question is how strong correlations the real unit property poses on the two Galois group hierarchies. The asymmetry between positive and negative energy states would have interpretation in terms of the thermodynamic arrow of geometric time (see this) implied by the condition that either positive or negative energy states correspond to state function reduced/prepared states with well defined particle numbers and minimum amount of entanglement. For more details see the new chapter Infinite Primes and Motives or the article with same title. 
Infinite primes and motives
In algebraic geometry the notion of variety defined by algebraic equation is very general: all number fields are allowed. One of the challenges is to define the counterparts of homology and cohomology groups for them. The notion of cohomology giving rise also to homology if Poincare duality holds true is central. The number of various cohomology theories has inflated and one of the basic challenges to find a sufficiently general approach allowing to interpret various cohomology theories as variations of the same motive as Grothendieck, who is the pioneer of the field responsible for many of the basic notions and visions, expressed it. Cohomology requires a definition of integral for forms for all number fields. In padic context the lack of wellordering of padic numbers implies difficulties both in homology and cohomology since the notion of boundary does not exist in topological sense. The notion of definite integral is problematic for the same reason. This has led to a proposal of reducing integration to Fourier analysis working for symmetric spaces but requiring algebraic extensions of padic numbers and an appropriate definition of the padic symmetric space. The definition is not unique and the interpretation is in terms of the varying measurement resolution. The notion of infinite has gradually turned out to be more and more important for quantum TGD. Infinite primes, integers, and rationals form a hierarchy completely analogous to a hierarchy of second quantization for a supersymmetric arithmetic quantum field theory. The simplest infinite primes representing elementary particles at given level are in oneone correspondence with manyparticle states of the previous level. More complex infinite primes have interpretation in terms of bound states.
This construction would realize thge number theoretical, algebraic geometrical, and topological content in the construction of quantum states in TGD framework in accordance with TGD as almost TQFT philosophy, TGD as an infiniteD geometry, and TGD as a generalized number theory visions. This picture leads also to a proposal how padic integrals could be defined in TGD framework.
pAdic physics is interpreted as physical correlate for cognition. The so called Stone spaces are in oneone correspondence with Boolean algebras and have typically 2adic topologies. A generalization to padic case with the interpretation of p pinary digits as physically representable Boolean statements of a Boolean algebra with 2^{n}>p>p^{n1} statements is encouraged by padic length scale hypothesis. Stone spaces are synonymous with profinite spaces about which both finite and infinite Galois groups represent basic examples. This provides a strong support for the connection between Boolean cognition and padic spacetime physics. The Stone space character of Galois groups suggests also a deep connection between number theory and cognition and some arguments providing support for this vision are discussed. For details see the new chapter Infinite Primes and Motives. 
Finding the roots of polynomials defined by infinite primesInfinite primes identifiable as analogs of free single particle states and bound manyparticle states of a repeatedly second quantized supersymmetric arithmetic quantum field theory correspond at n:th level of the hierarchy to irreducible polynomials in the variable X_{n} which corresponds to the product of all primes at the previous level of hierarchy. At the first level of hierarchy the roots of this polynomial are ordinary algebraic numbers but at higher levels they correspond to infinite algebraic numbers which are somewhat weird looking creatures. These numbers however exist padically for all primes at the previous levels because one one can develop the roots of the polynomial in question as powers series in X_{n1} and this series converges padically. This of course requires that infinitep padicity makes sense. Note that all higher terms in series are padically infinitesimal at higher levels of the hierarchy. Roots are also infinitesimal in the scale defined X_{n}. Power series expansion allows to construct the roots explicitly at given level of the hierarchy as the following induction argument demonstrates.
For background see the chapter TGD as a Generalized Number Theory III: Infinite Primes and for the pdf version of the argument the chapter NonStandard Numbers and TGD. 
NonStandard Numbers and TGDI had opportunity to read articles of Elemer Rosinger about possible physical applications of nonstandard numbers and it was natural to compare these numbers with the generalization of real numbers inspired by the notion of infinite primes. I dediced to attach the commentary as a new chapter to "Physics as a Generalized Number Theory". The abstract gives a rough overall view about the commentary. The chapter represents a comparison of ultrapower fields (loosely surreals, hyperreals, long line) and number fields generated by infinite primes having a physical interpretation in Topological Geometrodynamics. Ultrapower fields are discussed in very physicist friendly manner in the articles of Elemer Rosinger and these articles are taken as a convenient starting point. The physical interpretations and principles proposed by Rosinger are considered against the background provided by TGD. The construction of ultrapower fields is associated with physics using the close analogies with gauge theories, gauge invariance, and with the singularities of classical fields. Nonstandard numbers are compared with the numbers generated by infinite primes and it is found that the construction of infinite primes, integers, and rationals has a close similarity with construction of the generalized scalars. The construction replaces at the lowest level the index set Λ=N of natural numbers with algebraic numbers A, Frechet filter of N with that of A, and R with unit circle S^{1} represented as complex numbers of unit magnitude. At higher levels of the hierarchy generalized possibly infinite and infinitesimal algebraic numbers emerge. This correspondence maps a given set in the dual of Frechet filter of A to a phase factor characterizing infinite rational algebraically so that correspondence is like representation of algebra. The basic difference between two approaches to infinite numbers is that the counterpart of infinitesimals is infinitude of real units with complex number theoretic anatomy: one might loosely say that these real units are exponentials of infinitesimals. For details see the new chapter NonStandard Numbers and TGD. 
Generalization of thermodynamics allowing negentropic entanglement and a model for conscious information processingCosta de Beauregard considers a model for information processing by a computer based on an analogy with Carnot's heat engine (see this). I am grateful for Stephen Paul King for bringing this article to my attention in Time discussion group and also for inspiring discussions which also led to the birth of this section. As such the model Beauregard for computer does not look convincing as a model for what happens in biological information processing. Combined with TGD based vision about living matter, the model however inspires a model for how conscious information is generated and how the second law of thermodynamics must be modified in TGD framework. The basic formulas of thermodynamics remain as such since the modification means only the replacement S→ SN, where S is thermodynamical entropy and N the negentropy associated with negentropic entanglement. This allows to circumvent the basic objections against the application of Beauregard's model to living systems. One can also understand why living matter is so effective entropy producer as compared to inanimate matter and also the characteristic decomposition of living systems to highly negentropic and entropic parts as a consequence of generalized second law. I do not bother to type further and give instead a link to the article Generalization of thermodynamics allowing negentropic entanglement and a model for conscious information processing at my homepage and also to the chapter Negentropy Maximization Principle. 