During last weeks I have been writing a new chapter Quantum Adeles. The key idea is the generalization of p-adic number fields to their quantum counterparts and they key problem is what quantum p-adics 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 p-adics and quantum adeles really are?
What quantum p-adics are?
The first guess is that one obtains quantum p-adics from p-adic 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 p1<p. In the decomposition of coefficients to primes p1<p these primes are replaced with quantum primes assignable to p.
One could pose the additional condition that coefficients are smaller than pN and decompose to products of primes l<pN mapped to quantum primes assigned with q= exp(i2π/pN). The interpretation would be in terms of pinary cutoff. For N=1 one would obtain the counterpart of p-adic numbers. For N>1 this correspondence assigns to ordinary p-adic integer larger number of quantum p-adic integers and one can define a natural projection to the ordinary p-adic integer and its direct quantum counterpart with coefficients ak<p in pinary expansion so that a covering space of p-adics 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 p-adics defined in power series of pN with coefficients an<pN and expressed as products of quantum primes l<pN. 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 p-adics.
- The first additional idea is that one replaces p-adic integers with wave functions in the covering spaces associated with the prime factors l of integers n. This delocalization would give a genuine content for the attribute "quantum" as it does in the case of electron in hydrogen atom.
The natural physical interpretation for these wave functions would be as cognitive representations of the quantum states in matter sector so that momentum, spin and various internal quantum numbers would find cognitive representation in quantum Galois degrees of freedom.
One could talk about self-reference: the unseen internal degrees of freedom associated with p-adic integers would make it possible to represent physical information. Also the ratios of infinite primes reducing to unity give rise to similar but infinite-dimensional number theoretical anatomy of real numbers and leads to what I call Brahman=Atman identity.
- Second additional idea is to replace numbers with sequences of arithmetic operations that is quantum sum +q and quantum product ×q represented as fundamental 3-vertices and to formulate basic laws of arithmetics as symmetries of these vertices give rise to additional selection rules from natural additional symmetry conditions. These sequences of arithmetics with sets of integers as inputs and outputs are analogous to Feynman diagrams and the factorization of integers to primes has the decomposition of braid to braid strands as a direct correlate. One can also group incoming integers to sub-groups and the hierarchy of infinite primes describes this grouping.
A beautiful physical interpretation for the number theoretic Feynman diagrams emerges.
- The decomposition of integers m and n of a quantum rational m/n to products of primes l correspond to the decomposition of two braids to braid strands labeled by primes l. TGD predicts both time-like and space-like braids having their ends at partonic 2-surfaces. These two kinds of braids would naturally correspond to the two co-prime integers defining quantum rational m/n.
- The two basic vertices +q and ×q correspond to the fusion vertex for stringy diagrams and 3-vertex for Feynman diagrams: both vertices have TGD counterparts and correspond at Hilbert space level direct sum and tensor product. Note that the TGD inspired interpretation of +q (direct sum) is different from string model interpretation (tensor product). The incoming and outgoing integers in the Feynman diagram corresponds to Hilbert space dimensions and the decomposition to prime factors corresponds to the decomposition of Hilbert space to prime Hilbert spaces as tensor factors.
- Ordinary arithmetic operations have interpretation as tensor product and direct sum and one can formulate associativity, commutativity, and distributivity as well as product and sum as conditions on Feynman diagrams. These conditions imply that all loops can be transformed away by basic moves so that diagram reduces to a diagram obtained by fusing only sum and product to initial state to produce single line which decays to outgoing states by co-sum and co-product. Also the incoming lines attaching to same line can be permuted and permutation can only induce a phase factor. The conjecture that these rules hold true also for the generalized Feynman diagrams is obviously extremely powerful and consistent with the picture provided by zero energy ontology. Also connection with twistor approach is suggestive.
- Quantum adeles for ordinary rationals can be defined as Cartesian products of quantum p-adics and of reals or rationals. For algebraic extensions of rationals similar definition applies but allowing only those p-adic primes which do not split to a product of primes or the extension. Number theoretic evolution means increasing dimension for the algebraic extension of rationals and this means that increasing number of p-adic primes drops from the adele. This means a selective pressure under which only the fittest p-adic primes survive. The basic question is why Mersenne primes and some primes near powers of two are survivors.
The connection with infinite primes
A beautiful connection with the hierarchy of infinite primes emerges.
- The simplest infinite primes at the lowest level of hierarchy define two integers having no common prime divisors and thus defining a rational number having interpretation in terms of time-like and space-like braids characterized by co-prime integers.
- Infinite primes at the lowest level code for algebraic extensions of rationals so that the infinite primes which are survivors in the evolution dictate what p-adic primes manage to avoid splitting. Infinite primes coding for algebraic extensions have interpretation as bound states and the most stable bound states and p-adic primes able to resist corresponding splitting pressures survive.
At the n:th level of the hierarchy of infinite primes correspond to monic polynomials of n variables constructed from prime polymomials of n-1 variables constructed from.... The polynomials of single variable are in 1-1 correspondence with ordered collections of n rationals. This collection corresponds to n pairs of time-like and space-like braids. Thus infinite primes code for collections of lower level infinite primes coding for... and eventually everything boils down to collections rational coefficients for monic polynomials coding for infinite primes at the lowest level of the hierarchy. In generalized Feynman diagrams this would correspond to groups of groups of .... of groups of integers of incoming and outgoing lines.
- The physical interpretation is in terms of pairs time-like and space-like braids having ends at partonic 2-surfaces with strands labelled by primes and defining as their product integer: the rational is the ratio of these integers. From these basic braids one can form collections of braid pairs labelled by infinite primes at the second level of hierarchy, and so on and a beautiful connection with the earlier vision about infinite primes as coders of infinite hierarchy of braids of braids of... emerges. Space-like and time-like braids playing key role in generalized Feynman diagrams and representing rationals supporting the interpretation of generalized Feynman diagrams as arithmetic Feynman diagrams. The connection with many-sheeted space-time in which sheets containing smaller sheet define higher level particles, emerges too.
- Number theoretic dynamics for ×q conserves the total numbers of prime factors so that one can either talk about infinite number of conserved number theoretic momenta coming as multiples of log(p), p prime or of particle numbers assignable to primes p: pn corresponds to n-boson state and finite parts of infinite primes correspond to states with fermion number one for each prime and arbitrary boson number. The infinite parts of infinite primes correspond to fermion number zero in each mode. The two braids could also correspond to braid strands with fermion number 0 and 1. The bosonic and fermionic excitations would naturally correspond the generators of super-conformal algebras assignable to light-like and space-like 3-surfaces.
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 time-like and space-like 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 prime-dimensional Hilbert spaces stable against further decomposition.
Second natural decomposition appearing in representation theory is into direct sums. This decomposition would take place for prime-dimensional Hilbert spaces with dimension l with dimensions anpn in the p-adic expansion. The replacement of an 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 pn-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.
- The enormous vacuum degeneracy of Kähler action implies that the normal derivatives of imbedding space coordinates both at space-like 3 surfaces at the boundaries of CD and at light-like wormhole throats are many-valued functions of canonical momentum densities. Two directions are necessary by strong form of holography implying effective 2-dimensionality so that only partonic 2-surfaces and their tangent space data are needed instead of 3-surfaces. This implies that space-time surfaces can be regarded as surfaces in local singular coverings of the imbedding space. At partonic 2-surfaces the sheets of the coverings co-incide.
- By strong form of holography there are two integers characterizing the covering and the obvious interpretation is in terms of two integers characterizing infinite primes and time-like and space-like braids decomposing into braids labelled by primes. The braid labelled by prime would naturally correspond to a braid strand and its copies in l points of the covering. The state space defined by amplitudes in the n-fold covering would be n-dimensional and decompose into a tensor product of state spaces with prime dimension. These prime-dimensional state spaces would correspond to wave functions in prime-dimensional sub-coverings.
- Quantum primes are obtained as different sum decompositions of primes l and correspond direct sum decompositions of l-dimensional state space associated with braid defined by l-fold sub-covering. What suggests itself strongly is a symmetry breaking. This breaking would mean the geometric decomposition of l strands to subsets with numbers of elements coming proportional to powers pn of p. Could anpn in the expression of l as ∑ akpk correspond to a tensor product of an-dimensional space with finite field G(p,n)? Does this decomposition to state functions localized to sub-braids relate to symmetries and symmetry breaking somehow? Why an-dimensional Hilbert space would be replaced with a tensor product of quantum-p1-dimensional Hilbert spaces? The proper understanding of this issue is needed in order to have more rigorous formulation of quantum p-adics.
- Number theoretical dynamics would therefore relate directly to the hierarchy of Planck constants. This would also dictate what happens for Planck constants in the two vertices. There are two options.
- For ×q vertex the outgoing particle would have Planck constant, which is product of incoming Planck constants using ordinary Planck constant as unit. For +q vertex the Planck constant would be sum. This stringy vertex would lead to generation of particles with Planck constant larger than its minimum value. For ×q two incoming particles with ordinary Planck constant would give rise to a particle with ordinary Planck constant just as one would expect for ordinary Feynman diagrams.
- Another possible scenario is the one in which Planck constant is given by hbar/hbar0= n-1. In this case particles with ordinary Planck constant fuse to particles with ordinary Planck constant in both vertices.
For both options the feed of particles with non-standard value of Planck constant to the system can lead to a fusion cascade leading to a generation of dark matter particles with very large value of Planck constant. Large Planck constant means macroscopic quantum phases assumed to be crucial in TGD inspired biology. The obvious proposal is that inanimate matter transforms to living and thus also to dark matter by this kind of phase transition in presence of feed of particles - say photons- with non-standard value of Planck constant.
The work with quantum p-adics 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 p-adic 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 p-adic physics of cognition. The optimistic inside me even insists that the basic mathematical structures of TGD are now rather well-understood. 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 p-adics. For N=1 it reduces essentially to ordinary p-adic 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.