Their construction is however structurally similar to a repeated second quantization of an arithmetic super-symmetry quantum field theory with states labeled by primes. An attractive identification of the hierarchy is in terms of the many-sheeted space-time. Also the abstraction hierarchy of conscious thought and hierarchy of n:th order logics naturally correspond to this infinite hierarchy. We ourselves are at rather lowest level of this hierarchy. Propositional logic and first order logic at best and usually no logic at all;-)
By generalizing from rational primes to hyper-octonionic primes one has good hopes about a direct connection with physics. The reason is that the automorphism group of octonions respecting a preferred imaginary unit is SU(3)subset G2 and physically corresponds to color group in the formulation of the number theoretical compactification stating equivalence of the formulations of TGD based on the identification of imbedding space with 8-dimensional hyperquaternions M8 and M4× CP2. The components of hyper-octonion behave like two color singlets and triplet and antitriplet. For a given hyper-octonionic prime there exists a discrete subgroup of SU(3) respecting the prime property and generating a set of primes at octonionic 6-sphere. For a given prime one can realize a finite number of color multiplets in this discrete space. The components in the hyper-complex subspace M2 remaining invariant under SU(3) can be identified as components of momentum in this subspace. M2 is needed for massless particles the preferred extremals of Kähler action assign this space to each point of space-time surface as space non-physical polarizations.
There are two kinds of infinite primes differing only by the sign of the "small" part of the infinite prime and for second kind of primes one can consider the action of SU(2) subgroup of SU(3) and corresponding discrete subgroups of SU(2) respecting prime property (note that this suggests a direct connection with the Jones inclusions of hyper-finite factors of type II1!). These representations give rise to two SU(2) multiplets and their orbital excitations identifiable as deformations of the partonic 2-surface. Four components of hyper-octonion remain invariant under SU(2) and have interpretation as momentum in M2 and electroweak charges. Therefore a pair of these primes characterizes standard model quantum numbers of particle if discrete wave functions in the space of primes are allowed. For color singlet particles single prime is enough. At the level of infinite primes one obtains extremely rich structure and it is possible to map the states of quantum TGD to these number theoretical states. Only the genus of partonic 2-surface responsible for family replication phenomenon fails to find an obvious interpretation in this picture.
The completely unexpected by-product is a prediction for the spectrum of quantum states and quantum numbers including masses so that infinite primes and rationals are not so unpractical as one might think! This prediction is really incredible since it applies to the entire hierarchy of second quantizations in which many particle states of previous level become particles of the new level (corresponding physically to space-time sheets condensed to a larger space-time sheet or causal diamonds inside larger causal diamond CD).
In zero energy ontology positive and negative energy states correspond to infinite integers and their inverses respectively and their ratio to a hyper-octonionic unit. The wave functions in this space induced from those for finite hyper-octonionic primes define the quantum states of the sub-Universe defined by given CD and sub-CDs. These phases can be assigned to any point of the 8-dimensional imbedding space M8 interpreted as hyper-octonions so that number theoretic Brahman=Atman identity or algebraic holography is realized! These incredibly beautiful infinite primes are both highly spiritual and highly practical just as a real spiritual person experienced directly Brahman=Atman state is;-).
A fascinating possibility is that even M-matrix- which is nothing but a characterization of zero energy state- could find an elegant formulation as entanglement coefficients associated with the pair of the integer and inverse integer characterizing the positive and negative energy states.
- The great vision is that associativity and commutativity conditions fix the number theoretical quantum dynamics completely. Quantum associativity states that the wave functions in the space of infinite primes, integers, and rationals are invariant under associations of finite hyper-octonionic primes (A(BC) and (AB)C are the basic associations), physics requires associativity only apart from a phase factor. The condition of commutativity poses a more familiar condition implying that permutations induce only a phase factor which is +/- 1 for boson and fermion statistics and a more general phase for quantum group statistics for the anyonic phases, which correspond to nonstandard values of Planck constant in TGD framework. These symmetries induce time-like entanglement for zero energy stats and perhaps non-trivial enough M-matrix.
- One must also remember that besides the infinite primes defining the counterparts of free Fock states of supersymmetric QFT, also infinite primes analogous to bound states are predicted. The analogy with polynomial primes illustrates what is involved. In the space of polynomials with integer coefficients polynomials of degree one correspond free single particle states and one can form free many particle states as their products. Higher degree polynomials with algebraic roots correspond to bound states being not decomposable to a product of polynomials of first degree in the field of rationals. Could also positive and negative energy parts of zero energy states form a analog of bound state giving rise to highly non-trivial M-matrix?
Also a rigorous interpretation of complexified octonions emerges in zero energy ontology.
- The two tips of causal diamond CD define two preferred points of M4. The fixing of quantization axes of color fixes in CP2 also a point at both light-like boundaries of CD. The moduli space for CDs is therefore M4× CP2 × M4++× CP2 and its M8 counterpart is obtained by replacing CP2 with E4 so that a space which correspond locally to complexified octonions is the outcome. p-Adic length scale hypothesis suggests very strongly a quantization of the second factor to a set of hyperboloids with light-cone proper time come as powers of 2. For other values of Planck constant rational multiples of these are obtained. This suggests quantization also for hyperboloids and CP2.
- An attractive hypothesis is that infinite-primes determine the discretization as Ga subset SU(2)subset SU(3) and Gb subset SU(3) orbits of the points of hyperboloid and CP2. The interpretation would be in terms of cosmology. The Robertson Walker space-time would be replaced with this discrete space meaning in particular that cosmic time identified as Minkowski proper time is quantized in powers of two. One prediction is quantization of cosmic redshift resulting from quantization of Lorentz boosts and has been indeed observed and extremely difficult to understand in standard cosmology. We would observe infinite primes directly!
I do not bother to type more. Interested reader can read the brief pdf file explaining all this in detail or read the chapter Physics as Generalized Number Theory III: Infinite Primes of "TGD as a Generalized Number Theory".