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how Infinite Primes could correspond to quantum states and
space-time surfaces
TGD
by Dr. Matti Pitkänen / January 13, 2010
Postal address:
Köydenpunojankatu 2 D 11
10940, Hanko, Finland
E-mail: [email protected]
URL-address: http://tgdtheory.com
(former address: http://www.helsinki.fi/~matpitka )
"Blog" forum: http://matpitka.blogspot.com/
I became conscious of infinite primes for almost 15 years ago. These numbers were the first
mathematical fruit of the TGD-inspired theory of Consciousness and define one of the most unpractical
looking aspects of Quantum-TGD.
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 nth 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 2 color singlets and triplet and anti-triplet. 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.
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There are 2 kinds of infinite primes differing only by the sign of the "small" part of the infinite
prime. For the 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. 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.
Also, a rigorous interpretation of complexified octonions emerges in Zero Energy Ontology. The 2
tips of causal diamond CD define 2 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 very strongly suggests 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 the Planck constant,
rational multiples of these are obtained. This suggests quantization also for hyperboloids and CP2 and
an attractive hypothesis is that infinite-primes determine the discretization as Ga subset SO(3)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 2. One prediction is quantization of cosmic redshift resulting from
2
quantization of Lorentz boosts and has been indeed observed and extremely difficult to understand in
standard Cosmology. We observe infinite primes directly!
I do not bother to type more. The 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.
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