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This Concept Map, created with IHMC CmapTools, has information related to: Infinite primes, INFINITE PRIMES 6. One can iterate the construction. a) Take finite and infinite primes ob- tained and define Dirac sea as product of them. Repeat the procedure.Now bound states cor- respond to polyno- mials of two vari- ables. b) One can repeat the construction again and again. c) Repeated se- cond quantization of arithmetic QFT is in questtion. Many- sheeted space-time strongly suggests that this procedure is physical. Even ga- laxy sized objects would at some scale represent elementa- ry particles., INFINITE PRIMES 5. Physical fermions and bosons form also bound states. Could infinite primes represent also them? a) X is formally like coordinate variable. On can form sums and products of infinite primes and interpret them as polynomials. b) The non-divisibility by infinite primes (not only finite) boils down the notion of prime polynomial property and one construct in- finite hierarchy of this kind of states having interpretation as infi- nite primes represent- ing bound states., INFINITE PRIMES 3. Construction gene- ralizes to give rep- resentation of ma- ny-fermion states. a) Divide X by any square free integer (product of first powers of primes) to get X1=X/n. b) Form the num- ber P1= X1+n. P1 mod mod p is non- vanishing for any finite prime p so that P1 is infinite prime. c) This number is analogous to many fermion states ob- tained by kicking the negative energy fermions labelled by primes p diding n to positive energy states. P1 repre- sents many-fer- mion state!, INFINITE PRIMES 3. Construction gene- ralizes to give rep- resentation of ma- ny-fermion states. a) Divide X by any square free integer (product of first powers of primes) to get X1=X/n. b) Form the num- ber P1= X1+n. P1 mod mod p is non- vanishing for any finite prime p so that P1 is infinite prime. c) This number is analogous to many fermion states ob- tained by kicking the negative energy fermions labelled by primes p diding n to positive energy states. P1 repre- sents many-fer- mion state!, INFINITE PRIMES 4. Infinite primes can represent also states containing bosons labelled by finite primes! a) Take integer m having not common factors with n. k:th power of prime p represents state consisting of k bosons labeled by prime p. Multiply X1 with it m to get P2= m*X1+n. P2 is infinite prime. b) Take an integer r having only prime factors appearing in n and multiply n with r to get P3= m*X1+ r*n. P3 is prime and represents a state in which there are bosons labelled by primes appear- ing in X1 and in n. c) This is the most general infinite prime representing Fock state of free bosons and fermions., INFINITE PRIMES 1. a) Original motivation from TGD inspired theory of conscious- ness. b) Generalizes the notion of infinite number by intro- ducing the notion of divisibility be- sides the notion of follower (n-->n+1) used by Cantor to construct infinite numbers. c) Construction has interpretation as repeated second quantization of super-symmetric arithmetic quan- tum field theory., INFINITE PRIMES 2. Basic observation: a) Form the product X of all finite primes. It is divisible by any prime b) Add to X +1 or -1. to get number P= X+1 or X-1. c) The number P mod p=+1 or -1 for any finite prime p. P is prime albeit infinite! d) Useful analogy: P is like Dirac sea for which all states label- led by finite primes are filled., INFINITE PRIMES 1. d) Construction can be also used to define infinite hierachy of real units as ratios of infinite rationals but having arbit- rarily complex number theoretic anatomy. This leads to a generalization of the notions of point and space. e) Space-time point with number theoretic anatomy wouldbe infinite- dimensional space and able to re- present quantum states of entire Universe! Number theoretic Brah- man=Atman or algebraic holo- graphy!
