Light hearted sketch for a TGD curriculum
This is just list. It would require enormous amount of explanation to make the notions correctly understood. Explaining mathematical notions to a mathematical layman is like walking in a minefield. It can happen at any moment;).
A. Classical TGD
1. Arguments leading to TGD.
Energy problem of general relativity: spacetime as 4D surface in M^4xS. Submanifold gravity. Deabstraction for the notion of spacetime as Riemann manifold. Need to generalize string model: string world sheets to 4D spacetime surfaces
2. Basic implications.
a) Geometrization of fields and quantum numbers. M^4xCP_2 unique choice for imbedding space.
b) Manysheeted spacetime. Spacetime quanta. Every object that I see defines a spacetime sheet: I literally see its boundary. Shape of the spacetime as 4D surface brings in new degrees of freedom allowing to geometrize all interactions, not only gravity. Fulfillment of dream of Einstein about classical physics as geometry.
c) Geometrization of classical fields. Oscillations of spacetime surface give rise to classical radiation fields for instance. No need to introduce them separately. Topological field quantization and the notion of field body/magnetic body. TGD view about classical fields different from Maxwellian view. Very important implications in biology.
Classical TGD is still standard mathematics although new for a physicists and also physical interpretation is new. Also the challenge of solving classical field equations leads to a lots of yetnonexisting mathematics.
B. Basic ideas of Quantum TGD.
Quantum TGD leads to a new mathematics.
a) Standard approaches to quantization does not work. Generalize Einstein's geometrization program: not only geometrization of classical physics but also that of quantum physics.
b) The world of classical worlds (WCW) as infiniteD space of spacetime surfaces obeying field equations plus additional conditions making them analogous to Bohr orbits. Bohr orbitology is an exact part of quantum physics, not only an approximation. This picture follows from mere General Coordinate Invariance which is also the basic pillar of GRT.
Physics is unique from the mathematical existence of WCW geometry. Already in the case of loop spaces geometry is unique. Now loops are replaced with 3surfaces and the constrants are even strong. M^4xCP_2 from mathematical existence of WCW geometry
c) Quantization without quantization! Quantum state corresponds to *classical* (not quantized) spinor fields in WCW. Fermionic statistics finds geometrization.
d) Zero energy ontology. Zero energy states and causal diamonds. How ZEO differs from positive energy ontology.
What problems it solves.
 Symmetries are crucial element of quantum TGD.
a) General coordinate invariance implies not only holography but strong form of holography
 Holography:
Lightlike 3surfaces carry information about quantum states. Spacetime surface provides fourdimensional classical correlates for quantum states. Also spacelike 3surfaces at the ends of causal diamonds provide representation of quantum states
 Strong form of holography:
Both lightlike and spacelike 3surfaces are enough to code physics. Conclusion: their intersections partonic 2surfaces are enough. Actually quite note: 4D tangent space data is also needed.
b) Conformal symmetries of string models generalize. Lightlike 3surfaces are metrically 2D and allow transformations preserving angles as symmetries. Conformal symmetry allows to have mathematically consistent theory. Dimension D=4 for spacetime unique. Also the lightcone boundary in M^4 (Minkowski space) has extended conformal symmetries. Hence imbedding space must have form M^4x S, S=CP_2 from standard model.
Equivalence Principle generalizes and finds a precise technical expression in terms of conformal symmetry.
C. Summary of key new physics notions
1. Spacetime as a 4D surface in M^4xCP_2. Shape of spacetime surface brings new degrees of freedom. Geometrization of classical fields and quantum numbers in terms of geometry of submanifold geometry. Manysheeted spacetime. Topological field qeuantization, field body, magnetic flux quanta, topological light rays.
This makes sense separately in real and various padic realms. pAdic spacetimes sheets, flux tubes, etc....
2. pAdic physics as correlate for cognition and intentions. Intention to action padic spacetime sheet to real one. Rational points as common to reality and various padicitities. Life in the intersection of real and padic worlds: cognition and matter. Primes near powers of 2 preferred primes. Mersenne primes. Number theoretic variant of Shannon entropy negative and identifiable as negentropy. Negentropic entanglement as correlate for experience of understanding as a rule. Negentropic entanglement key element of what it is to be living.
3. Zero energy ontology. Causal diamonds.
4. Hierarchy of Planck constants. For the simpleset option would follow from the vacuum degeneracy of Kahler action. Covering space of imbedding space M^4xCP_2 as effective tool of description. Dark matter as phases with nonstandard value of Planck constant. Dark matter at magnetic flux tubes forming macroscopic quantum phases controlling ordinary biomatter. The notion of magnetic body in biology.
D. New mathematics inspired by TGD
1. pAdic physics
Just for fun experimentation with the question what padic physics might be.
This lead to a discovery: one can calculate elementary particle masses using padic thermodynamics and assuming conformal symmetries.
pAdic numbers: basic properties. pAdic calculus.
pAdic physics as physics of cognition and intentionality. pAdic spacetime sheets as thought bubbles. Intersection of real and padic worlds corresponds to living matter: intersection of matter and mind! Number theoretic variant of Shannon entropy as negentropy. Negentropic entanglement fundamental in living matter.
!Generalization of number concept needed: fuse real number field and padic number fields to larger structure by gluing them along rationals.
This represents second piece of genuinely new mathematics.
2. Classical number fields and TGD
Classical number fields (reals, complex numbers, quaternions, and octonions) crucial for understanding the dimensions D=1,2,4,8 appearing in TGD.
Imbedding space seems to have purely number theoretical interpretation. Standard model symmetries have number theoretical meaning. WCW geometry is unique and it derives its uniqueness from number theory.
!More New mathematics: spacetime surfaces as quaternionic surfaces in some sense. Quaternionic means associative but noncommutative.
3. Infinite primes
The notion of infinite prime emerged from considerations related to consciousness theory. The construction of infinite primes can be seen as a repeated second quantization of a supersymmetric QFT with particles labelled by primes. This construction is hierarchical and the interpretation in terms of manysheeted spacetime is highly suggestive. Particles consisting of particles consisting of..... Hierarchy of abstractions: statements about statements about...... N:th order logics. There are many interpretations.
There are reasons to hope that quantum states could be labelled by infinite primes.
4. Number theoretic Brahman=Atman and algebraic holography.
5. Generalization of mathematical point by replacing it with Hilbert spaces. Fractal hierarchy of Hibert spaces having as their points Hilbert spaces having....
