Quantum Mathematics in TGD Universe

Some comments about quantum mathematics, quantum Boolean thinking and computation as they might happen at fundamental level.

1. One should understand how Boolean statements A→B are represented. Or more generally: How a computation like procedure leading from a collection A of math objects collection B of math objects takes place? Recall that in computations the objects coming in and out are bit sequences. Now one have computation like process. → is expected to correspond to the arrow of time.

   If fermionic oscillator operators generate Boolean basis, zero energy ontology is necessary to realize rules as rules connecting statements realized as bit sequences. Positive energy ontology would allow only statements A,B but not statements A→B about them. ZEO allows also to avoid restrictions due to fermion number conservation and its well-definedness.

   Collection A is at the passive boundary of CD and not changed in state function reduction sequence defining self and B is at the active one. As a matter fact, it is not single statement but a quantum superpositions of statements B, which resides there! In the quantum jump selecting single B at the active boundary, A is replaced with a superposition of A:s: self dies and re-incarnates as more negentropic entity. Q-computation halts.

   That both a and b cannot be known precisely is a quantal limitation to what can be known: philosopher would talk about epistemology here. The different pairs (a,b) in superposition over b:s are analogous to different implications of a. Thinker is doomed to always live in a quantum cognitive dust and never be quite sure of.

2. What is the computer like structure now? Turing computer is discretized 1-D time-like line. This quantum computer is superposition of 4-D space-time surfaces with the basic computational operations located along it as partonic 2-surfaces defining the algebraic operations and connected by fermion lines representing signals. Also string world sheets are involved. In some key aspects this is very similar to ordinary computer. By strong form of holography computations use only data at string world sheets and partonic 2-surfaces.
3. What is the computation? It is sequence of repeated state function reduction leaving the passive boundary of CD
   intact but affecting the position (moduli) of upper boundary of CD and also the parts of zero energy states there.
   It is a sequence of unitary processes delocalizing the active boundary of CD followed by localization but no reduction. This the counterpart for a sequence of reductions leaving quantum state invariant in ordinary measurement theory (Zeno etc). Commutation halts as the first reduction to the opposite boundary occurs. Self dies and re-incarnates at the opposite boundary. Negentropy gain results in general and can be see as the information gained in the computation. One might hope that the new self (maybe something at higher level of dark matter hierarchy) is a little bit wiser – at least statistically speaking this seems to be true by weak form of NMP!

4. One should understand the quantum counterparts for the basic rules of manipulation. ×,/,+, and – are the most familiar example.
   1. The basic rules correspond physically to generalized Feynman/twistor diagrams representing sequences of algebraic manipulations in the Yangian of super-symplectic algebra. Sequences correspond now to collections of partonic 2-surfaces defining vertices of generalized twistor diagrams.

   2. 3- vertices correspond to product and co-product for quantal stringy Noether charges. Geometrically the vertex – analog of algebraic operation – is a partonic 2-surface at with incoming and outgoing light-like 3-surfaces meet – like vertex of Feynman diagram. Co-product vertex is not encountered in simple algebraic systems, and is time reversed variant of vertex. Fusion instead of annihilation.
   3. This diagrammatics has a huge symmetry just like ordinary computations have. All computation sequences (note that the corresponding space-time surfaces are different!) connecting same collections A and B of objects produce the same scattering amplitude. This generalises the duality symmetry of hadronic string models. This is really gigantic simplification and the results of twistor Grassmann approach suggest that something similar is obtained there. This implication was so gigantic that I gave up the idea for years.
5. One should understand the analogs for the mathematical axioms. What are the fundamental rules of manipulation?
   1. The classical computation/deduction would obey deterministic rules at vertices. The quantal formulation cannot be deterministic for the simple reason that one has quantum non-determinism (weak form of NMP allowing also good and evil) . The quantum rules obey the format that God used when communicating with Adam and Eve: do anything else but do not the break the conservation laws. Classical rules would list all the allowed possibilities and this leads to difficulties as Goedel demonstrated. I think that chess players follow the “anti-axiomatics”.

   2. I have the feeling that anti-axiomatics – not any well-established idea, it occurred to me as I wrote this – could provide a more natural approach to quantum computation and even allow a new manner to approach to the problematics of axiomatisations. It is also interesting to notice a second TGD inspired notion – the infinite hierarchy of mostly infinite integers (generated from infinite primes obtained by a repeated second quantization of an arithmetic QFT) – could make possible a generalisation of Gödel numbering for statements/computations. This view has at least one virtue: it makes clear how extremely primitive conscious entities we are in a bigger picture!
6. The laws of physics take care that the anti-axioms are obeyed. Quite concretely:
   1. Preferred extremal property of Kähler action and Käler-Dirac action plus conservation laws for charges associated with super-symplectic and other generalised conformal symmetries would define the rules not broken in vertices.

   2. At the fermion lines connecting the vertices the propagator would be determined by the boundary part of Kahler-Dirac action. K-D equation for spinors and consistency consistency conditions from Kahler action (strong form of holography) would dictate what happens to fermionic oscillator operators defining the analog of quantum Boolean algebra as super-symplectic algebra.