### Quantum Mathematics in TGD Universe

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

- 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.

- 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.

- 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!

- One should understand the quantum counterparts for the basic rules of manipulation. ×,/,+, and - are the most familiar example.

- 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.

- 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.

- 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.

- 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.
- One should understand the analogs for the mathematical axioms. What are the fundamental rules of manipulation?

- 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".

- 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!

- 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".
- The laws of physics take care that the anti-axioms are obeyed. Quite concretely:

- 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.

- 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.

- 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.

## 10 Comments:

I must show my stupidity once again :P

One of the questions I have thought of lately is how a quantum number as aritmetic concept would look like, as instance inside a BH or as holography. If the 'computation' is made as something between boundaries this would be part of one boundary, but this is labelled by fractional aspects, that is a quantum phase. This decribes the uncertainty too, like a 'spagettified' number.

The complex numbers are a bit alike, but they are still ordered along a line. In a BH the 'numbers' are like fractional islands, but their displacement must still obey a law, as seen in the holography principle. This law must be a symmetry in my mind. It is like a quantum phase.

"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."

This statement talks of protected symmetries, like skyrmions, locally protected in hexagons. At least so, maybe even non-locally? In my mind it comes as a 'tunnelling' effect. I am still unsure about what that 'preferred' means, if it is a property of the symmetry itself?

If we look at symmetries they are also in a hierarchial order, with superradiant, as most non-local type, linear radiation, a dual and quasic structure. God herself? She is computating continously about every single detail that happen in the whole universe, in a holographic way, so her brain is a mirror of our own, also functions in the same way? This means our cortex and that symmetry are mirrors? Both has excellent supermemory :) through that symmetry.

If you can get a gist of what I say?

Look at this about quantum phase transitions. http://www.nature.com/ncomms/2015/150514/ncomms8111/full/ncomms8111.html

"By strong form of holography computations use only data at string world sheets and partonic 2-surfaces."

This can be compared to wavefunction and nodes in form of quasiparticles or Dirac fermions?

I answer to the latter question first.

Fermions in question are Dirac fermions. They are what I call fundamental fermions much like this in string model.

But fermions as a notion used in condensed matter physics are at a distance of light years conceptually.

They emerge as approximate notion after the replacement of many-sheeted space-time with that of GRT and even that of special relativity- approximation as in standard model.

One forgets everything about string world sheets and partonic 2-sufaces as basic structures and many-sheeted space-time, and speaks about spinors in Minkowski space!

Quite a dramatic coarse graining but one crucial thing is common: symmetries and conservation laws, which are the fundamental truths respected in the physical axiomatics or should one say anti-axiomatics.

To Ulla:

I should not perhaps use the word God. I try to use it with implicit ";-)". This because people associate with with all kinds of properties making him/her less abstract. By the way, the non-personal god could be NMP: principle rather than conscious entity;-).

Perhaps best manner to say this is to say that endless computation like process and and recreation is going on.

To Ulla:

Non-locality is present from beginning. The starting point of TGD was that particles corresponds to 3-dimensional surfaces rather than points.

The non-locallity is expressed by in Bell inequalities stating that particles in quantum mechanics have more correlations than they could have in any *purely local* theory relying on classical concept of probability (no interference of wave functions).

In TGD this fact has space-time correlates: strings connecting partonic 2-surfaces accompanied by flux tubes. TGD is a non-local theory but does not try to get rid of state function reduction like most the non-local theories such as Bohm's theory. TGD only removes un-necessary mystics- one might call it black magic- from quantum physics.

Einstein would be happy bout TGD- maybe he is ;-)-. Also Bohr would be relaxed snce he could reconsider whether it is really sensible to eliminate "ontology" from the vocabulary altogether and replace it with the attribute "crazy" as he did.

There is no unique reality but this does not mean that there would be no reality. There are superpositions of quantum realities replaced by new ones all the time as this universe is studying itself by recreating itself again and again and storing its findings as negentropic entanglement - the Akashic records;-).

Ye, The Akashic records... the prime string...

Condensed matter does not tell things that coarsly, in my mind.

"They emerge as approximate notion after the replacement of many-sheeted space-time with that of GRT and even that of special relativity- approximation as in standard model." Exactly, and this is the beautiful part of it, y get the curvature.

The 3-surfaces comes from quarks? Like this? http://www.sciencedirect.com/science/article/pii/037594749190738R

GRT is approximation to relativity: relates like classical thermodynamics to quantum thermodynamics.

Quarks correspond to 3-surfaces as all particles do. The 3-surfaces define classical geometric and topological correlates for their quantal properties. In string models strings should provide this description but to be honest, they do not!

The link talks about something different. 3-quark systems, not 3-surfaces.

This comment has been removed by the author.

I must formulate again.

Ye, but mesons as diquarks then? Higgs particle is a dipole. Why would the foundation be a 3-surface and then go back to a dipole?

Color confinement is what you think of?

http://ptp.oxfordjournals.org/content/65/5/1684.full.pdf

The hadronic quark bag system?

How does a 3 surface define a topological correlate (should be a dipole in this 2D world)? Is it the stability, the simple fact that they are seen? Odd numbers, asymmetry? Partons as 2-surfaces are related how?

http://www.physics.gla.ac.uk/ppt/lqcd.htm

How is this a scalar field? Geometry and topology are made of vectors?

A link would be fine, I have searched your many papers for this. It is now important for me to know the detailed structure.

To Ulla: I answer the questions that I understand.

Topology is science of shape. Coffee cup and "munkkirinkila" are topologically the same thing. Geometry takes also distances into account, not only shape. Coffee cup and coffee cup upside down are geometrically one and same thing.

Scalar field could be seen as a notion of differential geometry. Differential geometry is purely local geometry. Angles between vectors for instance. But not shapes.

About 3-surface as topological correlate of particle. Consider first 2-D surfaces. Take a plane and add small handles. You can move these handles around. They are like particles, particles as topological inhomogenities in 2-D world. Generalize this to 3-D case and you get something that I mean. Reduction of particles to space-time topology.

Geometry made of vector is putting apple and orange in the same bask.

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