In Thinking Allowed Original there was very interesting link added by Ulla about the possibility of non- associative quantum mechanics.
Also I have been forced to consider this possibility.
One must be of course very cautious in order to not draw too strong conclusions. Above one considers quantum physics at the level of single space-time sheet. What about many-sheeted space-time? Could non-associative physics emerge in TGD via many-sheeted space-time? To answer this question one must first understand what non-associativity means.
- The 8-D imbedding space of TGD has octonionic tangent space structure and octonions are non-associative. Octonionic quantum theory however has serious mathematical difficulties since the operators of Hilbert space are by definition associative. The representation of say octonionic multiplication table by matrices is possible but is not faithful since it misses the associativity. More concretely, so called associators associated with triplets of representation matrices vanish. One should somehow transcend the standard quantum theory if one wants non-associative physics.
- Associativity therefore seems to be fundamental in quantum theory as we understand it recently. Associativity is indeed a fundamental and highly non-trivial constraint on the correlation functions of conformal field theories. In TGD framework classical physics is an exact part of quantum theory so that quantum classical correspondence suggests that associativity could play a highly non-trivial role in classical TGD.
The conjecture is that associativity requirement fixes the dynamics of space-time sheets - preferred extremals of Kähler action - more or less uniquely. One can endow the tangent space of 8-D imbedding H=M4× CP2 space at given point with octonionic structure: the 8 tangent vectors of the tangent space basis obey octonionic multiplication table.
Space-time realized as n-D surface in 8-D H must be either associative or co-associative: this depending on whether the tangent space basis or normal space basis is associative. The maximal dimension of space-time surface is predicted to be the observed dimension D=4 and tangent space or normal space allows a quaternionic basis.
- There are also other conjectures (see this) about what the preferred extremals of Kähler action defining space-time surfaces are.
The "super(optimistic)" conjecture is that all these conjectures are equivalent.
- A very general conjecture states that strong form of holography allows to determine space-time surfaces from the knowledge of partonic 2-surfaces and 2-D string world sheets.
- Second conjecture involves quaternion analyticity and generalization of complex structure to quaternionic structure involving generalization of Cauchy-Riemann conditions.
- M8-M4× CP2 duality stating that space-time surfaces can be regarded as surface in either M8 or M4× CP2 is a further conjecture.
- Twistorial considerations select M4× CP2 as a completely unique choice since M4 and CP2 are the only spaces allowing twistor space with Kähler structure. The conjecture is that preferred extremals can be identified base spaces of 6-D sub-manifolds of the product CP3× SU(3)/U(1)× U(1) of twistor spaces of M4 and CP2 having the property that it makes sense to speak about induced twistor structure.
Non-associativity is an essentially linguistic phenomenon and relates therefore to cognition. p-Adic physics labelled by p-adic primes fusing with real physics to form adelic physics are identified as the physics of cognition in TGD framework.
- In non-associative situation brackets matter. A(BC) is different from (AB)C. From schooldays or at least from the first year calculus course one recalls the algorithm: when calculating the expression involving brackets one first finds the innermost brackets and calculates what is inside them, then proceed to the next innermost brackets, etc... In computer programs the realization of the command sequences involving brackets is called parsing and compilers perform it. Parsing involves decomposition of program to modules calling modules calling.... Quite generally, the analysis of linguistic expressions involves parsing. Bells start to ring as one realizes that parsings form a hierarchy as also do the space-time sheets!
- More concretely, there is hierarchy of brackets and there is also a hierarchy of space-time sheets, perhaps labelled by p-adic primes. B and C inside brackets form (BC), something analogous to a bound state or chemical compound. In TGD this something could correspond to a "glueing" space-time sheets B and C at the same larger space-time sheet. More concretely, (BC) could correspond to braided pair of flux tubes B and C inside larger flux tube, whose presence is expressed as brackets (..). As one forms A(BC) one puts flux tube A and flux tube (BC) containing braided flux tubes B and C inside larger flux tube. For (AB)C flux one puts tube (AB) containing braided flux tubes A and B and tube C inside larger flux tube. The outcomes are obviously different. A
- Non-associativity in this sense would be a key signature of many-sheeted space-time. It should show itself in say molecular chemistry, where putting on same sheet could mean formation of chemical compound AB from A and B. Another highly interesting possibility is hierarchy of braids formed from flux tubes: braids can form braids, which in turn can form braids,... Flux tubes inside flux tubes inside... Maybe this more refined breaking of associativity could underly the possible non-associativity of biochemistry: biomolecules looking exactly the same would differ in subtle manner.
- What about quantum theory level? Non-associativity at the level of quantum theory could correspond to the breaking of associativity for the correlation functions of n fields if the fields are not associated with the same space-time sheet but to space-time sheets labelled by different p-adic primes. At QFT limit of TGD giving standard model and GRT the sheets are lumped together to single piece of Minkowski space and all physical effects making possible non-associativity in the proposed sense are lost. Language would be thus possible only in TGD Universe! My nasty alter ego wants to say now something - my sincere apologies: in superstring Universe communication of at least TGD has indeed turned out to be impossible! If superstringy universe allows communications at all, they must be uni-directional!
p-Adic hierarchy is not the only hierarchy in TGD Universe: there is also the hierarchy of Planck constants heff=n× h giving rise to a hierarchy of intelligences. What is the relationship between these hierarchies?
- Could many-sheeted space-time of TGD provides the geometric realization of language like structures? Could sentences and more complex structures have many-sheeted space-time structures as geometrical correlates? p-Adic physics as physics of cognition would suggests that p-adic primes label the sheets in the parsing hierarchy. Could bio-chemistry with hierarchy of magnetic flux tubes added, realize the parsing hierarchies?
- DNA is a language and might provide a key example about parsing hierarchy. The mystery is that human DNA and DNAs of most simplest creatures do not differ much. Our cousins have almost identical DNA with us. Why do we differ so much? Could the number of parsing levels be the reason- p-adic primes labelling space-time sheets? Could our DNA language be much more structured than that of our cousins. At the level of concrete language the linguistic expressions of our cousin are indeed simple signals rather than extremely complex sentences of old-fashioned German professor forming a single lecture each. Could these parsing hierarchies realize themselves as braiding hierarchies of magnetic flux tubes physically and more abstractly as the parsing hierarchies of social structures. Indeed, I have proposed that the presence of collective levels of consciousness having hierarchy of magnetic bodies as a space-time correlates distinguishes us from our cousins so that this explanation is consistent with more quantitative one relying on language.
- I have also proposed that intronic portion of DNA is crucial for understanding why we differ so much from our cousins (see this and this). How does this view relate to the above proposal? In the simplest model for DNA as topological quantum computer introns would be connected by flux tubes to the lipids of nuclear and cell membranes. This would make possible topological quantum computations with the braiding of flux tubes defining the topological quantum computer program.
Ordinary computer programs rely on computer language. Same should be true about quantum computer programs realized as braidings. Now the hierarchical structure of parsings would correspond to that of braidings: one would have braids, braids of braids, etc... This kind of structure is also directly visible as the multiply coiled structure of DNA. The braids beginning from the intronic portion of DNA would form braided flux tubes inside larger braided flux tubes inside.... defining the parsing of the topological quantum computer program.
The higher the number of parsing levels, the higher the position in the evolutionary hierarchy. Each braiding would define one particular fundamental program module and taking this kind of braided flux tubes and braiding them would give a program calling these programs as sub-programs.
- The phonemes of language would have no meaning to us (at our level of self hierarchy) but the words formed by phonemes and involving at basic level the braiding of "phoneme flux tubes" would have. Sentences and their substructures would in turn involve braiding of "word flux tubes". Spoken language would correspond to a temporal sequence of braidings of flux tubes at various hierarchy levels.
- The difference between us and our cousins (or other organisms) would not be at the level of visible DNA but at the level of magnetic body. Magnetic bodies would serve as correlates also for social structures and associated collective levels of consciousness. The degree of braiding would define the level in the evolutionary hierarchy. This is of course the basic vision of TGD inspired quantum biology and quantum bio-chemistry in which the double formed by organism and environment is completed to a triple by adding the magnetic body.
- I have proposed that speech and music are fundamental aspects of conscious intelligence and that DNA realizes what I call bio-harmonies in quite concrete sense (see this and this): DNA codons would correspond to 3-chords. DNA would both talk and sing. Both language and music are highly structured. Could the relation of heff hierarchy to language be same as the relation of music to speech?
- Are both musical and linguistic parsing hierarchies present? Are they somehow dual? What does parsing mean for music? How musical sounds could combine to form the analog of two braided strand? Depending on situation we hear music both as separate notes and as chords as separate notes fuse in our mind to a larger unit like phonemes fuse to a word.
Could chords played by single instrument correspond to braidings of flux tubes at the same level? Could the duality between linguistic and musical intelligence (analogous to that between function and its Fourier transform) be very concrete and detailed and reflect itself also as the possibility to interpret DNA codons both as three letter words and as 3-chords (see this)?
See the new chapter Is Non-Associative Physics and Language Possible Only in Many-Sheeted Space-Time?.