Thursday, May 21, 2015

p-Adic physics as physics of cognition and imagination

The vision about p-adic physics as physics of cognition and imagination has gradually established itself as one of the key idea of TGD inspired theory of consciousness. There are several motivations for this idea.

The vision has developed fromthe vision about living matter as something residing in the intersection of real and p-adic worlds. One of the earliest motivations was p-adic non-determinism identified tentatively as a space-time correlate for the non-determinism of imagination. p-Adic non-determinism follows from the fact that functions with vanishing derivatives are piecewise constant functions in the p-adic context.

More precisely, p-adic pseudo constants depend on the pinary cutoff of their arguments and replace integration constants in p-adic differential equations. In the case of field equations this means roughly that the initial data are replaced with initial data given for a discrete set of time values chosen in such a manner that unique solution of field equations results. Solution can be fixed also in a discrete subset of rational points of the imbedding space. Presumably the uniqueness requirement implies some unique pinary cutoff. Thus the space-time surfaces representing solutions of p-adic field equations are analogous to space-time surfaces consisting of pieces of solutions of the real field equations. p-Adic reality is much like the dream reality consisting of rational fragments glued together in illogical manner or pieces of child's drawing of body containing body parts in more or less chaotic order.

The obvious looking interpretation for the solutions of the p-adic field equations would be as a geometric correlate of imagination. Plans, intentions, expectations, dreams, and cognition in general could have p-adic space-time sheets as their geometric correlates. A deep principle could be involved: incompleteness is characteristic feature of p-adic physics but the flexibility made possible by this incompleteness is absolutely essential for imagination and cognitive consciousness in general.

The original idea was that p-adic space-time regions can suffer topological phase transitions to real topology and vice versa in quantum jumps replacing space-time surface with a new one is given up as mathematically awkward: quantum jumps between different number fields do not make sense. The new adelic view states that both real and p-adic space-time sheets are obtained by continuation of string world sheets and partonic 2-surfaces to various number fields by strong form of holography.

The idea about p-adic pseudo constants as correlates of imagination is however too nice to be thrown away without trying to find an alternative interpretation consistent with strong form of holography. Could the following argument allow to save p-adic view about imagination in a mathematically respectable manner?

  1. Construction of preferred extremals from data at 2-surfaces is like boundary value problem. Integration constants are replaced with pseudo-constants depending on finite number pinary digits of variables depending on coordinates normal to string world sheets and partonic 2-surfaces.

  2. Preferred extremal property in real context implies strong correlations between string world sheets and partonic 2-surfaces by boundary conditions a them. One cannot choose these 2- surfaces completely independently. Pseudo-constant could allow a large number of p-adic configurations involving string world sheets and partonic 2-surfaces not allowed in real context and realizing imagination.

  3. Could imagination be realized as a larger size of the p-adic sectors of WCW? Could the realizable intentional actions belong to the intersection of real and p-adic WCWs? Could the modes of WCW spinor fields for which 2-surfaces are extandable to space-time surfaces only in some p-adic sectors make sense? The real space-time surface for them be somehow degenerate, for instance, consisting of string world sheets only.

    Could imagination be search for those collections of string world sheets and partonic 2-surfaces, which allow extension to (realization as) real preferred extremals? p-Adic physics would be there as an independent aspect of existence and this is just the original idea. Imagination could be realized in state function reduction, which always selects only those 2-surfaces which allow continuation to real space-time surfaces. The distinction between only imaginable and also realizable would be the extendability by using strong form of holography.

I have the feeling that this view allows respectable mathematical realization of imagination in terms of adelic quantum physics. It is remarkable that strong form of holography derivable from - you can guess, strong form of General Coordinate Invariance (the Big E again!), plays an absolutely central role in it.

See the article How Imagination Could Be Realized p-Adically?.

For a summary of earlier postings see Links to the latest progress in TGD.


Ulla said...

The strongest motivation is the vision about living matter as something residing in the intersection of real and p-adic worlds.

How do you know that not all matter follow this rule? I think it must be so, so we still doesn't know what differ ordinary and living matter. said...

Sorry for illogical statement and thank you for noticing it. My purpose was not to state it in this manner. The original vision - a couple of years ago or so- was that living matter can be said to reside in this intersection of realities and p-adicities- sensory and cognitive.

The questions are: what is living matter, sensory, cognitive, and intersection!

I have now mathematically sound answers to these questions in the conceptual framework provided by TGD (only in this framework of course!)

A more precise statement is that the core of this posting: both matter and cognition can be said to reside in this intersection: "genes of space-time" are in the intersection as 2-D string wold sheets and partonic 2-surfaces, one might say.

Imagined and real differ in that pure figment of imagination cannot be continued by holography to real space-time surface but can be done so to p-adic 4-surfaces because of pseudo constant phenomenon. This is mathematically sound definition: mathematics is full of this kind continuation problems of type "given boundary conditions can one find a solution of field equations".

Dream reality consisting of pieces glued together illogically could be seen as example of imagination. And of course, this posting too;-).

Ulla said...

Maybe (the tree of ) primes are not bent or curved in the same way as spacetime? said...

The primes appear at deeper level: at the level of number field (extension of rationals) for the parameters characterising the space-time surfaces. Space-time and space-time curvature are higher level concepts: relate to primes like sociology to elementary particle physics;-) .

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