Thursday, December 27, 2012

Scattering Amplitudes and the Positive Grassmannian

Perhaps I exaggerated a little bit in the previous posting, when I talked about declining theoretical physics. The work of Nima Arkani-Hamed and others represents something which makes me very optimistic and I would be happy if I could understand the horrible technicalities of their work. The article Scattering Amplitudes and the Positive Grassmannian by Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, and Trnka summarizes the recent situation in a form, which should be accessible to ordinary physicist. Lubos has already discussed the article.

All scattering amplitudes have on shell amplitudes for massless particles as building bricks

The key idea is that all planar amplitudes can be constructed from on shell amplitudes: all virtual particles are actually real. In zero energy ontology I ended up with the representation of TGD analogs of Feynman diagrams using only mass shell massless states with both positive and negative energies. The enormous number of kinematic constraints eliminates UV and IR divergences and also the description of massive particles as bound states of massless ones becomes possible.

In TGD framework quantum classical correspondence requires a space-time correlate for the on mass shell property and it indeed exists. The mathematically ill-defined path integral over all 4-surfaces is replaced with a superposition of preferred extremals of Kähler action analogous to Bohr orbits, and one has only a functional integral over the 3-D ends at the light-like boundaries of causal diamond (Euclidian/Minkowskian space-time regions give real/imaginary Chern-Simons exponent to the vacuum functional). This would be obviously the deeper principle behind on mass shell representation of scattering amplitudes that Nima and others are certainly trying to identify. This principle in turn reduces to general coordinate invariance at the level of the world of classical worlds.

Quantum classical correspondence and quantum ergodicity would imply even stronger condition: the quantal correlation functions should be identical with classical correlation functions for any preferred extremal in the superposition: all preferred extremals in the superposition would be statistically equivalent (see the earlier posting). 4-D spin glass degeneracy of Kähler action however suggests that this is is probably too strong a condition applying only to building bricks of the superposition.

Minimal surface property is the geometric counterpart for masslessness and the preferred extremals are also minimal surfaces: this property reduces to the generalization of complex structure at space-time surfaces, which I call Hamilton-Jacobi structure for the Minkowskian signature of the induced metric. Einstein Maxwell equations with cosmological term are also satisfied.

Massless extremals and twistor approach

The decomposition M4=M2× E2 is fundamental in the formulation of quantum TGD, in the number theoretical vision about TGD, in the construction of preferred extremals, and for the vision about generalized Feynman diagrams. It is also fundamental in the decomposition of the degrees of string to longitudinal and transversal ones. An additional item to the list is that also the states appearing in thermodynamical ensemble in p-adic thermodynamics correspond to four-momenta in M2 fixed by the direction of the Lorentz boost. In twistor approach to TGD the possibility to decompose also internal lines to massless states at parallel space-time sheets is crucial.

Can one find a concrete identification for M2× E2 decomposition at the level of preferred extremals? Could these preferred extremals be interpreted as the internal lines of generalized Feynman diagrams carrying massless momenta? Could one identify the mass of particle predicted by p-adic thermodynamics with the sum of massless classical momenta assignable to two preferred extremals of this kind connected by wormhole contacts defining the elementary particle?

Candidates for this kind of preferred extremals indeed exist. Local M2× E2 decomposition and light-like longitudinal massless momentum assignable to M2 characterizes "massless extremals" (MEs, "topological light rays"). The simplest MEs correspond to single space-time sheet carrying a conserved light-like M2 momentum. For several MEs connected by wormhole contacts the longitudinal massless momenta are not conserved anymore but their sum defines a time-like conserved four-momentum: one has a bound states of massless MEs. The stable wormhole contacts binding MEs together possess Kähler magnetic charge and serve as building bricks of elementary particles. Particles are necessary closed magnetic flux tubes having two wormhole contacts at their ends and connecting the two MEs.

The sum of the classical massless momenta assignable to the pair of MEs is conserved even when they exchange momentum. Quantum classical correspondence requires that the conserved classical rest energy of the particle equals to the prediction of p-adic mass calculations. The massless momenta assignable to MEs would naturally correspond to the massless momenta propagating along the internal lines of generalized Feynman diagrams assumed in zero energy ontology. Masslessness of virtual particles makes also possible twistor approach. This supports the view that MEs are fundamental for the twistor approach in TGD framework.

Scattering amplitudes as representations for braids whose threads can fuse at 3-vertices

Just a little comment about the content of the article. The main message of the article is that non-equivalent contributions to a given scattering amplitude in N=4 SYM represent elements of the group of permutations of external lines - or to be more precise - decorated permutations which replace permutation group Sn with n! elements with its decorated version containing 2nn! elements. Besides 3-vertex the basic dynamical process is permutation having the exchange of neighboring lines as a generating permutation completely analogous to fundamental braiding. BFCW bridge has interpretation as a representations for the basic braiding operation.

This supports the TGD inspired proposal (TGD as almost topological QFT) that generalized Feynman diagrams are in some sense also knot or braid diagrams allowing besides braiding operation also two 3-vertices. The first 3-vertex generalizes the standard stringy 3-vertex but with totally different interpretation having nothing to do with particle decay: rather particle travels along two paths simultaneously after 1→2 decay. Second 3-vertex generalizes the 3-vertex of ordinary Feynman diagram (three 4-D lines of generalized Feynman diagram identified as Euclidian space-time regions meet at this vertex). I have discussed this vision in detail here. The main idea is that in TGD framework knotting and braiding emerges at two levels.

  1. At the level of space-time surface string world sheets at which the induced spinor fields (except right-handed neutrino, see this) are localized due to the conservation of electric charge can form 2-knots and can intersect at discrete points in the generic case. The boundaries of strings world sheets at light-like wormhole throat orbits and at space-like 3-surfaces defining the ends of the space-time at light-like boundaries of causal diamonds can form ordinary 1-knots, and get linked and braided. Elementary particles themselves correspond to closed loops at the ends of space-time surface and can also get knotted (for possible effects see this).


  2. One can assign to the lines of generalized Feynman diagrams lines in M2 characterizing given causal diamond. Therefore the 2-D representation of Feynman diagrams has concrete physical interpretation in TGD. These lines can intersect and what suggests itself is a description of non-planar diagrams (having this kind of intersections) in terms of an algebraic knot theory. A natural guess is that it is this knot theoretic operation which allows to describe also non-planar diagrams by reducing them to planar ones as one does when one constructs knot invariant by reducing the knot to a trivial one. Scattering amplitudes would be basically knot invariants.

"Almost topological" has also a meaning usually not assigned with it. Thurston's geometrization conjecture stating that geometric invariants of canonical representation of manifold as Riemann geometry, defined topological invariants, could generalize somehow. For instance, the geometric invariants of preferred extremals could be seen as topological or more refined invariants (symplectic, conformal in the sense of 4-D generalization of conformal structure). If quantum ergodicity holds true, the statistical geometric invariants defined by the classical correlation functions of various induced classical gauge fields for preferred extremals could be regarded as this kind of invariants for sub-manifolds. What would distinguish TGD from standard topological QFT would be that the invariants in question would involve length scale and thus have a physical content in the usual sense of the word! Perhaps I exaggerated a little bit in the previous posting, when I talked about declining theoretical physics. The work of Nima Arkani-Hamed and others represents something which makes me very optimistic and I would be happy if I could understand the horrible technicalities of their work. The article Scattering Amplitudes and the Positive Grassmannian by Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, and Trnka summarizes the recent situation in a form, which should be accessible to ordinary physicist. Lubos has already discussed the article.

All scattering amplitudes have on shell amplitudes for massless particles as building bricks

The key idea is that all planar amplitudes can be constructed from on shell amplitudes: all virtual particles are actually real. In zero energy ontology I ended up with the representation of TGD analogs of Feynman diagrams using only mass shell massless states with both positive and negative energies. The enormous number of kinematic constraints eliminates UV and IR divergences and also the description of massive particles as bound states of massless ones becomes possible.

In TGD framework quantum classical correspondence requires a space-time correlate for the on mass shell property and it indeed exists. The mathematically ill-defined path integral over all 4-surfaces is replaced with a superposition of preferred extremals of Kähler action analogous to Bohr orbits, and one has only a functional integral over the 3-D ends at the light-like boundaries of causal diamond (Euclidian/Minkowskian space-time regions give real/imaginary Chern-Simons exponent to the vacuum functional). This would be obviously the deeper principle behind on mass shell representation of scattering amplitudes that Nima and others are certainly trying to identify. This principle in turn reduces to general coordinate invariance at the level of the world of classical worlds.

Quantum classical correspondence and quantum ergodicity would imply even stronger condition: the quantal correlation functions should be identical with classical correlation functions for any preferred extremal in the superposition: all preferred extremals in the superposition would be statistically equivalent (see the earlier posting). 4-D spin glass degeneracy of Kähler action however suggests that this is is probably too strong a condition applying only to building bricks of the superposition.

Minimal surface property is the geometric counterpart for masslessness and the preferred extremals are also minimal surfaces: this property reduces to the generalization of complex structure at space-time surfaces, which I call Hamilton-Jacobi structure for the Minkowskian signature of the induced metric. Einstein Maxwell equations with cosmological term are also satisfied.

Massless extremals and twistor approach

The decomposition M4=M2× E2 is fundamental in the formulation of quantum TGD, in the number theoretical vision about TGD, in the construction of preferred extremals, and for the vision about generalized Feynman diagrams. It is also fundamental in the decomposition of the degrees of string to longitudinal and transversal ones. An additional item to the list is that also the states appearing in thermodynamical ensemble in p-adic thermodynamics correspond to four-momenta in M2 fixed by the direction of the Lorentz boost. In twistor approach to TGD the possibility to decompose also internal lines to massless states at parallel space-time sheets is crucial.

Can one find a concrete identification for M2× E2 decomposition at the level of preferred extremals? Could these preferred extremals be interpreted as the internal lines of generalized Feynman diagrams carrying massless momenta? Could one identify the mass of particle predicted by p-adic thermodynamics with the sum of massless classical momenta assignable to two preferred extremals of this kind connected by wormhole contacts defining the elementary particle?

Candidates for this kind of preferred extremals indeed exist. Local M2× E2 decomposition and light-like longitudinal massless momentum assignable to M2 characterizes "massless extremals" (MEs, "topological light rays"). The simplest MEs correspond to single space-time sheet carrying a conserved light-like M2 momentum. For several MEs connected by wormhole contacts the longitudinal massless momenta are not conserved anymore but their sum defines a time-like conserved four-momentum: one has a bound states of massless MEs. The stable wormhole contacts binding MEs together possess Kähler magnetic charge and serve as building bricks of elementary particles. Particles are necessary closed magnetic flux tubes having two wormhole contacts at their ends and connecting the two MEs.

The sum of the classical massless momenta assignable to the pair of MEs is conserved even when they exchange momentum. Quantum classical correspondence requires that the conserved classical rest energy of the particle equals to the prediction of p-adic mass calculations. The massless momenta assignable to MEs would naturally correspond to the massless momenta propagating along the internal lines of generalized Feynman diagrams assumed in zero energy ontology. Masslessness of virtual particles makes also possible twistor approach. This supports the view that MEs are fundamental for the twistor approach in TGD framework.

Scattering amplitudes as representations for braids whose threads can fuse at 3-vertices

Just a little comment about the content of the article. The main message of the article is that non-equivalent contributions to a given scattering amplitude in N=4 SYM represent elements of the group of permutations of external lines - or to be more precise - decorated permutations which replace permutation group Sn with n! elements with its decorated version containing 2nn! elements. Besides 3-vertex the basic dynamical process is permutation having the exchange of neighboring lines as a generating permutation completely analogous to fundamental braiding. BFCW bridge has interpretation as a representations for the basic braiding operation.

This supports the TGD inspired proposal (TGD as almost topological QFT) that generalized Feynman diagrams are in some sense also knot or braid diagrams allowing besides braiding operation also two 3-vertices. The first 3-vertex generalizes the standard stringy 3-vertex but with totally different interpretation having nothing to do with particle decay: rather particle travels along two paths simultaneously after 1→2 decay. Second 3-vertex generalizes the 3-vertex of ordinary Feynman diagram (three 4-D lines of generalized Feynman diagram identified as Euclidian space-time regions meet at this vertex). I have discussed this vision in detail here. The main idea is that in TGD framework knotting and braiding emerges at two levels.

  1. At the level of space-time surface string world sheets at which the induced spinor fields (except right-handed neutrino, see this) are localized due to the conservation of electric charge can form 2-knots and can intersect at discrete points in the generic case. The boundaries of strings world sheets at light-like wormhole throat orbits and at space-like 3-surfaces defining the ends of the space-time at light-like boundaries of causal diamonds can form ordinary 1-knots, and get linked and braided. Elementary particles themselves correspond to closed loops at the ends of space-time surface and can also get knotted (for possible effects see this).

  2. One can assign to the lines of generalized Feynman diagrams lines in M2 characterizing given causal diamond. Therefore the 2-D representation of Feynman diagrams has concrete physical interpretation in TGD. These lines can intersect and what suggests itself is a description of non-planar diagrams (having this kind of intersections) in terms of an algebraic knot theory. A natural guess is that it is this knot theoretic operation which allows to describe also non-planar diagrams by reducing them to planar ones as one does when one constructs knot invariant by reducing the knot to a trivial one. Scattering amplitudes would be basically knot invariants.

"Almost topological" has also a meaning usually not assigned with it. Thurston's geometrization conjecture stating that geometric invariants of canonical representation of manifold as Riemann geometry, defined topological invariants, could generalize somehow. For instance, the geometric invariants of preferred extremals could be seen as topological or more refined invariants (symplectic, conformal in the sense of 4-D generalization of conformal structure). If quantum ergodicity holds true, the statistical geometric invariants defined by the classical correlation functions of various induced classical gauge fields for preferred extremals could be regarded as this kind of invariants for sub-manifolds. What would distinguish TGD from standard topological QFT would be that the invariants in question would involve length scale and thus have a physical content in the usual sense of the word!

For background see the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Physics as Infinite-dimensional Geometry" or the article Could N =2 or N =4 SUSY be a part of TGD after all?.

16 Comments:

At 8:44 AM, Anonymous Santeri Satama said...

To continue attempts to understand TGD, first a quote from Ekeland's 'Best of All Possible Worlds', with interesting parallels to Matti's vision; sorry it's in Finnish as I don't have English copy available:


"Tällä energian säilymisen ominaisuudella on geometrinen tulkinta, joka mahdollistaa liikkeiden määrittely ongelman yksinkertaistamisen. Ajatellaan suurta järjestelmää, jolla on lukuisia vapausasteita, sanokaamme n kpl, mikä tarkoittaa, että sen faasiavaruuden ulottuvuuksia on 2n. Kootaan tässä 2n-ulotteisessa avaruudessa kaikki tilat, joissa energia saa tietyn arvon, esimerkiksi 1: ne muodostavat eräänlaisen lehden, jossa on (2n - 1) ulottuvuutta, samanlaisen kuin tavallinen kaksiulotteinen pinta tavallisessa kolmiulotteiseessa tilassa. Tämä lehti voi sulkeutua kuin pallo tai torus, tai levitä äärettömiin kuten taso. Konstruoidaan sitten lehti, joka muodostuu kaikista järjestelmän tiloista, joissa energiataso on 2. Kyse on toisesta, joka on erillään ensimmäisestä ja joka ei leikkaa sitä, sillä energia ei voisi olla samalla kertaa 1 ja 2 samassa tilassa. Katsotaan sitten välissä olvevia energiatasoja 2,1:stä 2,9:ään, sitten s,01:stä 2,09:ään ja näin eteenpäin halki desimaalien: niitä on äärettömästi, ja jokainen niistä vastaa erillistä lehteä. Nämä kaikki lehdet täyttävät täysin faasiavaruuden, joka on näin kerrostunut eli stratifioitu: jokaisessa pisteessään faasiavaruus kohta yhden javain yhden rakennelehden (kyseisen kohdan energiatason). Tätä matemaatikot kutsuvat nimellä faasiavaruuden foliaatio eli lehditys, asian ymmärtämiseksi voi ajatella kirjan sivuja, tai proosallisemmin hienoikdsi siivuiksi leikattua ja sitten uudelleen koottua kinkkua.
(...)
Oletetaan faasiavaruudessa yksi rajattu alue, jonka reuna on faasiavaruuden kahteen osaan - sisäiseen ja ulkoiseen - jakava lehti: tätä kutsutaan matematiikassa hyperpinnaksi, joka on samanlainen kuin (kaksiulotteinen) pinta tavallisessa (kolmiulotteisessa) tilassa. Voidaan kirjoittaa puhtaasti geometrisella menetelmällä Eulerin-Lagrangen tyyppisiä yhtälöitä, jotka ovat hyperpinnalle ominaisia ja joita vastaavat kulkuradat eivät poistu siltä. Nämä yhtälöt eivät edellytä energian funktion määrittelemistä faasiavaruudessa, mutta jos toisaalta sellainen funktio on olemassa ja sillä on annettu hyperpinta energiatasona, silloin tähän energiatasoon kuuluvat liikeradat ovat yhtneväiset geometristä tietä löydettyjen ratojen kanssa."
Ekeland: Paras mahdollisista maailmoista - matematiikka ja kohtalo, pp 135-136.

 
At 8:45 AM, Anonymous Santeri Satama said...

Cf. With Matti:
"The mathematical aspects of p-adicization of quantum TGD are discussed. In a well-
de ned sense Nature itself performs the p-adicization and p-adic physics can be regarded as
physics of cognitive regions of space-time which in turn provide representations of real space-
time regions. Cognitive representations presumably involve the p-adicization of the geometry
at the level of the space-time and imbedding space by a mapping of a real space time region to a p-adic one. One can di fferentiate between two kinds of maps: the identifi cation induced by the common rationals of real and p-adic space time region and the representations of the external real world to internal p-adic world induced by a canonical identi cation type maps.
Only the identi cation by common rationals respects general coordinate invariance, and
it leads to a generalization of the number concept. Di fferent number fields form a book like
structure with number fi elds and their extensions representing the pages of the book glued
together along common rationals representing the rim of the book. This generalization forces
also the generalization of the manifold concept: both imbedding space and con figuration space
are obtained as union of copies corresponding to various number fields glued together along
common points, in particular rational ones. Space-time surfaces decompose naturally to real
and p-adic space-time sheets. In this framework the fusion of real and various p-adic physics
reduces more or less to to an algebraic continuation of rational number based physics to various
number fields and their extensions."
http://www.scienceoflife.nl/MPitkanen-TGD-Number-06-Piadic3.pdf

Ekeland continues by discussing multioscillators (cf. pendulums) and finding periodic solutions to Lagrange-Euler equations in convex hypersurfaces of two degrees of freedom and states resent proofs showing that they can can have only either 2 or infinity of periodic solutions of closed circuits.

So both approaches end up with book metaphor, where as Matti's approach goes much further and combines the space-time sheets into global book where they can communicate at the level of p-adic areas.

 
At 9:09 AM, Anonymous Santeri Satama said...



In the article discussed in above: "While a global picture is still missing, a huge amount of data has been generated..."

Here's my current attempt for a global vision. Matti has said that all p-adic areas get in some sense the value of one, and this to me sounds highly untrivial, though I'd like to comprehend better how exactly, maybe and hopefully Matti can explain. Any case, I vision that p-adic unity as the Center or Source of a ball-like "phase space", first surrounded by p-adic regions or layers and then the "hypersurface" consisting of rational areas shared by p-adics and reals, which binds together the space-time sheets of how Matti visions the book or books. This hypersurface divides the "phase space" of p-adics and reals (etc.?) into inner and outer regions, inner areas (at first glance!?) converging in p-adically infinite "singularity" and outer areas scattering as infinite extensions of reals. This view could also help to explain the interest in non-probable "imaginary" black hole -singularity objects presumably "out there" as an attempt and act of gnothi seauton.

 
At 9:17 AM, Anonymous Santeri Satama said...

Correction to previous post: non-probable -> non-provable

So, when you generalize from a simple harmonic oscillator to multioscillators, you still get only two countable results, either duality or infinity. Which I supposes also presupposes and requires infinite primes which can among other things describe various layers of the hypersurface I previously visioned?

Another question about causal diamonds: aren they multioscillators that Ekeland discusses? Also I expect that both points of the arrows must touch infinity; what is the relation of real and p-adic infinities here, if any?

 
At 5:02 AM, Anonymous Matti Pitkanen said...




To Santeri Satama:

Thank you for a nice summary of Ekelands vision. Ekeland describes basic notion of mechanics: energy constant surface of phase space at which particle orbit in phase space resides by energy conservation. This notion is central in thermodynamics.

Causal diamonds are *not dynamical systems* such as multi-oscillators. They are not energy constant surfaces in phase space. They *contain the dynamical systems* defined by space-time surfaces. At quantum level zero energy states have wave function in the moduli space of CDs (CD has position, it can be Lorentz boosted and rotated and translated). CD's size scale is assumed be quantized for number theoretic reasons and come as integer multiple of CP_2 size.

What matters in zero energy ontology is that causal diamonds have two *light-like* boundaries at which positive and negative energy parts of zero energy states are localized.The "upper" and "lower" ends of space-time surfaces (restaurants at the ends of the Universe are possible in TGD Universe!;-)) at the two light-like boundaries of CD: the analogs of initial and final state of a physical event. CDs contain the space-time surfaces in the superposition of preferred extremals (analogs of Bohr orbits) defining zero energy state.

The interpretation of CD suggested by TGD inspired theory of consciousness is as an imbedding space correlate for self, kind of spotlight of attention.


Light-likeness of the boundaries of CD implies a generalization of conformal symmetry at the boundaries: it is essential that one has 4-D Minkowski space x CP_2. These conformal symmetries form only half of the conformal symmetries of TGD. Also light-like 3-surfaces defining parton orbits possess generalized conformal symmetries. The 4-dimensionality of both M^4 and space-time surface are crucial [I would be happy if this would finally induce a "click" in the learned heads of colleagues ;-) although I know from bitter experience that although I can bring a horse to the fountain, I cannot force it to drink;-)].

Number theoretic vision inspires a speculative picture about a connection between the hierarchy of infinite primes and hierarchical structure of space-time sheets of many-sheeted space-time. Why I take this vision seriously is that the hierarchy of infinite primes is a number theoretical correlate for a hierarchy of second quantizations whereas the many-sheeted hierarchy is classical space-time correlate for it.

You mention frequencies. In the hierarchy of infinite primes frequencies labeling fermionic and bosonic oscillator operators are replaced by logarithms of primes. I remember that also in chaos theory frequencies comings as logarithms of primes appear: I do not however not the deeper mathematical reason.

 
At 6:34 PM, Anonymous Santeri Satama said...

Thanks for your response. Fully accepting my limitations in capacity of mathematical imagination, one can only hope that my simple reflections have some meaningful relation to your vision. And keep on questioning.

The CD as "spotlight of attention" sounds promising approach. Not necessarily limited to such movements, but the "pendulum" or "oscillator" of attention shifting between introspection and extrospection seems common experience, common enough to raise the question, if the introspective areas are characterized by p-adic areas and extrospective by real areas, can that be applied to CD's? E.g. describing the lower or "introspective" part or "light-like boundary" of CD p-adically and upper "extrospective" part by reals? Assuming those areas can be mapped on each in dynamic fashion. Or are the theory dependent and/or mathematical reasons denying such possiblity?

Or if the question above does not get even close, what is the TGD inspired vision of introspective and extrospective movements of attention? And in that vision, does mathematical contemplation and imagination belong to either category.

PS: You mentioned somewhere seeing "spots" long time after your Great Experience. Though you are not supporter of multiverse theories at least in standard form, I have hunch that those might represent other universes, perhaps characterized by other transcendentals than those revealed to us, but dunno. Maybe this hunch can be relevant to your question about WCW, maybe not, just thought maybe this would be worth mentioning.

 
At 10:21 PM, Anonymous Matti Pitkanen said...


I would tend to assign introspection and extrospective areas to conscious information about the space-time sheets of observer one one hand and on space-time sheets of external world.

The recent vision discussed in the posting about blackholes and blackhole evaporation is that the spacetime sheet defining "me" as something separate from the external world defines the line of generalized Feynman diagram (rather thick as compared with line of ordinary Feynman diagram!) and therefore has Eucdlian(!) signature of the induced metric with time and space in the same position. External world would correspond to Minkowskian signature.

This is just one possible proposal. Real space-time sheets correspond naturally to sensory experience and p-adic space-time sheets to cognition. You are thus saying that introspection-extrospection difference corresponds to sensory-cognitive dichotomy. I would argue that I have sensory experience from both external world and internal world so that I would disagree.

Mathematical imagination would naturally correspond to p-adic sector. The non-determinsm of p-adic differential equations suggests an interpretation in terms of the non-determinism of imagination.

p-Adic number fields have infinite number of algebraic extensions analogous to complex numbers and thus having arbitrarily large algebraic dimensions. Could it be that our ability to imagine higher dimensions corresponds to a geometric realization of these dimensions p-adically. Evolution would correspond to a gradual increase of the algebraic dimension.

We cannot imagine what it is to be sensorily higher-dimensional (at least I cannot;-)) and this could relate to the absence of higher algebraic extensions of reals.

You certainly noticed that I tried to cheat here!;-) Complex numbers definea 2-D algebraic extension of reals so that one might argue that we are able to see 8-D dreams in 4-D space-time!

My defense is following. The preferred extremals of Kaehler action define a generalization of 2-D complex structure to 4-D case in both Minkowskian and Eucdliian space-time regions. One can say that physical space-time is 2-D in "complex" sense just as complex plane is 1-D in complex sense. As also strong form of holography (saying the quantum states depend on partonic 2-surfaces and their tangent space data only) states.

Maybe you mean with "spots" the flow that I jheave perceived by closing my eyes in peaceful state of mind, say after having worked for few hours. The flow is to and from a kind of tunnel leading somewhere. I have considered the possibility that this flow is along magnetic flux tube connecting me somewhere.

Neuroscientists could certainly represent a trivializing and pathologizing explanation for this flow, and colleagues would list this experience as an additional strong support for the conjecture that I am completely mad as they have thought from the beginning!;-)

 
At 3:03 AM, Blogger Ulla said...

To die is also often experienced as a tunnel, and then the time forms a standing wave (?) allowing you to be in many times at the same time? It is like some kind of bosonic condensate?

Time can also cheat in emergency situations, giving you much more energy than would be possible, and also those 'lightnings in mind' of past times.

 
At 7:49 AM, Blogger Ulla said...

Look at this! http://www.hindawi.com/journals/jna/2010/794782/

The DNA repair enzyme has also a too high efficiency, like the photosynthesis electron tunnelling. Is DNA repair also activated from a quantum state? This uses 350-450 nm wavelength, which is the same as methionine or the start-signal for proteinsynthesis use. Can this be an indicium for the DNA-phantoms or 'dark' biology as you have stated, is in fact the mechanism?

 
At 8:22 AM, Anonymous Santeri Satama said...

Your bring up sensory experience. It's area where we run into problems of definition. What and how is sensing, and what experiences do we classify as sensual experiences? For example, Buddhist philosophy consider thinking just another sense. Wiki article (http://en.wikipedia.org/wiki/Sense) gives a list of many "internal" senses in addition to the five classical "external" senses, and also what it considers non-human senses.

I've also sometimes wondered, in connection to attention concentrating on "internal" body sense (not implying that it ends at skin level but can be also sensual experience of the whole "magnetic body"), how many dimensions such "internal" sensual experiences contain and is such a question even meaningful?

Assuming, based on phenomena of synaisthesia etc., that various senses are *filters* of holistic flow of information, rather than purely classical mechanisms of data-receiving and neurological reproduction and representation, is for example the experience of orthogonality (that we sense most directly with our sense of balance (and gravity towards Earth's gravity center) necessary for experience of dimensionality?

I cannot really imagine a ball containing more than three orthogonal axis, but i can imagine - and internally feel! - a ball consisting of infinity of directions emanating from the center and give that infinity of directions at least symbolic interpretation of n-dimensionality.

 
At 9:02 AM, Blogger Ulla said...

I continue with my finding, and mean not to interrupt the discussion. Sorry for this. I am excited as usual :)

http://www.ncbi.nlm.nih.gov/pubmed/20184295?dopt=AbstractPlus

.. adenine acts as an electrostatic bouncer that shoves the charge flow from flavin toward the DNA lesion that photolyase repairs. This explanation is provided by an explicit time-dependent quantum mechanical approach...The electron wave function dynamics accurately accounts for the previously proposed mechanism of transfer via the terminal methyl group of the flavin moiety present in the catalytic electron-donor cofactor, FADH(-), which also contains adenine.

adenine can replace methionine in a redox reaction? Can then also the mutagenesis be directed/controlled by pH-changes or wavelength changes of light?

http://www.ncbi.nlm.nih.gov/pubmed/9657679?dopt=AbstractPlus

 
At 2:51 AM, Anonymous Matti Pitkanen said...


To Santeri Satama:

Thank you for interesting questions and Happy New Year.

[SS] You bring up sensory experience. It's area where we run into problems of definition. What and how is sensing, and what experiences do we classify as
sensual experiences? For example, Buddhist philosophy consider thinking just another sense. Wiki article (http://en.wikipedia.org/wiki/Sense) gives a list of many "internal" senses in addition to the five classical "external" senses, and also what it considers non-human senses.

[MP] Certainly the definition of sensory experience requires some assumptions. It is very difficult to separate pure sensory qualia from the cognitive representations constructed from them. The sensory input must be decomposed to objects and this requires a lot of processing. Essentially standardized mental images are generated using a virtual sensory input to sensory organs which would be seats of primary sensory qualia (phantom limb is one basic objection which can be circumvented in TGD Universe).

Concerning qualia my basic assumptions are roughly the following. See http://tgdtheory.com/public_html/hologram/hologram.html#qualia .

a) Sensory qualia are the building bricks of sensory experience and since moment of consciousness corresponds to quantum jump and quantum states are labelled by quantum numbers, sensory qualia are characterized by increments of quantum numbers. Different kind of quantum numbers wold correspond to different kinds of basic qualia. For instance, colors would correspond to increments of color quantum numbers of quarks and gluons so that the term QCD color would not be a mere algebraic joke. This of course makes sense only if one accepts the almost-prediction of TGD about hierarchy of QCD like theories in various scales: in particular in the scales of living cell. If the people at LHC discover M_89 hadron physics, we are rather close to asking whether this radically new vision really makes sense.

b) Sensory experience contains also purely geometric information: vision, touch, and hearing do so.This information corresponds to 4-D geometry so that also dynamical information becomes geometric with this definition. I have talked about geometric qualia: angles, distances,... What is interesting, is that topologist Barbara Shipman found that honeybee dance has a mathematical description in terms of a flag manifolds associated with QCD color. Different choices of quantization axes (two kinds of them corresponding to the Cartan algebra of SU(2): color isospin and color hypercharge) for color form a flag manifold SU(3)/U(1)xU(1) and this space is involved with her model.

To be continued...

 
At 2:52 AM, Anonymous Matti Pitkanen said...


Continuation to the reply to Santeri:

[SS] I've also sometimes wondered, in connection to attention concentrating on "internal" body sense (not implying that it ends at skin level but can
be also sensual experience of the whole "magnetic body"), how many dimensions such "internal" sensual experiences contain and is such a question even meaningful?

[MP] A model for OBEs and various sensory illusions leads to the view that various sensations could relate to the movement or imagined movement (say the nasty feeling in stomach when one imagines falling down to nearby cliff or moving train illusion) involve relative motion of physical body and magnetic body whose conformation provides a representation for it and maybe also for the environment. Motion of magnetic body alone gives rise to imagined sensory experience and motion of biological body alone to purely real sensory experience. OBEs would relate to the relative motion of magnetic body. The perceptions would result from the change of the hologram defined by the radiation exchanged between magnetic body and biological body.

[SS] Assuming, based on phenomena of synaesthesia etc., that various senses are *filters* of holistic flow of information, rather than purely classical
mechanisms of data-receiving and neurological reproduction and representation, is for example the experience of orthogonality (that we sense most directly with our sense of balance (and gravity towards Earth's gravity center) necessary for experience of dimensionality?

I cannot really imagine a ball containing more than three orthogonal axis, but i can imagine - and internally feel! - a ball consisting of infinity of
directions emanating from the center and give that infinity of directions at least symbolic interpretation of n-dimensionality.

[MP] Angle pi/2 is special. Hilbert space is one basic example. More I am unable to say.
I have a strong temptation to interpret imagined higher-dimensional structures as p-adic constructs. As I probably already mentioned, p-adics allow infinite number of algebraic extensions and also non-algebraic extensions which are infinite-D. These are higher-D structures in the same sense as complex numbers are a 2-D structure. That we experience the world as 3-D and are not able to sensorily imagine higher dimensions could be simply due to the fact that we ourselves are 3-dimensional in real sense! Our cognitive me's can have arbitrarily high dimension. We are what we perceive;-). Buddhists would be correct when they interpret also cognition as sensory perception if cognitive perception is p-adic sensory perception.

 
At 3:01 AM, Anonymous Matti Pitkanen said...


To Ulla:

The finding is certainly interesting but I must honestly confess that it would require a considerable effort to get even gist of what "adenine can replace methionine in redox reaction" implies. Here I sense my biochemical limitations strongly. Happy New Year in any case!

 
At 8:40 AM, Blogger Ulla said...

Methionine is the starter of proteinsynthesis, and it seems it is started by blue light. Once we discussed this. I understand you have many other questions to think about.

I understood that this mechanism is maybe the one that gives the dark matter or magnetic flux (blue light) regulation.

Here a link to a PhD thesis about attention-awareness and how it is experienced. Usually there is no differentation, and then it is difficult to discuss this. Downloadable, in words

http://denisewilbanks.com/?attachment_id=169

An equally difficult thing to think of is what is a signal that cause the sensations. It comes down to what is the action principle, but then the qualias is just a different processing? Ramachandran uses brain areas for this interpretation http://www.youtube.com/watch?v=Rl2LwnaUA-k&feature=youtu.be

Are there brain areas that interpret different meanings for the same thing, like reals and complex numbers? Left versus right? Then there are sensations of brainless organisms (microtubuli also have a left and right side of the body) and the superorganisms which make problems. Sensings are interpreted in many steps and the most important ones are in the sense organ themselves? We use brain mostly to selection and projection, and in that way we create maybe meaning?

Happy new year. Still a few days left til the Chinese change, to the Snake. I feel this year will be the year of transformation :)

 
At 1:17 PM, Blogger Ulla said...

http://www.dailygalaxy.com/my_weblog/2013/01/-einsteins-emc2-may-breakdown-in-outer-space.html

 

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