The answer to the question became too long to serve as a comment so that I decided to add it as a blog posting.

**1. Huygens Principle**

Huygens principle can be assigned most naturally with classical linear wave equations with a source term. It applies also in perturbation theory involving small non-linearities.

One can solve the d'Alembert equation Box Φ= J with a source term J by inverting the d'Alembertian operator to get a bifocal function G(x,y) what one call's Green function.

Green function is bi-local function G(x,y) and the solution generated by point infinitely strong source J localised at single space-time point y - delta function is the technical term. This description allows to think that every point space-time point acts as a source for a spherical wave described by Green function. Green function is Lorentz invariant satisfies causality: on selects the boundary conditions so that the signal is with future light cone.

There are many kind of Green functions and also Feynman propagator satisfies same equation. Now however causality in the naive sense is not exact for fermions. The distance between points x and y can be also space-like but the breaking of causality is small. Feynman propagators

form the basics of QFT description but now the situation is changing after what Nima et al have done to the theoretical physics;-). Twistors are *the* tool also in TGD too but generalised to 8-D case and this generalisation has been one of the big steps of progress in TGD shows that M^{4}×CP^{2} is twistorially completely unique.

**2. What about Huygens principle and Green functions at the level of TGD space-time?**

In TGD classical field equations are extremely non-linear. Hence perturbation theory based Green function around a solution defined by canonically imbedded Minkowski space M^{4} in M^{4}×CP^{2} fails. Even worse: the Green function would vanish identically because Kahler action is non-vanishing only in fourth order for the perturbations of canonically imbedded M^{4}! This total breakdown of perturbation theory forces to forget standard ways to quantise TGD and I ended up with the world of classical worlds: geometrization of the space of 3-surfaces. Later zero energy ontology emerged and 3-surfaces were replaced by pairs of 3-surfaces at opposite boundaries of causal diamond CD defining the portion of imbedding space which can be perceived by conscious entity in given scale. Scale hierarchy is explicitly present.

Preferred externals in space-time regions with Minkowskian signature of induced metric decompose to topological light-rays which behave like quantum of massless radiation field. Massless externals for instance are space-time tubes carrying superposition of waves in same light-like direction proceeding. Restricted superposition replaces superposition for single space-time sheet whereas unlimited superposition holds only for the *effects* caused by space-time sheets to at test particle touching them simultaneously.

The shape of the radiation pulse is preserved which means soliton like behaviour: form of pulse is preserved, velocity of propagation is maximal, and the pulse is precisely targeted. Classical wave equation is "already quantized". This has very strong implications for communications and control in living matter . The GRT approximation of many-sheetedness of course masks tall these beauty as it masked also dark matter, and we see only some anomalies such as several light velocities for signals from SN1987A.

In geometric optics rays are a key notion. In TGD they correspond to light-like orbits of partonic 2-surfaces. The light-like orbit of partonic 2-surface is a highly non-curved analog of light-one boundary - the signature of the induced metric changes at it from Minkowskian to Eucldian at it. Partonic 2-surface need not expand like sphere for ordinary light-cone. Strong gravitational effects make the signature of the induced metric 3-metric (0,-1,-1) at partonic 2-surfaces. There is a strong analogy with Schwartscild horizon but also differences: for Scwartschild blackhole the interior has

Minkowskian signature.

**3. What about fermonic variant of Huygens principle?**

In fermionic sector spinors are localised at string world sheets and obey Kähler-Dirac equation which by conformal invariance is just what spinors obey in super string models. Holomorphy in hypercomplex coordinate gives the solutions in universal form, which depends on the conformal equivalence class of the effective metric defined by the anti-commutators of Kähler-Dirac gamma matrices at string world sheet. Strings are associated with magnetic flux tubes carrying monopole flux and it would seem that the cosmic web of these flux tubes defines the wiring along which fermions propagate.

The behavior of spinors at the 1-D light-like boundaries of string world sheets carrying fermion number has been a long lasting head ache. Should one introduce a Dirac type action these lines?. Twistor approach and Feynman diagrammatics suggest that fundamental fermionic propagator should emerge from this action.

I finally t turned out that one must assign 1-D massless Dirac action in induced metric and also its 1-D super counterpart as line length which however vanishes for solutions. The solutions of Dirac equation have *8-D light-like momentum* assignable to the 1-D curves, which are 8-D light-like geodesics of M^{4}×CP^{2}. The *4-momentum* of fermion line is time-like or light-like so that the propagation is inside future light-cone rather than only along future light-cone as in Huygens principle.

The propagation of fundamental fermions and elementary particles obtained as the composites is * inside* the future light-one, not only along light-cone boundary with light-velocity. This reflects the presence of CP_{2} degrees of freedom directly and leads to massivation.

To sum up, quantized form of Huygens principle but formulated statistically for partonic fermionic lines at partonic 2-surfaces, for partonic 2-surfaces, or for the masses quantum like regions of space-time regions - could hold true. Transition from TGD to GRT limit by approximating many-sheeted space-time with region of M^{4} should give Huygens principle. Especially interesting is 8-D generalisation of Huygens principle implying that boundary of 4-D future light-cone is replaced by its interior. 8-D notion of twistor should be relevant here.

## 64 comments:

Oh my God this Ramanujan is beautiful:

"Universal quadratic form

An integral quadratic form whose image consists of all the positive integers is sometimes called universal. Lagrange's four-square theorem shows that w^2+x^2+y^2+z^2 is universal. Ramanujan generalized this to aw^2+bx^2+cy^2+dz^2 and found 54 multisets {a,b,c,d} that can each generate all positive integers, namely,

{1,1,1,d}, 1 ≤ d ≤ 7

{1,1,2,d}, 2 ≤ d ≤ 14

{1,1,3,d}, 3 ≤ d ≤ 6

{1,2,2,d}, 2 ≤ d ≤ 7

{1,2,3,d}, 3 ≤ d ≤ 10

{1,2,4,d}, 4 ≤ d ≤ 14

{1,2,5,d}, 6 ≤ d ≤ 10

There are also forms whose image consists of all but one of the positive integers. For example, {1,2,5,5} has 15 as the exception. Recently, the 15 and 290 theorems have completely characterized universal integral quadratic forms: if all coefficients are integers, then it represents all positive integers if and only if it represents all integers up through 290; if it has an integral matrix, it represents all positive integers if and only if it represents all integers up through 15." http://en.wikipedia.org/wiki/Quadratic_form

There are 7 universal intervals for d for the limit a=1, b=(1-2) and c=(1-5), and their lengths are 7, 13, 4, 6, 8, 11 and 5. Three multiples of 2 and the 2 first prime pairs.

One possible physical interpretation of these intervals of positive integers is quantum number, and hence it could be said that there are 7 "quantum numbers" giving the "interval" (or field) of principal quantum number. Also, each interval of universal quadratic form can be considered a delta-function with discrete boundaries for coefficient d, and on rational line there are 5 center points of these seven intervals for d: 4 (for abc:111), 4½ (113, 122), 6½ (123), 8 (112, 125) and 9 (124); note also the symmetry 8=2x4 and 9=2x4½ (cf. Dirac "sky and sea"). Or, if you like, five complete pages in the "Good Book", glued together by coefficients of Ramanujan's formula.

Could it be shown that these 7 (pre)quantum numbers/intervals that generate the principal quantum number (1, 2, 3,...), also contain the other quantum numbers and their relations?

Amazing results. They might even have some exotic

application to physics some day.

http://www.fen.bilkent.edu.tr/~franz/mat/15.pdf

BTW, what do you mean by '3-surface', exactly? Does your definition include or allow also triangles on a plane?

The 15-theorem's limit 15 for the universal quadratic form (cf 8D) is a triangular number, first such with similar internal structure, with fully contained genuine triangular number (3).

By the way, universality is possible only in D=4 or higher. One cannot avoid association with space-time dimension.

a) The Dirac equation on fermion line (light-like geodesic characterise by 8-D light-like momentum) representing string boundary at the light-like parton orbit) leads to

8-D mass squared, which vanishes :

p_0^1-….-p_7^2=0 or

p_01^-..-p_3^2= m^2= p_4^2+…+p_7^2

Suppose that the number theoretical universality of TGD forces the four-momenta to have integer components or rational components in which case one can take out the common denominator and get similar situation again.

Mass squared is constant times integer by super-conformal invariance: eigenvalue of super-conformal scaling generator L_0. This condition requires that m^2 is integer.

By four-dimensionality p_4^+..p_7^2 indeed would have all non-negative integer values.

What about the Minkowskian variant p_0^1-..-p^3^2: is it also universal? My hunch is that in this signature universality is easier to satisfy. Probably also this problem has been discussed. Any information about this?

Again, the number 15 showing up in the present epoch is a convergence of "criticality"

http://arxiv.org/abs/1403.5227

criticality is such a funny term.. is it like on Star Trek when the captain is saying "faster" and Scotty from engeering replies they are already going at warp 9,99999 whatever... captain doesn't understand logarithm max out to 10?

--Stephen

The Grasmannian of quantum Pascal triangle (diamonds are the yang girls best friend...;) ) might be help-full here (ie., how does Ramanujan cognition work?):

http://math.ucr.edu/home/baez/qg-fall2007/pascal.html

A some-what related question, how do you know where your hand is, exPecially/at least if you don't quantum zeno by looking/thinking it?

By the way, the funniest and most revealing Ramanujan here was his only "mistake" that left out the gap of 15, which then became the 15-theorem and 290-theorem (290=1+17^2). (cf. self-avoiding and self-including walks).

Can you explain this phrase a little more, thx.

"This total breakdown of perturbation theory forces to forget standard ways to quantise TGD ..."

Still you talk of partition theory.

http://inspirehep.net/record/1297532/plots

Look at this. Is the CD-diamonds forming Dirac fermions in TGD? Can it be said so?

Stephen, love your comment. With very simple gardeners understanding, criticality = let it grow, and dont push the growth factors on the negative side when mulching. Or if and when you do, learn from overdoing it. On the other hand, who's to deny willing and accepting participation of being a growth factor, up to the state of harmony and balance, so dynamics...

Matti my friend, as our days are getting shorter and flying past faster, your question about universality of Minkowskian signature become more and more acute. I don't have the formal answer (you are better in that field than you think), but this I know: we can pose our questions and universe will answer, or allow us to find the answer by our selfs. Only hunch I have now, is that the three colors of algebraic quadrance in Normans' hyperbolic rational geometry are highly relevant and go deeper than Minkowski.

But the main point stays, this is all heuristic, and heureka will come when you are ready for it and the answer satisfies your deepest questions, that only you can ask for your self. Also according to TGD the whole of math reinvents itself from universal QJ to another, so with all empirical evidence you can Trust that universe answers your questions, as you are also a hologram containing all the Information aka Creation. In the way that gives you the deepest emotional satisfaction also in this life.

To Ulla:

you probably meant to say "perturbation theory". Perturbation theory is extremely general concept.

You make a guess for a solution. It is not quite correct but you can calculate corrections order by order in some small parameter. Cows are not spherical - not even for theoretical physicists, but physicist can start from a spherical cow as approximation and do next centuries perturbation theory to reproduce the correct shape from standard model;-).

In the sentence I referred to perturbation theory path integral developed by Feynman originally. The scattering amplitude is sum over amplitudes over all paths that particle can travel from A to B. Huygens principle says actually much the same classically and quantum theory brings only corrections to this as radiative corrections.

In TGD path would become four-surface connecting 3-surfaces A and B since particle is now 3-surface. This approach fails if one takes empty Minkowski space M^4 in M^4xCP_2 as the first approximation- "spherical cow". One cannot sum over all "paths" (space-time surfaces).

King is dead, and we must find a new king. The new king is WCW geometry. Path integral is replaced with functional integral over pairs of 3-surface A and B at opposite boundaries of WCW. The additional bonus is that functional integral is well-defined mathematically unlike path integral: this is the main victory of the new king. Of course, this also extends Einstein's geometrization program so that it applies to entire quantum theory and this is something really big.

Perturbation theory is universal approach to treat corrections to the sphericality of cows and one can of course develop perturbation theory for the functional integral in powers of Kaehler coupling strength alpha_K= g_^2/4*pi*h_eff. When h_eff is large it is small and large value of h_eff - the phase transition to dark matter phase - would save perturbation theory when it would not be possible otherwise.

What is however new that in topological sense there only one diagram. One can call it generalised Feynman diagram, or replacd "Feynman" with "twistor" or "knot" or "braid".

If I would replace simply with "TGD" , I would be regarded as a crackpot (CP) of second kind. If I would replace it with "Pitkänen" I would be regarded kind CP of first kind, the worst variety of crackpots. Now I am seen only as a CP of third kind and my friends and relatives have not so much to be ashamed of;-).

To Anonymous:

When I have mental image about my hand there, quantum Zenoing occurs as long as the mental image, one self in the hierarchy, exists.

The mental image about hand, exists as self identified as the period of state function reductions to fixed boundary of CD and leaving the part of zero energy state at that boundary invariant.

In ordinary quantum theory these state function reductions would do nothing for the state. Now they change the state at the second boundary of CD and even change the position of second boundary. This gives universally rise to the experience about flow of time since the average the distance between tips of CD increases. Funny to think that my hand would experience flow of time!

Standard neuroscience of course says that mental images are associated with *representations* of the external object in my brain rather than external objects as such, say my hand. To how high extent this is true is an interesting question. In TGD Universe flux tubes connect things together in all scales, in particular gravitational interaction involves them plus associated strings connecting partonic 2-surfaces. Attention creates flux tubes and presumably also associated strings.

Question: Are the flux tubes only between me and the representation about object or perhaps between me and the object? What happens when I look distant star by telescope?

Thanks to Anonymous for quantum Pascal triangle. It would be intereting to look this more closely.

Inclusions of HFFs and quantum groups characterised by quantum phases q= exp(i2pi/n)

are very interesting and quantum Pascal triangle is characterised by such a phase and gives quantum variant of binomial coefficient as a result. One can imagine quantum variants for integers characterising all kinds of combinatorial objects: Do quantum variants of objects make sense in some sense? Probably some mathematicians has pondered also this question.

Only few days ago I realised that in order to have "quantum quantum theory" as a tool to describe finite measurement resolution, it is better to have quantum variants of fermionic quantum anti-commutation relations for the induced spinors.

They have been formulated as I learned in five minutes from web.

These anticommutation relations however demand 2-D space/space-time! But just the well-definedness of em charge almost-forces 2-D string world sheets! And number theoretic arguments removes the "almost". In 4-D Minkowski space-time you do not get them!

In over-optimistic mood - officially allowed at morning hours - I can therefore conclude that the observation of anyons in condensed matter systems (assigned with 2-D boundaries) serves as a direct evidence for the localisation of induced spinors at 2-D surfaces and for large h_eff. I must however assume that also partonic 2-surfaces carry them- whether it is so has been an open question for a long time.

To Ulla about CD-diamonds and Dirac fermions: I would not say it in that manner.

Spinors and thus fermions are the from beginning. Space-time surfaces are correlates for what we call sensory, second quantised spinor fields are correlates for what we call Boolean cognition. Both are needed. I am because I sense and think. Descartes got half of it.

CDs are the imbedding space correlate of ZEO: zero energy states are associated with them and space-time surfaces are within CDs as also the induced spinor fields at string worlds sheets within CDs.

There are many levels in the geometric complex

and this does not make things easy for a layman.

In the lowest resolution you have space-time surfaces, imbedding space, WCW.

As you look at space-time surface in better resolution you begin to discern Euclidian and Minkowskian a regions, light-like orbits of partonic 2-surfaces between them, string world sheets and their 1-D boundaries at orbits of partonic 2-surfaces. You discover what are the space-time correlates of Feynman/ twistor/whatever diagrams. This is one part of geometrization and topologization of physics initiated by Einstein.

At imbedding level you find hierarchy of CDs required by ZEO, by non-determinism of Kaehler action, by cosmological facts, and by consciousness theory.

WCW decomposes to sub-WCWs associated with CDs, etc… This is very nice. Restriction to finite volume is not only an approximation, it is basic property of conscious experience: conscious experience does this for the physics: CD is the spotlight of consciousness.

There's a story behind the question "how do I know where my hand is":

An anthropologist was living with Siberian shamanistic tribe, and one day the hunters of the tribe came to the shaman, telling that the deer were not where they used to be (may troubled by oil company), and needles to say, if they would not find the deer, the tribe would be very hungry next winter. Shaman told the hunters to go and find the deer in another valley, they did and found them and tribe had food for next winter. The anthropologist who had been following these events asked the shaman, how did you know where the deer were? The shaman replied with a question: "how do you know where your hand is?"

The simplest level of math and dimensional/exponential logic tells us that this basic spatiotemporal "proprioseptic" knowing needs to be +1 dimensional field/space-time in relation to the object/intentional target (e.g. food) located. But in further analysis and number theoretic etc. geomatric analysis, the deductive mathematical logic is not independent from the mathematical language used. This is the most basic linguistic and relativistic truth, the measurement tools chosen also need to be holistically comprehended, and that is not an easy task for beings born and raised in this or that limiting paradigm (e.g. Cantorian "paradise", "real" number line etc.). Challenges are not meant to be easy, but challenging. Both layman and expert have their mutual handicaps and strengths, and expert skills can greatly benefit from layman intuitions and questions outside the axiomatic box in which the expert has been dogmatized. And laymen do not want to or need to do everything again ab ovo, but need to share and use the cumulated expert experience. In best situation the child like laymans open and anarchic curiosity and age old experts skills and wisdom are combined in one person.

So, in the spirit of socratic dialogue, can we establish and agree that by definition, any and all space-time surface is a 2D object, observable and observation event of which requires a 3D space? When I look at the ground and trees, I see 2D surfaces in 3D space, and this (external) observation event entails some kind of trigonometry to measure limits or boundaries. And when I observe these 3D and 4D observation fields (my "magnetic field body"), I'm thinking in a higher level of dimensional quadrance or quadrea.

All language, including number theories, is symbolic, ie. relation of parts and whole. Parts have meaning only in relation to whole. Ramanujan is a symbolic-holographic part of whole or God, thinking Godly thoughts with rare sensory directness. On Ramanujan level, the math is in the gut, not just learned rules in head. And Ramanujan cognition is not just "out there", but also inside each of us, like the hand and deer and the tribe are inside shamans cognitive n-dimensional field, where if question arises, answer arises...

Hmm. Cats. When a cat has the mental image of cat-self on the floor and cat-self on the table (in order to lay down on your keyboard), it does a rapid head movement of two eyes-vertical line up and down. Question: do we know and can we know what kind of trigonometry the cat is using, in order to jump with cat like grace?

PS: and for balance, a dog: https://www.youtube.com/watch?v=GhsNLvxYSNM

Orwin, is that you?

Matti, is there be a number theoretical link between the 4D quadrance universal coefficients and inversion transformation?

"The invariants for this symmetry in 4 dimensions is unknown", they can't be points, lines (strings) seem unknowable(?), so what about 'universal' areas/surfaces?

http://en.wikipedia.org/wiki/Inversion_transformation

To Anonymous about 4-D inversion.:

Conformal transformations in 4-D are conformal

transformations. They preserve angles and scale metric by local factor. Inner product given by Lorentz invariant metric transforms by this factor. The ratio of two inner products of tangent space vectors at point P given by : I= A.B/C.D is invariant under these transformations but this is tangent space invariants.

One obtains M^4 conformal transformations by starting from Poincare transformations which are linear and by combining them with inversion x^mu -->X^mu/X.X. Infinitesimal transformations

are characterised by extension of Poincare Lie algebra in which new Lie algebra generators are vectors (inversions at point differing infinitesimally from origin) and scalar (scaling).

I looked at the article on the transformations called conformal transformations and they look very different from conformal transformations and the authors do not actually show that the transformations are conformal transformations and they are not. Author just generalises the condition guaranteeing that Poincare transformation are isometries. Therefore there is not reason to expect that four-point invariant would exist .

There are no references in the article to the inversion transformation, which makes me really skeptic.

One can look for the conformal transformations by starting from complex case and then

look for a possible quaternionic generalisation.

In complex plane one can write z= u/v and conformal transformations act as u-->a*z+b, v-->c*z+d. From linearity at the level of C^2 follows the existence of the four-point invariant for Mobius transformations. The analogs of four point invariants for conformal transformation of M^4 do not exist but one can construct them in terms of 8-D twistors. In Minkowski space this kind of invariants do not exist.

Could quaternion analogs of Mobius transformations allow to generalize the four-point invariants from complex plane to 4-D quaternionic space? One would identify Q =Q_1/Q_2 as in complex case and act by linear transformations on Q_i. Here non-commutativity becomes the problem since one would like to have quaternion analyticity, which would mean Taylor or Laurent series with coefficients multiplying powers of Q from left or right but not both.

One must must specify whether the matrices for linear transformations act from right or left - say right. Also in the inversion (Q*a+b)/(Q*c+D) one must specify whether one divides from right or left. It seems that here one loses the idea about quaternion analyticity since one necessarily obtains terms which can be written as Q^n*k. The coefficients a,b,c,d should be real to avoid the problems. Projectivization Q=Q_1/Q_2 does not work by non-commutativity: one obtains terms of Q*a*Q. One should use biquaternions rather than their ratios to keep everything righ-linear. They would be rather analogous to 8-D linear representation of M^4 twistors.

I have a layman problem of comprehending Minkowski space and the expression "Euclidean and Minkowski regions" does not make sense. What kind of "space" would be divided into such different regions?

AFAIK Minkowski space is not fully algebraic, as it is based on transcendental angles and lengths and preserves and creates unnecessary complexities. Special Relativity follows naturally from quadratic form identity/symmetry of chromogeometry:

Euclidean:

blue dot product (Qb)

[x1, y1] ·b [x2, y2] ≡ x1x2 + y1y2,

Relativistic:

red dot product (Qr)

[x1, y1] ·r [x2, y2] ≡ x1x2 − y1y2

and

green dot product (Qg)

[x1, y1] ·g [x2, y2] = x1y2 + x2y1.

The identity of the threefold symmetry:

Qb^2 = Qr^2 + Qg^2

Relativistic observers can agree on observing shared dot products/2D-surfaces. Continuing linear algebra fully algebraically from this basic theorem of three colors of perpendicularity and their identity has a strong flavor and promise of "quantum quantum theory" where you don't run into complexities of transcendental Minkowski. Euclidean space contains/is given by the both inverses of relativistic spacetime.

In other words, following your notion, we're all in the Euclidean/quantum "black (w)hole", and relativistic red and green dot products/quadrances (cf red shift and green shift) just look like the "outside"...

Sorry, I failed to understand how the threefold symmetry would given Minkowskian inner product or shared inner product. What I get is x1y_2=-x_2y_1 from the threefold symmetry.

Link to chromogeometry

http://arxiv.org/pdf/0806.3617v1.pdf

The point was that threefold symmetry goes much deeper, on the level of unified theory. If I understood correctly, Minkowskian is just the 4D generalization of Qr.

Here's Norman's approach to linear algebra, looks very interesting:

https://www.youtube.com/playlist?list=PL01A21B9E302D50C1

Can it be shown, purely algebraically in (rational) linear algebra, that 8D Euclidean (Qb) is the sum of 4D relativistic and its inverse (Qr and Qg)? And could the two relativistic dot products be interpreted as CD? If so, the threefold symmetry would be very beautiful simplification of 8D imbedding space as the sum of two sides of CD.

To anonymous.

I looked at the article about rgb and found that what looks like product is not an ordinary product but something else which I do not understand.

From article O learn that the rgb identity is actually just the standard identity (x^2+y^2)^2= (x^2-y^2)^2+(2xy)^2 appearing in the construction of Pythagorean triangles whose sides have rational/integer lengths: x^2+y^2,x^2-y^2, 2xy. This is number theoretically very interesting.

A natural idea would be that p-adically the angles associated with Pythagorean triangles would be preferred. It however seems that preferred angles correspond to 2pi/n: actually roots of unity are p-adically well-defined: one cannot speak about angles but only sines and cosines of them. The notion of angle (actually phase) leads to the introduction of algebraic extensions of p-adic numbers: this makes also possible discrete Fourier analysis without which physics is not possible.

The three quadratic forms are 2-D length squared in Euclidian coordinates, Minkowski coordinates and Minkowski coordinates rotated by pi/2. I fail to see any connection with 4-D Minkowski space or with unification of interactions.

To Anonymous: About the lecture of Norman. I am going to be polemic now!;-). Nothing personal.

The geometry in lecture looks to me like elementary descriptive geometry- nothing bad in that as such. Linear algebra is also described. This is standard stuff which students of theoretical physics should learn during the first autumn- usually they do not.

I find it difficult to see how Norman's approach could lead to a unified theory. These concepts belong to the time before modern physics maybe even before Newton: there is no mention about even differential calculus!

Huge developments in mathematical consciousness have occurred after Newton. Modern physics involves partial differential equations, differential geometry, Hilbert spaces, advanced algebraic notions such as von Neumann algebras, group theory, topology, advanced number theory, ….. It is difficult for me to see how one could formulate it in terms of descriptive geometry by believing that rationals form a continuum.

Sorry for being polemic, but it is important to see the big picture about evolution of physics and mathematics, which has led to greatest revolution in consciousness that has happened on this planet. It is a pity that most scientists building curriculum vitae so often fail to realize this. There is no return to the times before Einstein and even less to the times before Newton.

This sounds frustrating but also professionals experience the same frustration: the understanding of individual remains nowadays infinitesimal. The miracle of science convinces me about existence of higher levels in the hierarchy of consciousness.

OK then, Let's look at the big picture :). The underlying philosophy and basic assumptions of science (mechanistic determinism) have been wrong since Descartes, and even today, hundred years after QM, the ruling paradigm is the cult of materialism and alienated objectivism, the consciousness of wannabe king of the hill chimp with a big stick. Sure, even blind chicken finds a kernel of a corn now and then, but as the whole of civilization revealed itself to be just this self-destructive collective psychosis we live entangled with, all the talk about evolution is just words, no actual evolving involved. Polemical enough big picture? ;-)

Looney crackpots like you are the most fun and interesting part of this field of magic. And same for Norman, to do math properly from the beginning and care for big audience aka fellow people in this day and age of ad hoc make believe "axioms" etc. scholastics about number of angels on needle point where irrationals converge, is a sure sign of a crackpot... ;)

Yes, if we give quadrance of general measurement theory (aka thermodynamics) any power, there is no going back to good old days of just shamanistic tribes, or just to Euclid and Euler, we have all these kernels too, they just don't fit together with all the acorns. The name of the game is to question basic assumptions, especially when in dead end.

Everybody can pick a stick and bullshit all they want, it's entertaining game but enough is enough, now back to business.

Threefold symmetry is a "higher level" Pythagoras theorem, and as it has been already shown without transcendental Minkowski that special relativity is rooted in Qr, that deserves a good look, and I believe my hypothesis that this level of holographic Pythagoran identity can help solve the riddle of unified theory is natural and justified. We can't say it cannot or can help without checking. As for the holographic levels or scales of Pythagoras' theorem, it's good to remember Bohm's philosophical notion of generative order.

The basic philosophy behind Norman's approach is finite measurement resolution ("does not believe in infinite sets" of Cantorian paradise), so it's best suited for that purpose, but does not alone lead to e.g. math of cognition and the fields and spacetimes in which mathematical observables take place.

The materialistic view is wrong for obvious reason: it neglects consciousness as acts of re-creation of Universe. The mathematics of behind materialistic view is however a wonderful creation and remains intact also when the world view is expanded.

I believe that finite measurement resolution is realised as properties of quantum states themselves. Rationals and their algebraic extensions are the tool to express the finite resolution classically. The fundamental geometric existence involves however both real and p-adic continua. Restriction to mere rationals is analogous to restricting to materialistic world view: both deny the transcendental.

Trascendentals is not mere mathematical spiritual luxury but extremely powerful tool. Without them physical theories would degenerate to computer programs.

Notions of continuity and discontinuity are at the heart of the matter, and they deserve careful philosophical thought. Ad hoc axiomatics serve the needs of applied engineering, but when we say that geometry and physics reduce to number theory, we are talking on different level. There are various continuities and discontinuities at various dimensional etc. exponential contexts, and their interrelations. Completeness is again different but related notion.

Notion of "real number line" is IMHO a misnomer and category error, as phenomenally it refers to 2D-fields of natural numbers with certain kind of exponential structure, not 1D-line like rational continuum. According to their exponential structure these natural number fields are _internal_ structures of integers and integer ratios. P-adic notion of exponentiality, its fields - or strings or braids?! start from the divine Whole One, platonic hen kai agathon (cf infinite primes in your language), divided into primes that multiplied together generate the natural numbers.

This is the big picture that I just figured out. Platonic anamnesis aka 'mathematical holography' is IMHO most basic axiomatic condition for reducing physics and theory of cognition to number theory. I haven't seen this number theoretic relation of whole and parts is not explained anywhere this simply and intuitively. And as math is these days usually taught and done in very misguiding language, this beautiful simplicity does not get often fully comprehended. The psychological mechanisms behind this obfuscation of Platonia are so obvious they need not to be pointed out.

Now we can see and say that the "p-adic" (One divided into primes is more intuitive language) unity is complete in the sense that it generates natural numbers. And parts _really_ make sense only in relation to holographic whole. 2D triangles and their algebraic relations ("irrationals") now make perfect sense p-adically, as line segments and their quadrants from hen kai agathon to rational number line. Their "real" and "complex" etc. approximations are quite literally the shadows on on the walls of Plato's cave, all converging to the infinitesimal that good bishop Berkeley ridiculed for good reasons, not the Ideas (Pythagoras' quadratic theorem, Ramanujan cognition etc) themselves.

Seen in this context, the dimensional "limit" of universal quadratic form and 15- and 290 theorems bring an interesting aspect of inherent discontinuity or economy to the picture. Which kind, exactly, and where?

And like the original Pythagoras' theorem, the Idea/Identity of threefold symmetry resides also in p-adic Platonia, outside the cave.

Of course also the Plato's cave is part of the whole, but to keep on evolving, we need to step once in a while outside the cave, clear out the spider webs and let new light create new shadows.

I agree with the philosophical picture but my big picture is different. I see different number fields as faces of a kaleidoscope.

p-Adics labelled by primes p=2,3,5,….(I call p "p-adic prime") plus their extensions are also completions of rationals containing their own kind of transcendentals as infinite series of powers of p not periodic after finite number digits as for rationals and infinite as real numbers. Number fields and their algebraic and possibly even non-algebraic extensions are to me a Big Book.

I see rationals as islands of order in sea of chaos. You want to throw away the sea and keep only the islands. To mer this is unrealistic. In physical systems you have not only the periodic orbits but

also the non-periodic ones and they dominate.

Rationals for me not all but only the back of Big Book. It is this Book in which I want to formulate physics - or at least TGD;-). Number theoretical universality - equations and formulas are expressed so that they make sense in any number field - is extremely powerful constraint as mass calculations based on p-adic thermodynamics demonstrated: p-adic temperature is quantized, the primes characterizing p-adic number fields (or briefly and somewhat confusingly "p-adic primes") characterise mass scales, etc…

[Here I must clarify: p-adic integers in R_p have just one prime, p unlike real integers and one does not have prime decomposition of p-adic integer

in p-adic sense and most of p-adic integers can be said to be infinite as reals].

The needed universal language would be very much analogous to tensor analysis, which is the manner to realize general coordinate invariance (, which by the way is even more powerful in TGD than in GRT).

One concrete application would be the interpretation of field equations and solutions for preferred externals in a manner making sense in any number field. p-Adic manifold as cognitive representation, cognitive chart would be second related notion. Third application would be the construction of scattering amplitudes in a manner not depending on number field: here vertices

as product and co-product of super-symplectic Yangian and amplitudes as computations freedom from a collection of algebra elements to another one might be the universal construction. Note the

Great Principle: Physics represents mathematics and not vice versa!!

What might be the tensor analysis of number theoretically universal physics?: maybe some-one is already writing a thesis about this;-).

Too bad Norman didn't include tensors in his lectures on linear algebra for beginners. :)

Internalizing the p-adic unity was a big step for me towards comprehending TGD, and breaking the barrier of real line was necessary for that, and what was big step for me might seem small step for your infinite primes... :)

Anyhow, I remembered this poem written a time ago in a book with some p-adic page numbers:

Pikkuinen kuuntelee avaruutta

missä satelliitti tähtiä metsästää,

nyt napsuu pajunlehtiin

irronneita sakaroita,

kiinnittää säteen kaukainen torni

kuin köyden läpi syvän pimean

missä syvän meren

oudot oliot

loihtivat omaa valoa

***

Have you ever tried imagining the kaleidoscope (Greek for 'beautiful/good form viewer') sea "1-adically"? Purely affine geometry comes formally closest to "1-adic".

To continue on the big picture, the internal Cauchy structures of rationals and roots also converge in one and same "point" or "number", called 'fluxon' or 'infinitesimal', and historically "the two wrongs that can make right", as Berkeley quipped on Newton and Leibnitz, can be taken as illogical or non-deductive heuristic that showed the path to the discovery of the p-adic unity of all-inclusive One. The whole narrative is a nice representation of

"causal diamond" on idea level. :)

The notion of "real numbers" in the form of Cauchy sequenses is archimedean (and they say p-adic field is non-archimedean), but the root sequenses converging to infinitesimal are "incomplete" only if by "axiom of choice" it is _decided_ that it is "Euclidean" in the sense of the fifth. affine/parallel-axiom. One could just as well and even better make not that choice and accept that the internal Cauchy sequences for rationals and algebraic roots converging at "infitesimal" point at infinity are non-euclidean and at least look if they are 'complete' in some definable non-euclidean sense. It is one thing to generate every possible combination of natural numbers and call that generative function complete (or universal quadrance ;)), but it quite another thing to call all the random noise created by simple combinatorics "transcendentals" and "unkowables" even there are no algebraic functions or other generative algorithms involved.

I came up with number theoretic metaphor for n-slit quantum experiment: p-adic unity as the source of light, and rationals as the cave entrances for cave man measurements of "real" Cauchy particles on the cave wall... how do you like that picture? ;)

Heh. So what kind of trigonometry cat uses to jump on the table? None, every cat knows there are only isosceles triangles, they just heff their heads up and down to bend the cave walls. :D

So, Berkeley ("to be is to be perceived") was at the root of Quantum Cat Jump in his theory of sensing, stressing the p-adic ultrametric of touch and body-sense, and "real" cave mechanic metrics of seeing: http://www.iep.utm.edu/berkeley/. As in ultrametric adelic all triangles are isosceles (again, cf. triangular numbers), by allowing "real" Cauchy field aka cave wall bend e.g. by scaling Planck, also Castaneda's assemblage point (http://www.prismagems.com/castaneda/donjuan8.html) can be given mathematical interpretation.

Posted to John Baez, cool to share also here:

Long time ago I started for some reason (platonic anamnesis?) meditating 3n+1 in cocentric way and doing Hilbert space meditation that way , and later I heard about Collatz conjecture aka 3n+1 problem and now I found that there is also a name for my meditation: http://en.wikipedia.org/wiki/Dehn_plane

Imagining an open ball like space where perpendicular dimension lines (defined by three points, "left" and "right" and "middle" where left and right cancel each other) all meet in the same middle point intuits that also the Collatz conjecture would be easily provable on non-archimedean Dehn level.

I can't do formal proofs as I'm just amateur philosopher of math, and google didn't bring up anything on Dehn approach to Collatz, so if this interests you, take a look.

My own approach to math is basically hippy "spiritual", or more exactly body-sense holography (gnothi seauton...;) ) , and I'm very happy that we can say "One Heart" also in the language of math: Dehn balls of Hilbert spaces in adelic ultrametric space of all p-adics combined, where every point of the ball is the center point. <3

https://www.youtube.com/watch?v=vdB-8eLEW8g

Matti, on a "practical" level, in the POSIX standard, "UNIX time" is the nix time (also known as POSIX time or Epoch time) is a system for describing instants in time, defined as the number of seconds that have elapsed since 00:00:00 Coordinated Universal Time (UTC), Thursday, 1 January 1970.

Let that simply be a point of reference relative to the time since the big bang 14.1413........... billion years...

then the triangle is always two points of time with the "fixed point at the origin" being the "big stretch" back at t=0, those being the 3 verticies of a 'triangle' . could that be the point at infinity added to the complex plane leading to topological considerations?

--Stephen

I would not like to bring in complex plane. This triangle almost degreasing to a long (the ratio of short and long is about 10^-9 ) line is hyperbolic rather than ordinary one. [The Mobius transformations by the way generalise also to the hyperbolic plane: imaginary unit satisfies now e^2=1].

"The solutions of Dirac equation have 8-D light-like momentum assignable to the 1-D curves..."

If physics reduce to number theory, and there is a 8-D limit of universal quadratic forms, and rational number lines can curve...?

Quadratic forms make sense in any dimension. D=4 is exception that with integer coefficients standard Euclidian length squared has all integer values. I dare guess that Minkowskian variant has also all values. This is the case.

From 8-D masslessness one would have

n_01^2-n_1^2-n_2^*2-n_3^2= m_0^2+m_1^2+m_2^2+m_3^2.

The right hand side has integer values and I guess that also the left hand side. When one writes this

in the form

n_01^2=n_1^2+n_2^*2+n_3^2+ m_0^2+m_1^2+m_2^2+m_3^2.

one realises that the condition is certainly satisfied. LHS is square of integer and even for integer values 7-D length squared as all integer values.

The identification of mass squared as conformal weight requires it to have integer spectrum and this is true if both M^4 and E^4 length squared have integer valued spectrum.

Sorry, can't follow your argument. How mass enters the picture, and what definition of mass is used above - Newton, SR, GR, some quantum or TGD specific ("p-adic length?")? In the last case, could you please explain TGD notion of mass in greater detail, or give link to the appropriate document?

General wisdom says that Minkowski is flat and does not do gravity, therefore GR "curved spacetime" is needed. Here's a grazy idea. The threefold symmetry suggests that to build complex plane and Riemannian manifolds from a number system, e.g. natural numbers, you need BOTH negative quadratic form (cf Minkowski) AND doubled quadratic form those "partonic" quadratic forms you get positive euclidean quadratic form and dimensions.

Very simply, e.g. in one dimension, if you only negate natural numbers without also doubling them, you end up with no numbers or just zero, instead of integers. :)

To see this most basic relation in mathematical cognition, you need some triangular point of observation: atomistic infinitesimal, cf. Minkowski/Qr and even more importantly inclusive adelic/infinite prime "point" of projective geometry. It is easy to see that planar geometries of Qr and Qg in Cartesian coordinates correspond to inner view and outer view, and perhaps 3D gravity - the only one we know by experience - is no more complex than the Euclidean special case of of 4D universal quadratic form.

Finite measurement resolution in terms of adelic point of observation/projection would most naturally be related to largest natural prime known to us. Such fmr would also assign Cauchy expansions of transcendental numbers finite length - meaning that largest known natural prime would have no measurable/computable internal Cauchy in that base???!

"meaning that largest known natural prime would have no measurable/computable internal Cauchy in that base???!"

-> Sorry, was thinking about internal Cauchy structure of pi in base of largest known natural prime.

p-Adic mass calculations assume that mass squared is thermal expectation value of mass squared over states for which mass squared is in suitable units integer n - conformal weight of the state by conformal invariance. p-Adically only integer valued spectrum actually makes sense.

p-Adic thermodynamics is highly unique if one assumes conformal invariance. Even more, it very probably does not exist without conformal invariance implying integer spectrum for conformal weights.

n=0 corresponds to the massless ground states and extremely small contributions with n=1,2.. make the expectation non-vanishing. State has non-vanishing p-adic mass squared which can be mapped to real mass squared. This is TGD view about Higgs mechanism.

One can look the situation at two levels. Imbedding space and space-time corresponding to inertial-gravitatuinal dichotomy behind Equivalence Principle.

a) Let us look it at imbedding space. In any case, one has p^2_iner=m^2=n. Inertial 4-momentum p=(p_0,0_1,0_2p_3) has four components. p^2_inert is inertial mass squared. The formulas p^2_inert -m^2=0 says 8-D mass squared vanishes in 8-D sense. m^2 is the analog of CP_2 momentum squared: eigenvalue of color Laplacian. Particle is massless in 8-D sense. This is generalisation of ordinary masslessnes and leads to 8-D variant of twistorialization making sense since M^4 and CP_2 are completely unique twistorially.

b) One can look the situation also at the level of space-time surface. Actually we look the situation at the boundary of string world sheet identifiable as fermion line in generalised Feynman/twistor/Yangian diagram. From the massless

*Dirac equation and its bosonic counterpart for induced spinor at the boundary of string world sheet one obtains that boundary is light-like geodesic in 8-D sense and the 8-D momentum associated with fermion is light like. Light-likeness in 8-D sense says

p^2_gr- p^2_E^4=0,

where 4-momentum p_gr=(n_0,n_1,n_2,n_3) could be called gravitational momentum and the E^4 vector p_E^4=(n_4,n_5,n_6,n_7) could be called E^4 4-momentum defining the gravitational dual of color quantum numbers.

I get finally to your question:

p_gr^2-(p_E^4)^2

gives the 8-D generalization of the quartic form and it allows solutions for all integers n= p_gr^2 as solutions.

The correspondence between inertial and gravitational in CP_2-E^4 case is not so straight-forward. It has as a direct physical counterpart as dualiity between descriptions of hadrons using SO(4) symmetry group and partons using color group SU(3): low energy description --high energy description. This duality generalizes to entire physics. Equivalence principle says p= p_gr.

The more you look at QFT, the uglier it looks, especially when compared to pure math. Stitches and patches of "length scales" in terms of orders of magnitude and "renormalization groups". Maybe the wiki on 'length scale' is just horrible, but also the concept itself does not make much sense, and with that card goes the house that it is supposed to support. Perhaps there is some connection between 15 degrees of freedom of conformal symmetry and the 15-theorem, that might guide us to the middle path between philosophically unsound notions of "individual" particles and fields with infinite degrees of freedom, but more likely the whole QFT, both its classical and algebraic versions, is doomed from the beginning to remain dead end. See e.g. this standard discussion: http://plato.stanford.edu/entries/quantum-field-theory/

TGD contains novel and beautiful deep ideas, but as long as its (non)formulation is tainted by confusing language and ill-defined concepts originating in QFT, reluctance against explicating it with formal rigour is more than understandable, but sadly does not give much hope of ability to communicate what is good and true in TGD. Most importantly, to our understanding on TGD theory of consciousness level there is no real conflict between platonist and fictionalist approaches to philosophy of mathematics. As also Einstein understood, when the math is beautiful enough, there is no question that the empirical evidence could fail to respond to the beauty. :)

QFT is an idealisation. Particles are made point like and the many-sheeted topologically non-trivial space-time is replaced with empty Minkowski space. This is quite a violent act from TGD point of view. On the other hand, N=4 SUSY has led to the Yangian symmetry and also discovery that scattering amplitudes might be extremely simple and obey quite unexpected symmetries such as Yangian symmetry so that this violent process seems to respect some of the key aspects.

The notions of resolution and length scale are to my view fundamental. Their recent formulation makes them to look like ugly ducklings.

Here von Neumann's mathematical vision would help. Hyperfirnite factors of type II_1 and their inclusion hierarchies would be the solution. Concrete representation would be in terms of

fractal hierarchies of breakings of conformal gauge symmetry assignable to supersympelectic and other key symmetry algebras of TGD possessing conformal structure (generators labelled by conformal weight).

Also hierarchy of Planck constants emerges in this manner and means generalisation of quantum theory and understanding of dark matter.

The relation to p-adic length scale hierarchy and negentropic entanglement I do not yet fully understand but certainly it is also very intimately connected to the other hierarchies.

I mentioned that I am trying to improve my understanding about the relationship of p-adic length scale hierarchy to that of Planck constants.

a) Negentropic entanglement corresponds to density matrix which is projector and thus proportional to mxm unit matrix. In ordinary quantum mechanics the reduction occurs to 1-D sub-space: projector is 1-D and to a ray of Hilbert space. In TGD projection occurs to a subspace with arbitrary dimension.

When the dimension of projecto is larger than one, the reduced is negentropically entangled and the number theoretic entanglement negentropy characterized by the largest prime power dividing m gives a negative entanglement entropy. Unitary entanglement gives in 2-particle case rise to this kind of situation. For systems with larger particle number the conditions are more demanding.

Two systems with h_eff=n*h and thus with n conformal equivalence classes of connecting space-time sheets could have this kind of entanglement- quantum superposition of pairs of sheets. Also quantum superpositions for pairs of *subsets* of space-time sheets would give negentropic entanglement. The entanglement coefficients would form unitary matrix.

b) It would be very natural to assume that the p-adic prime characterizing the system is the one giving rise to maximal entanglement negentropy. What looks like a problem is that the physical values of p-adic primes are very large. For electron p= M_127=2^127-1 is of order 10^38 and if the p-adicity for negentropic entanglement corresponds to that for electron then n should have p as a factor! For m in living matter values of order 10^12 appears and this is not possible.

To be continued..

This is a typical situation in which it is good to take a closer look. There are two systems: electron and electron pair. They are not same. And h_eff is associated with flux tubes connecting two systems.

Hence one could argue that the p-adic prime characterizing the magnetic flux tubes connecting two electrons is not same as that characterizing electrons but is much smaller? This would solve the problem.

The p-adic prime characterizing electron could assigned to the magnetic flux tubes connecting the wormhole throats of two wormhole contacts making electron (or any elementary particle)? This n would be really large! As a matter fact, this would mean that the quantum phase q=exp(i2*pi/n) is very near to unity and quantum quantum mechanics is very much like quantum mechanics. Note that n=1 and n=infty at opposite ends of spectrum give rise to ordinary quantum theory!

c) There is a further poorly understood problem. n =h_eff/h would correspond to the number of conformal equivalence classes of space-time surfaces connecting 3-surfaces at opposite boundaries of CD.

In ordinary classical n-furcation only single branch is selected. In second quantization n-furcation any number 0<m<=n of branches can be selected. 0<m<=n such space-time sheets would connect these 3-surfaces simultaneously. m would ike particle number. Now h_eff would be for a given m equal to h_eff= m*h.

This could relate to charge fractionization. What values of m are allowed? For instance, can one demand that m divides n?

I still don't comprehend the meaning - number theoretical structure - of 'p-adic length scale' - does that just refer to exponential orders of magnitude of p-adic integers? Or does the concept somehow relate to p-adic norm, p-adic distribution and p-adic measures? (http://en.wikipedia.org/wiki/P-adic_distribution)

What is the ratio of circle and it's diameter in adelic and p-adic context can it be given numerical value in a base p, e.g. 5?

Searching on that topic, this came up:

"Pi is troublesome number when one tries to generalize geometry to p-adic context. If one wants pi

one must allow infinite-D (in algebraic sense) of p-adic numbers meaning that all powers of pi multiplied by p-adic numbers are allowed. As such this is not catastrophe but if one tolerates only algebraic extensions then only the phases exp(i2pi/n) make sense. Only phases but not

angles. Something deep physically (distance measurement by interferometry)?

In light-hearted mood one might ask whether gravitation could save from this trouble and allow to speak about circumference of circle also in p-adic context. By replacing plane with a cone (this requires cosmic string;-)), 2pi defined as ratio of length of circle to its radius becomes k*2pi and could therefore be also rational."

Maybe you recognize the writer. :)

Yes, the notions of length, area, volume etc. are deeply physical, and it's easy to get confused when trying to define e.g. area in terms of length. Reduction of areas to lengths is not universal, the relation doesn't always commute.

Dehn plane seems very interesting way to think and look at these matters, also in terms of non-euclidean pi and possible "measure". And as space/field of "p-adics and reals glued together by _common rationals_" is by definition not "complete" in the usual Euclidean sense, it generates only repetitive Cauchy sequenses on both sides, it seems prudent to use notions of quadrance and spread (cf. inner products) instead of lengths and angles to think about pi and (pinary?) measures involved in this context.

Maybe there's a very good gut feeling behind the 'light-hearted' comment about gravity saving the situation... ;)

It seems that one of the main problems here is 'conformal' preserving of angles, especially if we want to proceed from non-archimedean quantum theory, e.g. adelic theory, to Archimedean classical metrics and measurements.

Transcendental angles do not solve the problem, rather they create it. Replacing conformality of angles with more general conformal spreads that get "quantum-numberish", spin-like values from 0 to 1 could make much more sense and clear out much of the linguistic and conceptual confusion.

Matti, the well defined notions of Pythagorean field and Pythagorean closure, which are closely related to TGD imbedding space, might help to find answers to your questions:

http://en.wikipedia.org/wiki/Pythagorean_field

Here can be found many hopefully useful remarks on filtration on Witt ring in relation to Euclidean and Pythagorean fields:

http://math.uga.edu/~pete/quadraticforms2.pdf

To Anonymous. p-Adic length scale involves

map of p-adic mass square do real one mediated by canonical identification Sum x_np^n --> Sum x^np^-n mapping reals to p-adics and vice versa.

p-Adic mass squared is of form

M^2_p= n_1p+ n_2p^2 =about n_1p

(because p^2 is very small in p-adic norm)

and is mapped to real mass squared

M^2_R =n_1/p+n_2/p^2+=about n_1/p

By uncertainty principle the real mass corresponds to length scale h/M propto sqrt(p) in suitable units.

To Anonymous: pi is well-defined in terms of geometry but becomes problematic in algebraic context unless one assumes real numbers.

In p-adic context one could try to define 2*p geometrically as a circumference of p-adic unit circle length. The problem is that p-adic unit circle is difficult to define!

Essentially the problem is that the notions of length, areas, etc.. are not possible to define as a purely p-adic notion. p-Adic physics is about cognition, not about sensory world in which one can quantify in terms of lengths, etc.. More concretely, length would require integral defining line length. Line should have ends but p-adic line does not have ends.

Metric makes sense as purely local notion also

p-adically. In particular, inner product involving *cosine* of angle can be defined in terms of p-adic metric. But not angles! The simplest angles 2pi/n assignable to roots of unity do not exist p-adically unless one introduces pi and its powers to extension.

One can argue that we always measure length ratios rather than directly angles. sines, cosines, tangents, inner products… That is sin(2*pi/n), etc… one can introduce by introducing algebraic extension of p-adic numbers containing n:th root of unity. This would mean discretisation in angle degrees of freedom and finite measurement resolution in angle (or rather phase degrees of freedom)

This allows also to define discrete Fourier analysis in angle degrees of freedom giving rise to the counterpart of integration. This is enough for doing physics if one accepts the notion of finite measurement resolution which emerges from basic TGD automatically.

Discretisation corresponds to the points of partonic 2-surface at which string ends are located. The localisation of spinors to string world sheets forced by well-definendess of em charge is behind this. As also the condition that generalisation of twistor structure requiring equivalence of octonionic gamma matrices with ordinary ones. Very deep mathematical connections with concrete physical meaning.

To Anonymous:

Conformal transformations preserve the angles in local sense: the *cosine of angle* is preserved. In p-adic contex metric makes sense and one can indeed define cosine of angle. Therefore also the notion of conformal transformation is well-defined.

The purely algebraic manner to define conformal transformations is as powers series and makes sense also p-adically.

One can say that p-adicization is possible for

differential (purely local) geometry but not for

global geometry (lenghs, areas…). The manner to

overcome this is to understand real and p-adic space-time surfaces as pairs. p-Adic preferred extremal is cognitive chart map of real preferred extremal and real preferred extemal vice versa (sating it loosely, a representation of thought as action).

An interesting question is whether all quantum jumps/state function reductions could actually occur as from real to p-adic and vice versa. Matter to thought o matte to thought... Sensory-cognitive-sensory-cognive. Cognition and intentionally would be the second half of existence present in all scales rather than only in brain: the other half would be sensory experience and action.

This picture would *force* finite measurement resolution number theoretically since the transition amplitudes between different number fields should involve only points of space-time which are in algebraic extension of rationals - in the intersection of reality and p-adicities.

At level of partonic 2-surfaces the parameters defining the representation of surface would be in this algebraic extension.

To Anomynous:

Pythagorean field has escaped my lazy attention. If Pythagoras had had Wikipedia, this would have saved the life of his pupil who started to produce strange talk about diagonal of unit square.

Pythagorean is an extension of rationals which contains also square roots of integers. In the real context infinite number of square roots are added

so that the extension is infinite-D.

In p-adic context only very few square roots of integers in the range [1,p-1] are needed. I still do not know whether sqrt[p] can be added. If so, then for p>2 the extension is 4-D and for p=2 8-D. During the first year of p-adic physics I wrote a little article about this and of course wondered about the connection with TGD.

Penrose made a reference to this little article in his book. Penrose is a bold man! This is one of the very few references to my work that colleagues have dared to make. The loss of scientific reputation is an extremely infective disease and can kill within few days;-)!

Your hypothesis for p-adic thermodynamics is far from convincing. First of all, according to local-global, there should be no problem with whole universe being conscious - and also fully loving - and the counter argument of "catastrophy for NMP" sounds like locally filtering psychological defense mechanism, or what Buddhists define as 'ignorance' or 'ego'.

Especially unconvincing is the whole idea of canonical identification, as well as the idea of p-adic logarithms, as p-adics by themselves are kinds of exponential inverse functions of real Cauchy logarithms, and/or vice versa. So instead of static canonical identification, both euclidean completions or rationals would rather pulsate like oscillator, and not unlike CD. :)

According to elementary math of periodic functions, the length of periodic phase of cosmic string is - for observers like us and our finite measurement resolution - the sum of all p-adic primes up to the biggest mersenne we have calculated. We can't calculate the exact length, as there are many "betweens" that we don't know at the moment, but Riemann hypothesis should give the rough idea.

And if we apply RH to primal temperature, doesn't that mean that cosmic thermodynamic loudness is exactly ½?! The metric measure of between 0/cold and 1/hot does not really matter, what is beautiful is that this sounds warm and homely octave. <3 :)

At the Middle Path point where h_eff evolves.

To Anomymous:

all of us have right to opinions, even non-educated ones.

The predictions from p-adic length scale hypothesis are impressive and this is enough for me. Especially so because one ends up with a beautiful vision about physics, which includes also cognition. Canonical identification (having more refined version involving cognitive/ length scale cutoffs) appears naturally in the formulation of p-adic manifold - or rather adelic manifold involving real and various p-adic manifolds as adelic structure.

I don't like bringing 'ego's or 'ignorance's and similar rhetoric stuff into physical argumentation. I have found that many people calling them spiritual use these coin words as kind of rhetoric weapons.

Neither do I want to base my thinking to empty statements such as "actually the is only ONE conscious entity". The fact is that there are multitude of conscious experiences, maybe also entities, and experience of unity is one particular experience. I want to understand consciousness, not to cherish one particular kind of experience - this belongs to priests.

I cannot understand what the rest of the argument means since they are not formulated with concepts that I am used to. I do not know what the motivation for introducing the length of cosmic string in this context is.

I can however tell that p-adic logarithm is not an nidea but completely well defined and standard mathematical notion existing for argument having p-adic norm below unity.

I do not know what you mean application of RH to primal temperature. You must have some theory behind it and TGD does not seem to be that theory.

Sorry, my way of learning TGD - or some emergent closely related theory - can be bothersome, but please bear with me. Creative tension aka inspiration has also it's destructive side, as some opinions need to be cleared out to make room for forming new opinions, which if succesful, lets us see also old opinions in new light.

Ego-ignorance rhetoric was not an ad hominem - or goes for me as much as for anyone - but intrinsic to a theory of consciousness, h_eff evolution and shannon entropy. Yes, there are multitude of conscious experiences and entities and therefore experience of unity should not be a priori excluded, if and when we want to understand consciousness as holistically as we can.

This is work in progress, as ideas keep revealing themselves, and now it feels that first idea worth mentioning is that in geometrical conscious experience senses are not spatially identical, but can be assigned dimensional character, first idea was that taste and smell without dimensional character, touch with point like or tangential character, sight 2D, hearing 3D and proprioseptic etc body-senses n-dimensional character. The point of this is that a TOE including consciousness cannot be based on only one sense phenomenal experience, e.g. sight, but needs to cover whole range. Not only mental images but also music, multidimensional feels, including feels of cold and warm, etc. are shared and teleported via (neg)entropic entanglement, and also without linguistic verification through classical channel. Much of this multitude of sentient experience stays locally subconsciouss, filtered out from local conscious experiences by entropic filters, but conscious experience can also disentangle from filters and "expand". The "unity" of experiences of unity can also have huge geometrical and scalar and sentient variety.

Continued:

This in mind, filtration on Witt ring is a natural place to start contemplate and feel out filtering entanglements and re-member what is filtered into subconscious. And in order to get thoughts more organized, we need a finite global measure to compare and organize local measures... hence numerical integer value for cosmic string/wave length.

Self-educated (in the full sense of TGD 'self' and beyond) intuitions don't equal non-educated opinions, and that finite value cosmic string of summing primes up to largest known Mersenne feels like the most natural candidate for the circumference of p-adic circle, which was searched for in the "light-hearted" comment above. This as such is not a new idea in terms of TGD.

Expanding the circumference into cone is the truly beautiful clue to beauty, when we give the planar circle geometric anatomy of Dehn plane, and identify Dehn plane with what you call partonic 2-surface meaning, if I understood correctly, the intersection of adelic space and real side Cauchy structures.

We can, I presume, imagine adelic unity as the perpendicular tip point of the cone above the intersecting point of Dehn plane where all dimension lines with rational anatomy meet, and limit the number of intersecting dimension lines by the number of primes up to largest known Mersenne, so that we can draw connecting p-adic lines - which can be interpreted directly as worm-hole flux tubes - to end points of Dehn plane dimension lines. So, according to Pythagoras theorem, the quadrance of p-adic line can be expressed in terms of quadrance of positive or negative side of rational line; and quadrance of Dehn plane meet point and adelic point of unity. So now we have non-archimedean triangels of adelic space that can be expanded into in some sense square "sheets" below the Dehn plane, in the "cave" of real extensions. With shifts between two notions of Dehn plane and other actions, the n-furcating geodesic anatomy of the "cone above" becomes very rich, and as the Dehn plane "filter" between above and below is rational, all the mappings between above and below are so far finite repeating wave lengths. And if we don't do Euclidean completeness action below or above at this stage, to my understanding the tip point of the cone below is the 'infinitesimal', and the surface of the cone below is hyperbolic. And if and when we do measurement in terms of completeness theorem in the cone below, it bursts open and scattering into particles happens.

And there is more, in the sense 'less is more': Assuming that Collatz conjecture is true, we can interprete the 3n+1 iteration as 3 spatial dimensions intersecting on Dehn plane and the +1 element as temporal component or dimension, perpendicular

to Dehn plane. In other terms, the circumference of Dehn plane as shown above can be filled by 3n+1 iteration leading always back to unity, and these 3n lengths would also be equal to three components of isosceles triangle. Hence, according to Collatz conjecture we can reduce the cones to tetrahedrons or bipyramid of 2 tetrahedrons.

There is also inverse possibility to interprete 3n as temporal dimensions, in other words observers moving up and down tetrahedral lines from unity to rational Dehn plane, and +1 as the spatial dimension partonic 2-surface.

This is what intention to understand TDG better and ability to self-learn only in very simple and rational - but possible not less high - math has produced up to this moment. Thank you for your patience.

http://apps.nrbook.com/bateman/Vol2.pdf#

Does this quartic of the elliptic theta function have anything to do with the "Higgs self-mass coupling" ?

http://apps.nrbook.com/bateman/Vol2.pdf#322

link updated with page number missing from previous comment

Does this quartic of the elliptic theta function have anything to do with the "Higgs self-mass coupling" ?

I see not connection.

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