I proposed that the replacement of h with heff = n× h= hgr= GMm/v0 could resolve the problem. It does not. I soo noticed that the typical size scale of string world sheet scales as hgr1/2, not as hgr= GMm/v0 as one might expect. The only reasonable option is that string tension behave as 1/hgr2. In the following I demonstrate that TGD in its basic form and defined by super-symmetrized Kähler action indeed predicts this behavior if string world sheets emerge. They indeed do so number theoretically from the condition of associativity and also from the condition that electromagnetic charge for the spinor modes is well-defined. By the analog of AdS/CFT correspondence the string tension could characterize the action density of magnetic flux tubes associated with the strings and varying string tension would correspond to the effective string tension of the magnetic flux tubes as carriers of magnetic energy (dark energy is identified as magnetic energy in TGD Universe).
Therefore the visit of string theory to TGD Universe remained rather short but it had a purpose: it made completely clear why superstring are not the theory of gravitation and why TGD can be this theory.
Do associativty and commutativity define the laws of physics?
The dimensions of classical number fields appear as dimensions of basic objects in quantum TGD. Imbedding space has dimension 8, space-time has dimension 4, light-like 3-surfaces are orbits of 2-D partonic surfaces. If conformal QFT applies to 2-surfaces (this is questionable), one-dimensional structures would be the basic objects. The lowest level would correspond to discrete sets of points identifiable as intersections of real and p-adic space-time sheets. This suggests that besides p-adic number fields also classical number fields (reals, complex numbers, quaternions, octonions are involved and the notion of geometry generalizes considerably. In the recent view about quantum TGD the dimensional hierarchy defined by classical number field indeed plays a key role. H=M4× CP2 has a number theoretic interpretation and standard model symmetries can be understood number theoretically as symmetries of hyper-quaternionic planes of hyper-octonionic space.
The associativity condition A(BC)= (AB)C suggests itself as a fundamental physical law of both classical and quantum physics. Commutativity can be considered as an additional condition. In conformal field theories associativity condition indeed fixes the n-point functions of the theory. At the level of classical TGD space-time surfaces could be identified as maximal associative (hyper-quaternionic) sub-manifolds of the imbedding space whose points contain a preferred hyper-complex plane M2 in their tangent space and the hierarchy finite fields-rationals-reals-complex numbers-quaternions-octonions could have direct quantum physical counterpart. This leads to the notion of number theoretic compactification analogous to the dualities of M-theory: one can interpret space-time surfaces either as hyper-quaternionic 4-surfaces of M8 or as 4-surfaces in M4× CP2. As a matter fact, commutativity in number theoretic sense is a further natural condition and leads to the notion of number theoretic braid naturally as also to direct connection with super string models.
At the level of modified Dirac action the identification of space-time surface as a hyper-quaternionic sub-manifold of H means that the modified gamma matrices of the space-time surface defined in terms of canonical momentum currents of Kähler action using octonionic representation for the gamma matrices of H span a hyper-quaternionic sub-space of hyper-octonions at each point of space-time surface (hyper-octonions are the subspace of complexified octonions for which imaginary units are octonionic imaginary units multiplied by commutating imaginary unit). Hyper-octonionic representation leads to a proposal for how to extend twistor program to TGD framework .
How to achieve associativity in the fermionic sector?
In the fermionic sector an additional complication emerges. The associativity of the tangent- or normal space of the space-time surface need not be enough to guarantee the associativity at the level of Kähler-Dirac or Dirac equation. The reason is the presence of spinor connection. A possible cure could be the vanishing of the components of spinor connection for two conjugates of quaternionic coordinates combined with holomorphy of the modes.
- The induced spinor connection involves sigma matrices in CP2 degrees of freedom, which for the octonionic representation of gamma matrices are proportional to octonion units in Minkowski degrees of freedom. This corresponds to a reduction of tangent space group SO(1,7) to G2. Therefore octonionic Dirac equation identifying Dirac spinors as complexified octonions can lead to non-associativity even when space-time surface is associative or co-associative.
- The simplest manner to overcome these problems is to assume that spinors are localized at 2-D string world sheets with 1-D CP2 projection and thus possible only in Minkowskian regions. Induced gauge fields would vanish. String world sheets would be minimal surfaces in M4× D1⊂ M4× CP2 and the theory would simplify enormously. String area would give rise to an additional term in the action assigned to the Minkowskian space-time regions and for vacuum extremals one would have only strings in the first approximation, which conforms with the success of string models and with the intuitive view that vacuum extremals of Kähler action are basic building bricks of many-sheeted space-time. Note that string world sheets would be also symplectic covariants.
Without further conditions gauge potentials would be non-vanishing but one can hope that one can gauge transform them away in associative manner. If not, one can also consider the possibility that CP2 projection is geodesic circle S1: symplectic invariance is considerably reduces for this option since symplectic transformations must reduce to rotations in S1.
- The fist heavy objection is that action would contain Newton's constant G as a fundamental dynamical parameter: this is a standard recipe for building a non-renormalizable theory. The very idea of TGD indeed is that there is only single dimensionless parameter analogous to critical temperature. One can of coure argue that the dimensionless parameter is hbarG/R2, R CP2 "radius".
Second heavy objection is that the Euclidian variant of string action exponentially damps out all string world sheets with area larger than hbar G. Note also that the classical energy of Minkowskian string would be gigantic unless the length of string is of order Planck length. For Minkowskian signature the exponent is oscillatory and one can argue that wild oscillations have the same effect.
The hierarchy of Planck constants would allow the replacement hbar→ hbareff but this is not enough. The area of typical string world sheet would scale as heff and the size of CD and gravitational Compton lengths of gravitationally bound objects would scale (heff)1/2 rather than heff = GMm/v0 which one wants. The only way out of problem is to assume T ∝ (hbar/heff)2. This is however un-natural for genuine area action. Hence it seems that the visit of the basic assumption of superstring theory to TGD remains very short. In any case, if one assumes that string connect gravitationally bound masses, super string models in perturbative description are definitely wrong as physical theories as has of course become clear already from landscape catastrophe.
Is super-symmetrized Kähler-Dirac action enough?
Could one do without string area in the action and use only K-D action, which is in any case forced by the super-conformal symmetry? This option I have indeed considered hitherto. K-D Dirac equation indeed tends to reduce to a lower-dimensional one: for massless extremals the K-D operator is effectively 1-dimensional. For cosmic strings this reduction does not however take place. In any case, this leads to ask whether in some cases the solutions of Kähler-Dirac equation are localized at lower-dimensional surfaces of space-time surface.
- The proposal has indeed been that string world sheets carry vanishing W and possibly even Z fields: in this manner the electromagnetic charge of spinor mode could be well-defined. The vanishing conditions force in the generic case 2-dimensionality.
Besides this the canonical momentum currents for Kähler action defining 4 imbedding space vector fields must define an integrable distribution of two planes to give string world sheet. The four canonical momentum currents Πkα= ∂ LK/∂∂α hk identified as imbedding 1-forms can have only two linearly independent components parallel to the string world sheet. Also the Frobenius conditions stating that the two 1-forms are proportional to gradients of two imbedding space coordinates Φi defining also coordinates at string world sheet, must be satisfied. These conditions are rather strong and are expected to select some discrete set of string world sheets.
- To construct preferred extremal one should fix the partonic 2-surfaces, their light-like orbits defining boundaries of Euclidian and Minkowskian space-time regions, and string world sheets. At string world sheets the boundary condition would be that the normal components of canonical momentum currents for Kähler action vanish. This picture brings in mind strong form of holography and this suggests that might make sense and also solution of Einstein equations with point like sources.
- The localization of spinor modes at 2-D surfaces would would follow from the well-definedness of em charge and one could have situation is which the localization does not occur. For instance, covariantly constant right-handed neutrinos spinor modes at cosmic strings are completely de-localized and one can wonder whether one could give up the localization inside wormhole contacts.
- String tension is dynamical and physical intuition suggests that induced metric at string world sheet is replaced by the anti-commutator of the K-D gamma matrices and by conformal invariance only the conformal equivalence class of this metric would matter and it could be even equivalent with the induced metric. A possible interpretation is that the energy density of Kähler action has a singularity localized at the string world sheet.
Another interpretation that I proposed for years ago but gave up is that in spirit with the TGD analog of AdS/CFT duality the Noether charges for Kähler action can be reduced to integrals over string world sheet having interpretation as area in effective metric. In the case of magnetic flux tubes carrying monopole fluxes and containing a string connecting partonic 2-surfaces at its ends this interpretation would be very natural, and string tension would characterize the density of Kähler magnetic energy. String model with dynamical string tension would certainly be a good approximation and string tension would depend on scale of CD.
- There is also an objection. For M4 type vacuum extremals one would not obtain any non-vacuum string world sheets carrying fermions but the successes of string model strongly suggest that string world sheets are there. String world sheets would represent a deformation of the vacuum extremal and far from string world sheets one would have vacuum extremal in an excellent approximation. Situation would be analogous to that in general relativity with point particles.
- The hierarchy of conformal symmetry breakings for K-D action should make string tension proportional to 1/heff2 with heff=hgr giving correct gravitational Compton length Λgr= GM/v0 defining the minimal size of CD associated with the system. Why the effective string tension of string world sheet should behave like (hbar/hbareff)2?
The first point to notice is that the effective metric Gαβ defined as hklΠkαΠlβ, where the canonical momentum current Πkα=∂ LK/∂∂α hk has dimension 1/L2 as required. Kähler action density must be dimensionless and since the induced Kähler form is dimensionless the canonical momentum currents are proportional to 1/αK.
Should one assume that αK is fundamental coupling strength fixed by quantum criticality to αK≈1/137? Or should one regard gK2 as fundamental parameter so that one would have 1/αK= hbareff/4π gK2 having spectrum coming as integer multiples (recall the analogy with inverse of critical temperature)?
The latter option is the in spirit with the original idea stating that the increase of heff reduces the values of the gauge coupling strengths proportional to αK so that perturbation series converges (Universe is theoretician friendly). The non-perturbative states would be critical states. The non-determinism of Kähler action implying that the 3-surfaces at the boundaries of CD can be connected by large number of space-time sheets forming n conformal equivalence classes. The latter option would give Gαβ ∝ heff2 and det(G) ∝ 1/heff2 as required.
- It must be emphasized that the string tension has interpretation in terms of gravitational coupling on only at the GRT limit of TGD involving the replacement of many-sheeted space-time with single sheeted one. It can have also interpretation as hadronic string tension or effective string tension associated with magnetic flux tubes and telling the density of Kähler magnetic energy per unit length.
Superstring models would describe only the perturbative Planck scale dynamics for emission and absorption of heff/h=1 on mass shell gravitons whereas the quantum description of bound states would require heff/n>1 when the masses. Also the effective gravitational constant associated with the strings would differ from G.
The natural condition is that the size scale of string world sheet associated with the flux tube mediating gravitational binding is G(M+m)/v0, By expressing string tension in the form 1/T=n2 hbar G1, n=heff/h, this condition gives hbar G1= hbar2/Mred2, Mred= Mm/(M+m). The effective Planck length defined by the effective Newton's constant G1 analogous to that appearing in string tension is just the Compton length associated with the reduced mass of the system and string tension equals to T= [v0/G(M+m)]2 apart from a numerical constant (2G(M+m) is Schwartschild radius for the entire system). Hence the macroscopic stringy description of gravitation in terms of string differs dramatically from the perturbative one. Note that one can also understand why in the Bohr orbit model of Nottale for the planetary system and in its TGD version v0 must be by a factor 1/5 smaller for outer planets rather than inner planets.
The condition that octonionic spinors are equivalent with ordinary spinors looks rather natural but in the case of Kähler-Dirac action the non-associativity could leak in. One could of course give up the condition that octonionic and ordinary K-D equation are equivalent in 4-D case. If so, one could see K-D action as related to non-commutative and maybe even non-associative fermion dynamics. Suppose that one does not.
- K-D action vanishes by K-D equation. Could this save from non-associativity? If the spinors are localized to string world sheets, one obtains just the standard stringy construction of conformal modes of spinor field. The induce spinor connection would have only the holomorphic component Az. Spinor mode would depend only on z but K-D gamma matrix Γz would annihilate the spinor mode so that K-D equation would be satisfied. There are good hopes that the octonionic variant of K-D equation is equivalent with that based on ordinary gamma matrices since quaternionic coordinated reduces to complex coordinate, octonionic quaternionic gamma matrices reduce to complex gamma matrices, sigma matrices are effectively absent by holomorphy.
- One can consider also 4-D situation (maybe inside wormhole contacts). Could some form of quaternion holomorphy allow to realize the K-D equation just as in the case of super string models by replacing complex coordinate and its conjugate with quaternion and its 3 conjugates. Only two quaternion conjugates would appear in the spinor mode and the corresponding quaternionic gamma matrices would annihilate the spinor mode. It is essential that in a suitable gauge the spinor connection has non-vanishing components only for two quaternion conjugate coordinates. As a special case one would have a situation in which only one quaternion coordinate appears in the solution. Depending on the character of quaternionion holomorphy the modes would be labelled by one or two integers identifiable as conformal weights.
Even if these octonionic 4-D modes exists (as one expects in the case of cosmic strings), it is far from clear whether the description in terms of them is equivalent with the description using K-D equation based ordinary gamma matrices. The algebraic structure however raises hopes about this. The quaternion coordinate can be represented as sum of two complex coordinates as q=z1+Jz2 and the dependence on two quaternion conjugates corresponds to the dependence on two complex coordinates z1,z2. The condition that two quaternion complexified gammas annihilate the spinors is equivalent with the corresponding condition for Dirac equation formulated using 2 complex coordinates. This for wormhole contacts. The possible generalization of this condition to Minkowskian regions would be in terms Hamilton-Jacobi structure.
Note that for cosmic strings of form X2× Y2⊂ M4× CP2 the associativity condition for S2 sigma matrix and without assuming localization demands that the commutator of Y2 imaginary units is proportional to the imaginary unit assignable to X2 which however depends on point of X2. This condition seems to imply correlation between Y2 and S2 which does not look physical.
To summarize, the minimal and mathematically most optimistic conclusion is that Kähler-Dirac action is indeed enough to understand gravitational binding without giving up the associativity of the fermionic dynamics. Conformal spinor dynamics would be associative if the spinor modes are localized at string world sheets with vanishing W (and maybe also Z) fields guaranteeing well-definedness of em charge and carrying canonical momentum currents parallel to them. It is not quite clear whether string world sheets are present also inside wormhole contacts: for CP2 type vacuum extremals the Dirac equation would give only right-handed neutrino as a solution (could they give rise to N=2 SUSY?).
Associativity does not favor fermionic modes in the interior of space-time surface unless they represent right-handed neutrinos for which mixing with left-handed neutrinos does not occur: hence the idea about interior modes of fermions as giving rise to SUSY is dead whereas the original idea about partonic oscillator operator algebra as SUSY algebra is well and alive. Evolution can be seen as a generation of gravitationally bound states of increasing size demanding the gradual increase of h_eff implying generation of quantum coherence even in astrophysical scales.
The construction of preferred extremals would realize strong form of holography. By conformal symmetry the effective metric at string world sheet could be conformally equivalent with the induced metric at string world sheets. Dynamical string tension would be proportional to hbar/heff2 due to the proportionality αK∝ 1/heff and predict correctly the size scales of gravitationally bound states for hgr=heff=GMm/v0. Gravitational constant would be a prediction of the theory and be expressible in terms of αK and R2 and hbareff (G∝ R2/gK2).
In fact, all bound states - elementary particles as pairs of wormhole contacts, hadronic strings, nuclei, molecules, etc. - are described in the same manner quantum mechanically. This is of course nothing new since magnetic flux tubes associated with the strings provide a universal model for interactions in TGD Universe. This also conforms with the TGD counterpart of AdS/CFT duality.
See the chapter Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds" of "Physics as infinite-dimensional geometry".
What about this? http://www.sciencealert.com/new-type-of-chemical-bond-discovered
The description of chemical bonds using magnetic flux tubes or associated strings (by generalisation of AdS/CFT duality about which I talk also as quantum classical correspondence) applies to all kinds of bondings - say bonds between nucleons in nuclear string model or colour magnetic bonds between quarks inside hadrons.
It is not possible to say anything about so specific as a particular kind of chemical bond: it is of course quite possible that also vibrating bonds are possible.
What is marvellous that by fractality the same extremely general description applies in all length scales.
Second big step forward is that h_eff is not a firm prediction of TGD and without it there is no hope of describing bound states: be they gravitational, chemical, hadronic, or nuclear. h_eff hierarchy is not only possible, it is necessary.
It is not an accident that bound states have been the fundamental weakness of quantum field theories: h_eff hierarchy makes possible to apply perturbative approach but for h_eff rather than h.
Depending on one's tastes one can interpreth_eff =n*h as effective Planck constant or as genuine.
I think this is a typo:
Second big step forward is that h_eff is not a firm prediction of TGD and without it there is no hope of describing bound states...
the first not shall not be there?
Yes, modern text editing programs know often think they knowsbetter. My intention was "is now" but editor decided "is not".
What happens with hbar and cosmological constant in an expanding universe? I know that c.c. is not in TGD, but as mainstream thinkers this lead to the incredible 'error' in the models, interpreted as dark energy? Not even Plancks constant can be constant any more, because it depends on relations?
The hierarchy of Planck constants means a hierarchy of bound states of increasing size. The longer the fermionic string/magnetic flux tube responsible for the bound state, the larger the value of h_eff.
The generation of bound states (gravitational and also other kinds) of increasing size is due to cosmic expansion, which at space-time level is due to the zero energy ontology involving causal diamonds whith are analogous to Big Bang followed by Big Crunch.
CDs themselves increase in size during the sequence of state function reductions to fixed boundary of CD as h_eff increases.
What is somewhat unexpected but understandable in terms of NMP is that the cosmic expansion automatically supports evolution as increase of h_eff and formation of more complex and bigger bound states.
What looks also strange that at every step in which h_eff increases, conformal symmetry breaks down and conformal some gauge degrees of freedom transform to physical ones. One goes off from quantum criticality. This is a spontaneous process, andmeans evolution.
Originally I thought that evolution would mean becoming critical. The solution of paradox is that thermodynamical criticality (second law) is not same as quantum criticality (NMP implying second law at the level of ensembles with entropy meaning different thing). Approaching thermodynamical criticality could be going off from quantum criticality. Complicated!
To Ulla: I have pondered cosmological constant a lot.
In TGD it does not appear at fundamental level but
in GRT description it is the only manner to describe the accelerated expansion. The GRT limit of TGD should give cosmological constant as a phenomenological parameter.
At more fundamental level the magnetic energy density of magnetic flux tubes carrying monopole fluxes and magnetic tension as negative pressure give rise to a description of dark energy. I am waiting for years that some colleague would finally decide to discover this explanation of dark energy;-).
Can it be that the existence of an ad hoc cosmological constant hide away the explanation for dark energy?
Cosmological constant term in Einstein's equations would modify them and actually mean that there would be no real vacuum energy density. This would save from negative pressure term.
Its introduction can of course can hide the source of dark energy.
Second interpretation is that Einstein equations in original form are true but the energy momentum tensor of matter contains term which gives rise to dark energy and negative "pressure". The latter interpretation is more flexible and for instance allows time dependent cosmological constant.
The resulting equations are same so that distinguishing between option is not easy. Vacuum energy density can be however modelled and cosmological constant corresponds to only particular density.
I believe that Einstein's equations- with or without genuine cosmological constant - must come from a "microscopic" theory (strange attribute in cosmology;-)) as also standard model. String models was one attempt in this direction but the sign of Lambda was wrong as many other things.
May I support the combination of a microscopic theory with the SM.
However IMHO GR will face a hard time without a strong equivalence principle and other stuff.
I think that the basic problem is that the followers of Einstein refuse to see the problems of GR. Loss of Poincare invariance- even the notion of broken invariance is broken in strict mathematical sense.
The idea about extending Poincare transformations to General Coordinate Invariance. which define gauge symmetry is simply wrong. Noether of course lurks background.
Equivalence Principle in strong sense - I understand it as the identity of gravitational and inertial masses and possibility to talk about them separately - is second key problem.
This may sound academic talk in ears of those who have not tried to build a unified theory extending GRT. When one does it, one learns how essential the understanding of these problems is and how big holes in the understanding actually exist and suggest that something is badly wrong in GRT.
A lot of sloppy thinking is allowed since one can always refer to Einstein or some other big name without arguing on basis of content.
"Square of CP2 radius fixed by quantum criticality." brings to mind the basic notion of quadrance that substitutes notion of length in purely algebraic rational geometry. The more general philosophical implication is that (dimensional) mathematical objecs cannot and not should be viewed atomistically, separate from the more higher dimensional observation space of a dimensional object, perhaps as simply as +1 dimension for observation space for any dimensional object. Which of course would also imply that the mathematical "imaginary" observation space of 8D imbedding space would be also have higher number of dimensions . The deep relation of imaginary number and +1 dimensionality is also intriguing. So no surprise that QM cannot do without notion of n-dimentional Hilbert space, and your suggestion about Hilbertian number theory. What is not at all clear is the meaning - theoretically contextualized and/or generalized (scale invariance?) - of "quantum criticality".
To make the main point as clearly as I can, all mathematical objects are "internal" objects by definition, inside Platonia of mathematical imagination aka "God's Eye". Hence, any mathematical object can be meaningful and fully comprehended only from the imaginary higher dimension of the object. The dynamics of combining empirical "here-and-now" participatory nature of "material" experiencing and holistic God's Eye of mathematical imagination in communicable way is the main inspiration and challenge of TGD, from what I have gathered so far. If fully accepted and rigidly applied, this approach has of course dynamic implications for any general measurement theory.
PS: the statement that "thermodynamics is the square root of general measurement theory" is of course fully expected and now self-evident implication of generalized notion of +1D "quadrancy" requirement to be able to view any and all mathematical objects. Kiitos ja anteeksi <3
To Anonymous: "Quantum theory is the square root of thermodynamics" is what I meant. This is not at all self-evident to me and implies generalisation of the notion of S-matrix replacing it with a collection of M-matrices between positive and negative energy parts of zero energy states. They can be seen "complex" square roots of densities matrices (real diagonal square root of density matrix multiplied by unitary S-matrix). M-matrices would organise into unitary U-matrix between zero energy states.
To Anomymous: "Square of CP2 radius fixed by quantum criticality". I am at all not sure whether I have said this;-). I talk always about ratios only: it is in the spine of every theoretical physicist! The values of dimensional quantities depend on units chosen! I might have said that the ratio of hbarG/R^2 which is dimensionless is be fixed by quantum criticality;-).
Imbedding space could indeed have infinite-dimensionality in number theoretic sense: these dimensions would be totally outside of physical measurement since they would correspond to number theoretical anatomy of numbers due to existence of infinite number of real units realised as ratios of infinite primes (and integers). As real numbers all these copies of real number would be identical.
"Seeing from higher dimension" certainly gives information about geometric object.But it also creates information. Circle is just circle. When its imbedded to 3-D space it can be knotted in arbitrarily complex manner. Same about space-times as surfaces: without imbedding one would have no standard model interactions, just gravitation! Individual without social environment is much less than individual with it!
"A circle is a circle is a circle" approach presupposes observation independent objectivist metaphysics, which any theory of consciousness can't do, while asking how - where and when etc - thought-observation events of mathematical objects happen. Physicists have age old bad habit of thinking "objectively", what interests in TGD is the challenge to consistently integrate consciousness at all levels of theory-forming. That means also letting go of the role of passive observation (in Greek: theory) and accepting dynamic participation in the drama (Greek: fronesis, sometimes translated as 'practical reason')
Searching for 'quantum criticality (QC)' I found this explanation: http://www.physics.rutgers.edu/qcritical/frontier3.htm
The notion of 'emergent matter' from QC superposition strongly suggests that dynamic participatory consciousness theory is involved in these observation events, in the form of mathematical imagination. But maybe you could expand on what you mean by 'quantum criticality', are you thinking about some kind of generalization that can be communicated, fits with other pieces of puzzle?
To Anonymous: Seeing observer as active part of the physical system rather than objective outsider is indeed the whole point of TGD view. I have worked hardly to understand what I might possibly mean with criticality;-). I find it sometimes difficult to understand what I am saying;-).
Criticality is intuitively clear notion: you are at saddle or top of the hill where big things can happen with very small perturbation. Returning back to criticality is what biosystems are able to achieve: homeostasis might be seen as this ability to not fall down where dead system would fall immediately.
My recent big surprise was that the increase Planck constant seems to happen spontaneously and generates potentially conscious information. Living systems are able to return to smaller h_eff: for instance, magnetic flux tubes connecting reacting biomolecules shorten because of reduction of h_eff and make possible reaction products to find each other. Maybe getting rid of Karma's cycle means letting go and allow the h_eff to increase;-).
"Top of the hill" metaphor associates with 'Law of diminishing returns', well known for gardeners, and ideal state of growth factors. In that sense quantum jump to higher degree of planck scale makes... sense ;).
"Simplest" or deepest mathematical generalization I can think of in this regard is the extra dimensional curve route from 1 to -1 in Euler's Identity, and imaginary-axis (rational axis = 0) of complex plane (by which I mean gaussian rationals, as I'm highly skeptical of "real" numbers). The poetic "coincidence" of square root of -1 being called "imaginary" is remarkable, letting go of karmic rational (or "real", if you insist) valuation and letting imagination flow freely.
I just found Weyl's little book on Symmetry, which begins with reference to argument about philosophical or theological debate about left and right between Leibnitz (relativism) and Descarters (reifying objectivism). What kind of symmetry break, if any kind, is the "identity" of square root 1, and extra dimensionality of square root of -1?
Re Einstein's followers, GRT and relativistic field theory(?), could this article on Noether's second(!) theorem and Weyl bear helpful meaning? http://www3.nd.edu/~kbrading/Research/WhichSymmetryStudiesJuly01.pdf
To believe on continuous number fields (both reals and various p-adic number fields and the various algebraic extensions of the latter) is to me like believing in God.
There is no way to prove that transcendentals exist (the term is certainly not an accent!) but without the assumption about their existence physics would become an aesthetic nightmare unless you happen to be a practical person loving to do numerics.
Imaginary unit is a represent ion of reflection or rotation by pi, as you wish. In Kahler geometry its action as purely algebraic entity is geometrized/ tensorized so that it does not look anymore so mystic.
I DO believe I am a God :). Continuity of rational number field is not only easy to prove, it is intuitively clear. The ability to believe otherwise is proof of limitless ability of self-deception. The idea that irrationals are "gaps" in relation to perfectly dense rational line is based on false expectation that irrationals can fit 1-dimensional line like natural numbers and rationals. A well known physics blogger once imagined a Hilbertian or n-dimensional number theory space where also irrationals find their natural and exact place.
Again, let's remember that the ability to THINK about observables like continuous line requires +1-dimensional "observation space" aka "mind", and let not get confused with definition before we know what we are doing when we think math.
Transcendental metanumbers are also very much loved and admired as very special (meta)rationals. Thanks for the Kähler hint for generalization of i.
I'm sure we can agree that thinking is a kind of fluid. Here's excellent clip on lagrangian vs. eulerian approaches to fluids like thinking etc: https://www.youtube.com/watch?v=zUaD-GMARrA
Lagrangian might get the girl, or punished for obsessive stalking, but eulerian has better change of gnothi seauton.
Remember Pythagoras Theorem and how it gave birth to relations that are not ratios of whole numbers, and do not fit the 1D-continuum of rational number _line_? Pythagoras theorem does not say anything about 1D-lengths (they don't exist for it) but 2D _areas_: quadrance1 + quadrance1 = quadrance3 (diagonal of isosceles square).
The claim that rational _line_ is not continuous is derived from fundamental category error of claiming that SQR2 is a "gap" missing from rational number line, even though there is nothing that can be reduced to 1D- length about fundamentally 2D object like diagonal of square. The information content of inherently 2D area cannot be contained by 1D number line, if we hold to the ideals of rigour and honesty.
I remember that we debated about this and I can only say that I disagree. Rationals and their algebraic extensions emerge at the level of cognition and measurement where one always has finite measurement/cognitive resolution. The intersection of cognitive and sensory worlds. At the level of geometry describing perceived world continuous number fields are indispensable for practical description and I think they really exist.
It's good to disagree! Notions of 'continuity', continuum' and 'field' and even 'number' are not that clear and require careful philosophical thinking. When we talk about "continuous number fields" of classical physics, are we talking about 'numbers' or just practical approximations (intervals) in some limited form of (all possible) cauchy-sequenses? And how do these approximate intervals of classical fields relate to the operators of quantum fields? Also, these approximate intervals of cauchy sequenses have a strong flavor of 'ontological uncertainty' about them. Or do you mean by "continuous number field" something else than 'all possible cauchy sequenses'?
PS: at the end of this lecture Norman comes close to an idea that is close to TDG: (classical) physical finite fields based on some high prime: https://www.youtube.com/watch?v=Y3-wqjV6z5E&list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf&index=24
I see the entire spectrum of number fields from finite fields serving as convenient approximation tools to rationals and their algebraic extensions, to reals, p-adic numbers, complex number, quaternions, and octonions as physically very natural structure of structure. To cut off everything after finite fields would destroy the whole TGD so that I am a little bit worried about such attempts;-).
There is a lot to destroy by keeping only finite fields. One can go even beyond reals as they are defined. One an introduce an infinite hierarchy of infinite primes/integers/rationals and form a hierarchy of ratios of these number behaving like real units. This implies that every point of real axis becomes infinite-D space which could represent entire Universe: algebraic holography or Brahman= Atman mathematically.
This all is of course something totally outside sensory perception and simple book keeping mathematics. What is however amazing that the hierarchy in question has very "physical" structure: supersymmetric arithmetic quantum theory second quantized again and again. Even more amazing, infinite primes analogous to bound states- something which is the Achilles heel of QFTs - emerge naturally so that infinite primes could pave the way to the construction of quantum theories and also TGD. Many-sheeted space-time with its hierarchical structure could directly correspond to it.
All this I would lose besides TGD if I would accept the view of Norman about numbers. My view is that one should not take the limitations of human thinking as criterion for what is possible mathematically. We have also the ability to imagine within confines of logic. For me this is the spiritual view as opposed to the "practical" view accepting only discrete structures, a view which -rather ironically - does not work in practice.
As pure mathematician, Norman has problem with the notion of infinite sets, and set theory as whole, and hence would not probably accept infinite primes either. Applied math of engineers can be more relaxed, but there is a category mistake if we identify approximations of applied math with rigorous algebraic definitions. As reals are defined, or rather not defined, (what do YOU mean be definition of reals?!) there is a serious logical problem of drawing ontological conclusions ("is") from epistemic limitations of applied math ("should").
Yes, we can and do postulate infinite fields of cauchy sequenses, create such imaginary number spaces with our minds (cf. "eulerian" in the video clip above), but their arithmetics cease to be algebraic as there is in some sense too _much_ information (cf. 1/3 and 0,333...), at least on the "real" side of comma. There is a kind of uncertainty principle _created_ by postulation of cauchy sequences on the "real" side, which would not necessarily emerge in purely algebraic approach.
When you say that "every point of real axis becomes infinite-D space", I'm reminded again that the space of cauchy sequenses, called "line" for no good reason, is _said_ to contain inherently multidimensional algebraic roots ("irrationals"), and mostly infinite sequenses that by some theorem are considered transcendentals. Giving up the invalid notion of category mistakes called "real _line_" as 1D-continuum does not mean abandoning the sea of cauchy sequenses and using that number space for what it is good for, it means that multidimensional algebraic relations such as SQR2 etc. are not _identified_ with its approximates in the cauchy sea. The "lagrangian" of an algebraic relation and the "eulerian" observation space of cauchy sea are not identified, but understood for their mutual purposes.
I agree with "ability to imagine within confines of logic". Notion of Real _line_ just does not fit the criteria of logic, anymore than square circle. Nothing logically solid can be constructed from an object that has not been logically derived, but by adding inherently multidimensional algebraic relations etc. to rational line, and calling the creation 'line'.
I believe you had good and solid motivations for searching "quantum math", and I hope this "disagreement" nourishes that interest. :)
Just a short comment about Cauchy sequences. Infinite primes are purely algebraic notion as also the infinitude of units obtained as ratios of infinite rationals. No limits are involved as for Cauchy sequences. In p-adic topology their norm is unity for any finite prime and also for smaller infinite primes. Infinity is is in the eye of topologist.
This notion of infinity is also different from that of Cantor which also relies on imagine infinite limits obtained by adding 1 again and again: n-->n+1. Now one has explicit formulas for infinite primes: no limiting procedures.
The notion generalises to algebraic extensions of rationals allowing also the notion of primeness.
My view is that no number field alone is enough: all are needed. Reals and all p-adic number fields are like pages of book glued along common rationals. Algebraic extensions give rise to more
pages so that book becomes rather thick. This structure is the natural starting point for physics which would describe also correlates of cognition and intention.
Yes, I also noticed infinite primes are not dependent from notion of real line, or even infinite sets of set theory, but can be classified as algebraic type.
The book metaphor for Cauchy number space brings to mind sea of white pages, where there can be found and written letters and words of symbolically meaningful interval approximations for algebraic relations. On the other hand, book metaphor does not reveal the "Russian doll" structure of Cauchy exponentiality and the zero state of Euler's Doubly Infinite Identity. Finnish expression for the essence of exponent 'kertoa itsellään' can be translated both as 'multiply by itself' and 'narrate by itself'. :)
Intuitively the exponential sea of all possible Cauchy sequenses has fractal dimensional structure, but my technical skills are not enough to tell if it can be assigned exact Hausdorff dimension value and if so, what. (http://en.wikipedia.org/wiki/Hausdorff_dimension)
As far as real numbers are considered they are equivalence classes of Cauchy sequences converging to the real number. All information about individual Cauchy sequences disappears in the definition of real number.
One can also consider the sequence of differences of two subsequent points in the sequence and the sequences could be regarded as a real space in obvious inner product if one allows arbitrarily large differences and all kinds of wanderings before the converge to the real number.
I assume in the above by "(definition of) real number" you mean: irrational number. :)
Cauchy sequenses of just negative exponents is metric space because "the continued fraction expansion of an irrational number defines a homeomorphism from the space of irrationals to the space of all sequences of positive integers". So as model for 'continuous field' Cauchy sequenses of negative powers is not more interesting than field of natural numbers.
As you know, the whole of Cauchy space that includes also p-adic positive exponents is much more interesting, and here's some interesting discussion on Euler's doubly infinite identity:
The second answer relating the identity to Sobolev-like space with Hilbert-space norm goes over my head, but hopefully not yours.
I meant all reals. One can of course have also Cauchy sequences which converge to rational.
What to my opinion matters is the convergence point of CSs. Not sequences as such: one identifies reals as equivalence class of CSs.
I am not a specialist in technical details of functional analysis, Sobolev, etc… I am more interested in the notion number itself from the point of view of physics and cognition. The technique of rigorization (I cannot avoid association with rigor mortis;-) is the task of mathematicians.
I did not understand your argument about CSs of negative powers. They are very special case about CS and the limit of power is usually 0, 1, or infinity depending on whether initial point is smaller, equal or larger than 1. For negative x one has limits 0,-infty or cycle hopping between -1 and +1.
Euler's double identify does not make sense to me except formally and I do not see it as a gateway to new math. Either (or both) of the two series involved does not converge for reals. It fails to converge also for p-adics. If x has norm smaller than 1 then 1/x has norm larger than one. If norm is 1 for both, then both series fail to converge.
So sorry, don't know why I thought of and called nominator exponents for real side decimal extensions as "negative". Probably because of I was also thinking of some kind of zero state ontology relation of Euler's doubly infinite identity (EDII), and I associated "determinant" p-adic side as 'positive' and "nominator" real side as 'negative'.
I believe that in Euler's mind EDII is/was intimately tied with his identity and proof concerning pi, e, i and 1 and 0. I inteprete EDII as number theoretical presentation of the maxim "as above, so below", and in that sense metamathematical equation of "if this arises, that arises". Or, inner spaces of nominator power series dividing one and "outer" inclusive spaces of determinant power series. This Russian doll relation between p-adic and real feels to me more significant than book metaphor.
As said, the (hindu-arabic) number fields for decimal (or any base of natural number) extensions for reals and also p-adics are homomorphic with field of all natural numbers, and the differences or meaningful relations are said to different notions of "distance". This is how - the space in which - we have been taught to think about numbers.
The notion of distance, is however, very problematic and baffles me, as it is inherently tied with notion of 'length', and as long as we do and think math inside some dimensional(orthogonal) coordinate system, notion of length is valid only in context of 1D-line, notion of area is valid only in context of 2D-plane, notion of volume is valid only in context of 3D-space, etc. This level or rigor (...mortis ;D) is IMHO highly non-trivial in order not to commit to category errors at most basic level, when questioning cognition and numbers in philosophically sincere way.
As dimensionality is inherent already at the level of exponential multiplying or dividing a number by itself, the 1D-notion of "distance" giving different meanings to (natural, real, p-adic etc.) homomorphic number fields, which we think of as 2D planes, needs to be reconsidered. I believe Norman is on the right track with quadrance as the most basic level of mathematical aka coordinated measuring for observers like us, and with the algebraic generalizations of quadrance into higher dimensions. This would mean in some sense letting go of karmic ties (investing more and more energy to self-deception) to what we have been authoritatively taught to believe and letting heff jump to a higher level of number theoretical self-comprehension. Greek 'a-letheia' means letting go of active ignorance of feeding energy to attachments and letting world come true by itself... in the baby steps of Tao... ;)
One way to think of EDII is that both sides converge to half of negative one. The symmetry of multiplication and division is deeper than this or that entangled property of this or that number system.
The simplest level of number theory is base one, and generating natural numbers in base one does not differentiate between multiplication and division, there is no either-or choice between inclusive whole 1 that gets divided and atomistic 1 that gets multiplied. Both movements generate identically endless strings of 1's, the platonic 'hen kai agathon'. This unity gets hidden and forgotten when we invent no-one (aka "zero") and start playing with base 2 and must invent rules to place 1 and 0 in a partially coherent way (e.g. boolean rules, peano axioms, etc.), and are faced with constant barrage of number theoretical choices similar to and including "is it a wave or particle", "which property of particle is entangled in this Bell measurement", etc. In the entangled state of base '0, 1' aka 2-dimensional base the choice of not choosing is no longer present.
This is what the that of EDII brought me now to contemplate, and you may consider this an empirical example of an untrained mathematical cognition having an observation event, an 'anamnesis'. What I really can't tell now if this was a movement of heff in the context or limit of base '1, 0', and if so, from 1 to 0 or from 0 to 1... :)
Farey sequences / series , has relations to Riemann hypothesis too
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