As one tries to develop a new mathematical notion and interpret it, one tends to forget the motivations for the notion. It is however extremely important to remember why the new notion is needed.
- In the case of quantum arithmetics Shnoll effect is one excellent experimental motivation. The understanding of canonical identification and realization of number theoretical universality are also good motivations coming already from p-adic mass calculations. A further motivation comes from a need to solve a mathematical problem: canonical identification for ordinary p-adic numbers does not commute with symmetries.
- There are also good e motivations for p-adic numbers? p-Adic numbers and quantum phases can be assigned to finite measurement resolution in length measurement and in angle measurement. This with a good reason since finite measurement resolution means the loss of ordering of points of real axis in short scales and this is certainly one outcome of a finite measurement resolution. This is also assumed to relate to the fact that cognition organizes the world to objects defined by clumps of matter and with the lumps ordering of points does not matter.
- Why quantum deformations of partonic 2-surfaces (or more ambitiously: space-time surfaces) would be needed? Could they represent convenient representatives for partonic 2-surfaces (space-time surfaces) within finite measurement resolution?
- If this is accepted there is not compelling need to assume that this kind of space-time surfaces are preferred extremals of Kähler action.
- The notion of quantum arithmetics and the interpretation of p-adic topology in terms of finite measurement resolution however suggest that they might obey field equations in preferred coordinates but not in the real differentiable structure but in what might be called quantum p-adic differentiable structure associated with prime p.
- Canonical identification would map these quantum p-adic partonic (space-time surfaces) to their real counterparts in a unique a continuous manner and the image would be real space-time surface in finite measurement resolution. It would be continuous but not differentiable and would not of course satisfy field equations for Kähler action anymore. What is nice is that the inverse of the canonical identification which is two-valued for finite number of pinary digits would not be needed in the correspondence.
- This description might be relevant also to quantum field theories (QFTs). One usually assumes that minima obey partial differential equations although the local interactions in QFTs are highly singular so that the quantum average field configuration might not even possess differentiable structure in the ordinary sense! Therefore quantum p-adicity might be more appropriate for the minima of effective action.
The conclusion would be that commutative quantum deformations of space-time surfaces indeed have a useful function in TGD Universe.
- If this is accepted there is not compelling need to assume that this kind of space-time surfaces are preferred extremals of Kähler action.
Consider now in more detail the identification of the quantum deformations of space-time surfaces.
- Rationals are in the intersection of real and p-adic number fields and the representation of numbers as rationals r=m/n is the essence of quantum arithmetics. This means that m and n are expanded to series in powers of p and coefficients of the powers of p which are smaller than p are replaced by the quantum counterparts. They are quantum quantum counterparts of integers smaller than p. This restriction is essential for the uniqueness of the map assigning to a give rational quantum rationals.
- One must get also quantum p-adics and the idea is simple: if the pinary expansions of m and n in positive powers of p are allowed o become infinite, one obtains a continuum very much analogous to that of ordinary p-adic integers with exactly the same arithmetics. This continuum can be mapped to reals by canonical identification. The possibility to work with numbers which are formally rationals is utmost importance for achieving the correct map to reals. It is possible to use the counterparts of ordinary pinary expansions in p-adic arithmetics.
- One can defined quantum p-adic derivatives and the rules are familiar to anyone. Quantum p-adic variants of field equations for Kähler action make sense.
- One can take a solution of p-adic field equations and by the commutativity of the map r=m/n→ rq=mq/nq and of arithmetic operations replace p-adic rationals with their quantum counterparts in the expressions of quantum p-adic imbedding space coordinates hk in terms of space-time coordinates xα.
- After this one can map the quantum p-adic surface to a continuous real surface by using the replacement p→ 1/p for every quantum rational. This space-time surface does not anymore satisfy the field equations since canonical identification is not even differentiable. This surface - or rather its quantum p-adic pre-image - would represent a space-time surface within measurement resolution. One can however map the induced metric and induced gauge fields to their real counterparts using canonical identification to get something which is continuous but non-differentiable.
- One can take a solution of p-adic field equations and by the commutativity of the map r=m/n→ rq=mq/nq and of arithmetic operations replace p-adic rationals with their quantum counterparts in the expressions of quantum p-adic imbedding space coordinates hk in terms of space-time coordinates xα.
- This construction works nicely if in the preferred coordinates for imbedding space and partonic (space-time) surface itself the imbedding space coordinates are rational functions of space-time coordinates with rational coefficients of polynomials (also Taylor and Laurent series with rational coefficients could be considered as limits). This kind of assumption is very restrictive but in accordance with the fact that the measurement resolution is finite and that the representative for the space-time surface in finite measurement resolution is to some extent a convention. The use of rational coefficients for the polynomials involved implies that for polynomials of finite degree WCW reduces to a discrete set so that finite measurement resolution has been indeed realized quite concretely!
Consider now how the notion of finite measurement resolution allows to circumvent the objections against the construction.
- Manifest GCI is lost because the expression for space-time coordinates as quantum rationals is not general coordinate invariant notion unless one restricts the consideration to rational maps and because the real counterpart of the quantum p-adic space-time surface depends on the choice of coordinates. The condition that the space-time surface is represented in terms of rational functions is a strong constraint but not enough to fix the choice of coordinates. Rational maps of both imbedding space and space-time produce new coordinates similar to these provided the coefficients are rational.
- Different choices for imbedding space and space-time surface lead to different quantum p-adic space-time surface and its real counterpart. This is an outcome of finite measurement resolution. Since one cannot order the space-time points below the measurement resolution, one cannot fix uniquely the space-time surface nor uniquely fix the coordinates used. This implies the loss of manifest general coordinate invariance and also the non-uniqueness of quantum real space-time surface. The choice of coordinates is analogous to gauge choice and quantum real space-time surface preserves the information about the gauge.
18 comments:
Dear Matti,
So thanks for last reply, for understanding more about definition of index M : N, I read pieces of some books about subfactors and at end I am perplexed :(.
I am thankful if it is possible for you to guide me about some questions:
at
M:N =dimN(L2(M)) = TrŃ (idL2(M)),
I know trace of an operator A as Tr(A), but what does it means when we add a space(or an algebra)B to corner of it as TrB(A) ? and a similar question for dimB(A) ?
Why does TrŃ (idL2(M)) related to dimension of that space?
Dear Hamed,
my mathematical understanding of inclusions is rudimentary. I do not understand how Jones arrives his result about spectrum of the dimensions for inclusion. I just accept the results a a pragmatic physicists: what else can I do;-)? It would be a good challenge to translate the results of Jones to language that simple physicists can understand.
I cannot guarantee whether I have understood correctly your questions. To do so I would need links to the texts where these notations appear.
One should learn what various notations and definitions mean. I collected some examples to refresh my own memories. My understanding is not of practicing mathematician but of physicist who tries to understand these notions physically. My guiding line is the notion of finite measurement resolution and somehow loose notion of "factor" space M/N with fractional dimension.
1. What does one mean with factor?
The commutator algebra N' of algebra N appears in the definition of factors. By definition factors are such that N''=N holds true. This means that the only operator of N commuting with all operators of N is unit operator in B(H). This is known as irreducibility in representation theory. For instance, finite-D sub-algebras N of entire B(H) fail to satisfy this condition since the commmutator of sub-algebra is infinite-D and its commutator contains also the projector to it besides N.
2. What does one mean with factor of type I and II?
Algebra can be represented in terms of its projectors and in quantum theory one typically decomposes the operators representing observables to direct sums of projectors multiplied by eigen values. Factor of type I has minimal projectors- projectors into 1-D sub-spaces. Spin eigenstate would correspond to minimal projector.
Factors of type II_1 have no minimal projectors- all projectors are infinite-dimensional in algebraic sense. The projectors finite-D with dimension not larger than 1 when one uses TR instead of Tr to define the dimension of projector. This is definitely something new. State function reduction would always give projection to infinite-D subspace of state space rather than 1-D ray. The interpretation would be in terms of finite measurement resolution. The complex ray would be replaced with N-ray and one would have non-commutative state space.
3. What does one mean with L^2(M)?
L^2(M) is Hilbert space in which M acts as sub algebra of bounded operators of H acts. I identify L^2(M) as the space of elements x which are expressionble as sum of projectors of M in such a manner that xx^dagger has finite trace so that they have also interpretation as elements of M.
See pages 10 and 42 of http://tgdtheory.com/public_html/pdfpool/vNeumann.pdf .
4. What Tr_B(A) means? I would guess that it means dimension of the space formed by B-orbits (B-rays) in algebra A.
|M_N|= dim_N(L^2M) would be dimension for the space of N-orbits of L^2M. I have talked loosely about N-orbits of elements of M.
To be continued...
More FAQs.
5. What does one mean with inclusion?
How inclusion differs from homomorphism respecting product and sum of operators? The difference is due to the presence of beta= |M:N| as normalization factor in the inclusion x--(x,x)/beta so that the product of images is by a factor 1/beta smaller than the image of product. In standard representation for the inclusion sequence on starts from tensor powers of algebra which could correspond to Clifford algebra for infinite-dimensional space representable as infinite tensor power for Clifford algebra for n-D space. Note that factors of type II are algebraic fractals in very precise sense.
Concrete representation: One has N=nxn matrices, M= n^2xn^2 matrices, M_1= n^4xn^4 matrices and so on...
nxn matrices x that is N are included to M as
diag(x,x)/beta.
n=2 gives the simplest represention. In this case one starts from 2-x2 matrices.
This formula is a good manner to memorize what one means with inclusion. The inclusion is homomorphism for beta=1 that is for the minimum value n=3 of n. For n--->infty the normalization factor approaches 1/beta=1/4.
In TGD framework one can identify tensor product as tensor product of finite-dimensional fermionic oscillator operator algebras.
6. What does one mean with (quantum) trace?
I denote the definition of trace implying unit trace for unit matrix by TR and ordinary trace by Tr. TR(Id)=1 for M is achieved by defining the trace as
TR(x)==Tr(x)/Tr(Id).
For P_N interpreted as element of N included in M in the above discussed manner, one obtains TR(P)= 1/beta.
From this one can define the effective dimension of M/N to be beta =|M:N| by using the fact that trace for effective tensor product M=Nx M/N is product of traces and Tr(Id_M)=1.
7. What does one mean with Trace in commutator algebra N' (..)?
N' is commutator algebra of N and M can be expressed as product of N and N'. The trace for unit operator of M equals to 1 and is product of TR(P_N) and TR(P_N') and this fixes TR(P_N') to be |M:N|. I interpret you question as question about where this formula comes from.
%%%%%%%%%%%%%%%%%% '
The text in the following pages of http://tgdtheory.com/public_html/pdfpool/vNeumann.pdf might help (or confuse even more;-)).
*"Basic definitions": page 10
*"Basic results about sub-factors ": page 16
*"The fundamental construction and Temperley Lieb algebras": page 22.
*Basic definitions about factors can be found at page 42. "Basic facts about factors".
I find "Ultimate L interesting".
http://www.newscientist.com/article/mg21128231.400-ultimate-logic-to-infinity-and-beyond.html
Gary Ehlenberger
Now studying Woodin's papers.
Dear Gary,
nice to hear about you. I read also this article and wrote even a new chapted to "Physics as Generalized Number Theory" comparing different views about infinity.
http://tgdtheory.com/public_html/tgdnumber/tgdnumber.html#infsur
Dear Matti,
Thank you very much. :)
I listed some questions. Don’t rush to answer (in particular, third), if they took a long time for you. So Thanks.
1-Which is correct? I didn’t understand it from your answer of my question about entropy:
“The dimension of M/N is equal with index of inclusion which for Jones inclusions is algebraic number in the range 1 to 4” or “dimension of M/N is the inverse of index of the inclusion in the range 1 to 1/4 ”
Then dim(M)=dim(M/N)+dim(N)?(it is interesting for me that dimension of M and N are infinite and dimension of M/N is finite!)
2- “At The topological condensation of space-time sheet to a larger space-time sheet, N would correspond to the condensing space-time sheet and M to the system consisting of both space-time sheets”
Is it correctly? : N is a von Neumann algebra and generated by all observables (operators) which can be measured within a condensing space-time sheet and in a similar way for M?or operators of M/N are observables which can be measured within a condensing space-time sheet?
If it is not correct, then what do operators of N and M or M/N characterize?
3- Can one distinguish between three kinds of infinite dimensional space? :
1-State space (or the "associated Hilbert space" of the system) in standard quantum mechanics and in TGD correspond to infinite dimensional space of 3-surfaces of M4*cp2 (?) (vectors of Hilbert space is correspond to the 3-sufaces) and
psi function is superposition of eigen states but psi function replaced with M matrix in TGD then M matrix is superposition of 3-sufaces! Certainly not!??
2-In QFT we can correspond to each vector of the state space, Creation and annihilation operators that create or annihilate them from or to vacuum. Space of creation and annihilation operators, for a fermion field, is infinite-dimensional. In TGD it is infinite-dimensional Clifford algebra.
At “The space M/N would correspond to the operators creating physical states modulo measurement resolution” your purpose of the operators creating physical states is like creation operators in QFT?
3- In standard quantum mechanics, one can correspond to each observable an operator and space of all operators are infinite dimensional. What is correspondence of them in TGD?
How are the three kinds of infinite dimensional space very entangling together in TGD? In other words what is relation between them?
Thank you for excellent questions. I try to answer first to the first two questions.
1-Which is correct? I didn’t understand it from your answer of my question about entropy: “The dimension of M/N is equal with index of inclusion which for Jones inclusions is algebraic number in the range 1 to 4” or “dimension of M/N is the inverse of index of the inclusion in the range 1 to 1/4 ”
Then dim(M)=dim(M/N)+dim(N)?(it is interesting for me that dimension of M and N are infinite and dimension of M/N is finite!)
[MP] I am Terribly sorry. I see that there is a stupid typo!!
dim(M)=dim(M/N)*dim(N)!!!
This is the correct formula. + appears for direct sum, not tensor product (obtained by replacing matrix elements with matrices).
This is a formal identification of dimension based on the idea that for ordinary tensor product dimension is product of dimensions and one can generalize definition of dimension by defining it as trace replaced with "quantum trace" which equals ti 1 for M! M is in this sense 1-dimensional and N has dimension smaller than 1!!
**************************************************
2- “At The topological condensation of space-time sheet to a larger space-time sheet, N would correspond to the condensing space-time sheet and M to the system consisting of both space-time sheets”
Is it correctly? : N is a von Neumann algebra and generated by all observables (operators) which can be measured within a condensing space-time sheet and in a similar way for M?or operators of M/N are observables which can be measured within a condensing space-time sheet?
If it is not correct, then what do operators of N and M or M/N characterize?
*This is one possible interpretation for measurement resolution. Note that I do not mention anything about ZEO so that it must be very early interpretation. The idea is that the condensed space-time sheet is regarded as a point like particle so that all its degrees of freedom relating to its shape are below measurement resolution. M would correspond to degrees of freedom associated with both space-time sheets I will also discuss different interpretation terms of CDs and sub-CDs.
Dear Hamed,
your third question consists of three parts and I organize my answer in the same way.
3- Can one distinguish between three kinds of infinite dimensional space?
1- State space (or the "associated Hilbert space" of the system) in standard quantum mechanics and in TGD correspond to infinite dimensional space of 3-surfaces of M4*cp2 (?) (vectors of Hilbert space is correspond to the 3-sufaces) and psi function is superposition of eigen states but psi function replaced with M matrix in TGD then M matrix is superposition of 3-sufaces! Certainly not!??
[MP] As mathematical objects Hilbert space is just Hilbert space possibly having additional structures (separable or non-separable is one attribute). Physically Hilbert space property is not enough. For instance, one has usually preferred state basis - eigenstates of measurement observables often belonging to symmetry algebra and of Hamiltonian.
Hilbert space emerges in TGD in many manners.
a) The space of 3-surfaces is infinite-D Kahler manifoeld. As a manifold and consists of pieces which are Hilbert spaces but not in quantum mechanical sense!
This kind of local piece correspond to the tangent space at one particular point (now 3-surface) providing for WCW a local coordinatization (just like plane gives coordinatization for Earth's surface locally) is Hilbert space. WCW Kahler metric defines inner product and hermitian conjugation and can be regarded as Hilbert space. In the same spirit the tangent space of 2-D manifold is 1-D complex Hilbert space.
b) Ordinary3-D wave function can be *formally* regarded as super-positiion of points of 3-space representing completely localized states. In the same spirit M-matrix can indeed be regarded as analog of wave function in the space of 3-surfaces formed by disjoint unions of 3-surfaces defined by ends of space-time surfaces at light-like boundaries of CD.
The formal basis would consists of wave functions localized to single 3-surface of this kind and is of course quite too singular.
c) More precisely: M-matrix can be regarded as WCW spinor field with components of infinite-D WCW spinor consisting of fermionic Fock states localizable at partonic 2-surface by effective 2-dimensionality.
Again finite-D analogy helps: the analog of M-matrix would consist of products of spinor fields with arguments localized at the boundaries of CD. These would be obtained by making 3-surfaces point like so that only cm degrees of freedom would remain.
The basic rule is simple: to get from TGD to "wave mechanics" replace 3-surfaces with point-like objects and many-fermion states with components of Dirac spinor.
%%%%%%%%
Dear Hamed,
second part of your third question concerning various kinds of Hilbert spaces.
2- In QFT we can correspond to each vector of the state space, Creation and annihilation operators that create or annihilate them from or to vacuum. Space of creation and annihilation operators, for a fermion field, is infinite-dimensional. In TGD it is infinite-dimensional Clifford algebra.
“The space M/N would correspond to the operators creating physical states modulo measurement resolution” your purpose of the operators creating physical states is like creation operators in QFT?
[MP] Finite measurement resolution for distance measurement means "dropping" away all operators which create states with very high momenta.
a) One cannot detect excitations which are too "wiggly" and they are eliminated from theory and express their presence only via the dependence of effective couplings constants on the resolution space. This kind of momentum cutoff is necessary in interacting QFTs not only in order to get rid of UV divergences but also because we simply have finite measurement resolution. Already weak length scale 10-17 meters as resolution scale requires LHC!
One does not actually drop these invisible degrees of freedom away but "integrates over them" in path integral so that their actual presence is seen in the dependence of the couplings of the resulting effective QFT on cutoff scale.
Casimir effect is one example about the physical effect coming from degrees of freedom not manifestly visible (vacuum quantum fluctuations of em field).
b) In zero energy ontology all states have vanishing total quantum numbers. Uncertainty Principle suggests how one should do translate length scale cutoff to ZEO.
One can i add to zero energy state associated with given CD details: obviously they correspond to sub-CDs. Add to the state zero energy states associated with sub-CDs corresponding to various scales down to CP_2 scale. This is like adding to a field pattern wiggles in various scales. Very fractal procedure.
c) Algebraically this means that one operates with operators in N to the zero energy state. Momentum cutoff means "integrating out" over all sub-CDs below some minimal size scale. This procedure corresponds to the replacement of M with M/N and should yield in good approximation a finite-D Hilbert space but having a fractal dimension given by |M:N|. Fractal dimension would be tell about the actual presence of the otherwise invisible fluctuations. I actually discussed this adding of details procedure in twistor context for some time ago.
Dear Hamed,
below the answer to the third part of your third question.
Your question:
3- In standard quantum mechanics, one can correspond to each observable an operator and space of all operators are infinite dimensional. What is correspondence of them in TGD? How are the three kinds of infinite dimensional space very entangling together in TGD? In other words what is relation between them?
[MP] The following would be a concise answer to your question about the relation.
Spinor fields solving Dirac equation for hydrogen atom (or harmonic oscillator potential to make the analogy even better) are replaced in TGD by WCW spinor fields: also they are completely classical at WCW level. Fermionic second quantization at space-time level provides gamma matrix algebra at WCW level! M corresponds to the WCW spinor fields allowing all sub-CD:s within CD. N corresponds to WCW spinor fields with all sub-CD size up to size defining smallest sub-CD size. M/N is obtained by "integrating" over all sub-CD sizes associated with N.
I hope that the addition of details is not confusing.
a) In TGD framework the simple rules of wave mechanics fail. Canonical quantization fails completely for Kaehler action and also path integral approach. The basic reason is that canonical momentum densities and time derivatives for imbedding space coordinates are not in 1-1 correspondence. [This actually leads to the interpretation of what hierarchy of Planck constants means.]
b) The correspondence between TGD and wave mechanics comes from the analogy with harmonic oscillator in 3-D. There are three oscillator operators and all states are created by using them. In infinite-D case there is infinite number of different harmonic oscillators just as in free quantum field theory, where they are labelled by 3- momenta. They correspond to a bosonic Kac-Moody type algebras creating physical states and acting as complexified vector fields of WCW defining its isometries (partial derivatives are replaced with covariant ones). Bosonic operators generate the analogs of partial waves of ordinary spinor field in WCW orbital degrees of freedom.
c) Spinors in 3-D case are replaced with WCW spinor fields in TGD. WCW Spinor at given point of WCW (3-surface) is created by fermionic oscillator operators. WCW spinor field by acting on vacuum state by bosonic KAc-Moody type algebra generators.
d) Structure is very similar to super-string model where bosonic oscillator operators are in rough sense in 1-1 correspondence with the 10-D target space coordinates for string world sheet and fermionic oscillator with "Gamma field" which correspond to second quantized Majorana spinor in super-string models. In TGD Gamma field corresponds to WCW gamma matrices expressible in terms of fermionic oscillator operators so that fermion statistics has purely geometric interpretation.
e) This picture as mathematical justification in the theory of factors. There is a theorem in the theory of factors stating there exists vacuum -"cyclic vector"- from which a dense set of states of Hilbert space can be generated. To me this says that quantum states can be created by using oscillator operators. *Bosonic "oscillator operators would correspond to Kac-Moody type algebra generators of Super-Kac-Moody type algebra.
I had a friend tell me that we are forced (if we consider it) to define reality in terms of information flows. Is this "true"? --Stephen C.
Depends on what one means with "define". In any case, Information and consciousness are at different ontological level than matter because they are "about. Matter and quantum states are not about anything. Very simple but extremely profound difference. Noticing this in time would save us from very very many useless papers;-).
Maybe "information is about but matter not" should be brainwashed in the brains of innocent children already in elementary school;-).
.Dear Matti,
Thanks a lot. You explained all that I needed to know for the questions. But please explain more about
“M corresponds to the WCW spinor fields allowing all sub-CD:s within CD. N corresponds to WCW spinor fields with all sub-CD size up to size defining smallest sub-CD size”
And why is a given point of WCW, a 3-surface!?
Something that makes me wonder is that you explain about the creatures of the magic world of TGD as you see in front of you:-). Your intuition is very admirable.
The last night, I dreamed about infinite dimensional geometry :-), but unfortunately I don’t remember any details!
I have also problems with details;-)! Especially so when their number is infinite! Rather frustrating! What I have learned that infinity is extremely relative notion. Finite measurement resolution suddenly makes WCW discrete or even reduces it to a finite set of 3-surfaces: this is like magic.
When I refer to a fixed point of WCW- a 3-surface- I am talking about WCW spinors- not spinor fields! This differences is very important! Also in 4-D case spinors are spinor fields restricted to a given point of space-time. This all is easy to understand if one consistently uses the analogies with finite-D case.
The geometrization of quantum theory allows to use only metric structure and its "square root", which is spinor structure. This is why geometrization program is so incredibly powerful in infinite-D context where the mere existence of geometry requires infinite-D symmetries. WCW spinor fields must represent physical states. As I explained, WCW spinors are WCW spinor fields at given point of WCW which is indeed 3-surface
WCW spinor fields represent zero energy states. What does this mean? What these 3-surfaces are. One can proceed by making gradually more accurate statements
a) First approximation. Consider only single CD. 3-surfaces are defined by the ends of the space-time surface at the boundaries of CD in the first approximation. They have therefore several disjoint components corresponding to space-like three surfaces at future and past ends of CD. So that one has time-non-locality which is something new.
b) Second approximation. One must include radiative corrections to zero energy state. They correspond to quantum fluctuations. Something pops out from vacuum and disappears. Zero energy state in smaller scale associated with sub-CD. This is like adding loop corrections to Feynman diagrams down to CP_2 scale. They correspond to addition of sub-CDs and creation of zero energy states associated with it using corresponding included algebra N.
Good guess: interaction vertices in generalized Feynman diagrams are surrounded by sub-CDs.
c) This view should be of course made much more precise and one should develop correspondence between the twistorial ideas, braid diagrammatic approach to Feynman diagrams, and inclusions of HFFs. This would be the next approximation.
See this attempt in twistorial context:
http://matpitka.blogspot.com/2012/01/proposal-for-twistorial-description-of.html
Precise quantitative formulas - Feynman rules: this is what should be constructed by gradually sharpening this vision. Two decades ago I regarded this a mission impossible but now I feel much more optimistic although I am not certainly quite too non-analytic thinker to achieve it. Left-brained thinker expressing ideas as formulas would be badly needed. Right brain and visual thinking is not enough;-)!
Dear Hamed,
just a brief comment. I noticed that WCW spinors fields as outcome of square root of Kahler geometry correspond in zero energy ontology directly to the complex square root of thermodynamics.
Complex square root of density matrix is matrix version of complex square root of real number. Hermitian square root of density matrix is analogous to square root of real positive number and S matrix to a phase factor allowed.
The commutativity of S-matrix and allowed hermitian square roots of density matrices correspond to the commutativity of the modulus and phase of complex number.
The requirement that hermitian matrices commute also would allow to extend the algebra to the analog of commutative Kac Moody type algebra involving also the powers of S. This condition would mean a choice of maximal set of commuting observables.
If one defines the product of the algebra generators as commutator this assumption is not necessary. Another option which does not require commutativity would be Jordan product AB+BA (indeed hermitian for A and B hermitian).
There are different types of number fields meeting at zero (the lightlike point in the spacetime). This zero must be the same as the vacuum, or where the Planck scale is born.
Can these lightlike spinors then use all other number groups, not reals, or a blend of every possible groups? Maybe the numbers are created there?
The adiabatic charachter of this point must then be important. This is made by phase changes? they add to the fields by superimposing waves? But what creates the waves in the first place?
Sabine tried to lay the gravity as an asymmetric force here. Gravity is linked to coupling constants. Why is the asymmetry then about 4%? Gravity is much lesser. There must be something else that force to the asymmetry too. Can you comment on this?
http://esciencecommons.blogspot.com/2012/01/epigenetis-zeroes-in-on-nature-vs.html
What gives gravitational collapse? Some asymmetry with perturbation too?
http://physics.aps.org/featured-article-pdf/10.1103/PhysRevLett.108.051103
This is linked to above question. Sabine has today some thinking about quantumfoam.
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