Sunday, March 24, 2013

Axioms of set theory from TGD viewpoint

Lubos made interesting comments related to the work of set theorist set theorist Paul Cohen. Cohen proved that axiom of choice (AC) cannot be proved nor disproved from Zermelo-Fraenkel axioms for set theory. He proved same also for continuum hypothesis (CH). Checking the definitions of various notions from Wikipedia led to an intellectual adventure of Sunday morning.

Axiom of choice from physics point of view

As a physicist I tend to share the skepticism of Lubos concerning AC. Stating it more formally: given a collection of sets indexed by some index set and not containing empty set, one can select from each set single element. What AC says that Cartesian product exists as a set. For finite sets there is no problem but already for a Cartesian power of reals one has a problem. Can one really choose from each factor R single element? Physicist might regard this kind of choice as too strong an idealization since there is always a finite accuracy involved. One can fix a given point of real axis only in finite accuracy since in the real world one must always perform a cutoff on the expansion of the real number in powers of 10, 2, or prime, or any natural number larger than one..

Physicist might prefer to weaken the axiom to a choice of a finite open ball around a point. This would lead from set theory to topology. One could return to set theory by postulating discrete topology in which each point is an open set. Physicist has however motivations to believe that physical world obeys less trivial topology and metric topology with the notion of distance defining the concept of nearness is natural for physicist. One can characterize the measurement resolution in terms of the radius of the ball replacing point. In p-adic context the situation is simpler than in real context since p-adic open balls are either disjoint or nested. Also the ordering, which is un-necessary luxury below measurement resolution, is lost in p-adic context. Since we want to minimize thinking, this is is a good reason for cognition to be p-adic;-).

TGD framework the notion of finite measurement resolution (cognitive resolution, numerical accuracy) plays a key role and has very powerful implications for the proposed mathematical framework of physics. For instance, the inclusions of hyper-finite factors relating closely to quantum groups often assigned to Planck scale represent finite measurement resolution at quantum level and at space-time and imbedding space level discretization become the counterpart for the finite measurement resolution at the level of geometry.

In p-adic context total disconnectedness implies difficulties with the notion of p-adic manifold and with integration and TGD suggests a notion of p-adic manifold based on coordinate charts mapping p-adic manifold to its real counterpart so one can induce real topology and well-orderedness of reals to p-adic context. Also here finite measurement resolution plays a key role.

Infinite primes and axioms of set theory

Cardinal numbers are defined as cardinalities of sets: two sets have the same cardinality if there is a bijection between them. Ordinals in turn can be seen as ordered sets of sets: set theoretically a given ordinal is the ordered sequence of ordinals defined by the cardinalities of the sets of smaller cardinality. The successor axiom of Peano arithmetics is essential here. Finite cardinals and ordinals can be identified but the notion of infinite ordinal is more refined notion than that of cardinal: this is obvious from the fact that infinite ordinals x and x+1 are not equivalent unlike corresponding cardinals.

Power set axiom states that the set of subsets of set exists. The cardinality of the power set is larger than the cardinality of set. In the case of natural numbers the cardinality of power set would be the cardinality of continuum and CH states that there are no cardinals between cardinality of natural numbers and cardinality of continuum postulated to that for the power set of natural numbers. CH can be extended to apply to any set and its power set. Cohen suggests that CH is not true although he only proves that both CH and not-CH are consistent with ZF axioms.

The notion of infinite prime provides number theoretic notion of infinity, which does not seem to reduce to the notions of infinite ordinals and cardinals. There is infinite hierarchy of infinite primes and infinite integers have a detailed number theoretic anatomy distinguishing them from cardinals and also "ordinary" ordinals. Infinity ceases to be some kind of limit and becomes something quite concrete expressed by an explicit formula having as a basic building brick the products of all primes at levels below a given level of hierarchy expressed as formal variables.

Physically this infinite hierarchy corresponds to a repeated second quantization of an arithmetic quantum field theory. The many-particle states of previous level become elementary particle states of the new level. Simple infinite primes correspond Fock states consisting of fermions and bosons labelled by prime valued "energy". There are also not so simple infinite primes analogous to bound states, which is rather interesting from the point of view of quantum field theories. Many-sheeted space-time could serve as a natural space-time correlate for this hierarchy in the sense that even galaxy sized objects could be seen in some aspects as elementary particles and proton could be also characterized as elementary fermion at particular level of the hierarchy.

The construction of infinite primes can be easily understood in terms of repeated second quantization.

  1. The Dirac vacuum is what one starts from and is at the first level of the hierarchy identified as the product X1=∏i pi of all ordinary primes. At the level n one has objects Xk, k=1,..n with Xk defined as the product of all primes belonging to levels m<k.


  2. At the lowest level simple infinite primes are obtained by dividing X with a square free integer m and adding to the result m: X→ X/m+/- m. The physical analogy is kicking of the fermions corresponding to prime factors of m from Dirac sea to positive energy states. After this one can add bosons (pm corresponds to m particles with "energy" p). One can also multiply X/m by an arbitrary integer n having no common factors with m: its decomposition to prime powers tells the numbers of bosons in corresponding modes. It is also possible to multiply m with integer r with any integer consisting of primes dividing m: also now the powers characterize the boson numbers in various modes. By construction nX/m and rm have no common prime factors so that the infinite integer is nmX/m+/- rm represents infinite prime. Depending on the sign factor one obtains two kinds of primes and X+/- 1 represent the simplest infinite primes and obviously differ by 2.


    Side remark: One can add to/subtract from X+/- 1 any integer and obtain an integer always divisible by some prime (any prime factor in the prime decomposition of X). One obtains therefore infinitely long range of infinite integers containing no primes. Analogous theorem for finite integers states that the range defined by the numbers n! +m, 0≤ m<n+1 contains no primes.

How do infinite primes and integers relate to the axioms of set theory? I am not a set theorist and can make only some observations of a dilettante. In the following I restrict the consideration to the infinite primes at the lowest level so that they have only ordinary primes below them.

The notion of infinity emerges in two different manners in the construction of infinite primes.

  1. How the infinite size of a given infinite prime relates to the infinite cardinals and ordinals? To me the relationship is not obvious. The number theoretical anatomy of infinite primes and the hierarchy of Dirac seas Xn defined by products of primes at various levels seem to distinguish the notion of infinite natural number from infinite ordinals and cardinals. In particular, infinite prime at a given level (first level now) has infinite number predecessors at the same level. Hence bi-directional ordering seems to replace ordinary uni-directional ordering of natural numbers. Does this mean something genuinely new or can it be reduced to set theory? Certainly it is difficult to imagine a set having number of elements given by infinite prime: infinite prime is more like energy than a number of elements.

  2. What is the number of infinite primes/integers/rationals at given level of hierarchy and how it relates to the notions of infinite cardinal and ordinal? The number of infinite primes at given level depends on what one allows. Consider the construction of infinite primes at the first level. The numbers n and m could be chosen to be finite. In this case the number of simple infinite primes would be the number of finite rationals n/rm. This implies denumerably of simple infinite primes (cardinality is same as for natural numbers).

    If one allows n or m to have infinite number of factors, the choices of m correspond to the set of subsets of finite primes and the resulting set is not denumerable and larger than the set of natural numbers (by CH it would be same as the cardinality of reals). One can argue that since the states with infinite fermion or boson number have infinite energy as physical states, they should be excluded so that the simple infinite primes would form a denumerable set. This result might generalize also to not so simple infinite primes, and might hold true at all levels of the hierarchy. Finite energy would correspond to denumerability. One can of course, ask "What about the entire Universe": should it correspond to the limit when the hierarchy level approaches infinite or to infinite particle numbers.

Infinite primes relate in an interesting manner to AC and so called well-ordering theorem often thought to be one of the holy truths of set theory.
  1. Well-ordering theorem states that every set can be well-ordered. The existence of well-ordering of course depends on axiomatics. For real numbers well-ordering is natural and generalizes to Cartesian powers of reals in obvious manner. In p-adic number fields well-ordering is not possible if one stays with the restrictions of p-adic topology. As already mentioned, canonical identification map to reals can induce well-ordering of positive reals to p-adic context.

  2. To my opinion the status of AC is not a mere academic question even for physicist. Banach-Tarski paradox illustrates this. If AC is true, one can decompose sphere into parts and re-arrange them by using only rotations and translations, which are volume preserving operations, so that the volume of the resulting sphere is two times larger than the original! The operation involves a construction of non-measurable sets having no well-defined volume and an infinite number of choices. This result would suggest that set theory alone is not enough for the needs of physics. The notion of measurable set is needed.

  3. Well-ordering theorem is implied by AC. By definition set X is well-ordered by a strict total order if ever nonempty subset of X has a least element. This seems to be essential for the induction hypothesis generalized from naturals to infinite context: in induction the first step is to show the triviality of the theorem for n=0 and then show it for arbitrary n using induction hypothesis. For infinite positive integers at given level the least element does not however exist. Consider the infinite integers at the first level of the hierarchy. Infinite primes nmX/m -rm are always positive but there is no smallest infinite prime of this kind. In fact, also the integers X-n have no lower limit just as X+n do not have any upper limit. One can have a least element only by allowing a jump to a lower level, and this limit is of course 0. Unless one is ready to make this jump, well-ordering theorem is not true and therefore also AC cannot hold true. Maybe one could replace the uni-directional induction by bi-directional induction for infinite integers.


19 Comments:

At 12:24 PM, Blogger Hamed said...

Dear Matti,

Partonic surface is in the intersection of space time surface with lower or upper boundary of CD*Cp2. light like 3-surfaces are light like orbits of partonic surface and the signature of induced metric changes at light like 3-surfaces. This is identified as wormhole throat and one can continue it to another space time sheet. This continuation identified as wormhole contact. It is a 4D surface and has Euclidean signature of metric.

Some questions about light like surfaces:
Intersections of light-like 3-surfaces with delta(M4)*Cp2 is partonic surface. I can’t imagine light like 3-surface separate from delta(M4)*Cp2. In my imagination light like orbit of partonic surface is only in the direction of delta(M4)*Cp2. What is my mistake?
Light like 3-surfaces are light like random. randomness is only for microscopic objects. For a macroscopic object, light like orbit of partonic surface isn’t random?

 
At 12:44 PM, Blogger Stephen said...

I'm sure there is a joke in here somewhere :)

http://hosted.ap.org/dynamic/stories/E/EU_ODD_FINLAND_HOTEL_SLEEPER?SITE=WFAA&SECTION=DEFAULT&TEMPLATE=ENTERTAINMENT.html

 
At 11:34 PM, Anonymous matti Pitkanen said...



Dear Hamed,

thank you for interesting and challenging questions.

[Hamed] Partonic surface is in the intersection of space time surface with lower or upper boundary of CD*Cp2. light like 3-surfaces are light like orbits of partonic surface and the signature of induced metric changes at light like 3-surfaces. This is identified as wormhole throat and one can continue it to another space time sheet. This continuation identified as wormhole contact. It is a 4D surface and has Euclidean signature of metric.

[MP] This is true: I usually write partonic 2-surface to avoid mis-understanding.


[Hamed] Some questions about light like surfaces:
Intersections of light-like 3-surfaces with delta(M4)*Cp2 is partonic surface. I can’t imagine light like 3-surface separate from delta(M4)*Cp2. In my imagination light like orbit of partonic surface is only in the direction of delta(M4)*Cp2. What is my mistake?


[MP] You force the partonic 2-surface to move along delta M^4! There is no need for this!

Let us look first what you imagine. Your light-like 3-surface in direction of delta M^4 . It is obtained by putting CP_2 coordinates constant: this would be just a piece of delta M^4. When you allow CP_2 coordinates to vary you get space like 3-surfaces since you get additional space-like contribution

Delta g_alphabeta= s_klpartial_alpha s^kpartial_beta s^l

to the indued 3-metric making it space-like.


One can imagine many examples of light-like 3-surfaces not restricted to delta M^4. The following are examples that first come in mind but you could exercise by trying to find more examples.

a) To get the simplest possible example of light-like 3-surface as an orbit of 2-surface let us take M^1xS^2 in M^4. M^1 is time axis and S^2 a two-sphere of radius r in E^3. Deform this surface to S^1 subset CP_2, S^1 a geodesic circle: Phi= omega*t. The line element of the induced metric is

ds^2 = (1-R^2 omega^2)dt^2-r^2dOmega^2

For omega=1/R you get just -r^2 dOmega^2 so that time direction becomes light-like.


b) In M^4 you can take any 2-D surface and allow it to expand with light-velocity in local normal direction. You get expanding light-front, a light-like 3-surface at which metric of M^4 is of course not degenerate.

c) It is is easy to construct light-like 3-surfaces with degenerate four-metric. Just construct a surface for which induced metric changes signature from Minkowskian to Euclidian. Certainly such surfaces exist!

For instance, you can also consider CP_2 type vacuum extremal and deform it. For vacuum extremal having 1-D M^4 projection light-likeness for this projection states

m_kldm^k/dtdm^l/dt=0

Here t is curve parameter , which is arbitrary function of CP_2 coordinates. The light likeness condition states that the induce metric is not affected at all and is just CP_2 metric.

A deformation is obtained by assuming m_kldm^k/dtdm^l/dt>0. The positive contribution to length squared tends to make induce metric Minkowskian and for large enough deformation this can happen in certain region of CP_2 type vacuum extremal so that light-like 3-surface is generated.


d) You can also consider a very simple 4-surface restricted to M^4xS^1, S^1 a geodesic circle parameterised by angle coordinate Phi. Phi= Phi(t) gives induced metric [1-R^2omega^2 (t),-1,-1,-1]
The metric becomes degenerate at omega(t)=1/R. Now light-like surface is t=constant surface: this is somewhat counter-intuitive.

If Euclidian space-time regions replace blackholes in TGD, this would be the simplest possible toy-model for the formation of black hole! For the allowed imbed dings of critical and overcritical RW cosmologies this kind of falling to Euclidian metric actually takes place in finite delta M^4 proper time.tric is [1,-1,-1,-1].

 
At 11:57 PM, Anonymous matti Pitkanen said...



[Hamed] Light like 3-surfaces are light like random. randomness is only for microscopic objects. For a macroscopic object, light like orbit of partonic surface isn’t random?

[MP] Interesting and difficult question. I try to clarify to myself - I hope also to you - what I have meant when I have spoken about light-like randomness. Let us first forget the "macroscopic"
and talk only about randomness.

a) I have mostly talked about randomness as property of CP_2 type *vacuum externals*. Their M^4 projections are light-like and light-likeness gives rise to the counterparts of stringy Virasoro conditions suggesting that possible loss of randomness means breaking of conformal invariance or gauge condition breaking this symmetry. The non-vacuum deformations of CP_2 type vacuum extremals are assignable to lines of generalised Feynman diagrams.


One might argue that by classical energy momentum conservation the random orbit for non-vacuums deformations consist of pieces of light-like geodesics in measurement resolution used. p-Adic particle massivation would be understood as resulting from the fact that in p-adic length scale the net motion is effectively along time-like geodesic due to zig zag motion with light velocity implying that the net motion is with velocity smaller than light-velocity. One can however argue that deformed CP_2 type vacuum extremal interacts with its Minkowskian environment exchanging four-momentum: does this correspond to the ordinary interactions by classical forces?


b) The randomness of CP_2 type vacuum extremals is not ordinary thermodynamical randomness. I have assigned it p-adic thermodynamics for mass squared (not energy), a description that should be equivalent with its real variant: the equivalence implies that the number theoretic quantisation of temperature and mass scales for p-adic thermodynamics induces corresponding quantisation for real temperature and mass scales.

c) Of course, light-like randomness is not complete randomness: the emergence of Virasoro conditions means that there is huge conformal symmetry behind in complete conflict with naive expectations.



How random these light-like 3-surfaces can be? The guiding principle is strong form of holography implied by strong form of general coordinate invariance. At quantum level partonic 2-surfaces and their tangent space data dictate the physics. The basic conditions is that two random orbits between partonic 2-surfaces at the opposite ends of CD having same tangent space data are physically equivalent? Also two space-like 3-surfaces at the end of CD "connecting" partonic 2-surface with fixed tangent space data are physically equivalent. One can imagine several realisations for these conditions.

a) Light-like randomness could mean that one has gauge invariance. An infinite number of light-like 3-surfaces (space-like 3-surfaces at ends of CD), which are physically equivalent have same value of Kahler function and thus giving rise to preferred extremal with same value of Kahler action.

b) The dynamics of both light-like 3-surfaces and space-like 3-surfaces at the ends of CD is completely deterministic for *preferred* externals. Hence knowing par tonic 2-surfaces and corresponding boundary data fixes everything. One can of course ask whether this kind of complete determinism could be an outcome of fixing the gauge! If so, the first and second option would be equivalent!

[Note that conservation laws do not force determinism since energy momentum is assigned with entire 3-surface rather than partonic 2-surface.]

c) Or could it be that the only randomness that is left is the randomness of the orbits of string ends over which one must functionally integrate. This would reduce the gauge invariance to stringy conformal invariance assignable to the ends of string world sheets.

You can try to to argue which option - if any - is correct or try to invent more options;-)!

 
At 11:59 PM, Anonymous matti Pitkanen said...


Dear Hamed,

Still about your question.

[Hamed]Light like 3-surfaces are light like random. randomness is only for microscopic objects. For a macroscopic object, light like orbit of partonic surface isn’t random?

[Matti] In previous answer I did not say anything about "macroscopic". The addition of "macroscopic" makes situation even more difficult. We are talking about what I would call generalized Feynman diagrams for which line could be defined as Euclidian regions of space-time or as their "boundaries" at which the signature of the induced metric changes to Minkoswskian.

Is it possible to speak about "macroscopic" at all in this context ? What "macroscopic" is taken to mean? If it means that one speaks only about statistical predictions of the theory, then one takes QFT limit and considers kinetic equations and all information about the microlevel is lost!

If one is willing to talk about macroscopic quantum coherence, the situation changes. For instance, the lines of generalised Feynman diagram with macroscopic size could correspond to phases with large value of hbar_eff. I have also considered the possibility of assigning even to ordinary macroscopic objects a line of generalised Feynman diagram as Euclidian region with shape of the body.

If so, and if one takes seriously the previous considerations, then situation remains open. Sorry;-)! Does non-vacuum extremal property remove the non-determisnim and light-like randomness? Do different random orbits correspond to gauge equivalent space-time geometries? Does gauge choice force determinism? Or could the gauge degeneracy be identified as conformal gauge symmetry and be localised at the ends of string world sheets at light-like orbits of partonic 2-surfaces.

More generally, randomness is property of vacuum extremals and also the surfaces having as CP_2 projection Lagrangian manifold are vacuum externals. They can be only limits of physical non-vacuum externals. WCW metric becomes degenerate at this limit so that they do not contribute to physics directly. For the deformations o non-vacuum extremals I expect randomness to be lost since otherwise classical conservation laws are lost. The loss need not be complete and 4-D spin glass degeneracy is highly suggestive and would allow to realize quantum classical correspondence also for quantum jumps sequences rather than only quantum states.

 
At 12:07 AM, Anonymous matti Pitkanen said...


To Stephen,

I would not be so sure;-). We are now in Finland.

I get continually offers to test all kinds of gadgets: doing this I would get the thing without paying anything. The problem is that I do not need this stuff and do not want to waste my time to test them. I even got an offer to test a travel to some sunny place!

In sharp contrast to this, I have found it absolutely impossible to get any funding for my whole-daily research work during these 35 years. Finland wants me to be consumer, not a person with brains.

 
At 10:57 AM, Blogger Stephen said...

Matti, I echo that sentiment.. I am quite dismayed at my surrounding culture too, and I do not care for gadgets even though I am required to use them to communicate with family and friends else they think I am being "unresponsive". Any way...you said

"If one is willing to talk about macroscopic quantum coherence, the situation changes. For instance, the lines of generalised Feynman diagram with macroscopic size could correspond to phases with large value of hbar_eff. I have also considered the possibility of assigning even to ordinary macroscopic objects a line of generalised Feynman diagram as Euclidian region with shape of the body."

so you mean like a convex hull around the macroscopic euclidean object?

 
At 9:28 PM, Anonymous matti Pitkanen said...


In accordance with fractality, macroscopic object would be a collection of space-time sheets of various sizes topologically condensed at each other (also the field accompanies macroscopic object).

A sheet with shape of macroscopic object would have Euclidian signature of induced metric and defined the line of generalized Feynman diagram (4-D when time coordinate is taken into account). Euclidian space-time sheets would be analogs of blackholes interiors in TGD.

Note the many "would"'s;-).

 
At 11:01 PM, Blogger Hamed said...

Dear Matti,

So thanks, I write my questions from your answers.
“Let us look first what you imagine. Your light-like 3-surface in direction of delta M^4 . It is obtained by putting CP_2 coordinates constant: this would be just a piece of delta M^4. When you allow CP_2 coordinates to vary you get space like 3-surfaces since you get additional space-like contribution”

Projection of this space-like 3-surface(at your answer) to M4 is a piece of delta M4? As I understand from your answer, by continuation of partonic 2-surface and allow CP2 coordinates vary too, one can make space like 3-surface and also light like 3-surface. But projection of both of them to M4 are piece of delta M4. As I remember intersection of the space like 3-surface and light like 3-surface is partonic 2-surface. They are basic and equivalent.

But there are two types of space like 3-surfaces? : The one is the intersection of space time surface with CD*CP2. Boundary of this space-like 3-surface is partonic 2-surface. The another is continuation of partonic surface by allowing CP_2 coordinates to vary as is discussed before.


There are misunderstandings for me in your example:
a)To get the simplest possible example of light-like 3-surface as an orbit of 2-surface let us take M^1xS^2 in M^4. M^1 is time axis and S^2 a two-sphere of radius r in E^3. Deform this surface to S^1 subset CP_2, S^1 a geodesic circle: Phi= omega*t. The line element of the induced metric is

ds^2 = (1-R^2 omega^2)dt^2-r^2dOmega^2


Induced metric of M4 to M^1xS^2 is easy but what is the role of a circle to this induced metric? In your example as you noted this circle is a deformation of two-sphere. I don’t understand “S1 subset CP2”. Do you mean S1 is a subspace of CP2? If so, what is your purpose to writing this?

 
At 12:00 PM, Anonymous Anonymous said...

http://lightsinthetexassky.blogspot.com/2013/03/cloaked-object-captured-on-security.html

Any theories? Freaks me out because I live there currently ..

--Stephen

 
At 10:11 PM, Anonymous matti Pitkanen said...


Hi Hamed,

thanks for questions. It is good to clarify thoroughly what has been said.



[Hamed] So thanks, I write my questions from your answers.
“Let us look first what you imagine. Your light-like 3-surface in direction of delta M^4 . It is obtained by putting CP_2 coordinates constant: this would be just a piece of delta M^4. When you allow CP_2 coordinates to vary you get space like 3-surfaces since you get additional space-like contribution”

Projection of this space-like 3-surface(at your answer) to M4 is a piece of delta M4? As I understand from your answer, by continuation of partonic 2-surface and allow CP2 coordinates vary too, one can make space like 3-surface and also light like 3-surface. But projection of both of them to M4 are piece of delta M4. As I remember intersection of the space like 3-surface and light like 3-surface is partonic 2-surface. They are basic and equivalent.

[MP] The difference between two is that light-like 3-surfaces are *orbits* (!!) of partonic 2-surfaces in light-like direction *transverse* to delta M^4! Their M^4 projection does not restrict to delta M^4 as it does for the space-like 3-surfaces at delta M^4xCP_2, which do not possess this extension! It is a pity that I cannot draw!


[Hamed] But there are two types of space like 3-surfaces? : The one is the intersection of space time surface with CD*CP2. Boundary of this space-like 3-surface is partonic 2-surface. The another is continuation of partonic surface by allowing CP_2 coordinates to vary as is discussed before.

[MP ]No! If "as discussed above" refers to deformations of light-like pieces of delta M^4xs, s point of CP_2, to CP_2 directions these notions are equivalent.


[Hamed] There are misunderstandings for me in your example:
a)To get the simplest possible example of light-like 3-surface as an orbit of 2-surface let us take M^1xS^2 in M^4. M^1 is time axis and S^2 a two-sphere of radius r in E^3. Deform this surface to S^1 subset CP_2, S^1 a geodesic circle: Phi= omega*t. The line element of the induced metric is

ds^2 = (1-R^2 omega^2)dt^2-r^2dOmega^2


Induced metric of M4 to M^1xS^2 is easy but what is the role of a circle to this induced metric?

[MP] I could have taken any curve of CP_2 but chose circle because the metric is so simple for it.

[Hamed] In your example as you noted this circle is a deformation of two-sphere. I don’t understand “S1 subset CP2”. Do you mean S1 is a subspace of CP2? If so, what is your purpose to writing this?

[Matti] S^2 is two sphere of E^3 subset M^4=M^1xE^3, *not* of CP_2!, CP_2 coordinates are constant for it. Then I make CP_2 coordinates non-constant but simplifying the situation assuming that they vary in S^1.

 
At 10:12 PM, Anonymous matti Pitkanen said...

No. I tend to believe that one can do without new physics here.

 
At 1:41 PM, Blogger Wes Hansen said...

I thought perhaps you might find this biological computation paper interesting: http://www.sciencedirect.com/science/article/pii/S0019103513000791.

Regards,
Wes Hansen

 
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