then you would have a twistor version of my view of string segments as local WorldLines in the Feynman PathIntegral picture.

Yes, the higher complex projective spaces are needed to make the connection to S^5. To make the Vazquez eigenvalue argument concrete, I’d like to start with C^3. Let A be a 3×3 complex Hermitian matrix with real (distinct) eigenvalues a1, a2, a3 and orthonormal eigenvectors v1, v2, v3. As the eigenvectors are of unit length, they lie on an S^5 in C^3.

Next, let us take the outer product of each eigenvector with itself, giving us the matrix set: v1(v1)*, v2(v2)* and v3(v3)*. (Note that such matrices are idempotent and have trace one,i.e., primitive idempotents). Taking the Jordan product of A with such matrices gives us:

Ao(vv*)=1/2(Avv*+vv*A)=1/2(avv*+avv*)=avv*

This tells us that the idempotents v1(v1)*, v2(v2)*, v3(v3)* are J(3,C) eigenmatrices for the operator A, with eigenvalues identical to the eigenvalues of their corresponding eigenvectors, namely, a1, a2, a3.

We now note that the outer product operation on each eigenvector actually induces a map from S^5 to CP^2, as 3×3 primitive idempotents can be regarded as points of CP^2. This allows us to map the two points in S^5 to CP^2, while preserving eigenvalues.

The aforementioned construction works for quaternion and octonion eigenvectors of H^3 and O^3 as long as the eigenvalues commute with the eigenvectors and the eigenvector entries lie in an associative subalgebra. The H^3 and O^3 constructions (S^11->HP^2 and S^15->OP^2, resp.) are not as clean because 3×3 Hermitian operators over H^3 and O^3 do not in general have a real spectrum. To see a simple counterexample, see Dray, Janesky and Manogue’s Octonionic Hermitian Matrices with Non-Real Eigenvalues. Fortunately, over the operator spaces J(3,H) and J(3,O), 3×3 Hermitian matrices over H and O do admit real eigenvalues for eigenmatrices of the form Q x Q=0 (i.e. Q in KP^2, K=R,C,H,O).

]]>Manin said “… the skies over the points of the Minkowski World are not all the lines in CP3, but only part of them, lying in a five-dimensional hypersurface …”.

If you map the twistor 5-dim hypersurface in CP3 to S5,

then you would have a twistor version of my view of string segments as local WorldLines in the Feynman PathIntegral picture.

Global WorldLines would follow, as kneemo said:

“… the continuous string would be recovered as a long polymer of projective space line segments …”.

Tony Smith

]]>Taking the projective space view, where S^2 ~ CP^1, the string bits correspond to line segments in projective space. With this interpretation, the continuous string would be recovered as a long polymer of projective space line segments.

]]>“… CP_2 type extremals whose small deformations serve as a model of elementary particles M^4 projection is light-like curve in M^4 …”.

and

this makes connections with Kea’s “… perspective … to look at worldsheets as twistor spheres, which replace points …”.

One of my favorite descriptions of such cpnnections is in the 1981 book Mathematics and Physics in which Yu. Manin says:

“… It is extremely important to … imagine the whole history of the Universe … as a complete four-dimensional shape, something like the “tao” of ancient Chinese philosophy.

The introduction of temporal dynamics is the next step. …

the natural structure for the absolute sky … at the point P0 … is the complex Riemann sphere … the complex projective line CP1 … the natural coordinates are complex numbers … they are always connected by a fractional-linear transformation …

each sky CP1 is simply embedded in … The “Penrose paradise” H = CP3 … the space of “projective twistors” …

the skies over the points of the Minkowski World are not all the lines in CP3, but only part of them, lying in a five-imensional hypersurface …

introduc[ing] additional … skies correspond[ing] to the missing lines in CP3 …[gives]… the compact complex spacetime of Penrose, denoted CM …

[with] the interpretation of the “tunnel effect” of quantum mechanics in terms of the classical evolution of a system in imaginary time …

What binds us to spacetime is our rests mass, which prevents us from flying at the speed of light, when time stops and space loses meaning.

In a world of light there are neither points nor moments of time;

beings woven from light would live “nowhere” and “nowhen”;

only poetry and mathematics are capable of speaking meaningfully about such things.

One point of CP3 is the whole life history of a free photon – the smallest “event” that can happen to light. …

The mysterious separation of the World into space and time is implicitly contained in the group … SL(2,C) ( or its generalization SL(4,C) [ = the Conformal Group Spin(2,4) ] ) …

In conclusion, I would like to say a few words about the theory of numbers … It is remarkable that the deepest ideas of number theory reveal a far-reaching resemblance to the ideas of modern theoretical physics. Like quantum mechanics, the theory of numbers furnishes completely non-obvious patterns for relationship between the continuous and the discrete (… p-adic numbers … classfield theory, which describes the relationship between prime numbers and the Galois groups of algebraic number fields). One would like to hope that this resemblance is no accident, and that we are already hearing new words about the World in which we live …”.

Tony Smith

PS – I apologize for the length of the quote, but I think that Manin’s book is a lot of fun and it may be hard to find nowadays.

]]>The thinnest string like objects have 2-D M^4 projection which is classical string orbit and holomorphic surface of CP_2 as CP_2 projection: geodesic sphere is simplest example. For all practical purposes these objects are one-dimensional. Their length can however be very long and I have dubbed them cosmic strings for this reason. The primordial cosmology according to TGD consists of this kind of strings.

One can deform these strings in M^4 directions so that M^4 projection becomes 4-D: magnetic flux tube is the result and fractal hierarchy of tubes with increasing thickness result.

*Galaxies are organized like pearls in necklace along this kind of strings forming a cosmic network.

*TGD based star model predicts that stars are mass concentrations around string like objects parallel to rotation axis.

*Hubble has found “elephant trunks”, strings forming double helices (around magnetic flux tubes), and directed to young high temperature stars: a clear signal that kind of metabolic energy receival is in question.

*The recent discovery of dust plasmoids behaving like primitive life forms involves double helices formed by dust particles of micron sized scale. There is even evidence for kind of genetic code: the value of twist angle of helix suffering bifurcations as function of radius of helix would define codons.

*Biology is full of molecular strings. DNA and aminoacid sequences are basic examples.

*Atomic nucleus according to TGD consists of nuclei consisting of flux tubes containing nuclear strings having quark and antiquark of MeV mass scale at their ends.

*Dark magnetic flux tubes carrying dark matter should populate universe in all scales.

Concerning comment of Tony Smith: in TGD the light-like direction of light-like 3-surfaces appearing as basic object is in somewhat similar role like time direction of Tony’s strings interpreted as world lines. The non-determinism makes the light-like direction genuine degree of freedom but with dynamics strongly constrained by the analog of conformal invariance in this direction which is not however gauge symmetry now. As matter fact, for so called CP_2 type extremals whose small deformations serve as a model of elementary particles M^4 projection is light-like curve in M^4 and lightlikeness implies Virasoro conditions of string model.

]]>Hi Tony. I agree that it is useful to think of point particles as strings, and I wish I had more time right now to look at your work. From my perspective it is very helpful to look at worldsheets as twistor spheres, which replace points.

]]>As you say, most conventional “… String theorists say that strings are tiny …”, but, as you also say, their lack of “… contact with reality might … have something to do with [your] lack of enthusiasm for [conventional] string theory …”.

My view (and in my model which does allow calculation of particle masses, force strengths, K-M parameters, cosmological ratios commonly known as Dark Energy : Dark Matter : Ordinary Mattter, etc) is:

Fundamental particles are points in spacetime.

Each point, over time, traces out a WorldLine in spacetime.

So, the natural fundamental physical interpretation of a “piece of string” is as a particle WorldLine.

Closed strings are WorldLines of loops (such as virtual loops in Feynman PathIntegral Sums, and such as composite spin-2 gravitons).

Open strings are the asymptotic incoming and outgoing particle WorldLines in the Feynman PathIntegral picture.

This works very nicely and physically realistically in plain old 26-dim Bosonic String Theory,

if you orbifold 8+8 = 16 dimensions to represent the 8 first-generation fermion particles and the 8 first-generation fermion antiparticles

and make an 8-dimensional spacetime plus Standard Model gauge bosons by stacking 8-branes,

all as described in my paper on the CERN CDS EXT preprint server known as EXT-2004-031.

It seems to me that such a WorldLine string theory might give a natural construction of a Bohm-type Quantum Potential,

as well as a useful way to physically visualize strings.

Tony Smith

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