S^5 ‘-> S^17 -> SP^2

should also make sense, with the caveat that SP^2 is singular. This might make for some interesting physics as, for example, singular G2 manifolds are used in the study of M-theory compactifications (e.g. hep-th/0606285).

]]>1) S^0 ‘-> S^2 -> RP^2

2) S^1 ‘-> S^5 -> CP^2

3) S^3 ‘-> S^11 -> HP^2

4) S^7 ‘-> S^23 -> OP^2

… One may wonder if the generalized Hopf fibrations have … application in the description of entangled qutrits. …”.

The corresponding sextonion sphere bundle would be

S5 to S17 to SP2

where S5 and S17 denote 5 and 17 spheres

and SP2 is sextonion projective 2-space.

Note the explicit appearance of S5, which also appears in the twistor stuff.

If qutrits are related to triality, with the middle spheres in the generalized sphere bundles being spheres in the 3-spaces R3, C3, H3, S3, and O3 (where here S3 is 3-dim space of sextonions),

then the material about triality in the Landsberg/Manivel paper at math/0402157 may be useful.

Tony Smith

]]>I actually meant S^23 -> OP^2. The S^15 is total space of the O^2 Hopf construction. So in the cases of R^3, C^3, H^3 and O^3 we recover the following sphere bundles over projective planes:

1) S^0 ‘-> S^2 -> RP^2

2) S^1 ‘-> S^5 -> CP^2

3) S^3 ‘-> S^11 -> HP^2

4) S^7 ‘-> S^23 -> OP^2

where KP^2 (K=R,C,H,O) is the base space and S^n (n=2,5,11,23) is the total space with fibers S^k (k=0,1,3,7), respectively.

Compare the above fibers to the fibers of the original Hopf fibrations:

1) S^0 ‘-> S^1 -> RP^1

2) S^1 ‘-> S^3 -> CP^1

3) S^3 ‘-> S^7 -> HP^1

4) S^7 ‘-> S^15 -> OP^1

where KP^1 (K=R,C,H,O) is the base space and S^m (m=1,3,7,15) is the total space with fibers S^p (p=0,1,3,7), respectively.

The original Hopf fibrations have been shown to describe **entangled qubits** in Bernevig and Chen’s Geometry of the 3-Qubit State, Entanglement and Division Algebras. One may wonder if the generalized Hopf fibrations have a similar application in the description of **entangled qutrits**.

As kneemo points out, the twistor construction involving S5 in CP3 can be seen in terms of eigenvectors that “… lie on an S5 in C3 …”,

and

it is fun to extend to quaternionic H3 and octonionic O3 constructions.

The O3 construction is related to J(3,O) which is the basis for the construction of my E6 model on the CERN CDS EXT preprint server known as EXT-2004-031 which has an 8-dimensional spacetime that gives an M4 x CP2 Kaluza-Klein structure in which the M4 4-dimensional physical spacetime is what is related to the Penrose twistor structures based on 6-real-dimensional CP3 that can be equivalently formulated in terms of the Conformal Group Spin(2,4) action on a 6-real-dimensional space with two-time signature (2,4).

Roughly,

the octonionic J(3,O) picture is for high-energy 8-dimensional spacetime

and

the complex CP3 twistors are for lower-energy dimensionally reduced 4-dimensional physical spacetime.

As kneemo says, the octonionic J(3,O) picture is related to the sphere S15 in OP2.

However,

there is a subtlety in the quaternionic J(3,H) picture in that the sphere S11 is not a sphere in 8-dimensional HP2.

When you ask whether there might be a natural 12-dimensional projective space in which the sphere S11 might naturally live, you might find the answer in an algebra known as the sextonions, described by Landsberg and Manivel in their paper “The Sextonions and E7(1/2)” on the arXiv at math/0402157 in which they say:

“… the intermediate Lie algebra between e7 and e8 satisfies some of the decomposition and dimension formulas of the exceptional simple Lie algebras. A key role is played by the sextonions, a six dimensional algebra between the quaternions and octonions. …

we investigate in some detail the geometry of the projective plane over S [the sextonions], which is a singular but close cousin of the famous four Severi varieties AP2, for A = R,C,H,O …

SP2 is singular, and a proper subvariety of the cone over HP2 …”.

Therefore the sphere S11 lives in 12-dimensional SP2 just like S15 lives in OP2.

My physical view is:

Octonion OP2 is for 8-dim high-energy unreduced spacetime;

Sextonion SP2 is for the 6-dim two-time conformal Spin (2,4) space in which conformal transformations act linearly

Penrose twistors are for 4-dim reduced physical spacetime M4 in which conformal transformations act non-linearly

Another nice thing about sextonion SP2 is that you can see that the 26-16 = 10 dimensions of string theory can be reduced at low energies into SP2 x CP2 where

SP2 is a 6-dim sextonionic two-time physical spacetime of Spin(2,4) = SU(2,2)

and

CP2 is a 4-dim internal symmetry space.

Then the SP2 can be mapped onto the conventional M4 4-dimensional spacetime on which Spin(2,4) Conformal Group action is non-linear.

If you look at the 4 special conformal transformations and 6 Lorentz transformations of Spin(2,4) in the way Irving Ezra Segal looked at them, you get a quantitatively correct picture of our universe that is now conventionally described as a Dark Energy making up (at the present time about 75 per cent of the stuff of our universe, or, equivalently, what Louise calls varying c (because special conformal transformations map light-cones to light-cones, and can change them from fat to thin, thus changing c which is the ratio of time-distance to space-distance).

I think that it is fun that so many superficially disparate things are really similar,

and that it is strange that so many human physicists bitterly disparage some things just because they appear to be superficially different (unfortunately, that phenomenon is not restricted to physics or physicists).

Tony Smith

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