Spacetime, vector bundles, and unification: the coming together of ideas which have been around for the last three decades or so

I have attached here Ed Witten's nice summary of `our knowledge of physics' :

The attempts at unification in the style of a Kaluza-Klein theory try to bring the vector bundle and the spacetime into the fold of a higher dimensional spacetime with its own metric, and the theory is then to be quantised.

In spirit, the octonionic theory does the same thing, but the formalism is not directly about `metric and higher dimensional spacetime'. Instead, one can construct the Dirac operator on a curved spacetime and find its eigenvalues; these are alternatives to the metric as observables of general relativity, and these eigenvalues are the entities used in our Kaluza-Klein programme, instead of the metric. We consider the eigenvalues of the Dirac operator on a higher dimensional spacetime. And these not only unify gravitation with the gauge fields of the vector bundle, but also unify the fermions with gauge fields and gravity. One quantum of this unified entity is named an atom of spacetime-matter, or an aikyon. Classical fields are recovered as `condensates' (i.e. macroscopic entanglements) of many aikyons.

Furthermore, we do not work on a higher dimensional curved Minkowski spacetime. We work on the square-root of Minkowski, a spinor spacetime, i.e. a twistor space labelled by the octonions. These define an 8D non-commutative space. We construct the Dirac operator on this octonionic space. Its eigenvalues are the dynamical variables which describe the aikyons. The act of quantisation consists of raising the eigenvalues to the status of matrices/operators. One matrix for every eigenvalue: this is the dynamical variable for the aikyon. Which matrix/operator? The very Dirac operator of which it is the eigenvalue. This is important, because in the quantum-to-classical transition, each matrix collapses to one of the eigenvalues, and from the collection of all the collapsed eigenvalues one again recovers the Dirac operator and hence the classical theory (on a 4D spacetime, because the collapse is a spacetime-dimension-reducing process, with classical gauge fields living on the 4D spacetime).

The matrix D for the aikyon has Grassmann numbers as its entries, and hence can be written as a sum of a bosonic matrix (even grade Grassmann) and a fermionic matrix (odd grade Grassmann). D plays the role of canonical momentum. Let us define a configuration variable Q whose time derivative (and hence velocity) is D. The Lagrangian of the aikyon is simply Trace[D^2] where Trace is matrix trace. The action is simply the time integral (Connes time) of this kinetic energy Tr[D^2]. This is nothing but Newton's free particle with a Ph. D. so to say Newton's 3D absolute Euclidean space has been replaced by curved octonionic space: the matrix-valued coordinate components D_i of D are analogous to metric tensor components, they determine the geometry of octonionic space, which encodes gauge fields of the standard model, gravity, and fermions. Newton's absolute time has been replaced by Connes time, the latter being a property of non-commuative geometries. This matrix-valued Lagrangian dynamics is Adler's trace dynamics: a pre-quantum pre-spacetime theory from which QFT on classical curved spacetime is emergent. The fundamental universe is made of a large number of aikyons.

To incorporate chiral fermions the octonionic space is generalised to split bioctonionic space, which is essentially the doubling of the octonionic space to 16D, with the second half being the parity reverse of the first, hence permitting chiral fermions to be introduced. The velocity Q-dot is used to define a new matrix variable q such that Q-dot = q_dot +q and the Lagrangian is rewritten in terms of q and further rewritten in terms of the bosonic and fermionic parts: q=q_B + q_F. Bosons and chiral fermions emerge. The dotted variables relate to gravity and to right-chiral fermions and are defined over the second half of the 16D space; the undotted ones relate to the standard model forces and to left chiral fermions and are defined over the first half of the 16D space. Left chiral fermions are eigenstates of electric charge; right chiral fermions are eigenstates of square-root mass: +sqrt{m} is matter; -sqrt{m} is antimatter. Our universe has only the former, having separated from antimatter in a breaking of scale-invariance (symmetry) in the very early universe. The Lagrangian prior to symmetry breaking is scale invariant. After the breaking, scale invariance is replaced by C, P, T: our universe violates CP and violates T. Together with the mirror antimatter universe which ours separated from, CPT is preserved. The mirror universe is a CP image and a T image of our universe.

The Lagrangian is assumed to have an E8 x E8 symmetry, with the first E8 symmetry being over the 8D half, and the second E8 over the parity reversed (split part) 8D half. The first 8D space has Euclidean signature and is equivalent to SO(10) space. The second 8D space has Lorentzian signature and is equivalent to SO(1,9) spacetime. We see the coming together of Witten's vector bundle and of spacetime into a unified entity, which is a physical reality. The vector bundle is not merely a mathematical construct; it is reality. The aikyon does not live in spacetime, but in E8 x E8 space: we could call it aikyon space. Only classical objects (these result from entanglement of many aikyons) live in spacetime (4D spacetime). Aikyons live in aikyon space, and the Higgs (implied naturally by the Lagrangian) couples left chiral and right chiral fermions.

Octonions are magical. Not only do they define spacetime and gauge field space, they also define elementary particles (SM fermions and bosons) and determine their properties such as quantisation of charge and mass. The octonionic coordinates and the Lagrangian work hand in hand in all this. Spinors made from Clifford algebras made from octonionic maps define quarks and leptons of the standard model. The first E8 branches as SU(3)_EuclideanSpace x SU(3)_ThreeGensLH x SU(3)_color X SU(2)_L x U(1)_Y. The second E8 branches as SU(3)_spacetime x SU(3)_ThreeGensRH x SU(3)_grav x SU(2)_R x U(1)_g Here SU(2)_R x U(1)_g lead to general relativity in the classical limit, whereas SU(3)_grav is new, and seems related to the conformal gravity modification of GR explicit in the Chamseddine-Connes spectral action principle (the heat kernel expansion of Tr[D^2]).

We recover the standard model and modified gravity in the emergent theory. The trace dynamics equations of motion, when reduced to an eigenvalue problem, give evidence for determining values of free parameters of the standard model.

Note that in unifying the vector bundle and the spacetime we never had to go to high energies. We have simply recast what we already know, onto a spinor spacetime, which when enlarged to higher dimensions, casts gauge fields and gravity into the aikyon space with E8xE8 symmetry. Quantum systems live in this aikyon space even at low energies. The quantum-to-classical transition that we observe around us all the time breaks E8xE8 because macroscopic objects are confined to 4D and their localisation is the very process which in the first place gives rise to the 4D classical spacetime and segregates the vector bundle (which is Euclidean space) from emergent spacetime.