The exceptional group E8 can arise in 16-dimensional Clifford space, a manifold whose tangent space is Clifford algebra Cl(1,3). The latter space besides an algebra is also a vector space V_{8,8} whose elements can be rotated into each other by means of the generators of SO(16) (or one of its non compact forms). The vectors of V_{8,8} generate Clifford algebra Cl(8,8) which contains the Lie algebra of E8 as a subspace. E8 thus arises from the fact that, just as in the spacetime M_4 there are r-volumes, r=0,1,2,3,3, generated by the the tangent vectors of the spacetime, there are R-volumes, R=0,1,2,...,16, in the Clifford space C, generated by the tangent vectors of C.

A novel view on the physical origin of E8   
Journal of Physics A: Mathematical and Theoretical 41 , 332001-10  (2008)
 

In 2002 I started to investigate a theory in which a 16-dimensional curved Clifford space provides a realization of Kaluza-Klein theory. No extra dimensions of spacetime are needed:  ``extra dimensions'' are in C-space.  We explore the spin gauge theory in C-space and
show that the generalized spin connection contains the usual 4-dimensional gravity and Yang-Mills fields of the U(1)xSu(2)xSU(3) gauge group. The representation space for the latter group is provided by 16-component generalized spinors composed of four complex 4-component spinors, defined geometrically as the members of four independent left minimal ideals of Clifford algebra.

These ideas appeared in the paper Kaluza-Klein Theory without Extra Dimensions: Curved Clifford Space Physics Letters B 614, 85-95 (2005)
See also 
Clifford Space as a Generalization of Spacetime: Prospects for Unification in Physics   hep-th/0411053
and 
Spin Gauge Theory of Gravity in Clifford Space: A Realization of Kaluza-Klein Theory in 4-Dimensional Spacetime  gr-qc/0507053   
International Journal of Modern Physics A, 21, 5905-5956 (2006)


 
 

I have found that the currently fashionable ideas about the Heisenberg picture (as being more fundamental than the Schroedinger picture) are quite in agreement with what I say about quantum mechanics and the (Lorentz) invariant evolution parameter $\tau$. One only needs to ``slightly" enlarge spacetime. Instead of 4-dimensional spacetime one has to take 16-dimensional Clifford space, i.e., the space of p-loops that enclose p+1-areas (0-loops being the ordinary points). [See the papers by C.Castro,  W. Pezzaglia, M. Pavsic,  and  A.Aurilia et.al..] Here I see a possible connection to the loop approach to quantum gravity.] In Clifford space we have the constrained theory, no evolution, Heisenberg picture, etc., whilst in spacetime we have a reduced, unconstrained, theory with evolution (as considered by Fock, Stueckelberg, Feynman, Horwitz, and others). In Clifford space (shortly  C-space) 16 components of momentum are constrained to a generalized mass shell, whilst in spacetime four components of the ordinary 4-momentum can be considered as being unconstrained (since we are considering a reduced theory, a reduced action). Upon Gupta-Bleuler quantization the constraint becomes the Klein-Gordon equation in C-space. Upon quantization of the reduced action we obtain the  Schroedinger equation in the reduced space (a subspace of which is 4-dimensional spacetime).  In fact, the Schroedinger equation in the reduced space is a  generalized Stueckelberg equation and it contains the ordinary Stueckelberg equation as a special case). Moreover the Fock-Schwinger proper time formalism is widely recognized as a very useful tool. It contains a ``fictitious" evolution parameter $\tau$. In my scheme the latter parameter is not fictitious, it is a genuine evolution parameter. The same evolution parameter would also appear in a generalization of the theory that would take into account the dynamics of  the metric tensor. 
This provides a resulution of the notorious ``problem of time'' in quantum gravity.

All the nice features of relativity (such as reparametrization invariance, ``block universe", Heisenberg picture, Lorentz transformations, etc.) still hold: not in spacetime, but in C-space. What is nice here is that we have not added extra dimensions to spacetime. We have merely considered the geometric structure, the Clifford bundle, that sits at every point of spacetime.

There exists a paper ``Quantum Entanglement in Time" (by C. Brukner at al.) eprint quant-ph/0402127   in which the authors have found a very strange connection between past and future: the very act of measuring the photon polarization a second time can affect how it was polarized earlier on (see also ``The weirdest link", New Scientist, 27 March 2004, p.32). They suggest that ``the difference between the spatial and temporal structure may ultimately be fundamental, or it may be an indication that we need a deeper theory in which the two need to be treated on a more equal footing (quantum field theory does not suffice in this sense).'' A ``deeper'' theory is considered in my works on the Stueckelberg theory briefly described above.  For more information see papers on Relativistic Dynamics   , book , and papers on Clifford algebra.   Namely, the essence of the Stueckelberg theory, as investigated in my works,  is that space and the so called coordinate time are indeed treated on a more equal footing.  In addition, in the Stueckelberg theory there occurs yet another time, the so called evolution time. Therefore, in the Stueckelberg theory two kinds  of   entanglement in time are theoretically possible: that due to Bruckenr at al. (which is now the entanglement in evolution time), and the entanglement in spacetime (which is analogous to the usual entanglement).

If we define momentum operator geometrically as gradient (derivative) ,  being basis vectors, then the Hamiltonian H (which is proportional to the square of p) is unambiguously defined even in curved space. There is no ordering ambiguity. (Class. Quant. Grav. Vol. 20 (2003) 2697-2714)

The expectation value for the momentum operator in curved space V_n is defined straightforwardly.
The expecation value for the momentum operator follows a geodetic curve V_n . 
Momentum operator is Hermitian with respect to the  scalar product in the definition of which enter the basis vectors  .

We have shown that -using the formalism of Clifford algebra- the intergation over a vector field in curved space is well defined : Vectors at different points  x of  V_n   are  all brought together into a chosen point x' by means of parallel transport along the geodesic joining x and x'. Usage of the generators of Clifford algebra as basis vectors enables to write such integral in a very compact notation (see paper). 

An illustration of such vector integral for the case of sphere is given here:   Html     Pdf     Ps    Ppt  . 
A demonstration of the transition to the case of  flat space  is given here:  Html   or Ppt .