[deleted]
Ben,
The best thing about these contests is they give an opportunity to brain storm on a range of possibilities. The questions concerning physical foundations, particularly with respect to cosmology and quantum gravity, require different ways of thinking. I sometimes think that our educations have a disservice. While of course a graduate student needs to know classical and quantum mechanics, electromagnetic fields and so forth, I sometimes think these cement in our thinking so as to prevent successful consideration of deeper problems.
I sometimes think that we often suffer from some of the difficulties seen in elementary students with basic mechanics and F = ma. Our brains are predisposed to thinking in certain ways, and though we may learn the breakthrough physics of the past, this learning often serves to foster thinking that is erroneous on deeper foundations.
Time evaluated from the Jacobi variational principle
δt = sqrt{m_iδx_iδx_i/(E-V)}
is related to a proper time, or an interval. I might then say that if we multiply by E-V on both sides we get
(E-V)δt = sqrt{m_iδx_iδx_i(E-V)}
where the left hand side appears to be a Lagrangian times an interval of time. This may then be written as
∫d^3 δt sqrt{-g}R = sqrt{m_iδx_iδx_i(E-V)}
We may then break out the Ricci scalar R = R_{ab}g^{ab} and the left hand side exhibits this symmetry. On the right hand side again there is symmetry with the interchange of δx_iδx_j δ_{ij}. This probably needs to be firmed up of course, but I think this captures the idea.
Causal dynamics on the other hand is ordered by events with the idea of building up geometry. So there are orderings such as x < y so that in some product we have xy = -yx. This seems to have some connection with Penrose tensor space theory, where for every symmetric tensor there is an antisymmetric tensor. The relationship between the two is a graded algebra similar to supersymmetry. The symmetric interchange between spatial coordinates in shape dynamics is similar to the symmetric interchange between boson fields. The antisymmetric interchange of events in causal sets is similar to the interchange between fermions ψ(x)ψ(y) = ψ(y)ψ(x). Hence a causal set is potentially identical in form to a Slater determinant. This then opens the door to a type of functor or category theory which maps elements of geometry to elements of field theory.
Fields on a Cauchy surface separated by spatial intervals define the "shapes." Intervals separated by null or timelike intervals define causal sets. The first of these is symmetric, while the next is antisymmetric. This is similar to Penrose's tensor space, which axiomatizes spaces. If you have a space in n dimensions one can represent the positive tensor dimension as ||| ...|•ε = 0, where | represents an element such as a vector or spinor and the set |||...| means an exterior product of these. The ε means a Levi-Civita symbol and this is a skew product. This can be seen equivalently as a skew symmetrization of the |||...| in a higher dimensional space. If this is zero, then the space of tensors is symmetric. This system however requires there to be the |||...|•g, where g is a symmetric tensor. Again this is equivalent to a symmetric trace in a higher dimensional space. The "dimension of these tensors" are n and -n respectively. They correspond to the symmetric and antisymmetric sets of tensors, which have a duality.
This duality between symmetric and skew symmetric elements, or for two tensors products of the sort
{ψ^a, ψ^b} = g^{ab}
[φ^a, φ^b] = ω^{ab}
involves supersymmetry. In the case of spacetime the generators of supersymmetry Q_a and \bar-Q_b construct Lorentz boosts
{Q_a,\bar-Q_b} = iσ^μ_{ab}∂_μ.
where the momentum boost operator p_μ = -i∂_μ constructs the Lorentz group. Meanwhile {Q_a,-Q_b} = 0. The anticommutator of the super generators seems to have a categorical relationship with the antisymmetry of causal nets. The rotation operator M^{μν} and the super-generator obey
[Q_a, M^{μν}] = 1/2σ_{ab}^{μν}Q_b,
and the commutator between the momentum p^μ and the generator Q_a is zero
[Q_a, p^μ] = 0
The relationship between the symmetry and antisymmetric approaches, say shape dynamics and causal set theory, might then have functors to Fermi-Dirac fields and boson fields, and a system which includes both might then have a graded Lie algebra with Grassmann generators that connect the two.
Cheers LC