CDT is similar to Loop Quantum Gravity (LQG) in a couple of respects. It seeks to quantize the gravitational degrees of freedom in a background independent and non-perturbative manner. One difference from the historical development of LQG is that LQG first used spin networks to create a non-dynamical structure, and then sometime later spin-foam models were developed which used a path-integral approach to evolve the networks. CDT incorporates the path integral at the outset as its “most important theoretical tool”.

The idea behind the path integral is to create superpositions of all the “virtual” paths or configurations which the spacetime degrees of freedom (metric field variables from General Relativity) can follow as time unfolds. This sum over all the possible configurations is the quantum spacetime.

The second key idea is to constrain the set of geometries which contribute to the sum to those which implement causality. There had been earlier approaches similar to CDT referred to as “Euclidean path-integral approaches to quantum gravity” which lacked this feature and did not exhibit the right dimensionality at the macro level.

Before discussing the causal constraint specifically, Loll et al. outline the general method for how the class of “virtual” paths should be chosen. As in quantum field theory, one needs some way to constrain or “regularize” the paths, so you don’t get wildly divergent outcomes. CDT's method in the context of spacetime geometry is to use “piecewise flat geometries” which are flat except for local subspaces where curvature is concentrated. The geometry used is a “triangulated” space (or Regge geometry). These “triangular” flat structures are “glued” together, so curvature only appears at the joints (they give a 2D pictorial example, to see how curvature arises when you glue these together). The motivation for this approach, the basic elements of which are not new, is that it is an economical way to build a discrete spacetime. Later, in finding a path integral, they will take the short distance cutoff to zero which they say will obtain a final theory which is not dependent on many of the arbitrary details which went into the construction.

So, with the geometries defined in this way: what ensemble of these should be included in the sum?

This is where causality comes into play. They again mention previous efforts which involved 4-dimensional Euclidean space, not 3+1 Lorentzian spacetime. It turns out there is not a relationship between a path integral for a Euclidean space vs. one for a Lorentzian space. So CDT needs to encode the causal Lorentzian structure right into the building blocks at the outset. If you impose appropriate causal rules, you can get four-dimensional spacetime to dominate at large scales. If this mirrors reality, then it is suggests it is the case in reality that causality at sub-Planckian scales is what is responsible for the existence of 4D spacetime.

So what are the causal rules in the CDT approach? “They are simply that each spacetime appearing in the sum over geometries should have a specific form. Namely, it should be a geometric object which can be obtained by evolving a purely spatial geometry

*in time*, in such a way that its spatial topology (the way in which space hangs together) is unchanged

*as a function of time*(emphasis added).” The authors have used computer simulations to model the nature of this spacetime at different scales, which lead to the four-dimensional shape emerging (it is not at this stage an analytical result). Now, one might naively question why getting four-dimensions to emerge when your microscopic building blocks were also four-dimensional (3+1) is a big deal, however, the result was far from predictable given the complex fluctuations and divergences generated by the quantum superpositions: in previous “Euclidean” versions, the dimensionality at higher scale would vary all over the place even if the building blocks were 4D-spatial.

The authors then discuss issues involving ongoing efforts to investigate other features of the spacetime model beyond dimensionality to see if they are consistent with gravity. They also discuss their hope that distinctively quantum gravitational cosmological predictions could be derived from the model.

An issue I have concerns the role of time in this model. It seems that a single time dimension is in place for all of the building blocks. This kind of global time directionality is philosophically less appealing compared to an approach which implements a strictly local time at the microscopic level (it also seems less consistent with relativity).

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