Causal Dynamical Triangulation & Causal Sets!
Two recent approaches to quantum gravity are less than 20 years old, and somewhat similar in approach. They quantize space-time itself, instead of space only, are background-independent, are non-perturbative, preserve causality, and reduce to Einstein's Theory of Gravity in the classical limit. Causal Dynamical Triangulation is basically a path-integral approach, requiring the usual sum over all possible paths, and so specific predictions using the approach require a numerical calculation, a Monte Carlo simulation. Space-time is divided into slices, called simplexes. Each simplex is flat but simplexes can be joined at the edges to reproduce any desired curvature. In the joining, the timelike edges are arranged so that time flows always forward along the timelike boundaries from tile to tile.
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Causal
Sets is an approach somewhat similar to Causal Dynamical
Triangulation, but more
general. Space-time is, on the smallest scale, sets of
points corresponding to events. A central problem of this
approach is to connect specificially chosen causal sets to
realistic space-times. Feynman's path integral approach to
quantum physics is a key feature of both of these efforts to
construct a theory of quantum gravity.
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The idea here is to start from a more general classical gravitational theory, which does not even involve space-time, and thus avoids quantization problems. Classical gravity can be described as a relational dynamical system, without ever appealing to space–time or its geometry. This description is the so-called shape dynamics description of gravity. The existence of relational first principles from which the shape dynamics description of gravity can be derived is a motivation to consider shape dynamics (rather than Einstein's theory of gravity) as the fundamental description of gravity, and a new starting point for quantization. The most important feature of this theory is the replacement of relativity of simultaneity with a more tractable gauge symmetry, namely invariance under spatial conformal transformations. The particular relational approach that underlies shape dynamics completely dispenses with space–time. This means that the shape dynamics description changes the foundations of gravity from space–time geometry to a relational dynamical system. Space–time and its geometry are, from the point of view of the shape dynamics description of gravity, merely effective concepts that describe physical experience. However, these effective descriptions may not exist as geometric objects in the strict mathematical sense. The shape dynamics formulation of gravity possesses a physical Hamiltonian that generates evolution of spatial conformal geometry. This disentangles the problem of time in quantum gravity: The gauge problem (the choice of foliation in the spacetime description) is replaced by the problem of finding spatial conformal geometries, leaving an evolution that is comparable to a system with a time-dependent Hamiltonian. The problem of time is suggested to be completely solved by restricting oneself to "objective observables," which are those observables that do not depend on any imagined external clock or rod.
In the current state of quantum gravity research, we seem to find no general agreement at all, anywhere, as to how to proceed. I wouldn't say that every possible avenue of research has been exhausted, far from it, but every new idea that has been proposed, particularly since the 1980s, has tended to peter out into disillusionment. It seems far from impossible that Einstein's original theory could be reformulated in different mathematical language, based on different symmetries and in a framework generally consistent with some form of quantum physics, resulting in what amounts to a quantum theory of gravity that retains classical features... and it seems equally far from impossible that some revolutionary re-imagining of field theory would eventually produce a quantum theory of space time. Only the future can reveal the future!
