I will quote some about Lorentzian path integral from
the most pedagogical paper I know----Renate Loll
----quote from "A Discrete History"----
The desire to understand the quantum physics of the gravitational interactions lies at the root of many recent developments in theoretical high-energy physics. By quantum gravity I will mean a consistent fundamental quantum description of space-time geometry (with or without matter) whose classical limit is general relativity.
Among the possible ramifications of such a theory are a model for the structure of space-time near the Planck scale, a consistent calculational scheme to compute gravitational effects at all energies, a description of (quantum) geometry near space-time singularities and a non-perturbative quantum description of four-dimensional black holes. It might also help us in understanding cosmological issues about the beginning (and end?) of our universe, although it should be said that some questions (for example, that of the “initial conditions”) are likely to remain outside the scope of any physical theory.
I guess anyone interested in this thread already has realized this: one of the unusual things about this approach is there are no coordinates.
That was the headline on Regge's 1961 paper that set things up for Renate Loll and friends------"General Relativity Without Coordinates".
they can consider the space Geom(M) of all spacetime geometries on some manifold-----each geometry is described by listing interconnections between uniformsized simplexes, some kind of computer data structure.
that is a point in Geom(M), it is real elementary barebones
there is no "gauge" or chaff of arbitrary choice (as when things are presented using coordinates)
and that barebones reality is what the quantum mechanics is about
I have always appreciated the spareness of LQG----it doesnt seem to have anything in it that isn't needed to describe a quantum theory of 4D spacetime. But to get started, LQG does employ a differentiable manifold and connections thereon. That takes in a batch of arbitrary mathematical equipage (physically meaningless "gauge" accessory) which then has to be factored out later. But I thought that LQG kept gauge to a bare miniumum. After all, how could one ever get started without an underlying smooth manifold?
Systems of coordinates are an arbitrary physically meaningless choice but how do you get started without them?
Well the framework for Lorentzian path integral, or DT, is even more stripped down nitty. No coordinate system. It seems right. have to go, will try to get back to this later.