I was conjecturing a Δ(x,z) formed from retarded and advanced potentials with a common point y to the future (timelike or lightlike separation) of both x and z. The propagation from x to y in the retarded case and from y and z in the advanced case is replaced by the commutator. This of course would need to be derived by different means of course. Yet quantum gravity implies a nonlocality of fields, and this seems like a possible way to derive this. The trick is to insure that no information or quantum bits are communicated between x and z. My conjecture here of course could be wrong.
The replacement with theta functions is a way of possibly treating black hole singularities. In that case the evolution equations contain a singularity and are meromorphic, rather than unitary. The theta function is
θ(z,t) = sum_{n=-∞}^∞exp(πin^2t 2πinz)
which takes the place of unitary functions such as e^{-iL}. I will not right now go into a complete discussion on this, but these modular functions I think in general replace unitary functions
One way to see a motivating reason for this is with the following metric with 1 - 2m/r = e^u. so then
ds^2 = e^udt^2 - e^{-u)dr^2 dΩ^2.
We now have to get dr from
dr = -2me^u/(1 - e^u)^2du.
Now the metric is
ds^2 = e^udt^2 -2m[e^u/(1 - e^u)^4]du^2 dΩ^2.
The singularity is at u = ∞, where the dt term blows up, and the horizon coordinate singularity at u = 0 is obvious in the du term. My rational was that the singularity had been removed "to infinity" in these coordinates. This makes the black hole metric similar to the Rindler wedge coordinates, which does not contain a singularity. In the accelerated frame or Rindler wedge there is singularity. The treatment of the Schwarzschild metric in the near horizon approximation Susskind uses is one where the singularity is sufficiently removed so that field in the Rindler wedge may be continued across the horizon without concerns. In this metric of mine the singularity is at infinity so the analytic functions for fields in the Rindler wedge are replaced with meromorphic functions with a pole at infinity.
The singularity at infinity causes trouble with the end point of the radiance process for it has to "move in" from infinity. The final quantum process of a black hole is a problem not well known in any coordinates. By the time the black hole is down to its last 10^4 or 10^3 Planck mass units the black hole itself is probably quantum mechanical. In my coordinates (assuming they are unique to me, which is not likely) the singularity at infinity may not have to "move" from infinity. There may be some nonlocal physics which causes its disappearance without having to move at all. This nonlocality is a correspondence between states interior to a black hole and those on the stretched horizon. The Susskind approach does not consider the interior, and he raises this as a question towards the end of his book "The Holographic Principle."
Unitarity is represented by a complex function e^{-iHt} and so forth, which is analytic. The loss of unitarity does not mean there is a complete loss of everything; in particular quantum information can still be conserved. A simple analytic function of this sort describes standard quantum physics. Gravity as we know is given by a hyperbolic group, such as SO(3, 1) ~ SL(2,C), where the latter has a map to SL(2,R)^2. The functions over these groups have posed difficulties for quantum gravity, for they are explicitly nonunitary. The trick of performing a Wick rotation on time or with τ = it is a way of recovering the compact groups we know in quantum physics.
It does turn out I think that we can think directly about quantum gravity by realizing that the SL(2,R) is related to a braid group with Z --- > B --- > PSL(2,Z), and that the braid group is contained in SL(2,R). Braid groups have correspondence with Yang-Baxter relations and quantum groups. The group SL(2,Z) is the linear fractional group, which is an elementary modular form. An elementary modular function is
f(z) = sum_{n=-∞}^{n=∞}c(n)e^{-2πi nz}
which in this case is a Fourier transform. In this case we are safely in the domain of standard QM and QFT. In general modular functions are meromorphic (analytic everywhere but infinity) and analytic condition is held on the upper half of the complex plane.
Of particular interest to me are the Eisenstein series of modular functions or forms. These define an integer partition function, which is an acceptable partition function or path integral for a stringy black hole. I include a graphic here illustrating an Eisenstein function. This has a certain self-similar structure to it, or what might be called an elementary form of a fractal. In this picture unitarity is replaced with modularity. In this more general setting the transformation do no promote a field through time by some operator, but that the operator simply computes the number of states or degrees of freedom in a way that is consistent. Unitarity is then a special case of this, which happens to fit into our standard ideas of causality.
The Eisenstein series describes a partition function or path integral for a black hole. The theory is not one of unitary evolution, but simply one of counting states or degrees of freedom on the horizon. In effect physics is more general than unitarity, where unitarity is a necessary condition to describe the semi-classical states in postulate #2.
Your essay went up by quite a bit when I scored it last night. I think it went back down some. To increase your community rating it is best to read other essays and leave commentaries there. You will of course find a whole host of essays that are nonsense. I try to down vote those to clear them from the upper ranks. Comments left on the blog site of other reasonable authors, with some attention drawn to your essay, can help raise your rating. If you "demolish" bad essays higher up it is of course best not to write anything. There is a bit of the "art of the schmooze" involved with this.
Cheers LC
