My post on the “physics of decimals” a couple of weeks back caught physics blogger Luboš Motl’s attention, with predictable results. Mostly, this led to a rather unproductive debate about semantics, but he did bring up one thing that I think deserves some further clarification.
In my post, I asked you to imagine asking a genie for the full consequences of quantum field theory. Short of genie-based magic, is this the sort of thing I think it’s at all possible to know?
In a word, no.
The world is messy, not the sort of thing that tends to be described by neat exact solutions. That’s why we use approximations, and it’s why physicists can’t just step in and solve biology or psychology by deriving everything from first principles.
That said, the nice thing about approximations is that there’s often room for improvement. Sometimes this is quantitative, literally pushing to the next order of decimals, while sometimes it’s qualitative, viewing problems from a new perspective and attacking them from a new approach.
I’d like to give you some idea of the sorts of improvements I think are possible. I’ll focus on scattering amplitudes, since they’re my field. In order to be precise, I’ll be using technical terms here without much explanation; if you’re curious about something specific go ahead and ask in the comments. Finally, there are no implied time-scales here: I’ll be rating things based on whether I think they’re likely to eventually be understood, not on how long it will take us to get there.
Let’s begin with the most likely category:
Probably going to happen:
Mathematicians characterize the set of n-point cluster polylogarithms whose collinear limits are well-defined (n-1)-point cluster polylogarithms.
The seven-loop N=8 supergravity integrand is found, and the coefficient of its potential divergence is evaluated.
The dual Amplituhedron is found.
A general procedure is described for re-summing the L-loop coefficient of the Pentagon OPE for any L into a polylogarithmic form, at least at six points.
We figure out what the heck is up with the MHV-NMHV relation we found here.
Likely to happen, but there may be unforeseen complications:
N=8 supergravity is found to be finite at seven loops.
A symbol bootstrap becomes workable for QCD amplitudes at two or three loops, perhaps involving Landau singularities.
Something like a symbol bootstrap becomes workable for elliptic integrals, though it may only pass a “physicist” level of rigor.
Analogues to all of the work up to the actual Amplituhedron itself are performed for non-planar N=4 super Yang-Mills.
Quite possible, but I’m likely overoptimistic:
The space of n-point cluster polylogarithms whose collinear limits are well-defined (n-1)-point cluster polylogarithms that also obey the first entry condition and some number of final entry conditions turns out to be well-constrained enough that some all-loop all-point statements can be made, at least for MHV.
The enhanced cancellations observed in supergravity theories are understood, and used to provide a strong argument that N=8 supergravity is perturbatively finite.
All-multiplicity analytic QCD results at two loops, for at least the simpler helicity configurations.
The volume of the dual Amplituhedron is characterized by mathematicians and the connection to cluster polylogarithms is fully explored.
A non-planar Amplituhedron is found.
Less likely, but if all of the above happens I would not be all that surprised:
A way is found to double-copy the non-planar Amplituhedron to get an N=8 supergravity Amplituhedron.
The enhanced cancellations in N=8 supergravity turn out to be something “deep”: perhaps they are derivable from string theory, or provide a novel constraint on quantum gravity theories.
Various all-loop statements about the polylogarithms present in N=4 are used to make more restricted all-loop statements about QCD.
The Pentagon OPE is re-summed for finite coupling, if not into known functions than into a form that admits good numerics and various analytic manipulations. Alternatively, the sorts of functions that the Pentagon OPE can sum to are characterized and a bootstrap procedure becomes viable for them.
Irresponsible speculations, suited to public talks or grant applications:
The N=8 Amplituhedron leads to some sort of reformulation of space-time in a way that solves various quantum gravity paradoxes.
The sorts of mathematical objects found in the finite-coupling resummation of the Pentagon OPE lead to a revival of the original analytic S-matrix program, now with an actual chance to succeed.
Analytic all-loop QCD results.
Magical genie land:
Analytic finite coupling QCD results.