This is the way in which all physical theories get formulated--- you first acquire certainty regarding the behavior of many special cases you have some experimental data or theoretical insight about, then you try to formulate a precise theory which extends these heuristic laws to a precise understanding, and when you succeed in matching the heuristic laws (when they apply) and you can predict everything consistently and correctly, you are done.
In fundamental physics, one has a pretty solid understanding of phenomena which are not quantum gravitational, because we have a precise fundamental theory of relativistic quantum fields. So the most significant things that are not fully understood at the precise level are the class of insights deriving from Hawking radiation and black hole classical behavior. These are semi heuristic, because there are puzzles that are not yet fully resolved within string theory. This class includes:
- The near horizon behavior of semiclassical black holes: an observer falling into a black hole sees nothing special when crossing the horizon, and this has not been rigorously demonstrated in string theory, it is only rigorously true classically. Now some people claim that it can't work in quantum gravity, that black holes are "firewalls". I read the arguments, and I find them uncompelling, because they mix assumptions about the semiclassical behavior at late times with full quantum observations on the Hawking radiation which are restricted when you measure the semiclassical state at late times, so the argument smells fishy and doesn't stop smelling fishy even with later clarifications, although understanding what is going wrong is important, and this is at the heuristic level, because we can't reconstruct black hole interiors completely from quantum gravity scattering data.
- The related holographic principle and black hole complementarity: this is also heuristic for semiclassical thermal black holes, although it is precise in AdS/CFT, for certain extremal black holes.
- Cosmological horizon entropy: the entropy of the cosmological horizon is not even in-principle understood in string theory, and it is a major clue to understanding how to do quantum gravity in deSitter space, because it is understood at the heuristic level--- it's the same as black hole entropy.
- Rindler horizons: The behavior of strings on a Rindler background is nontrivial, even though this is just Minkowski space. Quantum fields are completely understood on Rindler, but string theory on Rindler is harder, because you don't have an S-matrix (everything falls into the Rindler horizon).
For string theory, we only have heuristic guidelines regarding compactification and SUSY breaking. These heuristics are summarized in guidelines about what topologies and matter configurations give rise to which gauge fields and representations, and where one should expect to find the standard model. These are relatively well understood, since there are many explored vacua, but there are always surprises.
In cases where we know the fundamental laws without serious doubt, there are still cases where we understand things only at the heuristic level. The following are in QCD:
- Confinement and Regge theory: we know the QCD confines to Regge trajectories for mesons and topological soliton Baryons, but this formulation is not mathematically linked to QCD by any rigorous path.
- Pomerons: This is related--- we know that high energy scattering is dominated by Pomeron exchange, and this is heuristic only, because we can't relate this to QCD high energy diffractive scattering except in certain regimes which are not completely diffractive.
- Chiral perturbation theory: Again related, this is the low-energy approximation to QCD, and the parameters are from assumptions on low energy condensates. I suppose I shouldn't include this, because you can extract chiral data from lattice data in principle exactly, but I had something else in mind for what constitutes a full understanding--- it would mean that any chiral configuration can be mapped to a QCD configuration, and you could do the path integral for QCD in two steps--- as a path integral over long wavelength chiral configurations plus an additional path integeral over QCD in the given chiral background. Nobody did this, although it shouldn't be hard.
- Instanton fluid: there is a class of semiheuristic models that give the QCD vacuum as a dense instanton fluid, and this is not 100% understood. It's essentially the same problem as before--- condensates and confinement.
- Quark condensates: there are condensate models where we can semi-heuristically calculate the effects on hadrons using the SVZ sum-rules (QCD sum rules), but the condensate values again are not derived from QCD, although they can be measured from experiment. In principle, lattice gives them, but this is not enough--- you want to know the values and effects with more insight.
There are no doubts QCD resolves these questions, but the exact best way to make these heuristic things precise is unclear.
There are condensed matter systems where the understanding is heuristic in the same way--- the most famous and the one I like best is probably:
- HighTc: The condensate state of the HighTc superconductor can be described phenomenologically using a D-wave superconducting condensate. Getting this D-wave from the fundamental interactions has not been done in a universally convincing way, although there is no doubt that the fundamental theory will do it.
There are many other such things, this is usually all the open problems people work on. Historically, if you look at any phenomenon that was understood, it was understood heuristically before it was understood precisely, and learning the heuristic stuff is an important prerequisite for getting certainty about the final explanation, because physical theories are evolved by common sense, they don't emerge fully formed from nothing.