The S-matrix is the oracle physicists turn to in times of crisis

If we hit an actual wall, then that could be the missing part of the unified theory, some boundary that needs to get applied to the equations of both that makes them explicitly inapplicable in the regimes where today we just observe they don't work.

Of course, we might also hit an "engineering" wall, where we don't have any explicit boundary, but it's just too hard to perform the experiments that would help us understand how these theories relate. That would mean that the current status quo will remain the best we can do - two separate theories which we know are wrong, but where for any experiment we can perform, one or the other predicts the results almost perfectly.

QFTs are effective field theories: they describe interactions at a particular energy scale, smoothing over whatever unknown physics is going on at lower (or sometimes higher) length scales. This makes the strengths of interactions scale-dependent. So in QED, for instance, the fine structure "constant" is actually a few percent larger at the GeV scale than it is at zero energy. Fortunately for us, we only need to measure the fine-structure constant at finitely many energies to calculate its value at any energy scale we like. But this is just because QED happens to be well-behaved. When we try to apply the same techniques to the gravitational constant, we end up with infinitely many free parameters for reasons which are too technical to explain here. Any intro QFT textbook will probably have a rough heuristic argument (I used Srednicki), but we don't actually know with certainty that there's no clever way to get all those higher order terms to cancel - we just haven't found one and have no reason to expect one.

(Also not a physicist). GR works and has been tested experimentally on a large scale, but it doesn't always work on a small scale. QM works, has been tested experimentally on a small scale, but it doesn't always work on a large scale.

They just give different predictions in some very small or very large cases. Obviously they can't both be right: if one theory predicts A and the other theory predicts B then we can just do the experiment and one of them will turn out to be wrong. Unfortunately, they only differ in extreme circumstances that are hard to recreate, and we were not able to perform an experiment that will tell us which is right (or how to tweak them to make them compatible with each other).

Fun fact: There is a thing called a cosmological constant problem. It's the disagreement between the observed values of the cosmological constant (using GR) and a theoretical value suggested by QM. The observed value is... 10**120 times smaller than QM predicts it should be [1]. Now this is what I call a fundamental incongruence :).

[1] This is also called "probably the worst theoretical prediction in the history of physics".

I'm not a physicist, just a layman interested in this topic. So, if you're looking for very rigorous explanations, I can't help.

That said, my understanding is this: if you try to calculate the trajectory of small particles using the equations of GR, and then compare that to measurements, you'll find that GR is dead wrong. Conversely, if you try to calculate the motion of stellar objects in strong gravitational fields using the equations of QM or QFT, you'll again get results that don't match observations at all.

Further, for more complex mathematical reasons that I can't personally follow, if you just try to introduce the GR space-time deformation terms into QM equations, you can't compute anything because the equations diverge.

There are four profound problems:

1. The first is philosophical. GR is classical, with smooth continuous fields, locality and determinism*. QM has superposition, entanglement, non-locality and indeterminism. GR seems more intuitive, but QM seems more of an irreversible revolution, so slight win.

2. The second is philosophical. GR provides equations for how spacetime is warped by matter and energy (background independent). QM assumes a static background 'stage' of spacetime (wavefunction is a function of spaces, Schrodinger Equation is a wave equation wrt time, so background dependent). GR seems deeper, QM seems superficial. GR wins this hands down, QM has to change or be extended to make it background-independent (perhaps see String Theory).

3. The third is philosophical. QM has a 'measurement problem'. Wave: the Schrodinger Equation is linear, unitary, time reversible and (hence) information preserving. Particle: measurement is discrete, discontinuous, non-linear and information destroying. So there is either a bizarre mismatch between the two parts of the theory, or you strike down measurement, and propose the Many Worlds interpretation, which punts all the measurement problems into other causally disjoint 'worlds', without a clear way to establish probabilities. This is purely a problem with incomplete/incoherent explanation by QM. The fact that physicists cannot agree on an interpretation is damning (is the wavefunction real? is Hilbert space real? what is measurement?)

4. The last is practical. QFT is just Fourier Analysis where it true, and a terrible hack everywhere else: double quantization, renormalization of infinities, energy cut-offs, virtual particles, incomplete Standard Model (neutrinos?), why 3 generations, what is the vacuum (dark energy).

My scorecard is 40:60 draw in QM's favor, GR total win, and two QM no-contest own-goal losses. So a clear GR win, in the sense that a new integrated theory of Quantum Gravity will have:

- a copy of GR, even if it is emergent from discrete substrate

- radically overhauled version of QM (philosophical, explanatory and model details), even as predictions don't change much, and some important philosophical revolutions are preserved

My own personal pure guess is that the next level substrate is discrete. The two signposts are: holography, dimensional reduction and AdS/CFT-ishness; and the huge clue that quantum entanglement seems to create relativistic spacetime (but perhaps not proper time).

The main incompatibility is that QM predicts that all fields have a zero point energy associated with each mode. However, GR predicts zero point energy will cause spacetime to curve. However there are infinite modes in QM, so now using this in GR leads to all spacetime instantly collapsing. However we exist so this collapse hasn't happened. The solution is the elusive Quantum Gravity, something that physicists from Einstein, Feynman haven't been able to solve.

I won't have any Last Thursdayists under my roof.