Two theories that each work perfectly, and a scale where they demand incompatible things of the same event
There is no crisis in physics. That is the strange part. Quantum field theory predicts the magnetic moment of the electron to twelve digits. General relativity predicted gravitational waves a century before anyone detected them, and then got the waveform right. Neither theory is in trouble. The trouble is that they cannot both be applied to the same event, and there are events where you have no choice.
The conflict is not a clash of formulas that could be patched with a better approximation. It runs deeper than that, and it shows up in three different disguises that turn out to be the same problem wearing different clothes. Push a measurement to short enough distances and the apparatus collapses into a black hole. Try to quantise the gravitational field the way we quantised the electromagnetic one and the perturbation series stops converging, needing a new experimental input at every order. Ask what state the universe is in and you find that quantum mechanics wants a clock outside the system, while general relativity insists there is no outside.
This lesson makes each of those concrete, because the shape of the problem determines the shape of the answer. Loop quantum gravity is not one more attempt to fix the convergence of a series. It is a bet that the third disguise is the real one, and that a theory of gravity has to be built without a stage from the very first line.
Drag the resolution down and watch a completely ordinary thought experiment destroy itself. Two marks, one probe photon of the right wavelength, and one consequence nobody put in by hand. Nothing on this canvas uses quantum gravity; it uses Heisenberg and Schwarzschild, each in its own domain.
The ratio in the third box is exactly 2(ℓP/Δx)2, so it grows a hundredfold for every factor of ten you gain in sharpness. It reaches one at Δx = √2 ℓP ≈ 2.29e-35 m, and past that point the experiment answers a different question than the one you asked. Nothing here is quantum gravity yet — it is just quantum mechanics and general relativity, each used exactly as its own textbook says, arriving together at a length below which “distance” stops having an operational meaning.
The Planck length is not a new constant of nature. It is the only length you can build out of the three constants that were already there: Newton’s constant, the speed of light, and Planck’s. Max Planck noticed this in 1899, a quarter century before quantum mechanics existed, and thought of it as a set of units no civilisation could disagree about. What he did not know was that the combination marks a boundary.
The demonstration above is the reason it is a boundary. Resolving a separation demands a probe of comparable wavelength, that wavelength fixes an energy, that energy gravitates, and the ratio of the resulting horizon to the separation you were after is 2(ℓP/Δx)2. The exponent is what does the damage: the harder you look, the faster the horizon grows, and there is no clever apparatus that escapes the arithmetic because the argument never mentions the apparatus. Below roughly a Planck length, “the distance between these two things” is not a quantity that any experiment can be built to report.
An operationally meaningless quantity is a warning about the mathematics. The continuum says there are uncountably many distinct points inside any region, each a separate place where a field could take a separate value. Every one of those degrees of freedom is a place infinities can hide. If distance below a certain scale cannot be measured even in principle, then the continuum is describing structure that is not there — and the whole strategy of this module is to build a theory in which that structure was never assumed.
The same failure in the language of Feynman diagrams. Pick a theory, dial the energy, and watch the loop expansion either stay under control or stop being an expansion at all. Fermi’s theory of beta decay is on the menu because it had this exact disease and was cured — which is the most encouraging fact in this lesson.
The bars shrink for now. They stop shrinking at 10^19 GeV.
Structurally identical to Fermi theory, with the breakdown pushed out to 10¹⁹ GeV. The superficial degree of divergence is 2 + 2L, so loop L needs a counterterm with 2L + 2 derivatives — a fresh curvature invariant every single order.
Split the metric into a fixed background plus a small ripple, quantise the ripple, and you get a perfectly sensible theory of gravitons that reproduces Newton’s law and the bending of starlight. It is a genuine quantum theory of gravity, and it works. It just does not work all the way down. Newton’s constant carries mass dimension minus two, so the dimensionless quantity controlling the expansion has to be GE2, and by the time you reach the Planck energy every term in the series is the same size as every other one.
The bookkeeping is unusually cruel. The superficial degree of divergence of an L-loop graviton diagram grows as 2 + 2L, which means each order needs a counterterm with more derivatives than the last, and each of those counterterms comes with a coefficient no theory determines. There is a famous near-miss: for pure gravity with no matter, the one-loop divergences turn out to be proportional to the field equations and can be transformed away, a result of ’t Hooft and Veltman in 1974. The reprieve lasted a decade. In 1986 Goroff and Sagnotti pushed the calculation to two loops and found a divergence that does not vanish, with the unforgettable coefficient 209/2880. And adding matter of any kind ruins even the one-loop result.
Here is why that is a diagnosis rather than a death sentence. Fermi’s 1933 theory of beta decay had a coupling with the same mass dimension and the same sickness, with the series failing near 300 GeV. The theory was not wrong. It was the low-energy shadow of something with more structure, and the W boson turned up comfortably below the scale where the shadow stopped making sense. A non-renormalisable theory is a theory announcing that it is an approximation to something else. The question is what.
There is one more thing worth noticing about splitting the metric into background plus ripple: it needs a background. That move is exactly the thing general relativity spent a century arguing you are not entitled to, which brings us to the third disguise.
Three regimes, one scene. Start with a quantum field on a rigid grid, where every standard technique applies. Let the grid curve but stay fixed, and the techniques still work — this is the regime that gives us Hawking radiation. Then let the grid become one of the things in superposition, and watch what quantum mechanics was silently leaning on.
Ordinary quantum field theory. The grid is given in advance and never responds. States live on a slice of constant t, evolution is unitary in that t, and the energy of a mode is defined by how fast its phase turns against that clock. Every one of those sentences quietly refers to the grid.
Every sentence in quantum mechanics smuggles in a background. A state is defined on a slice of constant time. Evolution is generated by a Hamiltonian that pushes the state forward in that time. A particle’s energy is how fast its phase turns against that clock. Two operators commute when they are attached to points at spacelike separation, which presumes you already know which separations are spacelike. Remove the fixed spacetime and each of those definitions has nothing left to refer to.
General relativity removes it. The theory’s Hamiltonian does not generate evolution in a physical time, because relabelling the time coordinate is a gauge transformation and no observable can depend on it. What you get instead is a constraint: on physical states, the Hamiltonian gives zero. There is no evolution equation for the universe as a whole, only a condition that the state must satisfy. This is the Wheeler-DeWitt equation, and the puzzle of how a static equation can describe a world in which things visibly happen has been called the problem of time since the 1960s.
The going answer is that time was never fundamental. Physical questions are relational: not “what is the field doing at time seven” but “what is this field doing when that one reads seven”. One subsystem plays the clock, and the correlations between subsystems carry everything observable. It sounds like a philosophical retreat, but it is a technical requirement, and it is the reason the theory in this module is built out of graphs rather than out of fields on a grid. A graph has no coordinates for a diffeomorphism to move around. Only its combinatorics survive, and only what survives can be physical.