How can parameters in QFT depend on scale or energy?

Question

Often, in theoretical physics you hear things like "the coupling constants depend on the energy scale". One random example:

When this dependence on the scale is studied carefully, we find out that the dimensionless constants such as the fine-structure constant aren't really dimensionless, or aren't really constant. They run – depend on the energy – in a slower way and this running is the reason why the limit is singular if we want the theory to work at arbitrarily high energies or short distance scales.

But how is this even possible? The energy should not appear in the Lagrangian or equations of motion (EOM): it is a property of solutions to the EOM. Consider for example the wave equation, $$\frac{1}{v^2}\frac{\partial^2u}{\partial t^2}-\frac{\partial^2u}{\partial t^2}=0.$$ I feel like this quote would be equivalent to saying that $v$ depends on energy, i.e. $v=v(E)=v(E(u(x,t))$. This would mean the couplings depend in some complicated (and nonlinear!) way on the fields themselves. So how does this work in QFT?

Answer 1

It's subtle, enigmatic, and a core feature of QFT.

The highly counterintuitive linkage between dimensionless couplings and a dimensionfull energy scale is called dimensional transmutation, and the scale Λ at which the coupling diverges is a feature of the theory not apparent in the Lagrangean.

It is a strictly quantum phenomenon, (so don't expect any hint of it in the equations of motion!), due to the remarkable conspiracy ("renormalization group") of virtual degrees of freedom interacting systematically and driving this picture.

QFT texts have a field day detailing this mechanism; and, physically, the strong interactions, described by the QCD QFT are all but explained by it. It was only discovered and understood in the early 1970s, and surprised even the crustier experts!

Given such a Λ, one normalizes with it the scale of each energy μ probed by the theory, so that for different energies the corresponding quantum effective couplings are different, and not really coupling constants at all...