I was reading [this stack overflow answer](https://physics.stackexchange.com/a/474126/26466) which mentioned the “trace anomaly” in Quantum Chromodynamics. I tried reading [the wikipedia page](https://en.wikipedia.org/wiki/Conformal_anomaly) but it’s pretty much entirely over my head. I took physics in college but all of the math jargon of quantum mechanics is beyond me.
In: Physics
Alright. I’ve taken some QM, but not enough to be deeply familiar with Noether’s theorem. This isn’t quite the blind leading the blind, but it’s probably the best pseudo-layman description you’ll get.
Noether’s theorem is a theorem from mathematical physics. Essentially, she proved that anything which works by reasonable derivative rules, and which behaves the same if you shift a quantity by a given amount, MUST have a conservation law. That is, it *must* be the case that some quantity, like energy or momentum, is a fixed amount that can never go up or down, it just gets moved from one place to another. Noether’s theorem is what proves that energy must be conserved, because physics is time-invariant (you can run the same experiment now, or a year from now, or a year in the past, and the results will be equivalent). Linear momentum is conserved because it doesn’t matter if you shift an experiment n units to the left or to the right.
In the language of *field theory* rather than discrete particles, however, you don’t have a conserved *quantity,* you instead have conserved *flow*–the field intensifies in some places and thins out in other places. This conserved flow–the fact that the total amount-of-stuff never goes up or down, it just gets pushed around from place to place–is called the “Noether current.”
A seemingly-reasonable expectation of the laws of physics is that they should work the same at all *scales.* That is, no matter how big or small you look at it, the universe should behave essentially the same way. That’s a symmetry, “scale” or “dilation” symmetry. And, because you’ve found a (possible) symmetry, Noether’s theorem works bidirectionally–if *and only if* there’s a symmetry, there must be an associated conserved quantity.
The problem is, scale invariance results in a particular kind of symmetry which results in a universe where there are no particles that have mass, *or* a universe where all particles have a continuous *spectrum* of mass, not discrete specific masses. That’s…pretty obviously *wrong.* We know that. We can see particles with discrete mass every day. Hence, there must be a flaw. Either Noether’s theorem is wrong (which is pretty unlikely, given it’s a mathematical proof), or one of the assumptions required to make it true is wrong, or our initial assumption that the laws of physics are always scale-symmetric was wrong.
We’re unlikely to reject Noether’s theorem, and it *seems* that all of the assumptions required to apply it are in fact true, so we’re left with having to assume that, at some scale, the laws of physics DO behave differently–some kind of quantum-field-theory “current” (=flow of *new* stuff) must occur.
“Trace anomaly,” “scale current,” and other terms are the name for this spontaneous symmetry break responsible for the discrete-mass particles we observe. In order to properly understand the stuff involved, you would need to learn tensor algebra and other advanced mathematics that are a bit beyond me as well, so there’s really not much more to be said.
While quantum chromodynamics (QCD) expressed via its Lagrangian density for quark-physics looks clasically like a conformal field theory, a quantum anomaly in the trace of its energy momentum tensor (conformal anomaly) breaks conformal invariance as a quantum field theory.
Latest Answers