Quantum scarring refers to a phenomenon where the eigenstates of a classically chaotic quantum system have enhanced probability density around the paths of unstable classical periodic orbits. The instability of the periodic orbit is a decisive point that differentiates quantum scars from the more trivial observation that the probability density is enhanced in the neighborhood of stable periodic orbits. The latter can be understood as a purely classical phenomenon, a manifestation of the Bohr correspondence principle, whereas in the former, quantum interference is essential. As such, scarring is both a visual example of quantum-classical correspondence, and simultaneously an example of a (local) quantum suppression of chaos.
A classically chaotic system is also ergodic, and therefore (almost) all of its trajectories eventually explore evenly the entire accessible phase space. Thus, it would be natural to expect that the eigenstates of the quantum counterpart would fill the quantum phase space in the uniform manner up to random fluctuations in the semiclassical limit. However, scars are a significant correction to this assumption. Scars can therefore be considered as an eigenstate counterpart of how short periodic orbits provide corrections to the universal spectral statistics of the random matrix theory. There are rigorous mathematical theorems on quantum nature of ergodicity, proving that the expectation value of an operator converges in the semiclassical limit to the corresponding microcanonical classical average. Nonetheless, the quantum ergodicity theorems do not exclude scarring if the quantum phase space volume of the scars gradually vanishes in the semiclassical limit.
On the classical side, there is no direct analogue of scars. On the quantum side, they can be interpreted as an eigenstate analogy to how short periodic orbits correct the universal random matrix theory eigenvalue statistics. Scars correspond to nonergodic states which are permitted by the quantum ergodicity theorems. In particular, scarred states provide a striking visual counterexample to the assumption that the eigenstates of a classically chaotic system would be without structure. In addition to conventional quantum scars, the field of quantum scarring has undergone its renaissance period, sparked by the discoveries of perturbation-induced scars and many-body scars (see below).
Scar theory
The existence of scarred states is rather unexpected based on the Gutzwiller trace formula, which connects the quantum mechanical density of states to the periodic orbits in the corresponding classical system. According to the trace formula, a quantum spectrum is not a result of a trace over all the positions, but it is determined by a trace over all the periodic orbits only. Furthermore, every periodic orbit contributes to an eigenvalue, although not exactly equally. It is even more unlikely that a particular periodic orbit would stand out in contributing to a particular eigenstate in a fully chaotic system, since altogether periodic orbits occupy a zero-volume portion of the total phase space volume. Hence, nothing seems to imply that any particular periodic orbit for a given eigenvalue could have a significant role compared to other periodic orbits. Nonetheless, quantum scarring proves this assumption to be wrong. The scarring was first seen in 1983 by S. W. McDonald in his thesis on the stadium billiard as an interesting numerical observation. They did not show up well in his figure because they were fairly crude "waterfall" plots. This finding was not thoroughly reported in the article discussion about the wave functions and nearest-neighbor level spacing spectra for the stadium billiard. A year later, E. J. Heller published the first examples of scarred eigenfunctions together with a theoretical explanation for their existence. The results revealed large footprints of individual periodic orbits influencing some eigenstates of the classically chaotic Bunimovich stadium, named as scars by E. J. Heller.