In quantum mechanics, quantum scarring is 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 that have subsequently paved the way towards emerging concepts within the field, such as antiscarring and quantum birthmarks.

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, Eric 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 Heller.

A wave packet analysis was a key in proving the existence of the scars, and it is still a valuable tool to understand them. In the original work of Heller, the quantum spectrum is extracted by propagating a Gaussian wave packet along a periodic orbit. Nowadays, this seminal idea is known as the linear theory of scarring. Scars stand out to the eye in some eigenstates of classically chaotic systems, but are quantified by projection of the eigenstates onto certain test states, often Gaussians, having both average position and average momentum along the periodic orbit. These test states give a provably structured spectrum that reveals the necessity of scars. However, there is no universal measure on scarring; the exact relationship of the stability exponent ${\displaystyle \chi }$ to the scarring strength is a matter of definition. Nonetheless, there is a rule of thumb: quantum scarring is significant when ${\displaystyle \chi <2\pi }$, and the strength scales as ${\displaystyle \chi ^{-1}}$. Thus, strong quantum scars are, in general, associated with periodic orbits that are moderately unstable and relatively short. The theory predicts the scar enhancement along a classical periodic orbit, but it cannot precisely pinpoint which particular states are scarred and how much. Rather, it can be only stated that some states are scarred within certain energy zones, and by at least by a certain degree.

The linear scarring theory outlined above has been later extended to include nonlinear effects taking place after the wave packet departs the linear dynamics domain around the periodic orbit. At long times, the nonlinear effect can assist the scarring. This stems from nonlinear recurrences associated with homoclinic orbits. A further insight on scarring was acquired with a real-space approach by E. B. Bogomolny and a phase-space alternative by Michael V. Berry complementing the wave-packet and Hussimi space methods utilized by Heller and L. Kaplan.

As well as there being no universal measure for the level of scarring, there is also no generally accepted definition of it. Originally, it was stated that certain unstable periodic orbits are shown to permanently scar some quantum eigenfunctions as ${\displaystyle \hbar \rightarrow 0}$, in the sense that extra density surrounds the region of the periodic orbit. However, a more formal definition for scarring would be the following: A quantum eigenstate of a classically chaotic system is scarred by a periodic if its density on the classical invariant manifolds near and all along that periodic is systematically enhanced above the classical, statistically expected density along that orbit.

Most of the research on quantum scars has been restricted to non-relativistic quantum systems described by the Schrödinger equation, where the dependence of the particle energy on momentum is quadratic. However, scarring can occur in a relativistic quantum systems described by the Dirac equation, where the energy-momentum relation is linear instead. Heuristically, these relativistic scars are a consequence of the fact that both spinor components satisfy the Helmholtz equation, in analogue to the time-independent Schrödinger equation. Therefore, relativistic scars have the same origin as the conventional scarring introduced by E. J. Heller. Nevertheless, there is a difference in terms of the recurrence with respect to energy variation. Furthermore, it was shown that the scarred states can lead to strong conductance fluctuations in the corresponding open quantum dots via the mechanism of resonant transmission.