Nuclear Symmetry Energy in Strongly Interacting Matter: Past, Present and Future

The concept of symmetry under various transformations of quantities describing basic natural phenomena is one of the fundamental principles in the mathematical formulation of physical laws. Starting with Noether's theorems, we highlight some well-known examples of global symmetries and symmetry breaking on the particle level, such as the separation of strong and electroweak interactions and the Higgs mechanism, which gives mass to leptons and quarks. The relation between symmetry energy and charge symmetry breaking at both the nuclear level (under the interchange of protons and neutrons) and the particle level (under the interchange of u and d quarks) forms the main subject of this work. We trace the concept of symmetry energy from its introduction in the simple semi-empirical mass formula and liquid drop models to the most sophisticated non-relativistic, relativistic, and ab initio models. Methods used to extract symmetry energy attributes, utilizing the most significant combined terrestrial and astrophysical data and theoretical predictions, are reviewed. This includes properties of finite nuclei, heavy-ion collisions, neutron stars, gravitational waves, and parity-violating electron scattering experiments such as CREX and PREX, for which selected examples are provided. Finally, future approaches to investigation of the symmetry energy and its properties are discussed.