Abstract: |
Spontaneous symmetry breaking means that only a subgroup H of the original symmetry group G is manifest in its ground state, an important concept relevant to cosmology, astrophysics, condensed matter physics, particle and nuclear physics, and even in chemistry and biology. In the original conception by Nambu and later developed further by Goldstone and others, Lorentz invariance fixes the low-energy long-distance behavior of the system uniquely by the metric of the coset space G/H. In particular, there is one massless particle called Nambu-Goldstone boson for each generator of g not contained in h. However, it has been known for a long time that this result does not necessarily hold in systems with finite temperature and/or density, namely when the system is not Lorentz invariant, including simple magnets around us. We formulate Nambu-Goldstone bosons without Lorentz invariance and find that we need to specify additional data of the coset space G/H beyond its metric, namely its presympletic structure. |