of symmetry breaking is dynamical symmetry breaking. In the Lagrangian setting of Quantum field theory (QFT), the Lagrangian is a functional of quantum fields which is invariant under the action of a symmetry group . However, the vacuum expectation value formed when the particle collapses to a lower energy may not be invariant under . In this instance, it will partially break the symmetry of , into a subgroup . This is spontaneous symmetry breaking. Within the context of gauge symmetry however, SSB is the phenomenon by which gauge fields 'acquire mass' despite gauge-invariance enforcing that such fields be massless. This is because the SSB of gauge symmetry breaks gauge-invariance, and such a break allows for the existence of massive gauge fields. This is an important exemption from Goldstone's Theorem, where a Goldstone Boson can gain mass, becoming a Higgs Boson in the process. Further, in this context the usage of 'symmetry breaking' while standard, is a misnomer, as gauge 'symmetry' is not really a symmetry but a redundancy in the description of the system. Mathematically, this redundancy is a choice of trivialization, somewhat analogous to redundancy arising from a choice of basis. Spontaneous symmetry breaking is also associated with phase transitions. For example in the Ising model, as the temperature of the system falls below the critical temperature the symmetry of the vacuum is broken, giving a phase transition of the system. Explicit symmetry breaking In explicit symmetry breaking (ESB), the equations of motion describing a system are variant under the broken symmetry. In Hamiltonian mechanics or Lagrangian Mechanics, this happens when there is at least one term in the Hamiltonian (or Lagrangian) that explicitly breaks the given symmetry. In the Hamiltonian setting, this is often studied when the Hamiltonian can be written . Here is a 'base Hamiltonian', which has some manifest symmetry. More explicitly, it is symmetric under the action of a (Lie) group . Often this is an integrable Hamiltonian. The is a perturbation or interaction Hamiltonian. This is not invariant under the action of . It is often proportional to a small, perturbative parameter. This is essentially the paradigm for perturbation theory in quantum mechanics. An example of its use is in finding the fine structure of atomic spectra. Examples Symmetry breaking can cover any of the following scenarios: The breaking of an exact symmetry of the underlying laws of physics by the apparently random