Invariance of an action, equation or configuration under a translation. For a continuous spatial translation symmetry, Noether's theorem supplies the conserved momentum; spacetime translation gives the canonical stress-energy tensor.
A discrete symmetry has transformations separated from the identity rather than connected to it by continuously varying symmetry parameters.
A continuous symmetry belongs to a group whose parameters can vary continuously. Spontaneous breaking of such a symmetry can produce Goldstone bosons.
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In physics, symmetry refers to a property or characteristic of a system that remains invariant under certain transformations. This can involve spatial transformations, such as translations, rotations, and reflections, as well as other types such as time reversibility or particle interchange. Symmetry plays a critical role in understanding physical laws and phenomena, often leading to conservation laws and simplifying complex problems.