In
mathematics, 'continuous symmetry' is an intuitive idea corresponding to the concept of viewing some
symmetries as
motions, as opposed to e.g.
reflection symmetry, which is invariance under a kind of flip from one state to another. It has largely and successfully been formalised in the mathematical notions of
topological group,
Lie group and
group action. For most practical purposes continuous symmetry is modelled by a ''group action'' of a topological group.
The simplest motions follow a
one-parameter subgroup of a Lie group, such as the
Euclidean group of
three-dimensional space. For example
translation parallel to the ''x''-axis by ''u'' units, as ''u'' varies, is a one-parameter group of motions. Rotation around the ''z''-axis is also a one-parameter group.
Continuous symmetry has a basis role in
Noether's theorem in
theoretical physics, in the derivation of
conservation laws from symmetry principles, specifically for continuous symmetries. The search for continuous symmetries only intensified with the further developments of
quantum field theory.
See also
★
Infinitesimal transformation
★
Sophus Lie
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