Continuous symmetry
In
motions, as opposed to discrete symmetry, e.g. reflection symmetry
, which is invariant under a kind of flip from one state to another. However, a discrete symmetry can always be reinterpreted as a subset of some higher-dimensional continuous symmetry, e.g. reflection of a 2-dimensional object in 3-dimensional space can be achieved by continuously rotating that object 180 degrees across a non-parallel plane.
Formalization
The notion of continuous symmetry has largely and successfully been formalised in the mathematical notions of
group action
. For most practical purposes, continuous symmetry is modelled by a group action of a topological group that preserves some structure. Particularly, let be a function, and G is a group that acts on X; then a subgroup is a symmetry of f if for all .
One-parameter subgroups
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.
Noether's theorem
Continuous symmetry has a basic role in
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
- Goldstone's theorem
- Infinitesimal transformation
- Noether's theorem
- Sophus Lie
- Motion (geometry)
- Circular symmetry
References
- Barker, William H.; Howe, Roger (2007). Continuous Symmetry: from Euclid to Klein. American Mathematical Society. ISBN 978-0-8218-3900-3.