In
mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some
symmetries as
motion
In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and m ...
s, as opposed to
discrete symmetry
In mathematics and geometry, a discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square' ...
, e.g.
reflection symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.
In 2D the ...
, 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
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
,
Lie group and
group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
. 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
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
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
Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
in
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, 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
*
Goldstone's theorem
*
Infinitesimal transformation
In mathematics, an infinitesimal transformation is a limiting form of ''small'' transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space. This is conventionally represented by a 3×3 s ...
*
Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
*
Sophus Lie
*
Motion (geometry)
In geometry, a motion is an isometry of a metric space. For instance, a plane equipped with the Euclidean distance metric is a metric space in which a mapping associating congruent figures is a motion. More generally, the term ''motion'' is a s ...
*
Circular symmetry
References
*
{{DEFAULTSORT:Continuous Symmetry
Symmetry
Lie groups
Group actions (mathematics)