Circle group

The circle group or Torusgruppe is in mathematics, a group that summarizes the rotations around a fixed point in two-dimensional space ( a plane ) and describes the sequential execution of these rotations. Such rotation can be uniquely described by an angle, the sequential execution of two rotations corresponds exactly to the rotation about the sum of the two angles of the individual spins. One full turn is thereby identified again with no rotation.

Definition over angle

Starting from the idea as a group, the angle with addition, can the circuit group defined as a factor group, that is, two elements that differ by an integer, are identified with each other ( an integer number of full revolutions in intuition ). If you want to draw a direct relation to angles in radians, the definition is also possible.

Example: If you compare the elements of the circle group by representatives represent about as real numbers between zero (inclusive) and one ( exclusively ), it follows, for example:

This design is possible because - as each subgroup, there is abelian - a normal subgroup of is. There is also completed, in turn, is also a topological group inherits the properties such as local compactness and Metrisierbarkeit of.

As a Lie group

The circle group can be defined equivalently as a special orthogonal group, that is, as a set of real matrices of the form

For which it holds with the matrix multiplication as a group link. These are just the rotation matrices in two-dimensional space ( in the -dimensional space ). By means of the coordinates can be any such group element conceived on the unit circle in the two-dimensional plane as a point - the condition says precisely that lies on this circle. The circle - also called 1- sphere - forms a one-dimensional differentiable manifold, as with any such matrix group linking to the structure of the manifold is compatible, so the circle group is a Lie group.

It can be seen that even the group is compact because of the unit circle is a compact subset of the plane.

Since the unit circle can be thought of as a subspace of the real numbers even as a Riemannian submanifold, we obtain an exponential map from the tangent space at the point in the circle group. One identifies with this choice of the Riemannian metric, the elements of the tangent space to canonical way with the real numbers, is a surjective homomorphism even so is a one-parameter group.

The Lie algebra consists of the matrices of the form

Wherein the Lie bracket is defined by the commutator, thus always the same. The exponential map in terms of the theory of Lie groups is given by the Matrixexponential and corresponds exactly to the exponential in terms of the Riemannian geometry.

By means of the exponential map is the group of real numbers with the addition of just the universal covering group of the circle group. It can be concluded that the fundamental group of the circle is the group of integers under addition.

As a unitary group

Alternatively, you can define when the group or the unitary transformations on the one-dimensional vector space of complex numbers, the circle group. These transformations can be concretely as matrices with one entry, that is represented by complex numbers with the usual multiplication:

Applies with the Euler's formula

The figure, which is the imaginary unit is interpreted as Einheitstangentialvektor on the site is just the exponential map. In the Gaussian plane, the multiplication can be understood with just a rotation through the angle. The Lie algebra is in this description of the group from the imaginary numbers.

Characters

The harmonic analysis considers unitary representations of locally compact topological groups, namely continuous homomorphisms of the group into the unitary group on a Hilbert space equipped with the strong operator topology. Based on the generalized Fourier transform of functions on the group means of the irreducible representations of the group is defined. Play a special role in the one-dimensional representations, ie representations in the circle group called characters. These are always irreducible. From the lemma of Schur follows, conversely, that every irreducible, strongly -continuous unitary representation of a locally compact abelian topological group, that is one-dimensional a character. For the abelian case, the Fourier transform is thus reduced to a functional on the characters.

Periodic functions and Fourier series

Periodic functions can be defined as functions on the circle line. If one considers the topological structure, one obtains a natural concept of continuity, one notices also the group structure on the Hair measure a natural Integrierbarkeitsbegriff (alternatively simply on Riemannian manifolds or the Lebesgue integral on the real numbers restricted by the integral term on a closed interval ) and in compliance with the differentiable structure is also a natural Differenzierbarkeitsbegriff.

Since the circle group is abelian, the abstract Fourier transform is given only by characters on the circle group itself. One can show that every character on the circle group is differentiable, thus follows from the Homomorphieeigenschaft

The argument function call. From the periodicity of the function follows that the derivative at the identity element is an integer multiple of its needs, the characters are thus given by

These form an orthonormal basis of the space of square integrable complex-valued functions on the circle group (assuming the measure of all is on standardized), that is, any square-integrable periodic function can be represented by its Fourier transform, which is called in this case Fourier expansion, the inverse transformation can be, since there are only countably many characters that represent a number, the so-called Fourier series. Elementary, that is, without the use of sets of harmonic analysis as the set of Peter -Weyl or the Pontryagin duality, followed by the completeness of the set of Stone - Weierstrass.

Occurrence in physics

In quantum field theory Lagrangians occurring often contain a global gauge symmetry in the shape of the circle group, that is, multiplied to a field at any location with an element of the circle group regarded as a complex number, the Lagrangian and thus the effect will remain unchanged. The Noethertheorem associated to this symmetry provides a conserved quantity, which can be often seen as ( in particular electric ) charge, as well as a locally obtained, that is the equation of continuity sufficient current. The invariance of the Lagrangian means nothing else than that it depends only on the magnitude squares of the respective complex field sizes ( in quantum field theory the fields are finally seen as operator valued distributions, in this case it comes to the square of the amount of their respective operators, ie. of an operator ). Such a gauge symmetry occurs in quantum electrodynamics.