District group

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The execution of rotations one after the other corresponds to the addition of angles, here: 150 ° + 270 ° = 420 ° = 60 °

In mathematics , the circle group or torus group is a group that summarizes the rotations around a fixed point in two-dimensional space (a plane) and describes the execution of these rotations one after the other . Such a rotation can be clearly described by an angle , the execution of two rotations one after the other corresponds to the rotation around the sum of the two angles of the individual rotations. A full turn is again identified with no turn. ${\displaystyle \mathbb {S} }$ ${\displaystyle \mathbb {T} }$

Definition by angle

Based on the idea of ​​a group of angles with addition, the circle group can be defined as a factor group, i.e. two elements that differ by an integer (in the illustration an integer number of full revolutions) are identified with one another. If you want to draw a direct reference to angles in radians , the definition is also possible. ${\displaystyle \mathbb {S} :=\mathbb {R} /\mathbb {Z} }$ ${\displaystyle \mathbb {S} :=\mathbb {R} /(2\pi \mathbb {Z} )}$

Example: If the elements of the circle group are represented by representatives , for example as real numbers between zero (including) and one (excluding), the result is, for example:

${\displaystyle 0{,}3+0{,}4=0{,}7\,,\quad 0{,}3+0{,}8=0{,}1\,,\quad 0{,}4+0{,}6=0{,}0}$ (the pre-comma part is omitted).

This construction is possible because - like every subgroup , since it is abelian - is a normal subgroup of . Since it is also closed , there is also a topological group that inherits properties such as local compactness and metrisability from . ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {R} /\mathbb {Z} }$ ${\displaystyle \mathbb {R} }$

As a Lie group

The circle group can equivalently be defined as a special orthogonal group , i.e. H. as a set of real matrices of the form ${\displaystyle SO(2)}$

${\displaystyle {\begin{pmatrix}a&b\\-b&a\end{pmatrix}}}$,

for which applies, with the matrix multiplication as a group link. These are precisely the rotation matrices in two-dimensional space ( in -dimensional space). By means of the coordinates , each such group element can be understood as a point on the unit circle in the two-dimensional plane - the condition says that it lies on this circle. The circle - also called 1-sphere - forms a one-dimensional differentiable manifold , as with every such matrix group, the connection with the structure of the manifold is compatible, therefore the circle group forms a Lie group . ${\displaystyle a^{2}+b^{2}=1}$ ${\displaystyle SO(n)}$ ${\displaystyle n}$ ${\displaystyle (a,b)}$ ${\displaystyle a^{2}+b^{2}=1}$ ${\displaystyle (a,b)}$

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

Since the unit circle as a subspace of the real numbers can even be understood as a Riemannian submanifold , one obtains an exponential mapping from the tangential space in the point into the circle group. If one identifies the elements of the tangent space in a canonical way with the real numbers with this choice of the Riemannian metric, then it is even a surjective homomorphism , i.e. it becomes a one-parameter group . ${\displaystyle \operatorname {exp} }$ ${\displaystyle (1,0)}$ ${\displaystyle \operatorname {exp} \colon \mathbb {R} \to \mathbb {S} }$ ${\displaystyle \mathbb {S} }$

The Lie algebra consists of matrices of the form

${\displaystyle {\begin{pmatrix}0&a\\-a&0\end{pmatrix}}}$,

where the Lie bracket is given by the commutator, i.e. it is always the same . The exponential mapping in the sense of the theory of Lie groups is given by the matrix exponential and corresponds exactly to the exponential mapping in the sense of Riemannian geometry. ${\displaystyle 0}$

By means of the exponential mapping, the group of real numbers with the addition is precisely the universal superposition group of the circle group. From this it can be concluded that the fundamental group of the circle is the group of whole numbers with addition.

As a unitary group

→ Main article : U (1)

Alternatively, the circle group can be defined as the group or the unitary transformations on the one-dimensional vector space of the complex numbers . These transformations can be concretely used as matrices with one entry, i.e. H. represent by complex numbers with the usual multiplication: ${\displaystyle U(\mathbb {C} )}$ ${\displaystyle U(1)}$

${\displaystyle \mathbb {S} =U(\mathbb {C} )=U(1)=\left\{c\in \mathbb {C} \mid |c|=1\right\}}$

With Euler's formula, the following applies

${\displaystyle \mathbb {S} =\left\{e^{\mathrm {i} \varphi }\mid \varphi \in \mathbb {R} \right\}}$.

The mapping , where the imaginary unit is interpreted as a unit tangential vector at the point , is precisely the exponential mapping . In the Gaussian plane of numbers , the multiplication with can be understood as a rotation around the angle . The Lie algebra in this description of the group consists of the imaginary numbers . ${\displaystyle \mathrm {i} \varphi \mapsto e^{\mathrm {i} \varphi }}$ ${\displaystyle \mathrm {i} }$ ${\displaystyle 1}$ ${\displaystyle e^{\mathrm {i} \varphi }}$ ${\displaystyle \varphi }$

Characters

→ Main article : Character (math)

Concept of character

The harmonic analysis considers unitary representations of locally compact topological groups, i.e. H. continuous homomorphisms from the group to the unitary group over a Hilbert space provided with the strong operator topology . Building on this, the generalized Fourier transformation of functions on the group is defined using the irreducible representations of the group. The one-dimensional representations play a special role, i. H. Representations in the circle group, called characters . These are always irreducible. From the Schur's Lemma follows, conversely, that every irreducible , strong-continuous unitary representation of an Abelian locally compact topological group, so a character is one-dimensional. For the Abelian case, the Fourier transformation is reduced to a functional on the characters.

Circle group characters

On the one hand the circle group is used to define the character, on the other hand the circle group also has characters. The characters of the circle group are precisely the continuous homomorphisms , and they can all be stated. Each character of has the shape for one . Therefore one can identify the amount of characters with . It is no coincidence that the set of characters has a group structure again; it is a special case of the more general Pontryagin duality . ${\displaystyle {\mathbb {T} }}$ ${\displaystyle {\mathbb {T} }\rightarrow {\mathbb {T} }}$ ${\displaystyle {\mathbb {T} }}$ ${\displaystyle \chi _{n}(z)=z^{n}}$ ${\displaystyle n\in {\mathbb {Z} }}$ ${\displaystyle \mathbb {Z} }$

Periodic functions and Fourier series

Periodic functions can be defined as functions on the circular line. If one observes the topological structure, one obtains a natural concept of continuity, one also observes the group structure, a natural integrability concept via the hair measure (alternatively also simply via the integral concept on Riemannian manifolds or the Lebesgue integral on the real numbers restricted to a closed interval) and taking into account the differentiable structure also a natural concept of differentiability.

Since the circle group is Abelian, the abstract Fourier transformation is given solely by characters on the circle group itself. One can show that every character on the circle group is differentiable, thus follows from the homomorphic property ${\displaystyle \chi \colon \mathbb {S} \to \mathbb {S} }$

${\displaystyle \chi (x)=\operatorname {exp} (\operatorname {arg} (x)(D\chi )(1))}$,

where denotes the argument function. From the periodicity of the function it follows that the derivative of the neutral element must be an integral multiple of , so the characters are given by ${\displaystyle \operatorname {arg} }$ ${\displaystyle \mathrm {i} }$

${\displaystyle \chi _{m}(x):=\operatorname {exp} (m\operatorname {arg} (x)\mathrm {i} )=x^{m},\ m\in \mathbb {Z} }$.

These form an orthonormal basis of the space of the square-integrable complex-valued functions on the circle group (provided the measure from whole is normalized to), i.e. H. Every square-integrable periodic function can be represented by its Fourier transform, which in this case is called the Fourier expansion ; the inverse transform can be represented as a series, the so-called Fourier series, since there are only countably many characters. Elementary, d. H. Without using sentences from harmonic analysis like the Peter-Weyl theorem or the Pontryagin duality , completeness follows from the Stone-Weierstrass theorem . ${\displaystyle L^{2}(\mathbb {S} )}$ ${\displaystyle \mathbb {S} }$ ${\displaystyle 1}$

Occurrence in physics

In the quantum field occurring Lagrangians often contain a global gauge symmetry in shape of the circular array, d. H. If you multiply a field at each point with an element of the circle group understood as a complex number, the Lagrangian and thus the effect remain unchanged. The Noether's theorem provides an associated to this symmetry conservation size , often as (in particular electrical can be considered) 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 only depends on the absolute squares of the respective complex field quantities (in quantum field theory the fields are finally understood as operator-valued distributions , in this case it is about the square of the absolute value of the respective operators, i.e. for one Operator ). Such a gauge symmetry occurs in quantum electrodynamics . ${\displaystyle A^{*}A}$ ${\displaystyle A}$

Individual evidence

1. ^ Mitsuo Sugiura: Unit Representations and Harmonic Analysis . 2nd Edition. North-Holland, Amsterdam 1990, ISBN 0-444-88593-5 , pp. 12 .