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For a set of shifts $H$ and a map $\theta$ into a circle group, this additive energy counts quadruples whose shift sums and frequency sums both agree. It is the additive energy of the graph of $\theta$ in the product abelian group.

For the block embedding $SU(m)\times SU(n)\subset SU(m+n)$ with $m,n\ge 2$, the centralizer is the block-scalar subgroup $C=\{\mathrm{diag}(z{I}_m,w{I}_n):|z|=|w|=1,z^m{w}^n=1\}$. Schur lemma makes the diagonal blocks scalar and kills the off-diagonal intertwiners, since the two blocks are inequivalent representations of the product group.

Put $d=\gcd (m,n)$, $a=m/d$, $b=n/d$. The map $(z,w)\mapsto z^a{w}^b$ has image the $d$th roots of unity and connected kernel $(z,w)=(e^{ib\theta },e^{-ia\theta })$. A Bezout identity completes $(a,b)$ to an invertible integer coordinate change on the two-dimensional torus, proving $C\cong U(1)\times {\mathbb{Z}}_d$. Thus the full centralizer is a circle group exactly when $m,n$ are coprime. For $m=n=2$, $(zw{)}^2=1$ gives two circles, a useful counterexample to the unrestricted connectedness claim.

The character group of a topological group that is an abelian group consists of its continuous group homomorphisms to the circle group, with pointwise multiplication. For a finite group, all characters are continuous and their values are roots of unity, so this agrees with the character group of a finite abelian group.

Points ${\theta }_r$ have circular spacing at least $\delta$ when $\Vert {\theta }_r-{\theta }_s\Vert \ge \delta$ for distinct $r,s$, where $\Vert t\Vert ={\min }_{k\in \mathbb{Z}}|t-k|$. The distance is measured on the circle group $\mathbb{R}/\mathbb{Z}$, so points near zero and one can be close.

For $X\ne 0$, the image of $t\mapsto e^{tX}$, with its intrinsic immersed Lie subgroup topology, is isomorphic to $(\mathbb{R},+)$ or the circle group. The kernel is respectively zero or $\tau \mathbb{Z}$ for a least positive period $\tau$. The case $X=0$ is the trivial group. An irrational winding in a two-dimensional torus need not be an embedded or closed subgroup, so its intrinsic topology must be distinguished from the subspace topology.

A compact group is a topological group whose underlying topological space is a compact Hausdorff space. The circle group is a compact group: the unit circle is a closed bounded subset of the complex plane, so the Heine-Borel theorem gives compactness. The additive group $(\mathbb{R},+)$ is a locally compact group and a Hausdorff space, but is not a compact group: the open cover $\{(-n,n):n\ge 1\}$ has no finite subcover. A finite group with the discrete topology is another compact group; compactness does not require a smooth manifold structure.

The great advantage of a compact group is that one can average over every element without choosing a preferred origin. Precisely, every locally compact group that is a Hausdorff space admits a nonzero left Haar measure $\mu$, a regular Borel measure finite on compact sets, unique up to a positive scalar. Left invariance means $\mu (gA)=\mu (A)$ for every Borel set $A$ and every $g\in G$. There is also a right Haar measure; in general left and right Haar measures can differ. On a compact group, $0<\mu (G)<\infty$, and normalizing to $\mu (G)=1$ gives a unique regular Borel probability measure invariant under both left and right translation. See Tornier, Haar Measures, Theorem 2.2 for existence and uniqueness, and Ram's notes on left and right Haar measure for the distinction.

For a finite group $G$, normalized Haar measure assigns each element mass $1/|G|$, so integration is simply $|G{|}^{-1}{\sum }_{g\in G}f(g)$. For the circle group, the corresponding integral is uniform angle averaging, computed below. On $(\mathbb{R},+)$, Lebesgue measure is a Haar measure, but its total mass is infinite: no positive scalar multiple makes it a probability measure. Thus the existence of Haar measure does not make every topological group a compact group.

Historically, Alfréd Haar introduced the general invariant-measure construction in Der Massbegriff in der Theorie der kontinuierlichen Gruppen, Annals of Mathematics 34 (1933), pp. 147-169. This bibliographic identification is also recorded in André Weil's archived seminar notes, Mesure de Haar, reference Haa33. The conceptual step is to let group symmetry itself determine the weights used in integration, rather than relying on coordinates in an ambient Euclidean space.

For $X=\left(\begin{array}{cc}a& b\\ c& -a\end{array}\right)$, the image of its matrix exponential is compact precisely when $a^2+bc<0$ or $X=0$. If $a^2+bc=-{\omega }^2<0$, then $e^{tX}=\cos (\omega t)I_2+\sin (\omega t)X/\omega$ and its image is conjugate to the circle group. A nonzero nilpotent generator or a generator with real nonzero eigenvalues produces an unbounded image.

A continuous unitary character is a continuous group homomorphism from a topological group to the unit circle with multiplication. For abelian groups, multiplication of continuous unitary characters is pointwise. Such continuous unitary characters must be distinguished from the traces of higher-dimensional group representations, called characters of a representation, and from characters of an algebra.

For every integer $n$, the map ${\chi }_n(z)=z^n$ is a continuous unitary character of the circle group: ${\chi }_n(zw)={\chi }_n(z){\chi }_n(w)$ and $|{\chi }_n(z)|=1$. By contrast, the two-dimensional group representation $\rho (z)=\mathrm{diag}(z,z^{-1})$ has character of a representation $\mathrm{tr}\rho (e^{i\theta })=2\cos \theta$. Its value at the identity element is $2$, so this trace is not a continuous unitary character.

Consider two real scalar fields $\mathbf{\varphi }=({\varphi }_1,{\varphi }_2)$ with positive $\lambda ,v$, standard kinetic terms, and the displayed scalar potential. At $h=0$, the circle group rotates one field into the other: this is a global symmetry in field theory, with the same rotation at every point in space. The vacuum manifold is the circle ${\mathbf{\varphi }}^2=v^2$. Choose $(v,0)$ and differentiate the scalar potential twice:The scalar mass matrix assigns a restoring force to the radial displacement, whereas its tangent is one of the Goldstone directions in the scalar mass matrix. Moving around the circle costs no potential energy; moving between neighbouring points in space can still cost gradient energy. Flat angular potential energy therefore means a soft collective excitation, not that every spatial pattern is free.

A positive source $h$ explicitly selects the positive ${\varphi }_1$ direction. The minimum is $(w,0)$, where $w>v$ is the positive solution of $\lambda w(w^2-v^2)=h$. The two eigenvalues of the scalar mass matrix there areFor example, $\lambda =v=1$ and $h=0.15$ give $w\simeq 1.06792$, $m_{\mathrm{radial}}^2\simeq 2.42138$ and $m_{\mathrm{angular}}^2\simeq 0.140459$. The light angular excitation becomes a Goldstone boson as $h\downarrow 0$, while the radial mass stays finite. The source changes both the preferred direction and the radius; expanding about the old point $(v,0)$ at nonzero $h$ would miss that shift.

For finite complex measures on the additive circle group, convolution is the pushforward measure of their product under addition modulo $2\pi$. EquivalentlyBounded tests are integrable against the product variation measure, so Fubini's theorem applies. Testing the factorization $e^{-in(x+y)}=e^{-inx}{e}^{-iny}$ proves the displayed Fourier coefficient of a measure on the circle identity. The convolution of two probability measures is again a probability measure. On these measures convolution preserves simultaneous convergence in the weak-star topology: multiply the converging coefficients and use their common unit variation bound and dense trigonometric tests.

A covering homomorphism is a surjective group homomorphism between topological groups whose underlying map is a covering map. Its group kernel is the discrete fibre above the identity element. Group translations identify every other fibre with a coset of that group kernel.

For example, the group homomorphism $\mathbb{R}\to S^1$, $t\mapsto e^{2\pi it}$, is a covering map from the additive real numbers to the circle group, with group kernel $\mathbb{Z}$. Small open arcs lift to disjoint intervals, while adding an integer moves between the sheets. A surjective group homomorphism alone need not be a covering map; the local-sheet condition is additional topological information.

In four spacetime dimensions, the pure local gauge anomaly for a circle group is proportional to the sum of cubed U(1) charges over left-handed Weyl fermions, counting a right-handed Weyl fermion through its left-handed charge conjugate. A consistent anomaly-free gauge theory requires this sum to vanish, absent an additional anomaly-cancelling mechanism. Cancellation of the mixed gauge-gravitational anomaly additionally requires ${\sum }_a{Q}_a=0$. If the same spectrum is restricted to a finite cyclic group, $q_a=Q_a\mathrm{mod}n$ implies ${\sum }_a{q}_a^3=0\mathrm{mod}n$ and, when imposed, ${\sum }_a{q}_a=0\mathrm{mod}n$. These elementary consequences alone do not classify intrinsic discrete gauge anomalies or guarantee an anomaly-free integer lift.

Suppose fields with integer U(1) charges $Q_i$ acquire nonzero vacuum expectation values, with at least one $Q_i\ne 0$. The stabilizer in the circle group is exactly ${\mathbb{Z}}_d$, where $d=\gcd (|Q_1|,\dots ,|Q_r|)$. Indeed, $e^{i\theta }$ preserves every condensate if and only if $e^{i{Q}_i\theta }=1$ for every $i$. The Bezout identity implies $e^{id\theta }=1$, and conversely $d\mid Q_i$ proves every $d$th root of unity preserves every condensate. Neutral fields impose no constraint; if all condensing fields are neutral, the whole circle group survives.

For a finite complex measure on the additive circle group $\mathbb{R}/(2\pi \mathbb{Z})$, its Fourier coefficients are integrals against the group characters $e^{-inx}$, $n\in \mathbb{Z}$. They obey $|\hat{\mu }(n)|\le \Vert \mu \Vert$, where the norm is the total variation norm of a measure. A probability measure has zeroth coefficient one. The normalized Haar measure $m(dx)=dx/(2\pi )$ has all nonzero-frequency coefficients zero. A bounded family of finite Radon measures converges in the weak-star topology exactly when its coefficients converge to those of the proposed limit: extend linear combinations to all continuous tests using the density of trigonometric polynomials and the common variation bound.

The scalar circle group action on ${\mathbb{C}}^{n+1}$ with the standard symplectic form has moment map $\mu =-|z{|}^2/2$ in the convention ${\iota }_{X_H}\omega =dH$. Reduction of the radius-$R$ sphere gives Complex projective space and the form ${\omega }_R=(i{R}^2/2)\partial \bar{\partial }\log (1+|w{|}^2)$. Its area on a complex projective line is $\pi R^2$. Thus $R=\sqrt{2}$ yields the Fubini-Study form normalized to projective-line area $2\pi$, while $R=1$ yields half that form. The Hopf fibration identifies the quotient.

An internal circle group action with spacetime-independent parameter $\alpha$. For canonical complex scalar fields, Noether's theorem gives $j^{\mu }=i{\sum }_j{q}_j{\phi }_j^{†}\overleftrightarrow{{\partial }^{\mu }}{\phi }_j$. The Noether charge of the particle annihilated by ${\phi }_j$ is $q_j$, and its antiparticle has charge $-q_j$.

For $\mathcal{L}={\partial }_{\mu }{\psi }^{\ast }{\partial }^{\mu }\psi -V(|\psi {|}^2)$, constant $\alpha$ preserves both terms. This is an internal symmetry of a classical field theory with circle group $U(1)$. A spacetime-dependent phase would introduce derivative terms, so the ordinary-derivative density has only the global symmetry. The associated current is the Noether charge of a complex scalar field construction; the orientation $e^{-i\alpha }$ fixes its overall sign.

If the Hamiltonian commutes with total $S^z$, simultaneous spin rotations $e^{-i\varphi S_{\mathrm{tot}}^z}$ form a circle group symmetry. In a state of definite total $S^z$, the transverse one-point averages vanish because $S^x\pm i{S}^y$ change that eigenvalue. A chosen classical orientation or symmetry-breaking limit requires separate care when constructing a linear spin-wave approximation.

A left Haar measure on a locally compact group that is a Hausdorff space is a nonzero regular Borel measure, finite on compact sets, satisfying $\mu (gA)=\mu (A)$ for every Borel set $A$. Such a measure exists and is unique up to a positive scalar. Right Haar measure instead satisfies $\mu (Ag)=\mu (A)$; the two need not coincide. On a compact group, normalization gives total mass one and invariance on both sides. See Tornier, Haar Measures, Definition 2.1 and Theorem 2.2.

For a finite group, normalized Haar measure is uniform counting; for the circle group, it is normalized angle integration on the circle. These examples turn symmetry into a rule for averaging.

This is a smoothly varying positive-definite Hermitian form on each fiber, without requiring a holomorphic base or bundle. A partition of unity glues local positive metrics into a global one. Applying smooth Gram-Schmidt to local frames then makes the transitions unitary, reducing the structure group of a vector bundle to $U(k)$. For a line bundle its unit vectors form a principal bundle for the circle group.