Given any Hausdorff, locally compact, abelian topological group \(G\), define a character \(\chi:G\to U(1)\) on \(G\) to be any continuous group homomorphism from \(G\) to \(U(1)\), that is for all \(g_1,g_2\in G\), \(\chi(g_1\cdot g_2)=\chi(g_1)\chi(g_2)\) (of course the notation here for the group operation is multiplicative, but additive groups are common too as examples below will demonstrate).
Example: Let \(G=\textbf R\) be the (Hausdorff, locally compact, abelian) additive group of real numbers (e.g. in physics, this might be the group of time translations or spatial translations along a particular direction). If \(\chi:\textbf R\to U(1)\) is a character on \(\textbf R\), then it has to satisfy the functional equation \(\chi(x+y)=\chi(x)\chi(y)\) which clearly has solution \(\chi(x)=e^{\lambda x}\) for some a priori \(\lambda\in\textbf C\). However, imposing the additional \(U(1)\)-requirement that \(|\chi(x)|=1\) forces \(\lambda\in i\textbf R\), i.e. to be of the form \(\lambda=ik\) for some \(k\in\textbf R\). Thus, all characters on \(\textbf R\) are indexed by a real number \(k\in\textbf R\) and are given by \(\chi_k(x)=e^{ikx}\). By contrast, if we had used \(G=\textbf Z\), then characters \(\chi:\textbf Z\to U(1)\) on \(\textbf Z\) would be of the form \(\chi_{\theta}(n)=e^{in\theta}\) where we note that \(\chi_{\theta+2\pi}(n)=\chi_{\theta}(n)\).
Example: Let \(G=U(1)\cong SO(2)\cong S^1\cong\textbf R/\textbf Z\) be the (Hausdorff, locally compact, abelian) multiplicative group of unit complex numbers in the complex plane \(\textbf C\). A character \(\chi:U(1)\to U(1)\) must now obey \(\chi(zw)=\chi(z)\chi(w)\). Clearly the solution is of the form \(\chi_n(z)=z^n\) for some a priori \(n\in\textbf C\). However, writing \(z=e^{i\theta}\), it becomes clear that \(z^n=e^{in\theta}\) can only remain in \(U(1)\) iff \(n\in\textbf R\). However, in this case there is even a further restriction on \(n\) which comes from the periodic nature of the circle group \(U(1)\), that is, because \(e^{2\pi i}=1\), we must have \(\chi(e^{2\pi i})=\chi(1)\) so \(e^{2\pi i n}=1\). This will be true iff \(n\in\textbf Z\). Notice that these characters \(\chi_n(\theta)=e^{in\theta}\) on \(U(1)\) correspond precisely to the representation-theoretic characters of the \(1\)D complex unitary irreducible representations \(\phi_n:U(1)\to GL(\textbf C)\) of \(U(1)\), namely \(\phi_n(e^{i\theta})(z):=e^{in\theta}z\) (or more commonly, by identifying \(GL(\textbf C)\cong\textbf C-\{0\}\) just \(\phi_n(e^{i\theta})=e^{in\theta}\) with \(\chi_n(\theta)=\text{Tr}(\phi_n(e^{i\theta}))=e^{in\theta}\), the point being to notice their quantization \(n\in\textbf Z\).
Example: Let \(G=C_n\) be the (Hausdorff, locally compact, abelian) cyclic group (using the discrete topology) of order \(|C_n|=n\). If one thinks of \(C_n\cong\textbf Z/n\textbf Z\), then one has \(n\) characters indexed by \(\ell=0,1,2,…,n-1\) of the form \(\chi_{\ell}(m)=e^{2\pi i\ell m/n}\). On the other hand, if one thinks of \(C_n\cong\{z\in\textbf C:z^n=1\}\) as the \(n\)-th roots of unity in the complex plane \(\textbf C\), so this time as a multiplicative group rather than the additive group \(\textbf Z/n\textbf Z\), then the \(n\) characters become \(\chi_{\ell}(z)=z^{\ell}\) again for \(\ell=0,1,2,…,n-1\).
This discussion naturally leads to the following definition:
Definition: Given a Hausdorff, locally compact, abelian topological group \(G\), the Pontryagin dual \(\hat G\) of \(G\) is defined to be the set of all characters on \(G\):
\[\hat G:=\text{Hom}(G\to U(1))\]
Because \(U(1)\) is abelian, clearly \(\hat G\) is also an abelian group. It is also easy to show that, upon endowing \(\hat G\) with the compact-open topology (i.e. the topology of local uniform convergence on compact subspaces), the Pontryagin dual \(\hat G\) also inherits the same topological properties as \(G\) namely that \(\hat G\) also becomes a Hausdorff and locally compact topological group (this is reassuring as otherwise how could they dualize each other? Also, it means we’re permitted to compute the double Pontryagin dual \(\hat{\hat G}\) of \(G\) whose isomorphism with \(G\) is the essence of Pontryagin duality). More precisely, as the “hat” notation suggests, \(G\) and \(\hat G\) will have a “Fourier-like” duality between them.
Example: As the examples above show, one has \(\hat{\textbf R}\cong\textbf R\), \(\hat{\textbf Z}\cong U(1)\), \(\hat{U(1)}\cong\textbf Z\), and \(\hat{C_n}\cong C_n\). Thus, both \(\textbf R\) and \(C_n\) are self-dual while \(\textbf Z\) and \(\textbf U(1)\) are Pontryagin duals of each other. More generally, it is easy to see how \(\hat{G_1\times G_2}\cong \hat G_1\times\hat G_2\) for any two Hausdorff, locally compact abelian topological groups \(G_1,G_2\), so by induction one also computes Pontryagin dualities such as \(\hat{\textbf R^n}\cong\textbf R^n\) for Euclidean spaces, \(\hat{\textbf Z^n}\cong U(1)^n\) for standard lattices, \(\hat{U(1)}^n\cong\textbf Z^n\) for torii, etc. In general, these examples suggest that for any Hausdorff, locally compact abelian topological group \(G\), one has the phenomenon of Pontryagin duality:
\[\hat{\hat{G}}\cong G\]
where the isomorphism is canonical (just like in linear algebra the double algebraic dual \(V^{**}\cong V\) by the canonical isomorphism \(\textbf v(L):=L(\textbf v)\), here the isomorphism is \(g(\chi):=\chi(g)\) where \(g\in G\cong\hat{\hat{G}}\) and \(\chi\in\hat G\)). Put more abstractly, the Pontryagin dualization functor \(\hat{}\) is involutory.
One of the main reasons for constructing all this machinery with the Pontryagin dual \(\hat G\) is to be able to do Fourier analysis on any Hausdorff, locally compact, abelian topological groups \(G\) in addition to the classical case \(G=\textbf R^n\). Let me explain.
In general, the idea is that given any absolutely integrable complex-valued function \(f\in L^1(G\to\textbf C)\) on a Hausdorff, locally compact, abelian topological group \(G\), the Fourier transform \(\hat f:\hat G\to\textbf C\) of \(f\) maps from the Pontryagin dual \(\hat G\) of \(G\) to \(\textbf C\). It is defined by integrating \(f\) over \(G\) against the (projectively) unique Haar measure \(\mu:\sigma_G\to[0,\infty])\) on \(G\), that is to say, the unique translationally invariant Radon measure \(\mu(gH)=\mu(H)\) for all Borel sets \(H\subseteq G\) and \(g\in G\) (note that \(H\) is not necessarily a subgroup of \(G\), and so \(gH\) cannot necessarily be interpreted as a coset of \(H\), though it is in that spirit).
\[\hat f(\chi):=\int_{g\in G}f(g)\overline{\chi(g)}d\mu(g)\]
for all characters \(\chi\in\hat G\) on \(G\). Note that because \(G\) is abelian so that \(gH=Hg\), the left-invariant and right-invariant Haar measures on \(G\) coincide. Also, one can express this more concisely using the (physicist’s convention for the) inner product inherited from the complex Hilbert space \(L^2(G\to\textbf C)\):
\[\hat f(\chi)=\langle\chi|f\rangle_{L^2(G\to\textbf C)}\]
where there is no need to subscript by the Haar measure \(\mu\) because it is understood to be unique once the (Hausdorff, locally compact, abelian) topological group \(G\) is fixed. One can then show that the inverse Fourier transform is given (via unitarity? I think?):
\[f(g)=\langle g|\hat f\rangle_{L^2(\hat G\to\textbf C)}\]
with \(g=g(\chi)\) given by the canonical isomorphism of Pontryagin duality (maybe? Should the ordering of \(g\) and \(\hat f\) be swapped?).
Example: If \(G=\textbf R^n\), then characters on \(\textbf R^n\) are complex plane waves \(\chi_{\textbf k}(\textbf x)=e^{i\textbf k\cdot\textbf x}\) indexed by angular wavevector \(\textbf k\in\hat{\textbf {R}^n}}\cong\textbf R^n\) so:
\[\hat f(\textbf k)=\int_{\textbf x\in\textbf R^n}f(\textbf x)e^{-i\textbf k\cdot\textbf x}d^n\textbf x\]
In quantum mechanics, the additional physics of the de Broglie relation \(\textbf p=\hbar\textbf k\) ties the position \(\textbf X\) and momentum \(\textbf P\) observables as roughly speaking living in groups that are Pontryagin-dual to each other.
Another (unrelated?) fact I discovered which I thought was interesting in this regard is that if \(f(\textbf x)=f(|\textbf x|)=f(r)\) is isotropic about the origin \(r=0\) (such as an \(s\) atomic orbital in a hydrogen atom), then its Fourier transform \(\hat f(\textbf k)=\hat f(|\textbf k|)=\hat f(k)\) will also be isotropic in Fourier space, so rather than performing an \(n\)-dimensional integral over \(\textbf R^n\) to compute the Fourier transform of \(f\), one can get away with performing a one-dimensional integral in the radial coordinate \(dr\) via a Hankel transform. Specifically, using the unitary version of the Fourier transform:
\[\hat f(k)=\frac{1}{(2\pi)^{n/2}}\int_{\textbf x\in\textbf R^n}f(\textbf x)e^{-i\textbf k\cdot\textbf x}d^n\textbf x\]
The hypervolume element is \(d^n\textbf x=r^{n-1}d\Omega dr\) and the dot product is \(\textbf k\cdot\textbf x=kr\cos\angle\textbf k,\textbf x\) so:
\[\hat f(k)=\frac{1}{(2\pi)^{n/2}}\int_0^{\infty}r^{n-1}f(r)\int_{S^{n-1}}e^{-ikr\cos\angle\textbf k,\textbf x}d\Omega dr\]
Using standard integral representations of the Bessel functions, the final result is thus:
\[\hat f(k)=k^{-(n/2-1)}\int_0^{\infty}f(r)J_{n/2-1}(kr)r^{n/2}dr\]
For instance, given a Coulombic potential\(\phi(r)=1/r\) in \(\textbf R^3\), the Fourier transform \(\hat{\phi}\) is (in a suitable sense):
\[\hat{\phi}(k)=\frac{\sqrt{2/\pi}}{k^2}\]
Example: If \(G=U(1)\), then characters on \(U(1)\) are quantized by integers \(n\in\textbf Z\) where \(\chi_n(\theta)=e^{in\theta}\). This means that any map \(f(\theta)\) on the circle has a \(2\pi\)-periodic Fourier series:
\[f(\theta)=\sum_{n=-\infty}^{\infty}\hat f_ne^{in\theta}\]
where the Fourier coefficients are given by the Fourier transform:
\[\hat f_n=\int_0^{2\pi}f(\theta)e^{-in\theta}d\mu(\theta)\]
where in this case the normalized Haar measure on \(U(1)\) is \(d\mu(\theta)=d\theta/2\pi\). Parameterizing by \(z=e^{i\theta}\) manifestly shows (via the generalized Cauchy integral formula) that the Fourier coefficients are just the coefficients of the Laurent series of \(f\) at the origin in the variable \(z=e^{i\theta}\).
Example: For \(G=\textbf Z\), the characters become \(\chi_{\theta}(n)=e^{in\theta}\) and the Fourier transform of a complex-valued “sequence” \((f_n)_{n=-\infty}^\infty\) now looks like:
\[\hat f(\theta)=\int_{n\textbf Z}f_ne^{-in\theta}d\mu(n)\]
Now comes the subtle point, which is that the appropriate Haar measure \(\mu\) to use for a discrete group like the integers \(\textbf Z\) is the counting measure \(\mu(H):=|H|\) that just assigns the cardinality \(|H|\) to any collection \(H\) of integers. Essentially, the discrete nature of the counting measure ends up converting the integral into a series so that:
\[\hat f(\theta)=\sum_{n=-\infty}^{\infty}f_ne^{-in\theta}\]
One then has the inverse Fourier transform for the original sequence \((f_n)_{n=-\infty}^\infty\):
\[f_n=\frac{1}{2\pi}\int_0^{2\pi}\hat f(\theta)e^{in\theta}d\theta\]