# More with prime accents

References:

The real Heisenberg group ${\mathrm{Hs}}_{n}\left(\mathbf{R}\right)$ is the analytic Cartesian product $\mathbf{T}×{\mathbf{R}}^{n}×{\mathbf{R}}^{n}$ of the group $\mathbf{T}$ of complex numbers of absolute value $1$ with two copies of the $n$-dimensional real row space ${\mathbf{R}}^{n}$, under the group law $\left(u,x\right)·\left(v,y\right)=\left(u\phantom{\rule{0.3em}{0ex}}v\phantom{\rule{0.3em}{0ex}}b\left(x,y\right),\phantom{\rule{0.6em}{0ex}}\phantom{\rule{0.6em}{0ex}}x+y\right)\phantom{\rule{0.6em}{0ex}},$ where, for $x=\left(x\prime ,\phantom{\rule{0.3em}{0ex}}x″\right)$, $y=\left(y\prime ,\phantom{\rule{0.3em}{0ex}}y″\right)$ in W = ${\mathbf{R}}^{n}×{\mathbf{R}}^{n}$, $b$ is the $\mathbf{T}$-valued bi-additive function on $W$ defined by $b\left(x,y\right)=\mathbf{e}\left(x\prime {}^{t}y″\right)\phantom{\rule{0.6em}{0ex}},$ with ${}^{t}\left(\right)$ denoting transpose and $\mathbf{e}\left(t\right)$ denoting $\phantom{\rule{0.1em}{0ex}}\mathrm{exp}\left(2\pi it\right)$. The Schrödinger representation $U$ of ${\mathrm{Hs}}_{n}\left(\mathbf{R}\right)$ is its representation in the Hilbert space ${L}^{2}\left({\mathbf{R}}^{n}\right)$ that sends $\left(t,x\right)$ (for $t\in \mathbf{T}$, $x\in W$) to the operator $\Phi \to \Phi \prime =U\left(t,\phantom{\rule{0.3em}{0ex}}x\right)\Phi$, where $\begin{array}{cc}\text{(1)}& \Phi \prime \left(z\right)=t\mathbf{e}\left(x″\phantom{\rule{0.3em}{0ex}}{}^{t}z\right)\phantom{\rule{0.3em}{0ex}}\Phi \left(z+x\prime \right)\phantom{\rule{1em}{0ex}}for\phantom{\rule{1em}{0ex}}t\in \mathbf{T}\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.6em}{0ex}}x\in W\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.6em}{0ex}}z\in {\mathbf{R}}^{n}\phantom{\rule{0.6em}{0ex}}\text{.}\end{array}$

Scratch area

Primes as operators: $x=\left(x\prime ,\phantom{\rule{0.3em}{0ex}}x″\right)$

Ascii primes inside mi: $x=\left(\mathrm{x\text{'}},\phantom{\rule{0.3em}{0ex}}\mathrm{x\text{'}\text{'}}\right)$

Primes inside msup $\phantom{\rule{2em}{0ex}}x=\left({x}^{\prime },\phantom{\rule{0.3em}{0ex}}{x}^{″}\right)\phantom{\rule{2em}{0ex}}x=\left({x}^{\prime },\phantom{\rule{0.3em}{0ex}}{x}^{″}\right)$

Compare: $\mathbf{e}\left(x\prime \phantom{\rule{0.3em}{0ex}}{}^{t}y″\right)$ with $\mathbf{e}\left(x″\phantom{\rule{0.3em}{0ex}}{}^{t}z\right)$

When one of the unicode primes is viewed outside of MathML, one should be able to see what is in the font: x′