More with prime accents

References:

• My earlier analysis of prime accents.
• Comments at w3.org: Go to http://lists.w3.org/Archives/Public/www-math/ and search for “rendering primes”.

The real Heisenberg group ${\mathrm{Hs}}_n\left(\mathbf{R}\right)$ is the analytic Cartesian product $\mathbf{T}\times {\mathbf{R}}^n\times {\mathbf{R}}^n$ 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 $$\left(u,x\right)·\left(v,y\right)=\left(uvb\left(x,y\right),x+y\right),$$ where, for $x=\left(x\prime ,x″\right)$, $y=\left(y\prime ,y″\right)$ in W = ${\mathbf{R}}^n\times {\mathbf{R}}^n$, $b$ is the $\mathbf{T}$-valued bi-additive function on $W$ defined by $$b\left(x,y\right)=\mathbf{e}\left(x\prime {}^{t}y″\right),$$ with ${}^t\left(\right)$ denoting transpose and $\mathbf{e}\left(t\right)$ denoting $\exp \left(2\pi it\right)$. 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 $\Phi \to \Phi \prime =U\left(t,x\right)\Phi$, where $$\begin{array}{cc}\text{(1)}& \Phi \prime \left(z\right)=t\mathbf{e}\left(x″{}^{t}z\right)\Phi \left(z+x\prime \right)fort\in \mathbf{T},x\in W,z\in {\mathbf{R}}^n\text{.}\end{array}$$

Scratch area

Primes as operators: $x=\left(x\prime ,x″\right)$

Ascii primes inside mi: $x=\left(\mathrm{x\text{'}},\mathrm{x\text{'}\text{'}}\right)$

Primes inside msup $x=\left(x^{\prime },x^{″}\right)x=\left(x^{\prime },x^{″}\right)$

Compare: $$\mathbf{e}\left(x\prime {}^{t}y″\right)$$ with $$\mathbf{e}\left(x″{}^{t}z\right)$$

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