If $[H,C]\ne 0$, then, in order for $H$ to generate dynamics that preserve ${\mathrm{\Gamma }}_{\mathrm{phys}}$, we need to impose additional constraints $[H,C]$, $[H,[H,C]]$ and so on. These constraints are independent of $C$ unless $\omega ([H,C])=0$ for all $\omega \in {\mathrm{\Gamma }}_C$. Systems with multiple constraint elements are outside of the scope of the present manuscript and will be discussed elsewhere.

For example, the two-dimensional kinematical system from the previous section could be subjected to $C=\frac{1}{2}{P}_1^2+\frac{1}{2}{P}_2^2+\frac{1}{2}{Q}_1^2+\frac{1}{2}{Q}_2^2+Q_1^2{Q}_2-\frac{1}{3}{Q}_2^3$, which comes from the Hénon-Heiles Hamiltonian and generates a classically nonintegrable flow.

This projection is the pullback of linear functionals under the inclusion $𝚤:{\mathcal{A}}_{\mathcal{O}}\hookrightarrow \mathcal{A}$, so that $\varphi (\omega )=\omega \circ ı$.

The notation of [5] was developed for a clock that in general may be valid only in a finite range and may have to be transformed once the limits are reached. The label $Z$ therefore signifies a “Zeitgeist” and fashionables in $\mathcal{F}$ fall out of fashion once the Zeitgeist changes because a new clock must be introduced.

In fact the proof of local gauge fixing Sec. 3.2 of [5] explicitly assumes the stronger condition $[Z,C]=i\hslash \mathbf{1}$; however, all steps of the proof carry over with minor changes if the weaker condition $[Z,C]=i\hslash a_t\mathbf{1}+(Z-t\mathbf{1})B_t$, with $[B_t,Z^{\prime }]=\{0\}$ is used instead.

Note that, in general, $\widehat{N}(B{)}^{*}\ne \widehat{N}(B^{*})$.

In the algebra of polynomials the only divisor of zero is 0 and only elements $\alpha \mathbf{1}$, with $\alpha \in \mathbb{C}$ are invertible.