  The Principle of Maximal Conformality (PMC) provides a systematic way to set
the renormalization scales order-by-order for any perturbative QCD process. The
resulting predictions are independent of the choice of renormalization scheme,
a requirement of renormalization group invariance. The Crewther relation, which
was originally derived for conformal theory, provides a remarkable connection
between two observables when the $\beta$ function vanishes. The "Generalized
Crewther Relation" relates these two observables for physical QCD with nonzero
$\beta$ function; specifically, it connects the non-singlet Adler function
($D^{\rm ns}$) to the Bjorken sum rule coefficient for polarized deep-inelastic
electron scattering ($C_{\rm Bjp}$) at leading twist. A scheme-dependent
$\Delta_{\rm CSB}$-term appears in the analysis in order to compensate for the
conformal symmetry breaking (CSB) terms from perturbative QCD. In conventional
analyses, this normally leads to unphysical dependence in both the choice of
the renormalization scheme and the choice of the initial scale at any finite
order. However, by applying PMC scale-setting, we can fix the scales of the QCD
coupling unambiguously at every order of pQCD. ...... Thus one obtains a new
generalized Crewther relation for QCD which connects two effective charges,
$\widehat{\alpha}_d(Q)= \sum_{i\geq1} \widehat{\alpha}^{i}_{g_1}(Q_i)$, at
their respective physical scales. This identity is independent of the choice of
the renormalization scheme at any finite order, and the dependence on the
choice of the initial scale is negligible. Similar scale-fixed commensurate
scale relations also connect other physical observables at their physical
momentum scales, thus providing convention-independent, fundamental precision
tests of QCD.
