D-brane scalar-position normalization ($X_r^I=2\pi {\alpha }^{\prime }⟨{\varphi }_r^I⟩$)

A commuting diagonal expectation value has one transverse eigenvalue vector for each brane. In the covariant derivative normalization $D_a\varphi ={\partial }_a\varphi +i[A_a,\varphi ]$, the off-diagonal gauge field has mass $m^2={\sum }_I({\varphi }_1^I-{\varphi }_2^I{)}^2$. The corresponding stretched string has mass $|X_1-X_2|/(2\pi {\alpha }^{\prime })$ in its massless oscillator sector. Equality fixes the displayed scalar-position normalization, up to a common coordinate origin. The trace describes center-of-mass motion and the eigenvalue difference describes separation.