= D-brane scalar-position normalization
{c}
{title2=$X_r^I=2\pi\alpha'\langle\phi_r^I\rangle$}
A commuting diagonal expectation value has one transverse eigenvalue vector for each brane. In the covariant derivative normalization $D_a\phi=\partial_a\phi+i[A_a,\phi]$, the off-diagonal gauge field has mass $m^2=\sum_I(\phi_1^I-\phi_2^I)^2$. The corresponding stretched string has mass $|X_1-X_2|/(2\pi\alpha')$ 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.
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