  The relation between the heavy-quark field in QCD and the corresponding field
in HQET is derived up to three loops, and to all orders in the large-$\beta_0$
limit. The corresponding relation between the QED electron field and the
Bloch--Nordsieck one is gauge invariant to all orders. We also prove that the
$\bar{\text{MS}}$ anomalous dimension of the QED electron field depends on the
gauge parameter only at one loop.
