  We discuss how electron-electron interactions renormalize the spin-orbit
induced anticrossings between different subbands in ballistic quantum wires.
Depending on the ratio of spin-orbit coupling and subband spacing,
electron-electron interactions can either increase or decrease anticrossing
gaps. When the anticrossings are closing due to a special combination of Rashba
and Dresselhaus spin-orbit couplings, their gap approaches zero as an
interaction dependent power law of the spin-orbit couplings, which is a
consequence of Luttinger liquid physics. Monitoring the closing of the
anticrossings allows to directly measure the related renormalization group
scaling dimension in an experiment. If a magnetic field is applied parallel to
the spin-orbit coupling direction, the anticrossings experience different
renormalizations. Since this difference is entirely rooted in electron-electron
interactions, unequally large anticrossings also serve as a direct signature of
Luttinger liquid physics. Electron-electron interactions furthermore increase
the sensitivity of conductance measurements to the presence of anticrossing.
