  The stability of the Luttinger liquid to small transverse hopping has been
studied from several points of view. The renormalization group approach in
particular has been criticized because it does not take explicitly into account
the difference between spin and charge velocities and because the interaction
should be turned-on before the transverse hopping if the stability of the
Luttinger liquid is a non-perturbative effect. An approach that answers both of
these objections is explained here. It shows that the Luttinger liquid is
unstable to arbitrarily small transverse hopping. The crossover temperatures
below which either transverse coherent band motion or long-range order start to
develop can be finite even when spin and charge velocities differ. Explicit
scaling relations for these one-particle and two-particle crossover
temperatures are derived in terms of transverse hopping, spin and charge
velocities and anomalous exponents. The renormalization group results are
recovered as special cases when spin and charge velocities are identical. The
results compare well with recent experiments presented at this conference.
Magnetic field effects are alluded to.
