  We show that one-dimensional quantum systems with gapless degrees of freedom
and open boundary conditions form a new universality class of quantum critical
behavior, which we propose to call ``bounded Luttinger liquids''. They share
the following properties with ordinary (periodic) Luttinger liquids: absence of
fermionic quasi-particle excitations, charge-spin separation, anomalous
power-law correlations with exponents whose scaling relations are parametrized
by a single coupling constant per degree of freedom, $K_{\nu}$. The values of
$K_{\nu}$ are independent of boundary conditions, but the representation of the
critical exponents in terms of these $K_{\nu}$ depends on boundary conditions.
We illustrate these scaling relations by exploring general rules for boundary
critical exponents derived earlier using the Bethe Ansatz solution of the 1D
Hubbard model together with boundary conformal field theory, and the theory of
Luttinger liquids in finite-size systems. We apply this theory to the
photoemission properties of the organic conductors $(TMTSF)_2X$, and discuss to
what extent the assumption of finite strands with open boundaries at the sample
surface can reconcile the experimental results with independent information on
the Luttinger liquid state in these materials.
