How? Luttinger Liquids are formed and studied ONLY unidimensionally, in LINES. They are LINEAR liquids.
Luttinger liquids were observed in two or more dimensions already. But the important point is, that Luttinger liquids form and remain stable at higher temperatures when they are aligned unidimensionally. And for explanation of this effect the general highdimensional approach would be already useful. Of course, many other approaches are possible: for example Casimir force has been predicted to be stronger between ends of needles (and weaker or even repulsive above cavities) (which also points to common aspects of quantum entanglement and quantum vacuum).