In three-dimensions these excitations are gapless for all wavevectors. However, one dimension is different. The shaded area in Figure (b) above shows the relevant excitations for on one-dimensional fermion gas. This Figure is taken from a seminal paper by Haldane, who emphasized the distinct difference from higher dimensions.

The fact that there is a well defined dispersion for low momenta, shown above means that density fluctuations are well-defined quasi-particles in the one dimensions. This is the basis of

**bosonisation**and the**Luttinger liquid**, discussed in Chapter 9.
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