  We consider a set of half-BPS operators in $\mathcal{N}=4$ super Yang-Mills
theory which are appropriate for describing single-particle states of
superstring theory on AdS${}_5 \times S^5$. These single-particle operators are
defined to have vanishing two-point functions with all multi-trace operators
and therefore correspond to admixtures of single- and multi-traces. We find
explicit formulae for all single-particle operators and for their two-point
function normalisation. We show that single-particle $U(N)$ operators belong to
the $SU(N)$ subspace, thus for length greater than one they are simply the
$SU(N)$ single-particle operators. Then, we point out that at large $N$, as the
length of the operator increases, the single-particle operator naturally
interpolates between the single-trace and the $S^3$ giant graviton. At finite
$N$, the multi-particle basis, obtained by taking products of the
single-particle operators, gives a new basis for all half-BPS states, and this
new basis naturally cuts off when the length of any of the single-particle
operators exceeds the number of colours. From the two-point function
orthogonality we prove a multipoint orthogonality theorem which implies
vanishing of all near-extremal correlators. We then compute all maximally and
next-to-maximally extremal free correlators, and we discuss features of the
correlators when the extremality is lowered. Finally, we describe a half-BPS
projection of the operator product expansion on the multi-particle basis which
provides an alternative construction of four- and higher-point functions in the
free theory.
