Large sets of baryon interpolating field operators are developed for use in lattice QCD studies of baryons with zero momentum. Because of the cubical discretization of space, the continuum rotational group is broken down to a finite point group. Operators are classified according to the irreducible representations of the double octahedral group. At first, three-quark quasi-local operators are constructed for each isospin and strangeness with suitable symmetry of Dirac indices. Nonlocal baryon operators are formulated in a second step as direct products of the quasi-local spinor structures together with lattice displacements. Appropriate Clebsch-Gordan coefficients of the octahedral group are used to form linear combinations of such direct products. The construction maintains maximal overlap with the continuum SU(2) group in order to provide a physically interpretable basis. Nonlocal operators provide direct couplings to states that have nonzero orbital angular momentum.

Monte Carlo simulations of nucleon and delta baryon spectra are carried out with anisotropic lattices of anisotropy 3.0 with $\beta=6.1$. Gauge configurations are generated by the Wilson gauge action in quenched approximation with space-time volumes $(1.6,\mbox{fm})^3\times 2.1,\mbox{fm}$ and $(2.4,\mbox{fm})^3\times 2.1,\mbox{fm}$. The Wilson fermion action is used with pion mass $\simeq 500,\mbox{MeV}$. The variational method is applied to matrices of correlation functions constructed using improved operators in order to extract mass eigenstates including excited states. Stability of the obtained masses is confirmed by varying the dimensions of the matrices. The pattern of masses for the low-lying states that we compute is consistent with the pattern that is observed in nature. Ordering of masses is consistent for positive-parity excited states, but mass splittings are considerably larger than the physical values. Baryon masses for spin $S \ge 5/2$ states are obtained in these simulations. Hyperfine mass splittings are studied for both parities. No significant finite volume effect is seen at the quark mass that is used.