We calculate the nucleon and delta excited state spectra from lattice QCD.

Operators which transform as irreducible representations of the lattice symmetry

group are used as bases for variational calculations. We compute matrices of corre-

lation functions between all the operators in the variational bases. From the time

dependence of the eigenvalues of these matrices, we extract energy eigenvalues. By

subducing the continuum SU(3) rotation group to the octahedral group, we can

identify the spins of the continuum states which correspond to the lattice states.

  In the nucleon spectrum calculation, we use 24^3 × 64 anisotropic lattices with

pion masses of 416 MeV and 576 MeV. The lattices have a spacing of about 0.1 fm

and an anisotropy of 3. We use the Wilson gauge and the Wilson fermion actions

with two flavors of dynamical light quarks. The low-lying spectrum has many of

the qualitative features of the physical spectrum and we are able to identify the

continuum states which correspond to several of the lattice states. This includes

one of the first observations of a spin- 5 state on the lattice.

  For the delta spectrum calculation, we use 16^3 × 128 anisotropic lattices. The

gauge action is the tree-level tadpole improved Wilson gauge action, while in the

fermion sector we use the clover action. The pion mass is about 390 MeV and the

anisotropy is 3.5. We have two flavors of dynamical light quarks as well as dynamical

strange quarks. To compute the correlation functions, we use the distillation method

in which operators are projected on the the low lying eigenmodes of the Laplacian

operator, allowing for an exact computation of all-to-all propagators between the

distilled source and sink operators. We are able to identify four low-lying states

with continuum delta states.