  The wave function in the quantum theory of the O(N) extended supersymmetric
particle model describes a massless free field with spin N/2. This quantum
theory is here exactly solved in terms of gauge fields in arbitrary even
dimensions using only the basic quantum operators which include graded external
differentials, trace operators, index structure operators and their duals. The
resulting equations for the gauge fields are of first (N odd) or second order
(N even) and are shown to be generalized (Fang)-Fronsdal equations which are
fully gauge invariant since they include compensator fields in a natural way.
Local gauge invariant actions are first derived in analogy with the derivation
by Francia and Sagnotti in the symmetric case. Then a minimal formulation is
given within which it is easy to set up gauge invariant actions and here
appropriate actions for the above equations are proposed.
  In a second part it is shown that there exist projection operators from the
states of the field strengths (wave functions) to Weyl states (Weyl tensors)
expressed only in terms of the trace operators and their duals for integer
spins or in terms of the gamma trace operators and their duals for half-integer
spins. These Weyl states are not just fully gauge invariant but also invariant
under generalized Weyl transformations. If one lets the equations be determined
by the actions defined to be the squares of these Weyl tensors then the theory
is conformally invariant and the equations are of order N for the gauge fields
defined as in the exact theory. In d=4 this higher order conformal theory fits
into the framework for conformal higher spin theories set up by Fradkin,
Tseytlin and Linetski.
