  As a model, the Pais-Uhlenbeck fourth order oscillator with equation of
motion $d^4q/dt^4+(\omega_1^2+\omega_2^2)d^2q/dt^2 +\omega_1^2\omega_2^2 q=0$
is a quantum-mechanical prototype of a field theory containing both second and
fourth order derivative terms. With its dynamical degrees of freedom obeying
constraints due to the presence of higher order time derivatives, the model
cannot be quantized canonically. We thus quantize it using the method of Dirac
constraints to construct the correct quantum-mechanical Hamiltonian for the
system, and find that the Hamiltonian diagonalizes in the positive and negative
norm states that are characteristic of higher derivative field theories.
However, we also find that the oscillator commutation relations become singular
in the $\omega_1 \to \omega_2$ limit, a limit which corresponds to a prototype
of a pure fourth order theory. Thus the particle content of the $\omega_1
=\omega_2$ theory cannot be inferred from that of the $\omega_1 \neq \omega_2$
theory; and in fact in the $\omega_1 \to \omega_2$ limit we find that all of
the $\omega_1 \neq \omega_2$ negative norm states move off shell, with the
spectrum of asymptotic in and out states of the equal frequency theory being
found to be completely devoid of states with either negative energy or negative
norm. As a byproduct of our work we find a Pais-Uhlenbeck analog of the zero
energy theorem of Boulware, Horowitz and Strominger, and show how in the equal
frequency Pais-Uhlenbeck theory the theorem can be transformed into a positive
energy theorem instead.
