Dirac Quantization of the Pais-Uhlenbeck Fourth Order Oscillator

Abstract

As a model, the Pais-Uhlenbeck fourth order oscillator with equation of motion $d^4q/dt^4+(ω_1^2+ω_2^2)d^2q/dt^2 +ω_1^2ω_2^2 q=0$ 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 $ω_1 \to ω_2$ limit, a limit which corresponds to a prototype of a pure fourth order theory. Thus the particle content of the $ω_1 =ω_2$ theory cannot be inferred from that of the $ω_1 \neq ω_2$ theory; and in fact in the $ω_1 \to ω_2$ limit we find that all of the $ω_1 \neq ω_2$ 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.