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Generalized Lee-Wick Formulation from Higher Derivative Field Theories

We study a higher derivative (HD) field theory with an arbitrary order of derivative for a real scalar field. The degree of freedom for the HD field can be converted to multiple fields with canonical kinetic terms up to the overall sign. The Lagrangian describing the dynamics of the multiple fields is known as the Lee-Wick (LW) form. The first step to obtain the LW form for a given HD Lagrangian is to find an auxiliary field (AF) Lagrangian which is equivalent to the original HD Lagrangian up to the quantum level. Till now, the AF Lagrangian has been studied only for N=2 and 3 cases, where $N$ is the number of poles of the two-point function of the HD scalar field. We construct the AF Lagrangian for arbitrary $N$. By the linear combinations of AF fields, we also obtain the corresponding LW form. We find the explicit mapping matrices among the HD fields, the AF fields, and the LW fields. As an exercise of our construction, we calculate the relations among parameters and mapping matrices for $N=2,3$, and 4 cases.