Infinitely many inequivalent field theories from one Lagrangian

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Logarithmic time-like Liouville quantum field theory has a generalized PT invariance, where T is the time-reversal operator and P stands for an S-duality reflection of the Liouville field ϕ. In Euclidean space the Lagrangian of such a theory, L=1/2(∇ϕ)²-igϕ(iaϕ), is analyzed using the techniques of PT-symmetric quantum theory. It is shown that L defines an infinite number of unitarily inequivalent sectors of the theory labeled by the integer n. In one-dimensional space (quantum mechanics) the energy spectrum is calculated in the semiclassical limit and the mth energy level in the nth sector is given by Eₘ,ₙ (m+1/2)²a²/(16n²).