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By analytically continuing the coupling constant g of a coupled quantum theory, one can, at least in principle, arrive at a state whose energy is lower than the ground state of the theory. The idea is to begin with the uncoupled g = 0 theory in its ground state, to analytically continue around an exceptional point (square-root singularity) in the complexcoupling-constant plane, and finally to return to the point g = 0. In the course of this analytic continuation, the uncoupled theory ends up in an unconventional state whose energy is lower than the original ground state energy. However, it is unclear whether one can use this analytic continuation to extract energy from the conventional vacuum state; this process appears to be exothermic but one must do work to vary the coupling constant g.

By analytically continuing the eigenvalue problem of a system of two coupled harmonic oscillators in the complex coupling constant g, we have found a continuation structure through which the conventional ground state of the decoupled system is connected to three other lower unconventional ground states that describe the different combinations of the two constituent oscillators, taking all possible spectral phases of these oscillators into account [7]. In this work we calculate the connecting structures for the higher excitation states of the system and argue that -in contrast to the fourfold Riemann surface identified for the ground state -the general structure is eight-fold instead. Furthermore we show that this structure in principle remains valid for equal oscillator frequencies as well and comment on the similarity of the connection structure to that of the single complex harmonic oscillator.

Quantum systems governed by non-Hermitian Hamiltonians with PT symmetry are special in having real energy eigenvalues bounded below and unitary time evolution. We argue that PT symmetry may also be important and present at the level of Hermitian quantum field theories because of the process of renormalisation. In some quantum field theories renormalisation leads to PT -symmetric effective Lagrangians. We show how PT symmetry may allow interpretations that evade ghosts and instabilities present in an interpretation of the theory within a Hermitian framework. From the study of examples PT -symmetric interpretation is naturally built into a path integral formulation of quantum field theory; there is no requirement to calculate explicitly the PT norm that occurs in Hamiltonian quantum theory. We discuss examples where PT -symmetric field theories emerge from Hermitian field theories due to effects of renormalisation. We also consider the effects of renormalisation on field theories that are non-Hermitian but PT -symmetric from the start.

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