This paper examines chains of N coupled harmonic oscillators. In isolation, the jth oscillator (1 ≤ j ≤ N ) has the natural frequency ω j and is described by the HamiltonianThe oscillators are coupled adjacently with coupling constants that are purely imaginary; the coupling of the jth oscillator to the (j + 1)st oscillator has the bilinear form iγx j x j+1 (γ real). The complex Hamiltonians for these systems exhibit partial PT symmetry; that is, they are invariant under i → −i (time reversal), x j → −x j (j odd), and x j → x j (j even). [They are also invariant under i → −i, x j → x j (j odd), and x j → −x j (j even).] For all N the quantum energy levels of these systems are calculated exactly and it is shown that the ground-state energy is real. When ω j = 1 for all j, the full spectrum consists of a real energy spectrum embedded in a complex one; the eigenfunctions corresponding to real energy levels exhibit partial PT symmetry. However, if the ω j are allowed to vary away from unity, one can induce a phase transition at which all energies become real. For the special case N = 2, when the spectrum is real, the associated classical system has localized, almost-periodic orbits in phase space and the classical particle is confined in the complexcoordinate plane. However, when the spectrum of the quantum system is partially real, the corresponding classical system displays only open trajectories for which the classical particle spirals off to infinity. Similar behavior is observed when N > 2.

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Relativistic PT -symmetric fermionic interacting systems are studied in 1+1 and 3+1 dimensions. The noninteracting Dirac equation is separately P and T invariant. The objective here is to include non-Hermitian PT -symmetric interaction terms that give rise to real spectra. Such interacting systems could be physically realistic and could describe new physics. The simplest such non-Hermitian Lagrangian density is L = L0 + Lint =ψ(i / ∂ − m)ψ − gψγ 5 ψ. The associated relativistic Dirac equation is PT invariant in 1+1 dimensions and the associated Hamiltonian commutes with PT . However, the dispersion relation p 2 = m 2 − g 2 shows that the PT symmetry is broken (the eigenvalues become complex) in the chiral limit m → 0. For field-theoretic interactions of the form Lint = −g(ψγ 5 ψ) N with N = 2, 3, which we can only solve approximately, we also find that if the associated (approximate) Dirac equation is PT invariant, the dispersion relation always gives rise to complex energies in the chiral limit m → 0. Other models are studied in which x-dependent PT -symmetric potentials such as ix 3 , −x 4 , iκ/x, Hulthén, or periodic potentials are coupled to the fermionic field ψ using vector or scalar coupling schemes or combinations of both. For each of these models the classical trajectories in the complex-x plane are examined. Some combinations of these potentials can be solved numerically, and it is shown explicitly that a real spectrum can be obtained. In 3+1 dimensions, while the simplest system L = L0 + Lint =ψ(i / ∂ − m)ψ − gψγ 5 ψ resembles the 1+1-dimensional case, the Dirac equation is not PT invariant because T 2 = −1. This explains the appearance of complex eigenvalues as m → 0. Other Lorentz-invariant two-point and four-point interactions are considered that give non-Hermitian PT -symmetric terms in the Dirac equation. Only the axial vector and tensor Lagrangian interactions Lint = −iψBµγ 5 γ µ ψ and Lint = −iψTµν σ µν ψ fulfil both requirements of PT invariance of the associated Dirac equation and non-Hermiticity. The dispersion relations show that both interactions lead to complex spectra in the chiral limit m → 0. The effect on the spectrum of the additional constraint of selfadjointness of the Hamiltonian with respect to the PT inner product is investigated.

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.

The way that progenitor cell fate decisions and the associated environmental sensing are regulated to ensure the robustness of the spatial and temporal order in which cells are generated towards a fully differentiating tissue still remains elusive. Here, we investigate how cells regulate their sensing intensity and radius to guarantee the required thermodynamic robustness of a differentiated tissue. In particular, we are interested in finding the conditions where dedifferentiation at cell level is possible (microscopic reversibility), but tissue maintains its spatial order and differentiation integrity (macroscopic irreversibility). In order to tackle this, we exploit the recently postulated Least microEnvironmental Uncertainty Principle (LEUP) to develop a theory of stochastic thermodynamics for cell differentiation. To assess the predictive and explanatory power of our theory, we challenge it against the avian photoreceptor mosaic data. By calibrating a single parameter, the LEUP can predict the cone color spatial distribution in the avian retina and, at the same time, suggest that such a spatial pattern is associated with quasi-optimal cell sensing. By means of the stochastic thermodynamics formalism, we find out that thermodynamic robustness of differentiated tissues depends on cell metabolism and cell sensing properties. In turn, we calculate the limits of the cell sensing radius that ensure the robustness of differentiated tissue spatial order. Finally, we further constrain our model predictions to the avian photoreceptor mosaic.