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The problem of constructing alternative Hamiltonian formulations in the extended Chern–Simons theory with higher derivatives is considered. It is shown that the fourth-order extended theory admits a four-parameter series of alternative Hamiltonians which can be bounded from below, even if the canonical energy of the model is unbounded from below.

Most general third-order $3d$ linear gauge vector field theory is considered. The field equations involve, besides the mass, two dimensionless constant parameters. The theory admits two-parameter series of conserved tensors with the canonical energy-momentum being a particular representative of the series. For a certain range of the model parameters, the series of conserved tensors include bounded quantities. This makes the dynamics classically stable, though the canonical energy is unbounded in all the instances. The free third-order equations are shown to admit constrained multi-Hamiltonian form with the zero-zero components of conserved tensors playing the roles of corresponding Hamiltonians. The series of Hamiltonians includes the canonical Ostrogradski's one, which is unbounded. The Hamiltonian formulations with different Hamiltonians are not connected by canonical transformations. This means, the theory admits inequivalent quantizations at the free level. Covariant interactions are included with spinor fields such that the higher-derivative dynamics remains stable at interacting level if the bounded conserved quantity exists in the free theory. In the first-order formalism, the interacting theory remains Hamiltonian and therefore it admits quantization, though the vertices are not necessarily Lagrangian in the third-order field equations., 19 pages

This work continues our investigation on the high resolution infrared 13C-enriched spectrum of methane recorded on a Bruker IFS-125HR Fourier transform spectrometer at 80 K in the 4970–6200 cm−1 range. In this paper we report the results of assignment and modelling of the line positions and intensities of 13CH4 in the range between 5471 and 5852 cm−1, corresponding to the middle part of Tetradecad. The analysis was performed using a non-empirical effective Hamiltonian (EH) derived from the ab initio potential energy surface (PES) by high-order Contact Transformations (CT). The EH parameters were empirically optimized during the treatment of experimental data. In total, 3297 lines belonging to nine cold bands of the Tetradecad of 13CH4 were assigned in the 5471–5852 cm−1 range up to Jmax = 11. The 3297 lines included correspond to 91% of the total line number in the region. Among them, 3251 line positions were fitted with an rms deviation of 1.33×10−3 cm−1. Measured line intensities for 1624 transitions were modeled using the effective dipole transition moments approach with an rms deviation of about 9%. Finally, the observed transitions were incorporated to fit simultaneously the 13CH4 Hamiltonian parameters for the {Ground state/Dyad/Pentad/Octad/Tetradecad} system and the dipole moment parameters for the {Ground state - Tetradecad} system.

Eight spectra of normal samples of CH4 in the 3760–4100 cm−1 region were recorded by using a Fourier transform spectrometer in Reims, France at long paths (202 m, 602 m, 1604 m and 1804 m) and different pressures. Additional spectra of 12CH4 covering the same region were obtained at 100 K and very low sample pressures, which was used for low-J line position identification. Line positions and intensities were retrieved by least squares curve-fitting procedures and analyzed using the effective Hamiltonian and the effective dipole. A new measured line list contains positions and intensities for 9000 features. Quantum assignments were made for more than 3500 new transitions, which represent ∼95% of the integrated line intensity observed in this region. The resulting list of lines is much more accurate than the list of lines HITRAN 2016. All assigned 5703 line positions were fitted to RMS standard deviations of 0.0018 cm−1. The sum of observed intensities between 3760 and 4100 cm−1 fell within 2% of the predicted value from ab initio variational calculations reported in the TheoReTS database ( http://theorets.univ-reims.fr ; http://theorets.tsu.ru ).

The positivity conditions of the relative entropy between two thermal equilibrium states ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{1}$ and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{2}$ are used to obtain upper and lower bounds for the subtraction of their entropies, the Helmholtz potential and the Gibbs potential of the two systems. These limits are expressed in terms of the mean values of the Hamiltonians, number operator, and temperature of the different systems. In particular, we discuss these limits for molecules that can be represented in terms of the Franck--Condon coefficients. We emphasize the case where the Hamiltonians belong to the same system at two different times $t$ and ${t}^{\ensuremath{'}}$. Finally, these bounds are obtained for a general qubit system and for the harmonic oscillator with a time-dependent frequency at two different times.