We present a complete derivation of the semiclassical limit of the coherent state propagator in one dimension, starting from path integrals in phase space. We show that the arbitrariness in the path integral representation, which follows from the overcompleteness of the coherent states, results in many different semiclassical limits. We explicitly derive two possible semiclassical formulae for the propagator, we suggest a third one, and we discuss their relationships. We also derive an initial value representation for the semiclassical propagator, based on an initial gaussian wavepacket. It turns out to be related to, but different from, Heller's thawed gaussian approximation. It is very different from the Herman-Kluk formula, which is not a correct semiclassical limit. We point out errors in two derivations of the latter. Finally we show how the semiclassical coherent state propagators lead to WKB-type quantization rules and to approximations for the Husimi distributions of stationary states. Conclusion 65Appendix A Calculating the Prefactor by the Determinantal Method 68Appendix B Proof of eq. (3.27) 73Appendix C Cancelation of first order terms in S + I 75 IntroductionSemiclassical approximations in phase space using coherent states have been discussed extensively for several decades. This attractive topic, a favorite of many theoretical physicists and chemists, turns out to be very difficult. In this contribution to its literature, we shall attempt to sort out and clarify the web of contradictions and inconsistencies that have characterized the recent state of the field. We shall do so for the simplest possible case, one-dimensional coordinate space, i. e. two-dimensional phase space. This is the case where it is relatively easy to check the semiclassical approximations. We have done work in higher dimensions as well, but we do not include it here, as it would only obscure the basic relationships and further lengthen the paper. The conclusions we have reached are stated in section 7, and the reader who is already familiar with the subject may jump to them now to get an overall view. Because the pitfalls are numerous, however, we shall follow a slower approach, a historical one in this introduction, and then a systematic and detailed one in the body of the paper.The study of semiclassical methods has two basic motivations. First, it provides approximations to quantum mechanical quantities in terms of classical ingredients. These approximations should be very good if the typical classical actions are much larger than Planck's constant. Interestingly, they are often fairly good even at very low quantum numbers. Second, semiclassical methods also help in understanding the quantum mechanical processes themselves, providing a more intuitive description. This description includes quantum mechanical interference, since both amplitudes and phases can be calculated semiclassically.The semiclassical approximation for the evolution operator, or propagator, in the coordinate representation has been known for more than 70 ...

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The typical avoided crossings for Hermitian quantum systems depending on parameters, the diabolic crossing scenario, are generalized to the non-Hermitian case, e.g. for resonances. Two types of crossings appear: for type I, the real parts show an avoided and the imaginary parts a true crossing of the eigenenergies, and for type II the opposite is found. A simple symmetric non-Hermitian twostate matrix Hamiltonian is analysed in detail. The diabolic point bifurcates into two exceptional ones on exceptional lines where the matrices are defective. The adiabatic transport of eigenvectors and eigenstates in parameter space is discussed in this generalized diabolic crossing scenario, in particular the geometric Berry phases for a cyclic variation of system parameters, depending on the topology of the closed curves with respect to the exceptional lines.

The COVID-19 pandemic has the potential to affect the human microbiome in infected and uninfected individuals, having a substantial impact on human health over the long term. This pandemic intersects with a decades-long decline in microbial diversity and ancestral microbes due to hygiene, antibiotics, and urban living (the hygiene hypothesis). High-risk groups succumbing to COVID-19 include those with preexisting conditions, such as diabetes and obesity, which are also associated with microbiome abnormalities. Current pandemic control measures and practices will have broad, uneven, and potentially long-term effects for the human microbiome across the planet, given the implementation of physical separation, extensive hygiene, travel barriers, and other measures that influence overall microbial loss and inability for reinoculation. Although much remains uncertain or unknown about the virus and its consequences, implementing pandemic control practices could significantly affect the microbiome. In this Perspective, we explore many facets of COVID-19−induced societal changes and their possible effects on the microbiome, and discuss current and future challenges regarding the interplay between this pandemic and the microbiome. Recent recognition of the microbiome’s influence on human health makes it critical to consider both how the microbiome, shaped by biosocial processes, affects susceptibility to the coronavirus and, conversely, how COVID-19 disease and prevention measures may affect the microbiome. This knowledge may prove key in prevention and treatment, and long-term biological and social outcomes of this pandemic.