7 ature. Let us summarize some important findings. (I) Al- though all groundstate quantities vary continuously with coupling and phonon frequency, the steady change in 1D is replaced with a rapid crossover in the adiabatic regime in 3D, which closely resembles a true phase transition. While no characteristic energy scale exists in 1D, it is set in 3D by the critical coupling λ$_{c}$ of the self-trapping transition in the static limit. (II) All quantities converge (or diverge) to the static limit in a simple and clear way. Nevertheless quantitative agreement is obtained only for very small phonon fre- quencies ω$_{0}$/t ≪ 0 . 1. (III) Only for such small phonon frequencies heavy and large polarons occur in 1D (e.g. ω$_{0}$ < 0 . 02 for M > 5 and R > 4). A large radius requires a small phonon fre- quency in any case. (IV) Small and large polarons have significantly different transport and optical properties. A large and heavy po- laron is characterized through a small Drude weight but large optical absorption. A small and heavy polaron has small Drude weight and small optical absorption. (V) The small-polaron approximation for the optical con- ductivity is strictly valid only for very large phonon fre- quencies, while in general pronounced structures distort the Gaussian envelope. The overall shape of the envelope however remains comparable also for smaller phonon fre- quencies. These findings suggest certain physical interpretations. First, with all necessary caution, it may make sense to think of a self-trapping transition in 3D even at finite phonon frequency. There is nothing comparable in 1D. Second, the notion of an itinerant polaronic quasiparticle is to be used only with care: The nature of the quasi- particle differs at large and small phonon frequency (cf. Sec. VI). Third, Holstein polarons tend to be small ob- jects. This should largely prevent polaron dissociation, which may occur in a kind of polaronic Mott transition if the phonon clouds overlap at finite polaron density. It also prevents instabilities in anisotropic systems with a preferred direction of motion, where only the large adia- batic quasi-1D polarons are destabilized by electron hop- ping perpendicular to this direction $^{48,49}$. In principle, our results allow for direct comparison with, and hence validation of, approximate polaron the- ories. We have not tried to do so here, since we want our results to tell their own tale. Our discussion should have shown that the numerical data itself do not prevent a physical interpretation. But the interpretation adheres to direct terms – the polaron is heavy or light, small or large –, and simple pictures – such as a phonon-dressed electron vs. an electron moving in the potential gener- ated by the lattice displacement. There is no necessity to invoke complicated constructions which do not find their counterpart in the few quantities truly characterizing the polaron. The present problem, i.e. the single Holstein polaron at zero temperature, is treated here exhaustively. Of course, it is possible to go to even smaller phonon frequencies, Our results support a particular picture of polaron for- mation, which is in agreement only with parts of the liter- For the Holstein model one is in the fortunate posi- tion that physical concepts or approximate theories can be checked against very accurate data provided by ad- vanced numerical calculations. In that way the valid- ity of important ideas, potentially with generalizations to less tractable situations, can be assessed, and some of the controversies in the polaron literature may be re- solved. In the present contribution we develop our phys- ical discussion along the lines indicated by such accurate data. The new quality of this contribution is our abil- ity to cover the entire coupling and frequency range, in particular the far adiabatic regime including heavy large polarons in 1D. This progress becomes possible by the new numerical method introduced here. A description of lattice polaron formation requires the treatment of the two different time-scales of electron and phonon dynamics, their mutual influence and accompa- nying renormalization, and the non-trivial quantum cor- relations that develop. The adiabatic regime is especially difficult, which is typical for many problems where one tries to perform the classical limit for some part of a system while keeping the full quantum dynamics of the remainder. To summarize, the change in the optical conductiv- ity with phonon frequency reflects very clearly the two different time scales realized in the polaron problem. De- pending on whether the electron or the phonons set the fastest time scale, we observe different physics. In the an- tiadiabatic regime we find a well-defined polaronic quasi- particle whose behavior is governed by local phonon ex- citations dressing the electron and the arising strong cor- relations between electron and phonon dynamics. Or we find, in the adiabatic regime, an electron moving in a phononic background whose properties depend only on the average electron motion, while stronger correlations are missing. These differences and their consequences for polaron transport and dynamics have been studied here in terms of the optical conductivity and f-sum rule. skewed Gaussian envelope of σ$_{reg}$ ( ω ) is a signature of antiadiabatic small polarons $^{46,47}$. Note that the inte- grated optical conductivity S$_{reg}$ is small in this regime. For intermediate phonon frequencies complicated struc- tures develop, with a contribution from different polaron bands whose width is no longer small compared to ω$_{0}$ . In the far adiabatic regime the basic mechanism for op- tical absorption has changed. It is now dominated by electronic transitions in the potential well generated by the lattice displacement. Of course, a phonon admixture is present for any finite ω$_{0}$ . Only in the static limit the phonon configuration remains unchanged during an elec- tronic transition, and the Franck-Condon effect is fully realized. VII. CONCLUSION AND OUTLOOK