# Berry, Tabor ## Level clustering in the regular spectrum \[Links: [DOI](https://doi.org/10.1098/rspa.1977.0140)\] \[Abstract: In the regular spectrum of an $f$-dimensional system each energy level can be labelled with $f$ quantum numbers originating in $f$ constants of the classical motion. Levels with very different quantum numbers can have similar energies. We study the classical limit of the distribution $P(S)$ of spacings between adjacent levels, using a scaling transformation to remove the irrelevant effects of the varying local mean level density. For generic regular systems $P(S) = e-s$, characteristic of a Poisson process with levels distributed at random. But for systems of harmonic oscillators, which possess the non-generic property that the ‘energy contours’ in action space are flat, $P(S)$ does not exist if the oscillator frequencies are commensurable, and is peaked about a non-zero value of $S$ if the frequencies are incommensurable, indicating some regularity in the level distribution; the precise form of $P(S)$ depends on the arithmetic nature of the irrational frequency ratios. Numerical experiments on simple two-dimensional systems support these theoretical conclusions.\] ## Summary - original work on the energy eigenvalues of a quantum system whose classical counterpart is integrable; origin of the [[0585 Berry-Tabor conjecture|Berry-Tabor conjecture]] - no level repulsion for quantum systems with integrable classical counterparts # Gibbons, Hawking ## Action integrals and partition functions in quantum gravity \[Links: [DOI](https://doi.org/10.1103/PhysRevD.15.2752)\] \[Abstract: One can evaluate the action for a gravitational field on a section of the complexified spacetime which avoids the singularities. In this manner we obtain finite, purely imaginary values for the actions of the Kerr-Newman solutions and de Sitter space. One interpretation of these values is that they give the probabilities for finding such metrics in the vacuum state. Another interpretation is that they give the contribution of that metric to the partition function for a grand canonical ensemble at a certain temperature, angular momentum, and charge. We use this approach to evaluate the entropy of these metrics and find that it is always equal to one quarter the area of the event horizon in fundamental units. This agrees with previous derivations by completely different methods. In the case of a stationary system such as a star with no event horizon, the gravitational field has no entropy.\] ## Summary - original work on [[0555 Gravitational path integral|gravitational path integral]] and the Euclidean computation of [[0004 Black hole entropy|black hole entropy]]