Resumo A integral de caminhos de Feynman coloca de saída a ação clássica no cerne da evolução quântica. Ao invés de apenas contrastar a multiplicidade de caminhos quânticos com a raridade das trajetórias clássicas, podemos nos valer do princípio variacional que as identificam para construir abrangentes aproximações semiclássicas para o operador de evolução. A transformada de Fourier de seu traço fornece então a densidade de níveis quânticos de energia como uma soma sobre as órbitas periódicas clássicas: a fórmula do traço de Gutzwiller. As órbitas peródicas também explicam as correlações entre os níveis, seguindo padrões universais de matrizes aleatórias se o sistema clássico for caótico. Ao reverso, discrepâncias entre os espectros universais e os semiclássicos desvendaram correlações entre órbitas periódicas, nunca vislumbradas ao longo da história da mecânica clássica.

Abstract The Feynman path integral places the classical action in the very core of quantum evolution. Instead of contrasting the multiplicity of quantum paths with the scarcity of classical trajectories, we can avail ourselves of the variational principle by which they are identified. This leads to wide ranging semiclassical approximations for the evolution operator. The Fourier transform of the trace of this operator then supplies the density of quantum energy levels as a sum over the classical periodic orbits: The Gutzwiller trace formula. The periodic orbits also explain the correlations between levels, following universal classes of random matrices, if the classical system is chaotic. Conversely, discrepancies between the semiclassical spectra and the universal random matrix spectra uncouvered correlations between periodic orbits that had never been suspectected in the long history of classical mechanics.

We define a coupling of two baker maps through a pi/2 rotation both in position and in momentum. The classical trajectories thus exhibit spiraling, or loxodromic motion, which is only possible for conservative maps of at least two degrees of freedom. This loxodromic baker map is still hyperbolic, that is, fully chaotic. Quantization of this map follows on similar lines to other generalized baker maps. It is found that the eigenvalue spectrum for quantum loxodromic baker map is far removed from those of the canonical random matrix ensembles. An investigation of the symmetries of the loxodromic baker map reveals the cause of this deviation from the Bohigas-Giannoni-Schmit conjecture.