Didactic derivation of Gutzwiller's trace formula from the path integral, with overview of its use in explaining random matrix theory statistics for quantum energy levels.
Semiclassical theory of transport
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abstract
We discuss the semiclassical approximation to transport problems in quantum chaotic systems. The figures of merit are moments of the transmission matrix and of the time delay matrix. After reviewing a few results obtained by treating these matrices are random matrices, we show how expressions for their elements in terms of sums over trajectories lead to diagrammatic formulations that correspond to perturbative calculations. This semiclassical approach agrees with random matrix theory when it should, and allows further elements to be incorporated, like tunnel barriers, superconductors, absorption effects. We also discuss how this approach can be encoded in matrix integrals, resulting in a powerful and versatile theory that is amenable to algebraic solutions.
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quant-ph 3years
2026 3verdicts
UNVERDICTED 3roles
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Review of random matrix theory application to quantum chaos, covering symmetry classes, eigenvalue statistics, unfolding, and correlation functions.
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Didactic derivation of Gutzwiller's trace formula from the path integral, with overview of its use in explaining random matrix theory statistics for quantum energy levels.
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Quantum graphs are presented as a paradigmatic model for quantum chaos, with the paper providing a didactical overview of foundational results and some recent developments.
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Quantum chaotic systems: a random-matrix approach
Review of random matrix theory application to quantum chaos, covering symmetry classes, eigenvalue statistics, unfolding, and correlation functions.