Quantum signatures of classical chaos, studied through the spectral statistics of bounded quantum systems whose classical counterparts are chaotic (stadium billiard, kicked rotor, hydrogen in a strong magnetic field, Sinai billiard). Core…
quantum-chaos
Wigner surmise + BGS conjecture: random-matrix universality of chaos
Wigner 1951 conjectured that the nearest-neighbor level-spacing distribution of unfolded eigenvalues of a large real-symmetric random…
Berry-Tabor 1977: integrable quantum systems show Poisson statistics
Berry-Tabor 1977 (Proc. R. Soc. A 356:375) argued that quantum counterparts of *integrable* classical systems have uncorrelated eigenvalues…
Maldacena-Shenker-Stanford 2016 universal chaos bound λ_L ≤ 2πk_BT/ℏ
Maldacena-Shenker-Stanford 2016 (JHEP 2016:106) proved a thermodynamic upper bound on the Lyapunov exponent extracted from the early-time…
Wigner GOE surmise moments: ⟨s⟩ = 1, ⟨s²⟩ = 4/π (exact)
Closed-form Gaussian-integral check of the Wigner GOE surmise P(s) = (π/2)·s·exp(-π·s²/4). Normalisation ∫₀^∞ P ds = 1 and unit mean ⟨s⟩ =…
Poisson level statistics: mean = variance = 1, P(0) = 1 (exact)
Closed-form check of the Berry-Tabor conjecture's Poisson spacing distribution. Normalisation: ∫₀^∞ exp(-s) ds = 1. First moment: ⟨s⟩ =…
MSS chaos bound saturation: λ_L/(2πT) = 1 at the SYK / black-hole point
Pinning statement of the Maldacena-Shenker-Stanford 2016 chaos bound at the saturation / reference point. In natural units ℏ = k_B = 1,…
Gutzwiller trace formula on integrable SHO: periodic-orbit action S(E) = 2*pi*E/omega via action-angle
Gutzwiller periodic-orbit trace formula on the integrable harmonic-oscillator limit. Setup: the semiclassical density of states is written…
Loschmidt echo in Fermi-golden-rule regime: M(t) = exp(-sigma^2 delta^2 t^2 / hbar^2)
Loschmidt echo / fidelity decay framework. Setup: prepare |psi>, evolve forward under H_0 for time t, reverse-evolve under H_1 = H_0 +…
GUE sine-kernel: R_2(s) = 1 - (sin(pi s)/(pi s))^2; level repulsion at s=0 and Poisson at s->inf
GUE random-matrix sine-kernel framework (Dyson 1962). Setup: for chaotic quantum systems with time-reversal symmetry broken (e.g.…
Theorem: SHO period-energy invariant T(E) * omega = 2*pi (exact for all E > 0)
Theorem (SHO period-omega product canonical): for the 1D harmonic oscillator H(q, p) = p^2/(2m) + m omega^2 q^2/2, the classical action…
Theorem: Loschmidt FGR Gaussian log-slope log M(t)/t^2 = -sigma^2*delta^2/hbar^2 (exact)
Theorem (Loschmidt FGR Gaussian-slope canonical): in the Fermi-golden-rule regime of Peres (1984), the Loschmidt echo M(t) = exp(-sigma^2…
Theorem: GUE R_2(s) limits R_2(0) = 0 (strong repulsion), R_2(s->inf) = 1 (decorrelation)
Theorem (GUE sine-kernel limit canonical): the GUE 2-level correlation R_2(s) = 1 - (sin(pi s)/(pi s))^2 satisfies the two universal…
BGS conjecture (1984)
Bohigas-Giannoni-Schmit 1984: quantized chaotic systems show GOE/GUE/GSE level-statistics from random-matrix-theory; classical-quantum…
Gutzwiller trace (1971)
M Gutzwiller 1971 semiclassical trace-formula; periodic-orbit sum -> spectral density; basis of quantum-chaos rigorous theory.
Scar states (Heller 1984)
E Heller 1984 wavefunction-scarring along unstable periodic-orbits; modern weak-localization vs strong scars Sridhar 1991 +…
Quantum kicked rotor (1979)
Casati-Chirikov-Izrailev-Ford 1979 quantum-kicked-rotor: dynamical-localization vs classical-diffusion; analog of Anderson-localization.
Loschmidt echo (Jalabert-Pastawski 2001)
Jalabert-Pastawski 2001 Loschmidt-echo decay-rate = classical Lyapunov-exponent; modern OTOC out-of-time-ordered correlator.
ETH (Deutsch-Srednicki)
Deutsch 1991 + Srednicki 1994 eigenstate-thermalization-hypothesis: chaotic-many-body eigenstates appear thermal; basis of…
BGS conjecture (1984)
O Bohigas-M-J Giannoni-C Schmit 1984 BGS-conjecture chaotic-RMT-spectrum; modern modern foundational text + 2024 ML-confirmed +…
Berry-Tabor (1977)
M Berry-M Tabor 1977 integrable-Poisson-spectrum; modern modern foundational text + 2024 modern semiclassical-trace-formulae chaotic-test.
Peres echo (1984)
A Peres 1984 + Jalabert-Pastawski 2001 Loschmidt-echo; modern modern foundational text + 2024 OTOC quantum-chaos + scrambling diagnostics.
SYK (Sachdev-Ye 1993 / Kitaev 2015)
S Sachdev-J Ye 1993 + A Kitaev 2015 SYK; modern modern foundational + holographic-NFL + 2024 conformal-IR + maximal-Lyapunov-chaos.
ETH (Srednicki 1994)
M Srednicki 1994 + Deutsch 1991 eigenstate-thermalization-hypothesis; modern modern foundational text + 2024 ML-test ETH chaotic many-body.
Scars (Heller 1984)
E Heller 1984 quantum-scars; modern modern foundational text + 2024 PXP-Rydberg quantum-many-body scars Lukin Bernien-Choi 2017.