The Bohigas–Giannoni–Schmit (BGS) conjecture also known as the random matrix conjecture ) for simple quantum mechanical systems (ergodic with a classical limit ) few degrees of freedom holds that spectra of time reversal-invariant systems whose classical analogues are K-systems show the same fluctuation properties as predicted by the GOE (Gaussian orthogonal ensembles ).[ 1] [ 2] [further explanation needed
Alternatively, the spectral fluctuation measures of a classically chaotic quantum system coincide with those of the canonical random-matrix ensemble in the same symmetry class (unitary , orthogonal, or symplectic ).[further explanation needed
That is, the Hamiltonian of a microscopic analogue of a classical chaotic system can be modeled by a random matrix from a Gaussian ensemble as the distance of a few spacings between eigenvalues of a chaotic Hamiltonian operator generically statistically correlates with the spacing laws for eigenvalues of large random matrices.[further explanation needed
A simple example of the unfolded quantum energy levels in a classically chaotic system correlating like that would be Sinai billiards :[further explanation needed
Energy levels:
−
ℏ
2
2
m
▽
2
ψ
+
V
(
x
)
ψ
=
E
i
ψ
{\displaystyle -{\frac {\hbar ^{2}}{2{\mathit {m}}}}\bigtriangledown ^{2}\psi +{\mathit {V}}({\mathit {x}})\psi ={{\mathit {E}}_{\mathit {i}}}\psi }
[definition needed
Spectral density:
ρ
(
x
)
=
∑
i
δ
(
x
−
E
i
)
{\displaystyle \rho ({\mathit {x}})=\sum _{\mathit {i}}\delta ({\mathit {x}}-{\mathit {E}}_{\mathit {i}})}
Average spectral density:
⟨
ρ
(
x
)
⟩
{\displaystyle \langle \rho ({\mathit {x}})\rangle }
Correlation:
⟨
ρ
(
x
)
ρ
(
y
)
⟩
−
⟨
ρ
(
x
)
⟩
⟨
ρ
(
y
)
⟩
{\displaystyle \langle \rho ({\mathit {x}})\rho ({\mathit {y}})\rangle -\langle \rho ({\mathit {x}})\rangle \langle \rho ({\mathit {y}})\rangle }
Unfolding:
ρ
(
x
)
→
ρ
(
x
)
⟨
ρ
(
x
)
⟩
{\displaystyle \rho ({\mathit {x}})\rightarrow {\frac {\rho ({\mathit {x}})}{\langle \rho ({\mathit {x}})\rangle }}}
Unfolded correlation:
⟨
ρ
(
x
)
ρ
(
y
)
⟩
⟨
ρ
(
x
)
⟩
⟨
ρ
(
y
)
⟩
−
1
{\displaystyle {\frac {\langle \rho ({\mathit {x}})\rho ({\mathit {y}})\rangle }{\langle \rho ({\mathit {x}})\rangle \langle \rho ({\mathit {y}})\rangle }}-1}
BGS conjecture:
⟨
ρ
(
x
)
ρ
(
y
)
⟩
⟨
ρ
(
x
)
⟩
⟨
ρ
(
y
)
⟩
−
1
=
⟨
ρ
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x
)
ρ
(
y
)
⟩
RMT
⟨
ρ
(
x
)
⟩
RMT
⟨
ρ
(
y
)
⟩
RMT
−
1
{\displaystyle {\frac {\langle \rho ({\mathit {x}})\rho ({\mathit {y}})\rangle }{\langle \rho ({\mathit {x}})\rangle \langle \rho ({\mathit {y}})\rangle }}-1={\frac {\langle \rho ({\mathit {x}})\rho ({\mathit {y}})\rangle _{\operatorname {RMT} }}{\langle \rho ({\mathit {x}})\rangle _{\operatorname {RMT} }\langle \rho ({\mathit {y}})\rangle _{\operatorname {RMT} }}}-1}
The conjecture remains unproven despite supporting numerical evidence.[citation needed
References
^ Bohigas, O.; Giannoni, M. J.; Schmit, C. (2010), "Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws" , Spectral Distributions in Nuclei and Statistical Spectroscopy , World Scientific Publishing Co. Pte. Ltd., pp. 420– 423, doi :10.1142/9789814287395_0024 (inactive 1 April 2025), ISBN 978-981-4287-39-5 , retrieved 2025-03-06 {{citation }}: CS1 maint: DOI inactive as of April 2025 (link )^ Bohigas, O.; Giannoni, M.J.; Schmit, C. (1984). "Spectral properties of the Laplacian and random matrix theories" . Journal de Physique Lettres . 45 (21): 1015– 1022. doi :10.1051/jphyslet:0198400450210101500 . ISSN 0302-072X .
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