Liouville Invariance in Quantum and Classical Mechanics
Alec Maassen van den Brink
and A. M. Zagoskin
Received January 28, 2002; accepted April 30, 2002
The density-matrix and Heisenberg formulations of quantum mechanics follow—for
unitary evolution—directly from the Schr
oodinger equation. Nevertheless, the sym-
metries ofthe corresponding evolution operator, the Liouvillian L¼i ½Á; H , need
not be limited to those ofthe Hamiltonian H. This is due to L only involving
eigenenergy diﬀerences, which can be degenerate even ifthe energies themselves are
not. Remarkably, this possibility has rarely been mentioned in the literature, and
never pursued more generally. We consider an example involving mesoscopic
Josephson devices, but the analysis only assumes familiarity with basic quantum
mechanics. Subsequently, such L-symmetries are shown to occur more widely, in
particular also in classical mechanics. The symmetry’s relevance to dissipative sys-
tems and quantum-information processing is brieﬂy discussed.
KEY WORDS: symmetry; superoperators; phase-space ﬂow.
PACS: 03.65.-w, 03.67.-a, 45.20.Jj, 74.50.þr
Density matrices streamline quantum statistics, by combining the calcula-
tion of quantum-mechanical expectation values and classical averaging over
a probability distribution of states, into a single trace operation.
aesthetically inclined might further appreciate that for a pure state j i, its
density matrix j ih j is uniquely deﬁned, unlike j i itself.
While the state vector obeys the Schro
dinger equation id
j i¼Hj i
h ¼ 1), the density matrix evolves according to the von Neumann (or
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Quantum Information Processing, Vol. 1, Nos. 1/2, April 2002 (# 2002)