Unstable Hamiltonian Systems and Scattering Theory
A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Non-equilibrium Phenomena".
Deadline for manuscript submissions: 15 September 2024 | Viewed by 60
Special Issue Editor
2. Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel
3. Department of Physics, Ariel University, Ariel 40700, Israel
Interests: relativistic quantum mechanics and quantum field theory; theory of classical and quantum unstable systems and chaos; quantum theory on hypercomplex Hilbert modules; complex projective spaces in quantum dynamics; relativistic statistical mechanics and thermodynamics; high-energy nuclear structure and particle physics
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Special Issue Information
Dear Colleagues,
Classical Hamiltonian systems, although described by smooth Hamiltonian equations, may sometimes develop unstable and chaotic trajectories such as period doubling and trajectory crossing, in nonlinear systems, leaving clear signatures on a Poincar´e surface constructed in the phase space. Such phenomena can occur in fluid mechanics, where smooth laminar flow can break into turbulence. The occurrence of such behaviour is observed even for relatively simple systems such as the Henon–Heiles oscillator, and systems with polynomial potentials. The trajectories may trace out beautiful pictures as seen in, for example, the book of Taylor. It has recently been shown that the criterion for instability on a manifold of curved space using geodesic deviation can be applied to Hamiltonian systems on a flat space using the following formula:
H =p2/2m+V(x)
by defining a conformal transformation
E = H = gij(x)pipj/2m
for which
g ij(x) = δ ij E/(E-V)
The computation of geodesic deviation for this conformally mapped space for a large number of simulated examples provides a reliable prediction of local stability and, often, of chaotic behaviour.
In the framework of quantum theory, we start with Gamow’s description (1928) of nuclear alpha particle emission with an ad hoc form of the Schrödinger equation with complex valued Hamiltonian, providing exact exponential decay and the essential semi-group property for the evolution operator.
Z(t1)Z(t2) = Z(t! + t2).
Wigner and Weisskopf [1930] formulated a more fundamental approach. Assuming an initial state ψ (the unstable state), after the Hamiltonian acts for a time t, the probability that the system remains in this initial state (survival probability) is as follows:
|(ψ, e−iHtψ)|2 = psurvival(t) = 1 − pdecay(t).
This decay law cannot be exponential for short times; however, for longer times, although not too long, the decay becomes exponential. For very long times, it becomes inversely polynomial. This behaviour may be studied by examining the analytic properties of the Laplace transform of psurvival(t), the resolvent
R(z) = (ψ,1\(z − H)ψ),
for which a pole in the second sheet at zp provides an exponential decay law for times that are not too short and not too long. For two-body decay, however, the matrix valued residues are not orthogonal, and the reconstituted time dependence of the decay does not therefore have the required semigroup property.
Lax and Phillips, based on the fundamental work of Foias and Nagy, have shown that the semigroup property of the evolution of an unstable system can be achieved by defining a foliation of the Hilbert space on time, with incoming and outgoing subspaces. The orthogonal complement of these subspaces then evolves as an exact semigroup. The underlying idea of Foias and Nagy was that one can think of a larger system in which the evolution is unitary, and the semigroup property is achieved via projection into a subsystem. This structure has been exploited to construct a viable description of the two-channel K0 particle decay, and a quantum field theoretical picture of emission and absorption for the large system and the subsystem.
Almost all of the particles listed in the Particle Data Booklet (CERN) are resonances, very short-lived objects with sharply distributed masses and definite charge and spin. Observation of these objects has in many cases given rise to new theories of the forces in Nature, such as the unitary symmetries, supersymmetry, and made evident, as in the pi meson system, the unexpected breaking of parity (mirror reflection) symmetry.
Contributions to the theory of open and unstable systems, both classically and quantum mechanically, would be welcome for this project.
Prof. Dr. Lawrence Horwitz
Guest Editor
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Keywords
- classical instability
- chaos
- quantum instability
- decaying systems
- semigroups
- symmetries
