BREAKING A QUANTUM SYMMETRY ON
THE
TABLETOP. A
recurrent theme in art and science, the concept of symmetry has become
a powerful scientific tool for the analysis of physical systems.
However, under special circumstances, a "quantum anomaly" occurs: the
laws of quantum physics break a system's apparent symmetry. After a
long search, a research group (Horacio Camblong, University of San
Francisco, camblong@usfca.edu,
and collaborators at Universidad Nacional de La Plata, Argentina) has
found a relatively simple example of a quantum anomaly: the interaction
of a polar molecule with an electron. A polar molecule, despite being
neutral, has a permanent separation of electric chargea dipole. This
dipole produces an electric field, which can capture electrons if it is
strong enough. Can such an arrangement exist as a stable ion, with its
"extra" electron? The researchers formulated the answer to this
question in the language of symmetry. In physics, symmetry means that a
system, such as the moleculeelectron arrangement, behaves the same
after you perform a change to it, such as stretching the molecule to
larger scales and making appropriate adjustments to other variables in
the system. At first glance, the electronmolecule interaction exhibits
a remarkable scale invariance: the system "looks" the same when viewed
from different scales in space and timeat least in a classical
physics description which treats the molecule as a dipole and the
electron as a point of charge. But this tidy picture breaks down with a
proper treatment of the system, as prescribed by quantum field theory.
A quantum field theory treatment requires the process of
renormalization, which removes certain mathematical infinities and
inconsistencies from the quantum approach. This process also makes the
molecule's energy levels discrete or quantized rather than continuous.
Examining the system this way, the researchers found that the scale
invariance broke down. In fact, a large body of existing evidence, both
experimental and numerical, supports their conclusion. While all other
known quantum anomalies occur at high energies (an example is chiral
symmetry in nuclear physics), the work suggests that quantum symmetry
breaking can occur at much lower energies, in the domain of interacting
electrons and molecules. (Camblong et al., Physical Review Letters, 26
November 2001.)
Authors: Klishevich, Sergey; Plyushchay, Mikhail
The nonlinear supersymmetry of onedimensional systems is investigated
in the context of the quantum anomaly problem. Any classical
supersymmetric system characterized by the nonlinear in the Hamiltonian
superalgebra is symplectomorphic to a supersymmetric canonical system
with the holomorphic form of the supercharges. Depending on the
behaviour of the superpotential, the canonical supersymmetric systems
are separated into the three classes. In one of them the parameter
specifying the supersymmetry order is subject to some sort of classical
quantization, whereas the supersymmetry of another extreme class has a
rather fictive nature since its fermion degrees of freedom are
decoupled completely by a canonical transformation. The nonlinear
supersymmetry with polynomial in momentum supercharges is analysed, and
the most general oneparametric Calogerolike solution with the second
order supercharges is found. Quantization of the systems of the
canonical form reveals the two anomalyfree classes, one of which gives
rise naturally to the quasiexactly solvable systems. The quantum
anomaly problem for the Calogerolike models is ''cured'' by the
specific superpotentialdependent term of order hbar2. The nonlinear
supersymmetry admits the generalization to the case of twodimensional
systems.
