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Modern Physics Letters B

ON SINGULAR ANYON WAVEFUNCTIONS

Johan Grundberg, T. H. Hansson, A. Karlhede, Jon Magne Leinaas

Published Mar 20, 1991• 28 citations
Publisher

Abstract

We show that quantum mechanics allows anyon wavefunctions that are singular as the particles approach each other. An extra real parameter, which determines the nature of this singular behaviour, is necessary to have a well defined quantum mechanical problem. In realistic models, anyons are expected to have an internal structure, corresponding to a smearing of the statistical flux. We study such models and find that, in generic cases, only regular wavefunctions are allowed.

DOI
10.1142/s0217984991000642

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We show that quantum mechanics allows anyon wavefunctions that are singular as the particles approach each other. An extra real parameter, which determines the nature of this singular behaviour, is necessary to have a well defined quantum mechanical problem. In realistic models, anyons are expected to have an internal structure, corresponding to a smearing of the statistical flux. We study such models and find that, in generic cases, only regular wavefunctions are allowed.
When dealing with singular potentials, such as point interactions, the term V ( x )ψ( x ) in Schrödinger's equation is, in general, mathematically ill-defined. The traditional way of dealing with this difficulty, by employing regularization techniques, has led to some ambiguous results for the δ'( x ) potential in the literature. Here we propose a mathematically consistent approach to deal with the one-dimensional version of this problem by considering from the beginning the distributional nature of the whole interaction term. We show that the interaction is univocally determined from two simple mathematical requirements on the interaction distribution, together with the physical requirement of probability flux conservation and considerations of symmetry.
We analyse transition potentials, i.e. potentials exhibiting limiting inverse-square behaviour V(r) to r approximately 0 alpha r -2 in non-relativistic quantum mechanics using the techniques of supersymmetry. For the range - 1/4 <or= alpha < 3/4 , the eigenvalue problem becomes ill defined (since it is not possible to choose a unique eigenfunction based on square integrability and boundary conditions). It is shown that supersymmetric quantum mechanics (SUSYQM) provides a natural prescription for a unique determination of the spectrum. Interestingly, our SUSYQM-based approach picks out the same 'less singular' wavefunctions as the conventional approach and thus provides a simple justification for the usual practice in the literature. Two examples (the Poschl-Teller II potential and a two-anyon system on the plane) have been worked out for illustrative purposes.
The symbol-matter problem of biological systems, posed clearly by H. H. Pattee, is followed up to its origin in a quantum mechanical framework. The two complementer descriptional mode (i.e. dynamical versus symbolic or quantum mechanical versus ''classical'') are defined as connected in a Copenhagen–type quasiclassical (h → 0) dynamics (internal time-dependent discrete projection operator formalism). The concept of natural symbol as constraint is described in the form of a quasiclassical wavefunction with classical measurement records as variables while the basic projectors are defined as a quasiclassical dynamical "bra" Hilbert vector acting on a quantum dynamical "ket" vector. The origin problem is touched necessarily upon, as a split between measurement record (so its quasiclassical wavefunction, natural symbol) and the quantum mechanical dynamical wavefunction. The concept of symmetry-restoring internal time process is introduced, following from compelling internal retrocausality of a time evolution (advocated several times elsewhere). While adopting basically the Copenhagen view, using some ideas of Everett and other aspects of non–orthodox formulations of quantum measurement, it is shown that the necessary imbedding of internal (-τ) time evolution into the split off external (+t) one and internal time ’’quantizations’’ (leading to a discrete total semigroup) results in a converging chain of quantum internal measurement transitions, similar to those introduced by H. P. Stapp in his symbolic (human mind-) model, ultimately leading to a model of an elementary, very special self-reproducing quantum automata, where the state transition functions are internally determined (programmed) from the very origin of their emergence. It is shown that only off-diagonal statistical matrix elements are to be used in the internal projectionally evoluting time–series (metabolism), ’’self’’-reproduction (diagonality) being the fixed point of the recursion of the internal state transitions of the proposed quantum mechanical automata model (’’metaphor’’).
An elementary introduction is given to the concept of anyons—quantum particles whose ‘statistics’ interpolate smoothly between those of bosons and fermions. Such particles—or quasi-particle excitations—can occur in two-dimensional systems in which the particles experience strong short-range mutual repulsions. These circumstances effectively mean that one has to apply quantum mechanics to a configuration space in which some points are excluded (i.e. it has ‘holes’)—namely, those points where two particles would coincide. The Aharonov—Bohm effect provides the simplest physical example of quantum mechanics in such a non-simply connected space, and is also fundamental to a simple model for anyons. A qualitative introduction is given to two currently proposed physical applications of anyons: high temperature superconductivity, and the (fractional) quantum Hall effect.
Intermediate statistics interpolating from Bose statistics to Fermi statistics are allowed in two dimensions. This is due to the particular topology of the two dimensional configuration space of identical particles, leading to non trivial braiding of particles around each other. One arrives at quantum many-body states with a multivalued phase factor, which encodes the any- onic nature of particle windings. Bosons and fermions appear as two limiting cases. Gauging away the phase leads to the so-called anyon model, where the charge of each particle interacts “à la Aharonov-Bohm” with the fluxes carried by the other particles. The multivaluedness of the wave function has been traded off for topological interactions between ordinary particles. An alternative Lagrangian formulation uses a topological Chern-Simons term in 2+1 dimensions. Taking into account the short distance repulsion between particles leads to an Hamiltonian well defined in perturbation theory, where all perturbative divergences have disappeared. Together with numerical and semi-classical studies, perturbation theory is a basic analytical tool at disposal to study the model, since finding the exact N-body spectrum seems out of reach (except in the 2-body case which is solvable, or for particular classes of N-body eigenstates which generalize some 2-body eigenstates). However, a simplification arises when the anyons are coupled to an external homogeneous magnetic field. In the case of a strong field, by projecting the system on its lowest Landau level (LLL, thus the LLL-anyon model), the anyon model becomes solvable, i.e., the classes of exact eigenstates alluded to above provide for a complete interpolation from the LLL-Bose spectrum to the LLL-Fermi spectrum. Being a solvable model allows for an explicit knowledge of the equation of state and of the mean occupation number in the LLL, which do interpolate from the Bose to the Fermi cases. It also provides for a combinatorial interpretation of LLL-anyon braiding statistics in terms of occupation of single particle states. The LLL-anyon model might also be relevant experimentally: a gas of electrons in a strong magnetic field is known to exhibit a quantized Hall conductance, leading to the integer and fractional quantum Hall effects. Haldane/exclusion statistics, introduced to describe FQHE edge excitations, is a priori different from anyon statistics, since it is not defined by braiding considerations, but rather by counting arguments in the space of available states. However, it has been shown to lead to the same kind of thermodynamics as the LLL-anyon thermodynamics (or, in other words, the LLL-anyon model is a microscopic quantum mechanical realization of Hal- dane’s statistics). The one dimensional Calogero model is also shown to have the same kind of thermodynamics as the LLL-anyons thermodynamics. This is not a coincidence: the LLL-anyon model and the Calogero model are intimately related, the latter being a particular limit of the former. Finally, on the purely combinatorial side, the minimal difference partition problem — partition of integers with minimal difference constraints on their parts — can also be mapped on an abstract exclusion statistics model with a constant one-body density of states, which is neither the LLL-anyon model nor the Calogero model. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.