Fred Cooper

Supersymmetry and quantum mechanics

In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable and an array of powerful new approximation methods for handling potentials which are not exactly solvable. In this report, we review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications. Exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials, shape invariance and operator transformations. Familiar solvable potentials all have the property of shape invariance. We describe new exactly solvable shape invariant potentials which include the recently discovered self-similar potentials as a special case. The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multisoliton solutions of the KdV equation are constructed. Approximation methods are also discussed within the framework of supersymmetric quantum mechanics and in particular it is shown that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials. Supersymmetry ideas give particularly nice results for the tunneling rate in a double well potential and for improving large N expansions. We also discuss the problem of a charged Dirac particle in an external magnetic field and other potentials in terms of supersymmetric quantum mechanics. Finally, we discuss structures more general than supersymmetric quantum mechanics such as parasupersymmetric quantum mechanics in which there is a symmetry between a boson and a para-fermion of order p.

Aspects of supersymmetric quantum mechanics

Abstract We review the properties of supersymmetric quantum mechanics for a class of models proposed by Witten. Using both Hamiltonian and path integral formulations, we give general conditions for which supersymmetry is broken (unbroken) by quantum fluctuations. The spectrum of states is discussed, and a virial theorem is derived for the energy. We also show that the euclidean path integral for supersymmetric quantum mechanics is equivalent to a classical stochastic process when the supersymmetry is unbroken (E0 = 0). By solving a Fokker-Planck equation for the classical probability distribution, we find Pc(y) is identical to |Ψ0(y)|2 in the quantum theory.

Solitons in the Camassa-Holm shallow water equation

Abstract We study the class of shallow water equations of Camassa and Hold derived from the Lagrangian, L = ∫ [ 1 2 ( ϕ x x x x − ϕ x ) ϕ t − 1 2 ( ϕ x ) 3 − 1 2 ϕ x ( ϕ x x ) 2 − 1 2 κ ϕ x 2 ] d x , using a variational approach. This class contains “peakons” for k=0, which are solitons whose peaks have a discontinuous first derivative. We derive approximate solitary wave solutions to this class of equations using trial variational functions of the form u(x,t)=ψx=A(t) expt[-β(t)|x-q(t)|2n] in a time-dependent variational calculation. For the case k=0 we obtain the exact answer. For k≠0 we obtain the optimal variational solution. For the variational solution having fixed conserved momentum P=∝1/2 (u2+ux2), dx, the solitons scaled amplitude, A/P1/2, and velocity, q ˙ / P 1 / 2 , depend only on the variable z = κ / P . We prove that these scaling relations are true for the exact soliton solutions to the Camassa-Holm equation.

Nonequilibrium quantum fields in the large-N expansion.

An effective action technique for the time evolution of a closed system consisting of one or more mean fields interacting with their quantum fluctuations is presented. By marrying large-[ital N] expansion methods to the Schwinger-Keldysh closed time path formulation of the quantum effective action, causality of the resulting equations of motion is ensured and a systematic, energy-conserving and gauge-invariant expansion about the quasiclassical mean field(s) in powers of 1/[ital N] developed. The general method is exposed in two specific examples, O([ital N]) symmetric scalar [lambda][Phi][sup 4] theory and quantum electrodynamics (QED) with [ital N] fermion fields. The [lambda][Phi][sup 4] case is well suited to the numerical study of the real time dynamics of phase transitions characterized by a scalar order parameter. In QED the technique may be used to study the quantum nonequilibrium effects of pair creation in strong electric fields and the scattering and transport processes in a relativistic [ital e][sup +][ital e][sup [minus]] plasma. A simple renormalization scheme that makes practical the numerical solution of the equations of motion of these and other field theories is described.

Supersymmetry and the Dirac Equation

Abstract We discuss in detail two supersymmetries of the 4-dimensional Dirac operator / kD 2 where / kD = ∂ − ieA , namely the usual chiral supersymmetry and a separate complex supersymmetry. Using SUSY methods developed to categorize solvable potentials in 1-dimensional quantum mechanics we systematically study the cases where the spectrum, eigenfunctions, and S -matrix of / kD 2 can be obtained analytically. We relate these solutions to the solutions of the ordinary massive Dirac equation in external fields. We show that whenever a Schrodinger equation for a potential V ( x ) is exactly solvable, then there always exists a corresponding static scalar field ϕ ( x ) for which the Jackiw-Rebbi type (1 + 1)-dimensional Dirac equation is exactly solvable with V ( x ) and ϕ ( x ) being related by V ( x ) = ϕ 2 ( x ) + ϕ ′( x ). We also discuss and exploit the supersymmetry of the path integral representation for the fermion propagator in an external field.

Compacton solutions in a class of generalized fifth-order Korteweg--de Vries equations

Solitons play a fundamental role in the evolution of general initial data for quasilinear dispersive partial differential equations, such as the Korteweg-de Vries (KdV), nonlinear Schrödinger, and the Kadomtsev-Petviashvili equations. These integrable equations have linear dispersion and the solitons have infinite support. We have derived and investigate a new KdV-like Hamiltonian partial differential equation from a four-parameter Lagrangian where the nonlinear dispersion gives rise to solitons with compact support (compactons). The new equation does not seem to be integrable and only mass, momentum, and energy seem to be conserved; yet, the solitons display almost the same modal decompositions and structural stability observed in integrable partial differential equations. The compactons formed from arbitrary initial data, are nonlinearly self-stabilizing, and maintain their coherence after multiple collisions. The robustness of these compactons and the inapplicability of the inverse scattering tools, that worked so well for the KdV equation, make it clear that there is a fundamental mechanism underlying the processes beyond integrability. We have found explicit formulas for multiple classes of compact traveling wave solutions. When there are more than one compacton solution for a particular set of parameters, the wider compacton is the minimum of a reduced Hamiltonian and is the only one that is stable.

Solving φ1,24 field theory with Monte Carlo

We study lattice g0φ4 field theory for all g0 and fixed renormalized mass M in one and two dimensions using Monte Carlo techniques. We calculate the dimensionless renormalized coupling constant gR = gRM4−d, where d is the dimension of space—time, at fixed small values of the lattice spacing a for various g0 and lattice sizes. Our results are in quantitative agreement with the analyses of high temperature and strong coupling series which rely on extrapolation from large to small lattice spacing.

Path integral formulation of mean-field perturbation theory

Abstract We develop a convenient functional integration method for performing mean-field approximations in quantum field theories. This method is illustrated by applying it to a self-interacting φ 4 scalar field theory and a J μ J μ four-Fermion field theory. To solve the φ 4 theory we introduce an auxiliary field χ and rewrite the Lagrangian so that the interaction term has the form χφ 2 . The vacuum generating functional is then expressed as a path integral over the fields χ and φ. Since the χ field is introduced to make the action no more than quadratic in φ, we do the φ integral exactly. Then we use Laplaces method to expand the remaining χ integral in an asymptotic series about the mean field χ 0 . We show that there is a simple diagrammatic interpretation of this expansion in terms of the mean-field propagator for the elementary field φ and the mean-field bound-state propagator for the composite field χ. The φ and χ propagators appear in these diagrams with the same topological structure that would have been obtained by expanding in the same manner a χφ 2 field theory in which χ and φ are both elementary fields. We therefore argue that by renormalizing these theories so that the mean-field propagators are equivalent, the two theories are described by the same renormalized Greens functions containing the same three parameters, μ 2 , m 2 , and g . The quartic theory is completely specified by the renormalized masses μ 2 and m 2 of the χ and φ fields. These two masses determine the coupling constant g = g ( μ 2 , m 2 ). The cubic theory depends on μ 2 and m 2 and a third parameter g 0 , g = g ( μ 2 , m 2 , g 0 ), where g 0 is the bare coupling constant. We indicate that g ( μ 2 , m 2 , g 0 ) ⩽ g ( μ 2 , m 2 ) with equality obtained only in the limit g 0 → ∞. When g 0 → ∞ the wave function renormalization constant for the χ field in the cubic theory vanishes, and the cubic theory becomes identical to the quartic theory. Our approach guarantees that all quartic theories have the same graphical topology in the mean-field approximation. To illustrate this we show that the mean-field expansion of the four-Fermion current-current interaction theory is renormalizable and reproduces the results of the usual vector meson theory. A coupling-constant eigenvalue condition is derived which could serve to distinguish current-current interactions from normal electrodynamics.

The dark matter problem and quantum gravity

Abstract We explicitly solve the one-loop renormalization group equations for a particular higher derivative quantum gravity and obtain the induced r -dependence of Newtons constant due to quantum fluctuations of the geometry. We then show how, by considering these effects, it may be possible to (a) reconcile nucleosynthesis bounds on the density parameter of the Universe with the predictions of inflationary cosmology, and (b) reproduce the inferred variation of the density parameter with distance. Our calculation can be interpreted as a computation of the contribution of gravitons to the (local) energy density of the Universe.

Particle production in the central rapidity region

We study pair production from a strong electric field in boost-invariant coordinates as a simple model for the central rapidity region of a heavy-ion collision. We derive and solve the renormalized equations for the time evolution of the mean electric field and current of the produced particles, when the field is taken to be a function only of the fluid proper time