LATIN-AMERICAN SCHOOL OF PHYSICS XXXV ELAF; Supersymmetries in Physics and Its Applications
744(2004); http://dx.doi.org/10.1063/1.1853196View Description Hide Description
The goal of these lectures is to present a practical introduction to the subject of supersymmetry (SUSY). We shall focus on the basic techniques needed to write down supersymmetric Lagrangians in four and higher dimensions, namely superspace, superfields and their actions.
744(2004); http://dx.doi.org/10.1063/1.1853197View Description Hide Description
In spite of the extraordinary success of the Standard Model (SM) supplemented with massive neutrinos in accounting for the whole huge bulk of phenomenology which has been accumulating in the last three decades, there exist strong theoretical reasons in particle physics and significant “observational” hints in astroparticle physics for new physics beyond it. My lecture is devoted to a critical assessment of our belief in such new physics at the electroweak scale, in particular identifying it with low‐energy supersymmetric extensions of the SM. I’ll explain why we have concrete hopes that this decade will shed definite light on what stands behind the phenomenon of electroweak symmetry breaking.
744(2004); http://dx.doi.org/10.1063/1.1853200View Description Hide Description
744(2004); http://dx.doi.org/10.1063/1.1853201View Description Hide Description
An elementary introduction is given to the subject of supersymmetry in quantum mechanics which can be understood and appreciated by any one who has taken a first course in quantum mechanics. We demonstrate with explicit examples that given a solvable problem in quantum mechanics with n bound states, one can construct n new exactly solvable Hamiltonians having n − 1, n − 2,…, 0 bound states. The relationship between the eigenvalues, eigenfunctions and scattering matrix of the supersymmetric partner potentials is derived and a class of reflectionless potentials are explicitly constructed. We extend the operator method of solving the one‐dimensional harmonic oscillator problem to a class of potentials called shape‐invariant potentials. It is worth emphasizing that this class includes almost all the solvable problems that are found in the standard text books on quantum mechanics. Further, we show that given any potential with at least one bound state, one can very easily construct one continuous parameter family of potentials having same eigenvalues and s‐matrix. The supersymmetry inspired WKB approximation (SWKB) is also discussed and it is shown that unlike the usual WKB, the lowest order SWKB approximation is exact for the shape‐invariant potentials and further, this approximation is not only exact for large quantum numbers but by construction, it is also exact for the ground state. Finally, we also construct new exactly solvable periodic potentials by using the machinery of supersymmetric quantum mechanics.
744(2004); http://dx.doi.org/10.1063/1.1853202View Description Hide Description
The Hamiltonian in Supersymmetric Quantum Mechanics is defined in terms of charges that obey the same algebra as that of the generators of supersymmetry in field theory. The consequences of this symmetry for the spectra of the component parts that constitute the supersymmetric system are explored. The implications of supersymmetry for the solutions of the Schrödinger equation, the Dirac equation, the inverse scattering theory and the multi‐soliton solutions of the KdV equation are examined. Applications to scattering problems in Nuclear Physics with specific reference to singular potentials which arise from considerations of supersymmetry will be discussed.
744(2004); http://dx.doi.org/10.1063/1.1853203View Description Hide Description
We review the higher‐order supersymmetric quantum mechanics (H‐SUSY QM), which involves differential intertwining operators of order greater than one. The iterations of first‐order SUSY transformations are used to derive in a simple way the higher‐order case. The second order technique is addressed directly, and through this approach unexpected possibilities for designing spectra are uncovered. The formalism is applied to the harmonic oscillator: the corresponding H‐SUSY partner Hamiltonians are ruled by polynomial Heisenberg algebras which allow a straight construction of the coherent states.
744(2004); http://dx.doi.org/10.1063/1.1853204View Description Hide Description
The supersymmetric method is a powerful method for the nonperturbative evaluation of quenched averages in disordered systems. Among others, this method has been applied to the statistical theory of S‐matrix fluctuations, the theory of universal conductance fluctuations and the microscopic spectral density of the QCD Dirac operator.
We start this series of lectures with a general review of Random Matrix Theory and the statistical theory of spectra. An elementary introduction of the supersymmetric method in Random Matrix Theory is given in the second and third lecture. We will show that a Random Matrix Theory can be rewritten as an integral over a supermanifold. This integral will be worked out in detail for the Gaussian Unitary Ensemble that describes level correlations in systems with broken time‐reversal invariance. We especially emphasize the role of symmetries.
As a second example of the application of the supersymmetric method we discuss the calculation of the microscopic spectral density of the QCD Dirac operator. This is the eigenvalue density near zero on the scale of the average level spacing which is known to be given by chiral Random Matrix Theory. Also in this case we use symmetry considerations to rewrite the generating function for the resolvent as an integral over a supermanifold.
The main topic of the second last lecture is the recent developments on the relation between the supersymmetric partition function and integrable hierarchies (in our case the Toda lattice hierarchy). We will show that this relation is an efficient way to calculate superintegrals. Several examples that were given in previous lectures will be worked out by means of this new method. Finally, we will discuss the quenched QCD Dirac spectrum at nonzero chemical potential. Because of the nonhermiticity of the Dirac operator the usual supersymmetric method has not been successful in this case. However, we will show that the supersymmetric partition function can be evaluated by means of the replica limit of the Toda lattice equation.