These potentials are modified versions of morse potentials and can also be related to modified forms of the radial harmonic oscillator and modified coulomb potential. Im reading wittens paper on supersymmetry and morse theory and am confused about the details of the instanton calculation which he uses to define a morse complex beginning at page 11 of the pdf. Once people started studying various aspects of susy qm, it was soon clear that this. The hamiltonian hierarchy method is used to get the real energy eigenvalues and corresponding wave functions. Division algebras and supersymmetry ii baez, john c. Can one also represent other hamiltonians as the absolute square of an operator and then construct their solutions algebraically. To make progress on these problems, physicists developed supersymmetric quantum mechanics, an application of the supersymmetry susy superalgebra to quantum mechanics as opposed to quantum field theory. We want to understand the groundstates of the model in various geometries with various boundary conditions. Witten writes down the relevant supersymmetric lagrangian and then states. A simple but often underappreciated fact is that the worldline theory of any spinning particle is supersymmetric, and hence is supersymmetric quantum mechanics, on the worldline. Another reference is ed wittens supersymmetry and morse theory this classical aspects of this story are really dodgy because we have fermions, so we need odd complex numbers which anticommute and things like that, and quantization is really strange, so if you wish you can just skip to the quantum parts. We show reflection symmetries of the theory are useful to select true vacuums. Supersymmetric theory of stochastic dynamics or stochastics sts is an exact theory of stochastic partial differential equations sdes, the class of mathematical models with the widest applicability covering, in particular, all continuous time dynamical systems, with and without noise. The remarkable result will be considered that using supersymmetric quantum mechanics one can derive the classical morse inequalities.
Motivated strongly by the unifica tion of the fundamental forces, supersymmetry is a formalism that describes a symmetry in the exchange of bosons with fermions. Barannikov, the framed morse complex and its invariants, advances in soviet math. Supersymmetry and supersymmetric quantum mechanics. Supersymmetry and morse theory school of natural sciences. Morse theory the simplest example of supersymmetric quantum mechanics is a system which is very well known in mathematics. The case of ordinary supersymmetric quantum mechanics corresponds to k 2. The students should learn the basics of supersymmetric quantum mechanics and understand some of the connections with mathematics, such as cohomology, topological invariants of riemannian manifolds and index theory. We propose our work tackling the problem from the perspective of ladder operators which later can be directed to the whole representation of the system. 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. Supersymmetric quantum mechanics, exactly solvable potentials. Supersymmetry is a relatively recent development in theoretical physics.
Ulb229cq975 c extended harmonic oscillator and parasupersymmetric quantum mechanics c. Although supersymmetric quantum mechanics might seem irrelevant, it has important mathematical and physical conse quences. The morse homology theorem, which says that the morse complex. A connection between fractional supersymmetric quantum mechanics and ordinary supersymmetric quantum mechanics is brie. A method identifying the morse potential according to the energy spectrum of diatomic molecule has been suggested. The text contains a large selection of examples, problems, and solutions that illustrate the fundamentals of susyqm and its applications. The relation to supersymmetric quantum mechanics is due to.
Pdf supersymmetry and morse theory semantic scholar. The idea is to study symmetry breaking in quantum mechanics to get a better understanding of this process and then draw conclusions for quantum eld theory qft. Irreps and offshell invariant actions of the nextended. Supersymmetric quantum mechanics susy qm is a powerful tool for generating new potentials with known spectra departing from an initial solvable one. At this present seibergwitten theory2 has attracted the attention of both theoretical physicists and mathematicians. A short history of the interaction between qft and topology. Fractional supersymmetric quantum mechanics as a set of. Its di cult to see how this method relates to everything else you might have heard about supersymmetry and high energy particle physics. Im reading wittens paper on supersymmetry and morse theory and am confused about the details of the instanton calculation which he uses to define a morse complex beginning at page 11 of the pdf witten writes down the relevant supersymmetric lagrangian and then states. Thus, in the early days, susy was studied in quantum mechanics as a testing ground for the nonperturbative methods of seeing susy breaking in. Recently developed supersymmetric in field theory has been successfully employed to make. In 2 we will discuss systems which obey the simplest supersymmetry algebra of l4. The main point is that homology is defined via hodges harmonic forms and critical points enter via stationary phase approximation to quantum mechanics. After all these systems are reduced to supersymmetric quantum mechanics, one might hope to achieve classi cation of the critical points of the height function by means of morse theory, cf.
In these lecture notes we will present some general formulas concerning susy qm of first and second order for onedimensional arbitrary systems, and we will illustrate the method through the trigonometric poschlteller potentials. This was perhaps the rst instance of using quantum theory to nd topological invariants. The simplest example of supersymmetric quantum mechanics is a system which is very well known in mathematics. After susy was introduced into quantum mechanics people started to real. The supersymmetric quantum mechanics with large number n of supersymmetries. Morse theory and supersymmetry uvafnwi universiteit van. Trigonometric poschlteller plus rosenmorse is a noncentral shaped invariance potential.
Here we will concentrate on the quantum mechanical properties. We approach the subject from the viewpoint of morse theory. This is a textbook on quantum mechanics which presents the traditional treatment of the schrodinger equation as well as the dirac equation more briefly. C extended harmonic oscillator and parasupersymmetric. An introduction to supersymmetry in quantum mechanical. Domain walls, black holes, and supersymmetric quantum mechanics. An introduction to supersymmetry in quantum mechanical systems. It is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian.
For a diatomic molecule system in morse potential the eigenfunctions and eigenvalues have been found by supersymmetric quantum mechanics sqm approach. A brief introduction to supersymmetric quantum mechanics november 28, 2007 in class, weve learned about the supersymmetric, or factorization, method for solving some quantum mechanical potentials. The presentation is based on the introduction of a generalized weylheisenberg algebra w k. A connection between fractional supersymmetric quantum mechanics and ordinary supersymmetric quantum mechanics is established in this letter. Supersymmetric quantum mechanics susy qm is a powerful tool for generating new potentials with known spectra departing. Grassmann algebras, clifford algebras and the nextendedd 1. The following are lecture notes on supersymmetric quantum mechanics and morse theory given in a minilecture series in the spring of 2019. For example, it has been used to demonstrate dynamical super symmetry breaking 51, to prove the atiyahsinger index theorem 6.
Another reference is ed wittens supersymmetry and morse theory this classical aspects of this story are really dodgy because we have fermions, so we need odd. For a brief derivation in the context of quantum eld theories one may consider the referenced literature e. Exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials, shape invariance and operator transformations. We will consider a couple of problems in dimensions, that is, supersymmetric quantum mechanics, by using supersymmetric path integrals, to illustrate the relationship between physics of the supersymmetric model and geometry of the background space which is some manifold in the form of euler characteristic of this manifold. After a considerable number of works investigating susy in field theory, con rmation. To make progress on these problems, physicists developed supersymmetric quantum mechanics, an application of the supersymmetry susy. The energy eigenvalues and eigenfunction of trigonometric poschlteller plus rosen morse non\ud central potential are analysis using supersymmetric quantum mechanics. Although ordinary supersymmetric quantum mechanics sqm was introduced more than 20 years ago, its extension as fractional sqm is still the object of numerous works. The mathematical formalism of supersymmetric quantum mechanics and the darboux transformation are used to determine the bound states for the morse anharmonic oscillator with an approximate rotational term.
Supersymmetry and morse theory 665 the results of 2 have an analogue for complex manifolds, which will be discussed in a separate paper. Division algebras and supersymmetry iii huerta, john, advances in theoretical and mathematical physics, 2012. Indeed, due to the energy scale to be of arbitrary origin the energy in quantum mechanics is not always positive. Trigonometric poschlteller plus \ud rosenmorse is a noncentral shaped invariance potential. Trigonometric poschlteller plus rosen morse is a noncentral shaped invariance potential. Central potential are analysis using supersymmetric quantum mechanics. Instanton effects and witten complex in supersymmetric. It was hoped that studying susys consequences in this simpler setting would lead to new understanding.
It is even possible to obtain exact results thanks to supersymmetry. For a compact space m, it has been shown by witten that supersymmetric quantum mechanics with the action is just hodgede rham theory of m. The students should learn the basics of supersymmetric quantum mechanics and understand some of the connections with mathematics, such as cohomology, topological invariants of riemannian manifolds and. Supersymmetry and supersymmetric quantum mechanics request pdf.
For the first time the taylorexpanded renormalized morse oscillator is studied within the framework of supersymmetric quantum mechanics theory. We will see that such systems have a very surprising connection with morse theory. Three lectures onfukayaseidelcategoriesand webbased formalism. The supersymmetric quantum mechanics theory and darboux. Domain walls, black holes, and supersymmetric quantum. Understanding the consequences of supersymmetry has proven mathematically daunting, and it has likewise been difficult to develop theories that could account for symmetry breaking, i.
Another reference is ed wittens supersymmetry and morse theory this classical aspects of this story are really dodgy because we have fermions, so we need odd complex numbers which anticommute and things. Pathintegral aspects of supersymmetric quantum mechanics. Instanton solutions or tunneling paths in this theory would be extrema of this lagrangian, written with a euclidean. The result has shown that morse potential is shape invariance. This deformed supersymmetric quantum mechanics arises as the pointparticle limit of the type ii. Three lectures onfukayaseidelcategoriesand webbased. The basis of supersymmetric quantum mechanics was set in theoretical particle physics.
Some of the implications of modern ideas in mathematics for supersymmetric theories are discussed. Supersymmetric theory of stochastic dynamics wikipedia. 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 parafermion of. We have written this book in order to provide a single compact source for undergraduate and graduate students, as well as for professional physicists who want to understand the essentials of supersymmetric quantum mechanics. On the work of edward witten university of california.
Supersymmetric quantum mechanics and solvable models. Here is the main di erence of susy between eld theory and quantum mechanics. Instantons in wittens supersymmetry and morse theory. Hence, we can use topological methods in particular, morse theory to classify possible bps states and supersymmetric vacua they connect. The book provides a single compact source for undergraduate and graduate students and professional physicists who want to understand the essentials of supersymmetric quantum mechanics susyqm. We recall the treatment of the harmonic oscillator by means of ladder operators a and a and pose the following question. This explains the reason why it can be applied to several different mathematical problems ranging from morse theory 3 to the index theorems. A brief introduction to supersymmetric quantum mechanics.
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