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Register a new userTwo-Dimensional Supersymmetry: From SUSY Quantum Mechanics to Integrable Classical Models
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Two known 2-dim SUSY quantum mechanical constructions - the direct generalization of SUSY with first-order supercharges and Higher order SUSY with second order supercharges - are combined for a class of 2-dim quantum models, which {it are not amenable} to separation of variables. The appropriate classical limit of quantum systems allows us to construct SUSY-extensions of original classical scalar Hamiltonians. Special emphasis is placed on the symmetry properties of the models thus obtained - the explicit expressions of quantum symmetry operators and of classical integrals of motion are given for all (scalar and matrix) components of SUSY-extensions. Using Grassmanian variables, the symmetry operators and classical integrals of motion are written in a unique form for the whole Superhamiltonian. The links of the approach to the classical Hamilton-Jacobi method for related flipped potentials are established.
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We consider the self-adjoint extensions (SAE) of the symmetric supercharges and Hamiltonian for a model of SUSY Quantum Mechanics in $mathbb{R}^+$ with a singular superpotential. We show that only for two particular SAE, whose domains are scale invariant, the algebra of N=2 SUSY is realized, one with manifest SUSY and the other with spontaneously broken SUSY. Otherwise, only the N=1 SUSY algebra is obtained, with spontaneously broken SUSY and non degenerate energy spectrum.
Superpotentials in ${cal N}=2$ supersymmetric classical mechanics are no more than the Hamilton characteristic function of the Hamilton-Jacobi theory for the associated purely bosonic dynamical system. Modulo a global sign, there are several superpotentials ruling Hamilton-Jacobi separable supersymmetric systems, with a number of degrees of freedom greater than one. Here, we explore how supersymmetry and separability are entangled in the quantum version of this kind of system. We also show that the planar anisotropic harmonic oscillator and the two-Newtonian centers of force problem admit two non-equivalent supersymmetric extensions with different ground states and Yukawa couplings.
Two planar supersymmetric quantum mechanical systems built around the quantum integrable Kepler/Coulomb and Euler/Coulomb problems are analyzed in depth. The supersymmetric spectra of both systems are unveiled, profiting from symmetry operators not related to invariance with respect to rotations. It is shown analytically how the first problem arises at the limit of zero distance between the centers of the second problem. It appears that the supersymmetric modified Euler/Coulomb problem is a quasi-isospectral deformation of the supersymmetric Kepler/Coulomb problem.
Out-of-time-order correlator (OTOC) $langle [x(t),p]^2 rangle $ in an inverted harmonic oscillator (IHO) in one-dimensional quantum mechanics exhibits remarkable properties. The quantum Lyapunov exponent computed through the OTOC precisely agrees with the classical one. Besides, it does not show any quantum fluctuations for arbitrary states. Hence, the OTOC may be regarded as ideal indicators of the butterfly effect in the IHO. Since IHOs are ubiquitous in physics, these properties of the OTOCs might be seen in various situations too. In order to clarify this point, as a first step, we investigate the OTOCs in one dimensional quantum mechanics with polynomial potentials, which exhibit butterfly effects around the peak of the potential in classical mechanics. We find two situations in which the OTOCs show exponential growths reproducing the classical Lyapunov exponent of the peak. The first one, which is obvious, is using suitably localized states near the peak and the second one is taking a double scaling limit akin to the non-critical string theories.
We elaborate on integrable dynamical systems from scalar-gravity Lagrangians that include the leading dilaton tadpole potentials of broken supersymmetry. In the static Dudas-Mourad compactifications from ten to nine dimensions, which rest on these leading potentials, the string coupling and the space-time curvature become unbounded in some regions of the internal space. On the other hand, the string coupling remains bounded in several corresponding solutions of these integrable models. One can thus identify corrected potential shapes that could grant these features generically when supersymmetry is absent or non-linearly realized. On the other hand, large scalar curvatures remain present in all our examples. However, as in other contexts, the combined effects of the higher-derivative corrections of String Theory could tame them.
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