Entanglement constrained by superselection rules

Bipartite entanglement may be reduced if there are restrictions on allowed local operations. We introduce the concept of a generalized superselection rule to describe such restrictions, and quantify the entanglement constrained by it. We show that ensemble quantum information processing, where elements in the ensemble are not individually addressable, is subject to the superselection rule associated with the symmetric group (the group of permutations of elements). We prove that even for an ensemble comprising many pairs of qubits, each pair described by a pure Bell state, the entanglement per element constrained by this superselection rule goes to zero for a large number of elements.     

PT-symmetrized supersymmetric quantum mechanics

Supersymmetry between bosons and fermions is modelled withinPT-symmetric quantum mechanics. A non-Hermitian alternative to the Witten’s supersymmetric quantum mechanics is obtained. Work supported by the grant Nr. A 1048004 of GA AS CR. Presented at the DI-CRM Workshop held in Prague, 18–21 June 2000.     

Non-commutative supersymmetric quantum mechanics

General non-commutative supersymmetric quantum mechanics models in two and three dimensions are constructed and some two- and three-dimensional examples are explicitly studied. The structure of the theory studied suggest other possible applications in physical systems with potentials involving spin and non-local interactions.     

Reducibility of supersymmetric quantum mechanics

We constructdeformed annihilation and creation, operators of the harmonic oscillator context in terms of the parity operator and realize, in that way the superalgebrasqm(2) of supersymmetric quantum mechanics. Moreover, this specific example is related to the physical application known as the Calogero problem. The reducibility, of supersymmetric quantum mechanics is then established for arbitraryodd superpotentials, but not for even ones. We also get (minimal) dynamical algebras in both cases, shedding new light on such physical quantities as the Runge-Lenz vector.     

Aspects of supersymmetric quantum mechanics

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 P_c(y) is identical to |Ψ₀(y)|² in the quantum theory.     

About fractional supersymmetric quantum mechanics

Fractional Euler-Lagrange equations are investigated in the presence of Grassmann variables. The fractional Hamiltonian and the path integral of the fractional supersymmetric classical model are constructed. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.     

F-theory Yukawa couplings and supersymmetric quantum mechanics

The localized fermions on the intersection curve Σ of D7-branes, are connected to a N=2 supersymmetric quantum mechanics algebra. Due to this algebra the fields obey a global U(1) symmetry. This symmetry restricts the proton decay operators and the neutrino mass terms. Particularly, we find that several proton decay operators are forbidden and the Majorana mass term is the only one allowed in the theory. A special SUSY QM algebra is studied at the end of the paper. In addition we study the impact of a non-trivial holomorphic metric perturbation on the localized solutions along each matter curve. Moreover, we study the connection of the localized solutions to an N=2 supersymmetric quantum mechanics algebra when background fluxes are turned on.     

Relativistic quantum mechanics of supersymmetric particles

The quantum mechanics of an electron in an external field is developed by Hamiltonian path integral methods. The electron is described classically by an action which is invariant under gauge supersymmetry transformations as well as worldline reparametrizations. The simpler case of a spinless particle is first reviewed and it is pointed out that a strictly canonical approach does not exist. This follows formally from the gauge invariance properties of the action and physically it corresponds to the fact that particles can travel backwards as well as forward in coordinate time. However, appropriate application of path integral techniques yields directly the proper time representation of the Feynman propagator. Next we extend the formalism to systems described by anticommuting variables. This problem presents some difficulty when the dimension of the phase space is odd, because the holomorphic representation does not exist. It is shown, however, that the usual connection between the evolution operator and the path integral still holds provided one indludes in the action the boundary term that makes the classical variational principle well defined. The path integral for the relativistic spinning particle is then evaluated and it is shown to lead directly to a representation for the Feynman propagator in terms of two proper times, one commuting, the other anticommuting, which appear in a symmetric manner. This representation is used to derive scattering amplitudes in an external field. In this step the anticommuting proper time is integrated away and the analysis is carried in terms of one (commuting) proper time only, just as in the spinless case. Finally, some properties of the quantum mechanics of the ghost particles that appear in the path integral for constrained systems are developed in an appendix.     

Functional integral in supersymmetric quantum mechanics

The solution of the square root of the Schrödinger equation for supersymmetric quantum mechanics is expressed in the form of series. The formula may be considered as a functional integral of the chronological exponent of the superpseudodifferential operator symbol over the superspace.     

Modified factorization method and supersymmetric quantum mechanics

We suggest a modified factorization scheme within a supersymmetric framework which affords a consistent treatment of a wide class of Schrödinger potentials. A consequence of this is the possibility of deriving a boson-fermion Hamiltonian with linear interaction.     

Hidden nonassociative structure in supersymmetric quantum mechanics

It is shown that the Hamilton equations in supersymmetric quantum mechanics can be presented in nonassociative form, where the Hamiltonian is decomposed into two nonassociative factors.     

A Golden-Thompson inequality in supersymmetric quantum mechanics

We establish a supersymmetric version of the Golden-Thompson inequality.Supported in part by the Department of Energy under grant DE-FG02-88ER25065.     

Twist deformations of the supersymmetric quantum mechanics

The $ mathcal{N} $ mathcal{N} -extended Supersymmetric Quantum Mechanics is deformed via an abelian twist which preserves the super-Hopf algebra structure of its Universal Enveloping Superalgebra. Two constructions are possible. For even $ mathcal{N} $ mathcal{N} one can identify the 1D $ mathcal{N} $ mathcal{N} -extended superalgebra with the fermionic Heisenberg algebra. Alternatively, supersymmetry generators can be realized as operators belonging to the Universal Enveloping Superalgebra of one bosonic and several fermionic oscillators. The deformed system is described in terms of twisted operators satisfying twist-deformed (anti)commutators. The main differences between an abelian twist defined in terms of fermionic operators and an abelian twist defined in terms of bosonic operators are discussed.     

Higher dimensional supersymmetric quantum mechanics and Dirac equation

We exhibit the supersymmetric quantum mechanical structure of the full 3+1 dimensional Dirac equation considering ‘mass’ as a function of coordinates. Its usefulness in solving potential problems is discussed with specific examples. We also discuss the ‘physical’ significance of the supersymmetric states in this formalism.     

Generalized Gelfand-Naimark-Segal construction for supersymmetric quantum mechanics

The consistent treatment of anticommuting parameters in quantum theories requires the introduction of the Hilbert Q module with a Q scalar product (where Q is infinite-dimensional Grassman-Banach algebra). The extended GNS construction for representations of Q algebras on such Q modules is given.     

Ground state structure in supersymmetric quantum mechanics

We present a rigorous analysis of the vacuum structure of two models of supersymmetric quantum mechanics. They are the quantum mechanics versions of the two-dimensional N = 1 and N = 2 Wess-Zumino quantum field models. We find that the N = 2 quantum mechanics has degenerate vacua. The space of vacuum states is bosonic, and its dimension is determined by the topological properties of the superpotential.     

Boolean transfer from coherent quantum logics to quantum logics with continuous superselection rules

Quantum logics with continuous superselection rules are shown to be Booleanvalued coherent quantum logics. Since modern set theory provides a transfer principle from standard mathematics to Boolean-valued mathematics, this makes it possible to transfer automatically well-known results on coherent quantum logics to quantum logics with continuous superselection rules. Many illustrations are given.