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Using Hamiltonian methods, we find six relativistic theories of nonlinear electrodynamics for which plane wave perturbations about a constant uniform background are not birefringent. All have the same conformal strong-field limit to Bialynicki-Birula (BB) electrodynamics, but only four avoid superluminal propagation: Born-Infeld (BI), its non-conformal ``extreme'' limits (electric and magnetic) and the conformal BB limit. The quadratic dispersion relation of BI is shown to degenerate in the extreme limits to a pair of linear relations, which become identical in the BB limit., Comment: 39 pages. V3: Expanded version with additional material

In the Hamiltonian formulation of chiral 2k-form electrodynamics, the 2k-form potential on the (4k+1)-space is defined up to the addition of either (i) a closed $2k$-form or (ii) an exact 2k-form, depending on the choice of chirality constraint. Case (i) is realized by the Floreanini-Jackiw 2D chiral boson (for k=0) and its Henneaux-Teitelboim generalisation to k>0. For all k, but focusing on the 6D case, we present a simple Lorentz-invariant Hamiltonian model that realizes case (ii), and we derive it from Siegel's manifestly Lorentz invariant Lagrangian formulation., Comment: 24 pages, minor corrections and further additional references in v3

We analyze the global symmetries of ${\cal N}=4$ supersymmetric mechanics involving $4n$-dimensional Quaternion-K\"ahler (QK) $1D$ sigma models on projective spaces $\mathbb{H}{\rm H}^n$ and $\mathbb{H}{\rm P}^n$ as the bosonic core. All Noether charges associated with global worldline symmetries are shown to vanish as a result of equations of motion, which implies that we deal with a severely constrained hamiltonian system. The complete hamiltonian analysis of the bosonic sector is performed., Comment: 0 + 27 pages, essential revision: sections renumbered, sect. 2 extended, sect. 3.2, a new affiliation, refs. [1], [3], [13], [16-18], [22] and some clarifying comments added; published version

The IR limit of a planar static D3-brane in AdS5 x S5 is a tensionless D3-brane at the AdS horizon, with dynamics governed by a strong-field limit of the Dirac-Born-Infeld action analogous to that found from the Born-Infeld action by Bialynicki-Birula. As in that case, the field equations are those of an interacting 4D conformal invariant field theory with an Sl(2;R) electromagnetic duality invariance, but the D3-brane origin makes these properties manifest. We also find an Sl(2;R)-invariant action for these equations., Comment: 24 pages. Contribution to a volume in memory of Peter Freund. V2 includes an additional reference, an expanded discussion of some issues, especially Lorentz invariance, and corrections of minor errors

Using the N=4, 1D harmonic superspace approach, we construct a new type of N=4 supersymmetric mechanics involving 4n-dimensional Quaternion-K\"ahler (QK) 1D sigma models as the bosonic core. The basic ingredients of our construction are {\it local} N=4, 1D supersymmetry realized by the appropriate transformations in 1D harmonic superspace, the general N=4, 1D superfield vielbein and a set of 2(n+1) analytic "matter" superfields representing (n+1) off-shell supermultiplets (4, 4, 0). Both superfield and component actions are given for the simplest QK models with the manifolds \mathbb{H}H^n = Sp(1,n)/[Sp(1) x Sp(n)] and \mathbb{H}P^n = Sp(1+n)/[Sp(1) x Sp(n)] as the bosonic targets. For the general case the relevant superfield action and constraints on the (4, 4, 0) "matter" superfields are presented. Further generalizations are briefly discussed., Comment: further minor corrections in eqs. (2.21), (4.24) and (A9)

The Souriau massless Galilean particle of "colour" $k$ and spin $s$ is shown to be the Galilean limit of the Souriau tachyon of mass $m = ik$ and spin $s$. We compare and contrast this result with the recent Galilean limit of the Nambu-Goto string and the Green-Schwarz superstring., Comment: 9 pages. v3 incorporates minor corrections and clarifications to v2, to accord with published version

The action for a massless particle in 4D Minkowski space has a worldline time-reversing symmetry corresponding to CPT invariance of the quantum theory. The analogous symmetry of the N-extended superparticle is shown to be anomalous when N is odd, in the supertwistor formalism this is because a CPT-violating worldline-Chern-Simons term is needed to preserve the chiral U(1) gauge invariance. This accords with the fact that no massless N=1 super-Poincar\'e irrep is CPT-self-conjugate. There is a CPT self-conjugate supermultiplet when N is even, but it has $2^{N+1}$ states when N/2 is odd (e.g. the N=2 hypermultiplet) in contrast to just $2^N$ when N/2 is even (e.g. the N=4 Maxwell supermultiplet). This is shown to follow from a Kramers degeneracy of the superparticle state space when N/2 is odd., Comment: 23 pp. Typos corrected, references added and Appendix expanded in v.2. Additional sign choices, simplified equations and clarifications included in v.3

We present a novel construction of the super-Pauli-Lubanski pseudo-vector for 4D supersymmetry and show how it arises naturally from the spin-shell constraints in the supertwistor formulation of superparticle dynamics. We illustrate this result in the context of a simple classical action for a "superspinning particle" of superspin 1/2. We then use an Sl(2;K)-spinor formalism for K=R,C,H to unify our 4D results with previous results for 3D and 6D (super)Pauli-Lubanski tensors., Comment: 23 pp. v4 is a modification of the rewritten v3 to include clarifications and additional references of the published version

Gauge-invariant twistor variables are found for the massive spinning particle with N-extended local worldline supersymmetry, in spacetime dimensions D=3,4,6. The twistor action is manifestly Lorentz invariant but the anticommuting spin variables appear exactly as in the non-relativistic limit. This allows a simple confirmation that the quantum N=2 spinning particle has either spin one or spin zero, and that N>2 is quantum inconsistent for D=4,6., Comment: 21 pages. Revised to correct minor errors, and to add subsection on massless limit. Equation numbering problem corrected in v.3

The generic action for an N-extended superparticle in D-dimensional Minkowski spacetime is shown to have "hidden" supersymmetries (related by "dualities" to the manifest supersymmetries) such that the full supersymmetry algebra is BPS-saturated; the exceptions (which include, trivially, the massless case) are those for which the manifest supersymmetry algebra is already BPS-saturated. Moreover, it is shown that any "non-BPS" superparticle action is a gauge-fixed version of the "BPS" superparticle action for which all supersymmetries are manifest. An example is the N=1 massive D=10 superparticle, which actually has N=2 supersymmetry and is equivalent to the action for a D0-brane of IIA superstring theory., Comment: 13 pages. Minor improvements plus new reference in v2

In the (super)twistor formulation of massless (super)particle mechanics, the mass-shell constraint is replaced by a "spin-shell" constraint from which the spin content can be read off. We extend this formalism to massive (super)particles (with N-extended spacetime supersymmetry) in three and four space-time dimensions, explaining how the spin-shell constraints are related to spin, and we use it to prove equivalence of the massive N=1 and BPS-saturated N=2 superparticle actions. We also find the supertwistor form of the action for "spinning particles" with N-extended worldline supersymmetry, massless in four dimensions and massive in three dimensions, and we show how this simplifies special features of the N=2 case., Comment: 41 pages. v2 has many additional references, and more details of "hidden" supersymmetries of massive superparticle actions. New format in v3, which corrects typos and includes alternative massive 4D superparticle action with only first-class constraints

We perform a light-cone gauge quantization of the Ramond closed spinning string in three spacetime dimensions (3D). The spectrum is Lorentz invariant and identical to that of the 3D Green-Schwarz closed superstring with ${\cal N}=2$ space-time supersymmetry, quantized in light-cone gauge., Comment: 15 pages. Revised to include discussion of modular invariance

We use the general $N = 1$ supersymmetric formulation of one dimensional sigma models on non trivial manifolds and its subsequent quantization to formulate the classical and quantum dynamics of the $ N= 2 $ supersymmetric charged particle moving on a sphere in the field of a monopole. The factorization method accommodated with the general covariance and it is used to integrate the corresponding system., Comment: 18 pages, latex209, no figures

We explain how fractional spin and statistics are relevant to (super)strings in a three-dimensional (3D) Minkowski spacetime., Comment: 5p. Contribution to the XVII European Workshop on String Theory 2011. Padua, Italy, 5-9 September 2011. v2 adds citations and corrects minor errors

The classical Green-Schwarz superstring action, with N=1 or N=2 spacetime supersymmetry, exists for spacetime dimensions D=3,4,6,10, but quantization in the light-cone gauge breaks Lorentz invariance unless either D=10, which leads to critical superstring theory, or D=3. We give details of results presented previously for the bosonic and N=1 closed 3D (super)strings and extend them to the N=2 3D superstring. In all cases, the spectrum is parity-invariant and contains anyons of irrational spin., Comment: 46 pages, v5 corrects more typos and minor errors

The minimal (${\cal N}=1$) superparticle in three spacetime dimensions (3D) is quantized. For non-zero mass it describes a spin-1/4 semion supermultiplet of "relativistic helicities" (-1/4, 1/4). The addition of a parity-violating Lorentz-Wess-Zumino term shifts this to $(\beta-1/4,\beta+1/4)$ for arbitrary $\beta$. For zero mass, in which case spin is not defined, the quantum superparticle describes a supermultiplet of one boson and one fermion., Comment: Contribution to proceedings of the SIGMA workshop "Supersymmetric Quantum Mechanics and Spectral Design", July 2010, Benasque, Spain

The Bogomolnyi vortex of the N=2 supersymmetric abelian-Higgs model in 2+1 dimensions is shown to be a ``semion'' of spin 1/4. Specifically, the effective superparticle action for one vortex is shown to describe, upon quantization, a parity self-dual centrally-charged `short' supermultiplet of ``relativistic helicities'' (-1/4,-1/4, 1/4,1/4)., Comment: 15 pp. Minor corrections and additional refs. in v2

The Nambu-Goto string in a 3-dimensional (3D) Minkowski spacetime is quantized preserving Lorentz invariance and parity. The spectrum of massive states contains anyons. An ambiguity in the ground state energy is resolved by the 3D N=1 Green-Schwarz superstring, which has massless ground states describing a dilaton and dilatino, and first-excited states of spin 1/4., Comment: 4 pages, two additional refs. in v2

We generalize previous results for the superplane Landau model to exhibit an explicit worldline N = 2 supersymmetry for an arbitrary magnetic field on any two-dimensional manifold. Starting from an off-shell N = 2 superfield formalism, we discuss the quantization procedure in the general case characterized by two independent potentials on the manifold and show that the relevant Hamiltonians are factorizable. In the restricted case when both the Gauss curvature and the magnetic field are constant over the manifold and, as a consequence, the underlying potentials are related, the Hamiltonians admit infinite series of factorization chains implying the integrability of the associated systems. We explicitly determine the spectrum and eigenvectors for the particular model with CP^1 as the bosonic manifold., Comment: 26 pages

We generalize an identity, first found by Bremner, for the action of three nested Nambu brackets., Comment: Added references and detailed proof of main result, conforming to journal version

We consider several ternary algebras relevant to physics. We compare and contrast the quantal versions of the algebras, as realized through associative products of operators, with their classical counterparts, as realized through classical Nambu brackets. In some cases involving infinite algebras, we show the classical limit may be obtained by a contraction of the quantal algebra, and then explicitly realized through classical brackets. We illustrate this classical-contraction method by the Virasoro-Witt example., Comment: Additional references

A Hilbert space metric is found for the SU(2|1)-invariant `superflag' Landau models, parametrized by integer 2N' and real number M, such that the Hilbert space norm is positive definite. The spectrum of these unitary super-Landau models is determined. The M=0 case yields a unitary Landau model on the supersphere SU(2|1)/U(1|1) with U(1) charge 2N=2N'+1. For the generic unitary superflag model, the manifest SU(2|1) symmetry is dynamically enhanced to SU(2|2); this is the `spherical' analog of the hidden worldline supersymmetry found previously in the planar limit., Comment: 51 pages. Minor changes. Some new references

We use the methods of PT-symmetric quantum theory to find a one-parameter family of ISU(1|1)-invariant planar super-Landau models with positive norm, uncovering an `accidental', and generically spontaneously-broken, worldline supersymmetry, with charges that have a Sugawara-type representation in terms of the ISU(1|1) charges. In contrast to standard models of supersymmetric quantum mechanics, it is the norms of states rather than their energies that are parameter-dependent, and the spectrum changes discontinuously in the limit that worldline supersymmetry is restored., Comment: 25 pages. Substantial revision to include discussion of classical worldline supersymmetry. Additional reference

We discuss supersymmetric biorthogonal systems, with emphasis given to the periodic solutions that occur at spectral singularities of PT symmetric models. For these periodic solutions, the dual functions are associated polynomials that obey inhomogeneous equations. We construct in detail some explicit examples for the supersymmetric pairs of potentials V_{+/-}(z) = -U(z)^2 +/- z(d/(dz))U(z) where U(z) = \sum_{k>0}u_{k}z^{k}. In particular, we consider the cases generated by U(z) = z and z/(1-z). We also briefly consider the effects of magnetic vector potentials on the partition functions of these systems., Comment: Changes are made to conform to the published version. In particular, some errors are corrected on pp 12-13

In previous papers we solved the Landau problems, indexed by 2M, for a particle on the ``superflag'' S U (2|1)/[U (1) x U (1)], the M = 0 case being equivalent to the Landau problem for a particle on the ``supersphere'' S U (2|1)/[U (1|1)]. Here we solve these models in the planar limit. For M = 0 we have a particle on the complex superplane C(1|1) ; its Hilbert space is the tensor product of that of the Landau model with the 4-state space of a ``fermionic'' Landau model. Only the lowest level is ghost-free, but for M > 0 there are no ghosts in the first [2M ]+1 levels. When 2M is an integer, the ([2M ] + 1)th level states form short supermultiplets as a consequence of a fermionic gauge invariance analogous to the ``kappa-symmetry'' of the superparticle., Comment: 23 pages, Note Added, References, Acknowledgments

Models of PT symmetric quantum mechanics provide examples of biorthogonal quantum systems. The latter incorporporate all the structure of PT symmetric models, and allow for generalizations, especially in situations where the PT construction of the dual space fails. The formalism is illustrated by a few exact results for models of the form H=(p+\nu)^2+\sum_{k>0}\mu_{k}exp(ikx). In some non-trivial cases, equivalent hermitian theories are obtained and shown to be very simple: They are just free (chiral) particles. Field theory extensions are briefly considered., Comment: 34 pages, 5 eps figures; references added and other changes made to conform to journal version

The Super Chern-Simons mechanics, and quantum mechanics of a particle, on the coset super-manifolds SU(2|1)/ U(2) and SU(2|1)/U(1)X U(1), is considered. Within a convenient quantization procedure the well known Chern-Simons mechanics on SU(2)/U(1) is reviewed, and then it is shown how the fuzzy supergeometries arise. A brief discussion of the super-sphere is also included., Comment: To appear in Proceedings of the International Workshop "Supersymmetries and Quantum Symmetries" (SQS'03, 24-29 July, 2003), Dubna, Russian Federation

We construct actions for non-relativistic strings and membranes purely as Wess-Zumino terms of the underlying Galilei groups., Comment: references added

We consider the quantum mechanics of a particle on the coset superspace $SU(2|1)/[U(1)\times U(1)]$, which is a super-flag manifold with $SU(2)/U(1)\cong S^2$ `body'. By incorporating the Wess-Zumino terms associated with the $U(1)\times U(1)$ stability group, we obtain an exactly solvable super-generalization of the Landau model for a charged particle on the sphere. We solve this model using the factorization method. Remarkably, the physical Hilbert space is finite-dimensional because the number of admissible Landau levels is bounded by a combination of the U(1) charges. The level saturating the bound has a wavefunction in a shortened, degenerate, irrep of $SU(2|1)$.

The `Chern-Simons Quantum Mechanics' of a particle on CP(n|m) is shown to yield the fuzzy descriptions of these superspaces, for which we construct the non-(anti)commuting position operators. For a particle on the supersphere CP(1|1) = SU(2|1)/U(1|1), the particle's wave-function at fuzziness level 2s is shown to be a degenerate irrep of SU(2|1) describing a supermultiplet of SU(2) spins (s-1\2, s)., Comment: 19 pp. Contribution to "Symmetries in Gravity and Field Theory", conference for Jose-Adolfo de Azcarraga's 60th birthday, June 2003, Salamanca, Spain. Minor corrections made and references included. Notation clarified, and additional reference in v.3

The standard quantum states of $n$ complex Grassmann variables with a free-particle Lagrangian transform as a spinor of SO(2n). However, the same `free-fermion' model has a non-linearly realized $SU(n|1)$ symmetry; it can be viewed as the mechanics of a `particle' on the Grassmann-odd coset space $SU(n|1)/U(n)$. We implement a quantization of this model for which the states with non-zero norm transform as a representation of $SU(n|1)$, the representation depending on the U(1) charge of the wave-function. For $n=2$ the wave-function can be interpreted as a BRST superfield., Comment: Latex, 14pp. Revised to include additional references. Further revision to incorporate further refs

We outline the description of Quantum Mechanics with noncommuting coordinates within the framework of star operation. We discuss simple cases of integrability., Comment: misprints corrected, additional references

Comparison between the exact value of the spectral zeta function, $Z_{H}(1)=5^{-6/5}[3-2\cos(\pi/5)]\Gamma^2(1/5)/\Gamma(3/5)$, and the results of numeric and WKB calculations supports the conjecture by Bessis that all the eigenvalues of this PT-invariant hamiltonian are real. For one-dimensional Schr\"odinger operators with complex potentials having a monotonic imaginary part, the eigenfunctions (and the imaginary parts of their logarithmic derivatives) have no real zeros., Comment: 6 pages, submitted to J. Phys. A

We present the application of the inverse scattering method to the design of semiconductor heterostructures having a preset dependence of the (conduction) electrons' reflectance on the energy. The electron dynamics are described by either the effective mass Schr\"odinger, or by the (variable mass) BenDaniel and Duke equations. The problem of phase (re)construction for the complex transmission and reflection coefficients is solved by a combination of Pad\'e approximant techniques, obtaining reference solutions with simple analytic properties. Reflectance-preserving transformations allow bound state and reflection resonance management. The inverse scattering problem for the Schroedinger equation is solved using an algebraic approach due to Sabatier. This solution can be mapped unitarily onto a family of BenDaniel and Duke type equations. The boundary value problem for the nonlinear equation which determines the mapping is discussed in some detail. The chemical concentration profile of heterostructures whose self consistent potential yields the desired reflectance is solved completely in the case of Schroedinger dynamics and approximately for Ben-Daniel and Duke dynamics. The Appendix contains a brief digest of results from scattering and inverse scattering theory for the one-dimensional Schroedinger equation which are used in the paper., Comment: 23 pages, 1 figure, to appear in Microelectronics Journal

We construct integrals of motion (IM) for the sine-Gordon model with boundary at the free Fermion point which correctly determine the boundary S matrix. The algebra of these IM (``boundary quantum group'' at q=1) is a one-parameter family of infinite-dimensional subalgebras of twisted affine sl(2). We also propose the structure of the fractional-spin IM away from the free Fermion point., Comment: 19 pages, LaTeX, no figures

We compute by means of the Bethe Ansatz the boundary S matrix for the open anisotropic spin-1/2 chain with diagonal boundary magnetic fields in the noncritical regime (Delta > 1). Our result, which is formulated in terms of q-gamma functions, agrees with the vertex-operator result of Jimbo et al., Comment: 9 pages, LaTeX, no figures