We give a new solution to a well-known problem, that of computing perturbed eigenvalues for quantum oscillators. This article is nearly self contained and begins with all the necessary algebraic tools to make the subsequent calculations. We define a new family of Lie algebras relevant to making computations for perturbed (anharmonic) oscillators, and show that the only two formally closed solutions are indeed harmonic oscillators themselves. Through elementary combinatorics and noncanonical forms of well-known Lie algebras, we are able to obtain a fully closed form solution for perturbed eigenvalues to first order.

Shape invariance condition in the framework of second-order supersymmetric quantum mechanics is studied. Two classes of solvable shape invariant potentials are consequently constructed, in which the parameters $a_0$ and $a_1$ of partner potentials are related to each other by translation $a_1 = a_0 + \alpha$. In each class, general properties of the obtained shape invariant potentials are systematically investigated. The energy eigenvalues are algebraically determined and the corresponding eigenfunctions are expressed in terms of generalized associated Laguerre polynomials. It is found that these shape invariant potentials are inherently singular, characterized by the $1/x^2$ singularity at the origin.

We provide the geometrical interpretation for the Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry invariance of the Lagrangian density of a four $(3+1)$-dimensional ($4$D) interacting U(1) gauge theory within the framework of superfield approach to BRST formalism. This interacting theory, where there is an explicit coupling between the U(1) gauge field and matter (Dirac) fields, is considered on a $(4,2)$-dimensional supermanifold parametrized by the four spacetime variables $x^{\mu}(\mu=0,1,2,3)$ and a pair of Grassmannian variables $\theta$ and $\bar\theta$ (with $\theta^{2}=\bar\theta^{2}=0$, $\theta\bar\theta+\bar\theta\theta=0$). We express the Lagrangian density and (anti-)BRST charges in the language of the superfields and show that (i) the (anti-)BRST invariance of the $4$D Lagrangian density is equivalent to the translation of the super Lagrangian density along the Grassmannian direction(s) ($\theta$ and/or $\bar\theta$) of the $(4,2)$-dimensional supermanifold such that the outcome of the above translation(s) is zero, and (ii) the anticommutativity and nilpotency of the (anti-)BRST charges are the automatic consequences of our superfield formulation.

We propose an operational method for the solution of differential equations involving vector products. The technique we propose is based on the use of the evolution operator, defined in such a way that the wealth of techniques developed within the context of quantum mechanics can also be exploited for classical problems. We discuss the application of the method to the solution of the Lorentz-type equations.

In quantum dynamical systems, a history is defined by a pair $(M,\gamma)$, consisting of a type $I$ factor $M$, acting on a Hilbert space $H$, and an $E_0$-group $\gamma = (\gamma_t)_{t\in \Bbb{R}}$, satisfying certain additional conditions. In this paper, we distort a given history $(M,\gamma)$, by a finite family $\mathcal{G}$ of partial isometries on $H$. In particular, such a distortion is dictated by the combinatorial relation on the family $\mathcal{G}$. Two main purposes of this paper are (i) to show the existence of distortions on histories, and (ii) to consider how distortions work. We can understand Sections 3, 4 and 5 as the proof of the existence of distortions (i), and the properties of distortions (ii) are shown in Section 6.

The auxiliary field method is a technique to obtain approximate closed formulae for the solutions of both nonrelativistic and semirelativistic eigenequations in quantum mechanics. For a many-body Hamiltonian describing identical particles, it is shown that the approximate eigenvalues can be written as the sum of the kinetic operator evaluated at a mean momentum $p_0$ and of the potential energy computed at a mean distance $r_0$. The quantities $p_0$ and $r_0$ are linked by a simple relation depending on the quantum numbers of the state considered and are determined by an equation which is linked to the generalized virial theorem. The (anti)variational character of the method is discussed, as well as its connection with the perturbation theory. For a nonrelativistic kinematics, general results are obtained for the structure of critical coupling constants for potentials with a finite number of bound states.

Free differential algebras (FDAs) provide an algebraic setting for field theories with antisymmetric tensors. The "presentation'' of FDAs generalizes the Cartan-Maurer equations of ordinary Lie algebras, by incorporating $p$-form potentials. An extended Lie derivative along antisymmetric tensor fields can be defined and used to recover a Lie algebra dual to the FDA that encodes all the symmetries of the theory including those gauged by the $p$-forms. The general method is applied to the FDA of $D=11$ supergravity: the resulting dual Lie superalgebra contains the M-theory supersymmetry anticommutators in presence of $2$-branes.

Introducing an extended Lie derivative along the dual of $A$, the $3$-form field of $D=11$ supergravity, the full diffeomorphism algebra of $D=11$ supergravity is presented. This algebra suggests a new formulation of the theory, where the $3$-form field $A$ is replaced by bivector $B^{ab}$, bispinor $B^{\alpha\beta}$, and spinor-vector $\eta^{a\beta}$ $1$-forms. Only the bivector $1$-form $B^{ab}$ is propagating, and carries the same degrees of freedom of the $3$-form in the usual formulation, its curl $\mathcal{D}_{[\mu} B^{ab}_{\nu]}$ being related to the $F_{\mu \nu a b}$ curl of the $3$-form. The other $1$-forms are auxiliary, and the transformation rules on all the fields close on the equations of motion of $D=11$ supergravity.

We obtain the low-lying energy-momentum spectrum for the imaginary-time lattice four-Fermi or Gross-Neveu model in $d+1$ space-time dimensions ($d=1,2,3$) and with $N$-component fermions. Let $0<\kappa\ll 0$ be the hopping parameter, $\lambda>0$ the four-fermion coupling, $m>0$ the bare fermion mass and take $s\times s$ spin matrices ($s=2,4$). Our analysis of the one and the two-particle spectrum is based on spectral representation for suitable two- and four-fermion correlations. The one-particle energy-momentum spectrum is obtained rigorously and is manifested by $\frac{sN}{2}$ isolated and identical dispersion curves, and the mass of particles has asymptotic value order $-\ln\kappa$. The existence of two-particle bound states above or below the two-particle band depends on whether Gaussian domination does hold or does not, respectively. Two-particle bound states emerge from solutions to a lattice Bethe-Salpeter equation, in a ladder approximation. Within this approximation, the $(\frac{sN}{2}-1)\frac{sN}{4}$ identical bound states have $\mathcal{O}(\kappa^0)$ binding energies at zero system momentum and their masses are all equal, with value $\approx -2 \ln \kappa$. Our results can be validated to the complete model as the Bethe-Salpeter kernel exhibits good decay properties.