J. D. M. Vianna

Galilean covariance and the Duffin-Kemmer-Petiau equation

We use a five-dimensional approach to Galilean covariance to investigate the non-relativistic Duffin-Kemmer-Petiau first-order wave equations for spinless particles. The corresponding representation is generated by five 6×6 matrices. We consider the harmonic oscillator as an example.

A study of confined quantum systems using the Woods-Saxon potential

We propose the Woods-Saxon (WS) potential to simulate spatial confinement. The great advantage of our methodology is that it enables the study of a wide range of systems and confinement regimes by varying two parameters in the model potential. To test the methodology we have studied the confined harmonic oscillator in two different regimes: when the confinement potential exhibits a sudden jump; and when the confinement is described by a smooth function. We have also applied the present procedure to a realistic problem, a confined quantum dot-atom. The numerical calculation is performed with the equally spaced discrete variable representation (DVR). Our results are in close agreement with those available in the literature, and we believe our method to be a good alternative for studying confined quantum systems.

Symmetry groups, density-matrix equations and covariant Wigner functions

A representation theory for Lie groups is developed taking the Hilbert space, say Hw, of the w∗-algebra standard representation as the representation space. In this context the states describing physical systems are amplitude wave functions but closely connected with the notion of the density matrix. Then, based on symmetry properties, a general physical interpretation for the dual variables of thermal theories, in particular the thermofield dynamics (TFD) formalism, is introduced. The kinematic symmetries, Galilei and Poincare, are studied and (density) amplitude matrix equations are derived for both of these cases. In the same context of group theory, the notion of phase space in quantum theory is analysed. Thus, in the non-relativistic situation, the concept of density amplitude is introduced, and as an example, a spin-half system is algebraically studied; Wigner function representations for the amplitude density matrices are derived and the connection of TFD and the usual Wigner-function methods are analysed. For the Poincare symmetries the relativistic density matrix equations are studied for the scalar and spinorial fields. The relativistic phase space is built following the lines of the non-relativistic case. So, for the scalar field, the kinetic theory is introduced via the Klein–Gordon density-matrix equation, and a derivation of the Juttiner distribution is presented as an example, thus making it possible to compare with the standard approaches. The analysis of the phase space for the Dirac field is carried out in connection with the dual spinor structure induced by the Dirac-field density-matrix equation, with the physical content relying on the symmetry groups. Gauge invariance is considered and, as a basic result, it is shown that the Heinz density operator (which has been used to develope a gauge covariant kinetic theory) is a particular solution for the (Klein–Gordon and Dirac) density-matrix equation.

A study of two-electron quantum dot spectrum using discrete variable representation method

A variational method called discrete variable representation is applied to study the energy spectra of two interacting electrons in a quantum dot with a three-dimensional anisotropic harmonic confinement potential. This method, applied originally to problems in molecular physics and theoretical chemistry, is here used to solve the eigenvalue equation to relative motion between the electrons. The two-electron quantum dot spectrum is determined then with a precision of at least six digits. Moreover, the electron correlation energies for various potential confinement parameters are investigated for singlet and triplet states. When possible, the present results are compared with the available theoretical values.

Galilean Duffin–Kemmer–Petiau algebra and symplectic structure

We develop the Duffin–Kemmer–Petiau (DKP) approach in the phase-space picture of quantum mechanics by considering DKP algebras in a Galilean covariant context. Specifically, we develop an algebraic calculus based on a tensor algebra defined on a five-dimensional space which plays the role of spacetime background of the non-relativistic DKP equation. The Liouville operator is determined and the Liouville–von Neumann equation is written in two situations: the free particle and a particle in an external electromagnetic field. A comparison between the non-relativistic and the relativistic cases is commented.

Non-commutative geometry and symplectic field theory

In this work we study representations of the Poincare group defined over symplectic manifolds, deriving the Klein-Gordon and the Dirac equation in phase space. The formalism is associated with relativistic Wigner functions; the Noether theorem is derived in phase space and an interacting field, including a gauge field, approach is discussed.

Poincaré-Lie algebra and relativistic phase space

In this work we investigate representations of the Poincare group taking as the representation space the Hilbert space of thermofield dynamics, a real-time formalism developed in quantum field theory at finite temperature. We concentrate our study on those representations that give rise to the notion of phase space, with direct application in kinetic theory. As a result, we show an alternative way to derive a relativistic Boltzmann equation, based on the notion of a propagator defined in phase space. The quantum counterpart of the approach is discussed through the notion of the Wigner function.

A configuration interaction model to investigate many-electron systems in cavities

A configuration interaction (CI) model to treat confined many-electron systems is presented. Our model proposes a spatially confined linear combination of configuration interaction (LCCI) functions, built from basis functions that do not satisfy confinement boundary conditions. As an application we have calculated total energies for the He ground state, assuming that the atom is enclosed within a spherical cavity with infinite potential walls. Comparisons with other results in the literature are made in order to verify our model.

On the generalized phase space approach to Duffin-Kemmer-Petiau particles

We present a general derivation of the Duffin-Kemmer-Petiau (D.K.P) equation on the relativistic phase space proposed by Bohm and Hiley. We consider geometric algebras and the idea of algebraic spinors due to Riesz and Cartan. The generators βμ(p) of the D.K.P algebras are constructed in the standard fashion used to construct Clifford algebras out of bilinear forms. Free D.K.P particles and D.K.P particles in a prescribed external electromagnetic field are analized and general Liouville type equations for these cases are obtained. Choosing particular values for the label p we classify the different types of the D.K.P Liouville operators.

Duffin–Kemmer–Petiau theory with minimal and non-minimal couplings

In this work we study the properties of the Duffin–Kemmer–Petiau (DKP) formalism in a general representation of β matrices. In particular, we analyze the conservation of the total angular momentum, and the effects of minimal and non-minimal prescriptions. The selection of the scalar and vector sectors of the formalism is done in a simple and consistent way by using operators independent of a specific choice of representation. Physical applications are explored, in which the minimal prescription reproduces the system in the presence of external magnetic field, and the non-minimal one leads to the DKP oscillator. We obtain and discuss the motion equations, eigenstates and energy spectrum of the different sectors of the theory for the two types of couplings.