Ademir Eugênio de Santana

Thermal quantum field theory - Algebraic aspects and applications

General Principles: Elements of Thermodynamics Elements of Statistical Mechanics Partition Function and Path Integral Interacting Fields Thermal Field Theory: Thermofield Dynamics (TFD) Thermal Oscillators: Bosons and Fermions Representation of the Thermo-Poincare Group Free Fields at Finite Temperature Thermal Interacting Fields Scattering Processes and Reaction Rates via TFD Applications to Quantum Optics: Thermal Quantum States of Field Mode Non-classical Properties of Thermal States Bipartite Systems and Thermal States Measure of Non-classicality and Temperature SU(2) and SU(1,1) States Thermal Confined Fields: About Confinement and Thermal Theories Casimir Effect for the Electromagnetic Field in Box Casimir Effect for Fermions Superconducting Transition Temperature in Films, Wires and Grains Critical Behavior of Type II Superconducting Films in a Magnetic Field First-Order Phase Transition in Superconducting Films Compactified GrossA-Neveu Model at T=0 Compactified GrossA-Neveu Model at Finite Temperature Applications to Open Systems: TFD, Wigner Functions and Kinetic Theory Schrodinger Approach, TFD and Nonequilibrium Systems Dressed State Approach to the Thermalization Processes.

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.

Lie groups and thermal field theory

Considering basic ingredients of the so-called thermofield dynamics associated with a Lie algebra, L, an algebra, LT, is derived using the first kind of solution of the modified Yang-Baxter equation. LT has a double algebraic structure and represents a new possibility of representations for Lie symmetries associated with the thermal phenomena. As an example, the Poincare group is studied.

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.

Galilean covariant Lagrangian models

We construct non-relativistic Lagrangian field models by enforcing Galilean covariance with a (4, 1) Minkowski manifold followed by a projection onto the (3, 1) Newtonian spacetime. We discuss scalar, Fermi and gauge fields, as well as interactions between these fields, preparing the stage for their quantization. We show that the Galilean covariant formalism provides an elegant construction of the Lagrangians which describe the electric and magnetic limits of Galilean electromagnetism. Similarly we obtain non-relativistic limits for the Proca field. Then we study Dirac Lagrangians and retrieve the Levy-Leblond wave equations when the Fermi field interacts with an Abelian gauge field.

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.

Generalized bogoliubov transformation for confined fields: applications for the casimir effect

The Bogoliubov transformation in thermofield dynamics, an operator formalism for the finite-temperature quantum field theory, is generalized to describe a field in arbitrary confined regions of space and time. Starting with the scalar field, the approach is extended to the electromagnetic field and the energy-momentum tensor is written via the Bogoliubov transformation. In this context, the Casimir effect is calculated for zero and nonzero temperature, and therefore it can be considered as a vacuum condensation effect of the electromagnetic field. This aspect opens an interesting perspective for using this procedure as an effective scheme for calculations in the studies of confined fields, including interacting fields.

Nonrelativistic Wave Equations with Gauge Fields

We illustrate a metric formulation of Galilean invariance by constructing wave equations with gauge fields. It consists of expressing nonrelativistic equations in a covariant form, but with a five-dimensional Riemannian manifold. First we use the tensorial expressions of electromagnetism to obtain the two Galilean limits of electromagnetism found previously by Le Bellac and Lévy-Leblond. Then we examine the nonrelativistic version of the linear Dirac wave equation. With an Abelian gauge field we find, in a weak field approximation, the Pauli equation as well as the spin—orbit interaction and a part reminiscent of the Darwin term. We also propose a generalized model involving the interaction of the Dirac field with a non-Abelian gauge field; the SU(2) Hamiltonian is given as an example.

Quantum fields in toroidal topology

Abstract The standard representation of c ∗ -algebra is used to describe fields in compactified space–time dimensions characterized by topologies of the type Γ D d = ( S 1 ) d × M D − d . The modular operator is generalized to introduce representations of isometry groups. The Poincare symmetry is analyzed and then we construct the modular representation by using linear transformations in the field modes, similar to the Bogoliubov transformation. This provides a mechanism for compactification of the Minkowski space–time, which follows as a generalization of the Fourier integral representation of the propagator at finite temperature. An important result is that the 2×2 representation of the real-time formalism is not needed. The end result on calculating observables is described as a condensate in the ground state. We initially analyze the free Klein–Gordon and Dirac fields, and then formulate non-abelian gauge theories in Γ D d . Using the S -matrix, the decay of particles is calculated in order to show the effect of the compactification.

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.