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We consider the massive vector N-component (λϕ 4 ) D theory in Euclidian space and, using an extended Matsubara formalism we perform a compactification on aThis allows us to treat jointly the effect of temperature and spatial confinement in the effective potential of the model, setting forth grounds for an analysis of phase transitions driven by temperature and spatial boundary. For d = 2, which corresponds to the heated system confined between two parallel planes (separation L), we obtain, in the large N limit at one-loop order, formulas for temperature-and boundary-dependent mass and coupling constant. The equation for the critical curve in the β × L plane is also derived.

The development of quantum theory on a torus has a long history, and can be traced back to the 1920s, with the attempts by Nordström, Kaluza and Klein to define a fourth spatial dimension with a finite size, being curved in the form of a torus, such that Einstein and Maxwell equations would be unified. Many developments were carried out considering cosmological problems in association with particles physics, leading to methods that are useful for areas of physics, in which size effects play an important role. This interest in finite size effect systems has been increasing rapidly over the last decades, due principally to experimental improvements. In this review, the foundations of compactified quantum field theory on a torus are presented in a unified way, in order to consider applications in particle and condensed matted physics. The theory on a torus Γ d D = (S 1 ) d × R D−d is developed from a Lie-group representation and c*-algebra formalisms. As a first application, the quantum field theory at finite temperature, in its real-and imaginary-time versions, is addressed by focussing on its topological structure, the torus Γ 1 4 . The toroidal quantum-field theory provides the basis for a consistent approach of spontaneous symmetry breaking driven by both temperature and spatial boundaries. Then the superconductivity in films, wires and grains are analyzed, leading to some results that are comparable with experiments. The Casimir effect is studied taking the electromagnetic and Dirac field on a torus. In this case, the method of analysis is based on a generalized Bogoliubov transformation, that separates the Green function into two parts: one is associated with the empty space-time, while the other describes the impact of compactification. This provides a natural procedure for calculating the renormalized energy-momentum tensor. Self interacting four-fermion systems, described by the Gross-Neveu and Nambu-Jona-Lasinio models, are considered. Then finite size effects on the hadronic phase structure are investigated, taking into account density and temperature. As a final application, effects of extra spatial dimensions are addressed, by developing a quantum electrodynamics in a five-dimensional space-time, where the fifth-dimension is compactified on a torus. The formalism, initially developed for particle physics, provides results compatible with other trials of probing the existence of extra-dimensions.

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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.

The concept of thermoalgebra, a kind of representation for the Lie-symmetries developed in connection with thermal quantum field theory, is extended to study unitary representations of the Galilei group for thermal classical systems. One of the representations results in the first-quantized Scho nberg formalism for the classical statistical mechanics. Furthermore, the close analogy between thermal classical mechanics and thermal quantum field theory is analysed, and such an analogy is almost exact for harmonic oscillator systems. The other unitary representation studied results in a field-operator version of the Scho nberg approach. As a consequence, in this case the counterpart of the thermofield dynamics (TFD) in classical theory is identified as both the first and second-quantized form of the Liouville equation. Non-unitary representations are also studied, being, in this case, the Lie product of the thermoalgebra identified as the Poisson brackets. A representation of the thermal SU(1, 1) is analysed, such that the tilde variables (introduced in TFD) are functions in a double phase space. As a result the equations of motion for dissipative classical oscillators are derived.

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