Fermion Field. In curved spacetime, the Dirac equation for a spin-1/2 fermion with an electromagnetic field [A.sub.[mu]] takes on the form

In fact, besides the usual ambiguity in the choice of the quantum representation for a fixed classical field description, time-dependent scalings of the

fermion field modes (after an appropriate mode decomposition) were also allowed, effectively modifying the original D'Eath and Halliwell parametrization of the system.

A modified form of local gauge invariance in which

fermion field phase is allowed to vary with each space point but not each time point, leads to the introduction of a new compensatory field different from the electromagnetic field associated with the photon.

Passos, "Aharonov-Bohm effect for a

fermion field in a planar black hole "spacetime"," The European Physical Journal C, vol.

In Section 3 we investigate localization of the zero mode of the

fermion field on the brane which is derived from a polynomial potential.

As the angular momentum J = -i[[partial derivative].sub.[phi]] + (s/2)[[sigma].sup.z] commutes with the H, it is possible to decompose the

fermion field as

where g is the positive coupling constant and the

fermion field [psi](x) has scale dimension 1/2 [10].

The next question is how to break the 3-3-1 gauge symmetry and provide with masses to the

fermion fields in each model.

As it is well known, Green's functions in the functional-integral approach are defined by means of the so-called generating functional with sources for the boson and

fermion fields, but the corresponding functional integrals cannot be evaluated exactly because the interaction part of the Hubbard Hamiltonian is quartic in the Grassmann

fermion fields.

which specifies the propagators of electromagnetic, Z-, [W.sup.[+ or -]]-boson, Higgs, and

fermion fields, and the interactions between these fields.

Matter fields (

fermion fields) are defined in the context of Mobius structures to be sections of vector bundles associated with Q.

The structure of the paper is the following: in Section 2, we introduce the Bogoliubov transformations in quantum field theory (QFT) and we study the energy-momentum tensor density for vacuum condensates of boson and

fermion fields. In Sections 3 and 4, we present the contribution given to the energy of the universe by thermal states, with reference to the Hawking and Unruh effects, by fields in curved space and by particle mixing phenomena.