International Journal of Modern Physics A

In 1938, Stueckelberg introduced a scalar field which makes an Abelian gauge theory massive but preserves gauge invariance. The Stueckelberg mechanism is the introduction of new fields to reveal a symmetry of a gauge-fixed theory. We first review the Stueckelberg mechanism in the massive Abelian gauge theory. We then extend this idea to the standard model, stueckelberging the hypercharge U(1) and thus giving a mass to the physical photon. This introduces an infrared regulator for the photon in the standard electroweak theory, along with a modification of the weak mixing angle accompanied by a plethora of new effects. Notably, neutrinos couple to the photon and charged leptons have also a pseudo-vector coupling. Finally, we review the historical influence of Stueckelberg's 1938 idea, which led to applications in many areas not anticipated by the author, such as strings. We describe the numerous proposals to generalize the Stueckelberg trick to the non-Abelian case with the aim to find alternatives to the standard model. Nevertheless, the Higgs mechanism in spontaneous symmetry breaking remains the only presently known way to give masses to non-Abelian vector fields in a renormalizable and unitary theory.

Gauge theory on the q-deformed two-dimensional Euclidean plane is studied using two different approaches. We first formulate the theory using the natural algebraic structures on , such as a covariant differential calculus, a frame of one-forms and invariant integration. We then consider a suitable star product, and introduce a natural way to implement the Seiberg–Witten map. In both approaches, gauge invariance requires a suitable "measure" in the action, breaking the E_q(2)-invariance. Some possibilities to avoid this conclusion using additional terms in the action are proposed.

We define curved five-dimensional (5D) space–time from the embedding of 5D surfaces in a 6D flat space. Demanding that the 6D coordinates satisfy a separation of variables form and that the 5D metric is diagonal, we obtain that each curved 5D surface contains 4D hyperboloid de-Sitter subspaces with maximally symmetry SO(4,1). Therefore, we define a very special form for the curved 5D surface where the extra-dimension is perpendicular to the 4D hyperboloid de-Sitter spaces. By relating to a minimally coupled scalar field with a potential which depends on the extra-dimension only, the curved 5D surface's form is satisfied. A mechanism by means of which the extra-dimension can be of a finite size, is found. The borders of the finite extra-dimension are obtained when the scalar field potential goes to infinity for certain finite values of the scalar field. The geodesic lines' equations show that a particle cannot cross such borders.

Fuzzy spaces are obtained by quantizing adjoint orbits of compact semi-simple Lie groups. Fuzzy spheres emerge from quantizing S² and are associated with the group SU(2) in this manner. They are useful for regularizing quantum field theories and modeling space–times by noncommutative manifolds. We show that fuzzy spaces are Hopf algebras and in fact have more structure than the latter. They are thus candidates for quantum symmetries. Using their generalized Hopf algebraic structures, we can also model processes where one fuzzy space splits into several fuzzy spaces. For example we can discuss the quantum transition where the fuzzy sphere for angular momentum J splits into fuzzy spheres for angular momenta K and L.

We study the causality violation in the nonlocal ϕ⁴-theory (as formulated by Kleppe and Woodard) containing a finite mass scale Λ. Starting from the Bogoliubov–Shirkov criterion for causality, we construct and study combinations of S-matrix elements that signal violation of causality in the one-loop approximation. We find that the causality violation in the exclusive process ϕ+ϕ→ϕ+ϕ grows with energy, but the growth with energy, (for low to moderate energies) is suppressed to all orders compared to what one would expect purely from dimensional considerations. We however find that the causality violation in other processes such as ϕ+ϕ→ϕ+ϕ+ϕ+ϕ grows with energy as expected from dimensional considerations at low to moderate energies. For high enough energies comparable to the mass scale Λ, however, we find a rapid (exponential-like) growth in the degree of causality violation. We suggest a scenario, based on an earlier work, that will enable one to evade a large theoretical causality violation at high energies, should it be unobserved experimentally.

We study the lowest-order modifications of the static potential for Born–Infeld electrodynamics and for the θ-expanded version of the noncommutative U(1) gauge theory, within the framework of the gauge-invariant but path-dependent variables formalism. The calculation shows a long-range correction (1/r⁵-type) to the Coulomb potential in Born–Infeld electrodynamics. However, the Coulomb nature of the potential (to order e²) is preserved in noncommutative electrodynamics.