Abstract
In this review, we revisit our approach to constructing an effective theory for Abelian and Non-Abelian gauge theories in 4D. Our goal is to have an effective theory that provides a simple classical picture of the main qualitatively important features of these theories. We set out to ensure the presence of the massless photons—Goldstone bosons in Abelian theory and their disappearance in the Non-Abelian case—accompanied by the formation of confining strings between charged states. Our formulation avoids using vector fields and instead operates with the basic degrees of freedom that are the scalar fields of a nonlinear σ-model. The Mark 1 model we study turns out to have a large global symmetry group-the 2D diffeomorphism invariance in the Abelian limit, which is isomorphic to the group of all canonical transformations in the classical two dimensional phase space. This symmetry is not present in QED, and we eliminate it by “gauging” this infinite dimensional global group. Introducing additional modifications to the model (Mark 2), we are able to prove that the “Abelian” version is equivalent to the theory of a free photon. Achieving the desired property in the “Non-Abelian” regime turns out to be tricky. We are able to introduce a perturbation that leads to the formation of confining strings in our Mark 1 model. These strings have somewhat unusual properties, in that their profile does not decay exponentially away from the center of the string. In addition, the perturbation explicitly breaks the diffeomorphism invariance. Preserving this invariance in the gauged model as well as achieving confining strings in Mark 2 model remains an open question.
Highlights
Understanding confinement in Non-Abelian gauge theories is a long standing theoretical problem
The Mark 1 model we study turns out to have a large global symmetry group-the 2D diffeomorphism invariance in the Abelian limit, which is isomorphic to the group of all canonical transformations in the classical two dimensional phase space
We have tried to follow the guide of 2 + 1 dimensional gauge theories and, based on several requirements, “guess” a theory of scalar fields that may emulate the effective theory of 3 + 1 dimensional gauge theories
Summary
Understanding confinement in Non-Abelian gauge theories is a long standing theoretical problem. The question arises if a similar description can be achieved in 3 + 1 dimensions One would like this description to encompass the Goldstone boson nature of photons in QED as well as an interpretation of confinement in terms of topological charges in Non-Abelian theories. As we will see, requiring the energy of a soliton in the Abelian regime to be finite puts a very strong restriction on possible forms of the kinetic term for the scalar fields This noncanonical kinetic term results in rather unusual properties of confining strings once the symmetry breaking perturbation is introduced. We make some comments on how this can be achieved, but the implementation is left for the future
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