An account is given of the new insight into the theory of magnetic monopoles originating from the work of 't Hooft (1976) and Polyakov (1974). Their magnetic monopole, associated with the conventional electromagnetic gauge group U(1), occurs as a finite-energy smooth soliton solution to an SU(2) gauge theory. A precise picture of its internal structure, the values of its magnetic charge and its mass are obtained. These new developments bring together previously unrelated fields of study, namely the Dirac monopole (with point structure) and the Sine-Gordon soliton in two-dimensional space-time. Properties of more general monopoles, associated with large gauge groups now thought to be relevant in physics, are discussed. Based on this new viewpoint, conjectures can be made about a future quantum theory of monopoles.

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Magnetic monopoles in gauge field theories

This paper reviews recent developments of effective Lagrangians and field algebras as means of treating chiral symmetry and partially conserved axial current (PCAC) for the study of elementary particle physics. The techniques employed are developed in considerable detail. As examples, we concentrate primarily on spin 0 and 1, linear and nonlinear realizations of $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ and $\mathrm{SU}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(3)$ and some of the significant predictions of the theory are derived. The paper contains an extensive discussion of an effective Lagrangian with nonets of real scalar, pseudoscalar, vector, and axial-vector mesons that illustrate the problems of broken $\mathrm{SU}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(3)$.

Effective lagrangians and field algebras with chiral symmetry.

Nonlinear realizations of space-time symmetries. Scalar and tensor gravity

The basic framework for discussing quantum mechanics on multiply connected spaces is presented using the covering space concept. The theorem of Laidlaw and DeWitt is rederived and extended to the case of field theory. It is pointed out that chiral dynamics is similar to Skyrme's nonlinear theory and forms another example of Finkelstein's kink idea. The possible existence of ' pi geons' is raised, and the fact that the pion manifold may be any one of the Clifford-Klein constant curvature space-forms, rather than just the whole three-sphere, is suggested. The related formalism for quantum mechanics on homogeneous spaces is given in general terms.

https://iopscience.iop.org/article/10.1088/0305-4470/5/7/004/pdf

Quantum mechanics and field theory on multiply connected and on homogeneous spaces

Invariance under the infinite-parameter generally covariant group is equivalent to simultaneous invariance under the affine and the conformal group. A nonlinear realization of the affine group (with linearization on the Poincare group) leads to a symmetric tensor field as Goldstone field. The requirement that the theory correspond simultaneously to a realization of the conformal group as well leads uniquely to the theory of a tensor field whose equations are Einstein’s. This shows that the theory of the gravitational field is the theory of spontaneous breaking of affine and conformal symmetries in the same way as chiral dynamics is the theory of spontaneous breaking of chiral symmetry. This analogy brings out new aspects of the role of gravitation in the theory of elementary particles.

Theory of dynamical affine and conformal symmetries as the theory of the gravitational field

Nonlinear transformations of meson fields under any chiral group are derived in which the fields appear as a local co-ordinate system on a differentiable manifold. The techniques are extended to include any multiplet of nucleon fields.

A group-theoretic approach to chiral transformations

Various aspects of nonlinear-group realizations are discussed and it is shown that the nonlinearSU3×SU3 chiral-invariant meson Lagrangian, and hence the meson-meson low-energy processes, can be expressed completely in terms of the canonical metric on the group space ofSU3.

Metric structures and chiral symmetries

The equivalence between the linear and nonlinear chiral transformations of the 0− mesons is demonstrated using a simple form of an embedding theorem of Palais.

The embedding of nonlinear meson transformations in a Euclidean space

Using meson transformations under chiralSU3×SU3 derived from general schemes, covariant derivatives are calculated within the framework of the tree approximation. These are used to find mesonmeson scattering lengths.

C. J. Isham

Papers

A group-theoretic approach to chiral transformations

Metric structures and chiral symmetries

The embedding of nonlinear meson transformations in a Euclidean space

Meson-meson scattering lengths from chiralSU 3×SU 3