Abstract
This last and very brief chapter highlights some open problems pertinent to the subject of the book. The main thrust of the book lies in the development of the effective field theory formalism for spontaneously broken symmetry. Accordingly, the list of open questions contains mostly technical issues whose resolution would further extend the validity of the results of the book. Some of the issues, related to the classification of actions of groups of manifolds, are purely mathematical. Others are more directly related to physical applications of the formalism.
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I will close the book by highlighting some technical issues that I could not have addressed properly. I label them as “open questions,” at the risk of revealing my own ignorance rather than a gap in the existing knowledge. Since the book revolves around developing a formalism for spontaneously broken symmetry, some of the items are purely mathematical. The first commandment of effective field theory (EFT) dictates that the effective Lagrangian must contain all operators that respect the symmetry of the system. Accordingly, an important ingredient of the EFT formalism is the classification of all possible Lagrangians consistent with the symmetry. This requirement lies behind most of the issues listed below.
The classification of group actions on a manifold in Chap. 7 relies heavily on the assumption that the isotropy subgroup H of the symmetry group G is compact. Any progress towards classification of group actions such that H is noncompact would be most welcome. This applies in particular to coset spaces \(G/H\) that are not reductive, for which many of the nice features of our standard nonlinear realization are lost.
FormalPara Global Existence of the Group ActionThe standard nonlinear realization of symmetry developed in Chap. 7 is restricted to a single local coordinate patch. Accordingly, the explicitly constructed action of the group G is limited to its elements near unity. Yet, the global existence of the group action on the whole manifold \(\mathcal {M}\), or coset space \(G/H\), has been implicitly assumed throughout the book. A better understanding of possible topological obstructions to extending the group action from a local coordinate patch to the entire manifold would be desirable. This issue is particularly pressing for Lie groups consisting of several connected components. As far as I know, the first attempt to systematically deal with such cases was made in [1]. However, more work is needed to establish a general, practically useful formalism.
FormalPara Group Action via Generalized Local TransformationsMost of the book focuses on symmetries realized by point transformations on a target space \(\mathcal {M}\) or its Cartesian product with the spacetime M. This allows one to use the mathematical language of group actions on a finite-dimensional manifold. However, we have also seen some physically relevant examples of generalized local symmetries, where the transformation of fields is allowed to depend on their derivatives. This suggests working directly in the infinite-dimensional space of fields as maps \(M\to \mathcal {M}\). With the knowledge of the symmetry group and the symmetry-breaking pattern, one may then still apply the agnostic nonlinear realization, mentioned in Chaps. 12 and 13. To what extent this exhausts all possible actions of the symmetry remains unclear. Yet, a setup that makes the implementation of generalized local symmetries systematic has long been in use in the theory of differential equations [2]. First attempts to apply this setup to EFT appeared very recently [3, 4]. Hopefully, they will help to place the treatment of generalized local symmetries on the same footing as that of point symmetries.
FormalPara Interplay of Nonlinear Realization and Inverse Higgs ConstraintsAs stressed in Chap. 13, the operational way of eliminating would-be Nambu–Goldstone (NG) fields that excite gapped modes using an inverse Higgs constraint (IHC) is optional. The EFT for solely genuine NG degrees of freedom, obtained by applying an IHC, may carry a realization of the symmetry by generalized local transformations. On the other hand, the EFT before the IHC is imposed is ambiguous in that its field content may depend on the choice of order parameter. Either way, the universality of the nonlinear realization of the symmetry may be compromised. Here, too, more work is needed to establish that the existing formalism(s) for spontaneously broken spacetime symmetry give(s) the most general effective Lagrangian.
FormalPara Nonlinear Realization of Translation SymmetryPossibly the largest gap in the narrative of the book is the treatment of spontaneously broken translation symmetry. The main challenge is to establish a unique local parameterization of given fields in terms of NG variables, one for each broken translation generator. Thus, the discussion in Chap. 13 is limited to order parameters spatially modulated just in one direction. A generalization of the formalism to ordered states of matter, modulated in several dimensions, is pending. The same obstacle hinders the application of the background gauge approach, which proved extremely efficient in case of internal symmetries (Chap. 8). Some concrete applications of this approach beyond the single example worked out in Chap. 13 can be found in [5]. However, a fully general background gauge formalism for construction of EFTs for spontaneously broken spacetime symmetry is not available as yet.
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Brauner, T. (2024). Some Open Questions. In: Effective Field Theory for Spontaneously Broken Symmetry. Lecture Notes in Physics, vol 1023. Springer, Cham. https://doi.org/10.1007/978-3-031-48378-3_16
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