Dynamical completion of the Isham–Salam–Strathdee nonlinear-realization interpretation of gravity: underlying GL(4,R) symmetry breaking and the Goldstone spectrum

Isham, Salam and Strathdee (Annals of Physics 62, 98, 1971) reinterpreted the vierbein formalism of general relativity as a nonlinear realization of $GL(4,\mathbf{R})$ spontaneously broken to the Lorentz group $O(3,1)$, while a companion conformal-group construction produced a metric, an affine connection, and curvature from two preferred fields, a scalar $\sigma$ and a vector $\phi_\mu$; the graviton and the other preferred fields appear as Goldstone-type modes. The construction is kinematic, and the authors themselves left open the physical mass of the $\sigma$-particle, the dynamics deciding whether the preferred vector $\phi_\mu$ is a massless gauge field or a massive vector, the viability of the non-Riemannian geometries with independent $\phi$, and the global validity of the local polar-type decomposition used to parametrize $GL(4,\mathbf{R})/O(3,1)$. Later work completed the kinematic correspondence (Borisov–Ogievetsky; Tomboulis) and substantially answered the spinor-coupling obstruction through infinite-dimensional representations, but the dynamical core remains open: no explicit $GL(4,\mathbf{R})$- (affine-) symmetric theory exhibiting the required spontaneous breaking is known, and the physical spectrum of the preferred fields has not been determined. The open problem asks for this dynamical completion — an underlying metric-free theory with demonstrated breaking to Lorentz/Poincaré, explicit symmetry-breaking terms fixing the masses of $\sigma$ and $\phi_\mu$, and the viability of the non-Riemannian sector — together with the global coset-parametrization question.

Background

When a continuous symmetry group $G$ of a relativistic quantum field theory is spontaneously broken to a subgroup $H$ (the vacuum is invariant under $H$ but not $G$), the low-energy physics is dominated by massless Nambu–Goldstone bosons. Coleman, Wess and Zumino showed that the general low-energy dynamics is organized by fields parametrizing the coset space $G/H$, on which $G$ acts nonlinearly — a 'nonlinear realization'. For internal symmetries this is a classification tool; for space-time symmetries it becomes geometric. If the broken generators include transformations of the coordinates themselves, the coset fields can serve as building blocks of geometry. Volkov and collaborators extended the construction to space-time symmetries, and Ivanov and Ogievetsky identified the 'inverse Higgs phenomenon': some candidate Goldstone fields can be eliminated algebraically in terms of derivatives of the others, so the physical Goldstone content can be smaller than the number of broken generators.

The source paper — Isham, Salam and Strathdee, 'Nonlinear realizations of space-time symmetries. Scalar and tensor gravity' (Annals of Physics 62, 98, 1971) — applied this to gravity in two parts. In the 'scalar' part, based on the conformal group (Lorentz transformations plus dilatations, i.e. scale transformations, and special conformal transformations), the nonlinear realization is governed by two 'preferred fields': a scalar $\sigma(x)$, from which a metric $g_{ab}=\eta_{ab}e^{-2\sigma}$ is built ($\eta_{ab}$ is the Minkowski metric), and a Lorentz vector $\phi_a(x)$, which defines an affine connection $\Gamma_{bc}{}^{a}=2(\delta_b^a\phi_c+\delta_c^a\phi_b-g_{bc}\phi^a)$ (the object that parallel-transports vectors, generalizing the Christoffel symbols). When $\phi_a=-\frac{1}{2}\partial_a\sigma$ this connection reduces to the Levi-Civita (Christoffel) connection and the geometry is Riemannian; a generic independent $\phi_a$ gives a non-Riemannian geometry whose signature is a nonzero antisymmetric part of the Ricci tensor, $R_{[ab]}\neq 0$. A current–field spectral argument (analysing the commutator of the dilatation current with $\sigma$) indicates that $\sigma$ is a massless Goldstone boson of spontaneously broken dilatations unless explicit symmetry breaking gives it a mass — a Goldstone mode lifted to finite mass by explicit breaking is called a pseudo-Goldstone boson. In the 'tensor' part, the group is $GL(4,\mathbf{R})$ — the group of invertible real $4\times 4$ matrices, i.e. the Jacobians of arbitrary smooth coordinate changes — spontaneously broken to the Lorentz subgroup $O(3,1)$. The coset $GL(4,\mathbf{R})/O(3,1)$ carries ten Goldstone fields, assembled through a polar-type decomposition of a matrix into a 'pseudosymmetric' and a Lorentz factor (well-defined only in a neighborhood of the identity) into a vierbein-like matrix $L_\mu{}^a$ — the vierbein, or tetrad, of general relativity, the frame field relating a general space-time frame to locally inertial Lorentz frames. The metric $g_{\mu\nu}=L_\mu{}^aL_\nu{}^b\eta_{ab}$ and its tensorial transformation law under coordinate changes then follow, so the kinematic framework of general relativity is re-derived as a nonlinear realization. Spinors couple through a spin connection determined algebraically by the coset fields, because no finite-dimensional linear representation of $GL(4,\mathbf{R})$ contains the half-integer-spin representations of the Lorentz group.

Later work established the kinematic correspondence firmly. Ogievetsky (1973) proved that the algebra of infinitesimal general coordinate transformations is the closure of the conformal and linear algebras, and Borisov–Ogievetsky (1974) showed that a combined nonlinear realization of the conformal and affine groups (the affine group being $GL(4,\mathbf{R})$ together with space-time translations) reproduces the framework of Riemannian geometry. Tomboulis (2011) redid the construction with the modern coset technique: the effective theory of $GL(4,\mathbf{R})$ spontaneously broken to $SO(3,1)$, written in terms of composite quantities that transform linearly under the full group, is exactly vierbein general relativity, with the Einstein–Hilbert action as the leading term and general covariance realized nonlinearly through the Goldstone fields. The correspondence the 1971 paper had only suggested is thus a theorem of the coset formalism. What the coset construction cannot supply — and what the source paper explicitly flagged as untouched — is dynamics: no Lagrangian, no symmetry-breaking mechanism, and no determination of the physical spectrum of the preferred fields follow from the symmetry pattern alone.

Problem Statement

The nonlinear-realization reinterpretation of gravity of Isham, Salam and Strathdee (1971) re-derives the metric, connection, curvature, and spinor coupling of general relativity as Goldstone/coset data of $GL(4,\mathbf{R})$ (and of the conformal group) spontaneously broken to the Lorentz group, but it supplies no dynamics: no Lagrangian, no symmetry-breaking mechanism, and no determination of the physical spectrum. The open problem is to complete this picture dynamically, as one coherent objective — turning the reinterpretation into a physical theory of gravity — through the following linked questions.

(1) Existence of the underlying theory: construct an explicit, consistent theory with global space-time symmetry under $GL(4,\mathbf{R})$ or a closely related affine group — the special-affine group (replacing $GL(4,\mathbf{R})$ by its unimodular subgroup $SL(4,\mathbf{R})$ of unit-determinant matrices) or the full affine group ($GL(4,\mathbf{R})$ together with space-time translations) — possessing no background metric, since $GL(4,\mathbf{R})$ admits no invariant metric, whose dynamics demonstrably breaks that symmetry spontaneously to the Lorentz (or Poincaré) subgroup, with the metric/vierbein emerging as composites of the resulting Goldstone fields; or prove that no such consistent theory exists.

(2) Spectrum of the preferred fields: within the nonlinear-realization effective theory, construct and analyze explicit symmetry-breaking terms and determine the physical mass spectrum — in particular whether the scalar Goldstone $\sigma$ of broken dilatations acquires a nonzero physical mass as a pseudo-Goldstone boson, and what dynamics decides whether the preferred vector $\phi_\mu$ behaves as a massless gauge field (as in a Weyl-type geometry, where a scale gauge connection accompanies the metric) or as a massive neutral vector.

(3) Non-Riemannian sector: determine whether the geometries with independent $\phi_a$ (nonzero antisymmetric Ricci tensor $R_{[ab]}$) admit viable phenomenology, or whether consistency and observation force the Riemannian constraint $\phi_a=-\frac{1}{2}\partial_a\sigma$.

(4) Global validity: settle the existence, uniqueness, and patching across coordinate charts of coset parametrizations of $GL(4,\mathbf{R})/O(3,1)$ over entire space-time manifolds, beyond the neighborhood of the identity in which the polar-type decomposition is defined.

The questions (1)–(3) are linked, since they ask whether the symmetry pattern can be dynamically realized and what its observable consequences are; (4) is a structural question that underpins any global formulation. As the source itself acknowledges, the quantization problem of gravity is not part of this program; and the spinor-coupling obstruction has been substantially answered through infinite-dimensional representations and the standard vierbein spin connection, so it is not part of the open core.

References

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