We study field theory models in the context of a gravitational theory without the cosmological constant problem (CCP). The theory is based on the requirement that the measure of integration in the action be not necessarily $\sqrt{-g}$ but be determined dynamically through additional degrees of freedom, like four scalar fields ${\phi }_a.$ We study three possibilities for the general structure of the theory. (A) The total action has the form $S=\int \Phi L{d}^{4}x$ where the measure $\Phi$ is built from the scalars ${\phi }_a$ in such a way that the transformation $\overrightarrow{L}L+\mathrm{const}$ does not affect the equations of motion. Then an infinite dimensional shift group of the measure fields (SGMF) ${\phi }_a$ by arbitrary functions of the Lagrangian density L, ${\phi }_a\to {\phi }_a{+f}_a\mathrm{}\left(L\right),$ is recognized as the symmetry group of the action up to an integral of total divergence. (B) The total action has the form ${S=S}_1{+S}_2,$ $S_1=\int \Phi L_1{d}^{4}x,$ $S_2=\int \sqrt{-g}{L}_2{d}^{4}x$ which is the only model different from (A) and invariant under the SGMF [but now with $f_a{=f}_a{(L}_1\left)\right].$ Similarly, now only $S_1$ satisfies the requirement that the transformation $L_1\to L_1+\mathrm{const}$ not affect the equations of motion. In both cases (A) and (B) it is assumed that ${L,L}_1{,L}_2$ do not depend on ${\phi }_a.$ (C) The action includes a term which breaks the SGMF symmetry. It is shown that in the first order formalism a constraint appears which allows us to solve the scalar field related to the dynamical measure degrees of freedom in terms of matter fields. The remarkable feature of the models discussed in this paper is that for all cases [(A), (B), and (C)], after the change of variables to the conformal Einstein frame, the classical field equations take exactly the form of general relativity (GR). Therefore the models are free from the well-known problem of the usual scalar-tensor theories in what concerns the classical GR tests. The only difference of the field equations in the Einstein frame from the canonical equations of the self-consistent system of Einstein’s gravity and matter fields is the appearance of the effective scalar field potential which vanishes in a true vacuum state (TVS) without fine-tuning in cases (A) and (B). To illustrate how the theory works, we present a few explicit field theory models where it is possible to combine the solution of the CCP with (1) the possibility for an inflationary scenario, and (2) spontaneously broken gauge unified theories (including fermions). In case (C), the breaking of the SGMF symmetry induces a nonzero energy density for the TVS. When considering only a linear potential for a scalar field $\phi$ in $S_1,$ the continuous symmetry $\overrightarrow{\phi }\phi +\mathrm{const}$ is respected. Surprisingly, in this case spontaneous symmetry breaking takes place while no massless (“Goldstone”) boson appears. We discuss the role of the SGMF symmetry for quantization and the possible connection of this theory with theories of extended objects.

For Einstein’s general relativity (GR) or the alternatives suggested up to date, the vacuum energy gravitates. We present a model where a new measure is introduced for integration of the total action in D-dimensional spacetime. This measure is built from D scalar fields ${\mathrm{\Phi }}_{\mathit{a}}$. As a consequence of such a choice of the measure, the matter Lagrangian ${\mathit{L}}_{\mathit{m}}$ can be changed by adding a constant while no gravitational effects, such as a cosmological term, are induced. Such a nongravitating vacuum energy theory has an infinite dimensional symmetry group which contains volume-preserving diffeomorphisms in the internal space of scalar fields ${\mathrm{\Phi }}_{\mathit{a}}$. Other symmetries contained in this symmetry group suggest a deep connection of this theory with theories of extended objects. In general the theory is different from GR although for certain choices of ${\mathit{L}}_{\mathit{m}}$, which are related to the existence of an additional symmetry, solutions of GR are solutions of the model. This is achieved in four dimensions if ${\mathit{L}}_{\mathit{m}}$ is due to fundamental bosonic and fermionic strings. Other types of matter where this feature of the theory is realized, are, for example, scalars without potential or subjected to nonlinear constraints, massless fermions, and point particles. The point particle plays a special role, since it is a good phenomenological description of matter at large distances. de Sitter space is realized in an unconventional way, where the de Sitter metric holds, but such de Sitter space is supported by the existence of a variable scalar field which in practice destroys the maximal symmetry. The only spacetime where maximal symmetry is not broken, in a dynamical sense, is Minkowski space. The theory has nontrivial dynamics in 1+1 dimensions, unlike GR. © 1996 The American Physical Society.

The gravitational field produced by a spherically symmetric "hedgehog" configuration in scalar field theories with global SO(3) symmetry (or higher) is studied in the limit in which these models become nonlinear $\sigma$ models. The same gravitational effect can be generated by a set of cosmic strings intersecting at a point, in the limit that one considers a continuous distribution of such intersecting strings in a spherically symmetric configuration (to be referred to as the "string hedgehog"). When the energy densities associated with the hedgehog are small, we obtain a static geometry, but for higher values, the resulting geometry is that of an anisotropic cosmology. The evolution of bubbles joining two phases, one of which contains a hedgehog (as defined above) is investigated. The role of such configurations in processes that lead to classical false-vacuum destabilization and in the evolution of inflationary bubbles is discussed. The generalization of our results to the gauged case, i.e., to magnetic-monopole hedgehogs, is discussed.