Stability of higher-dimensional Einstein-Yang-Mills theories on symmetric spaces

We study (4 + d)-dimensional Einstein-Yang-Mills theories with arbitrary gauge groups, G_YM. The theory is compactified on a d-dimensional symmetric coset space $\text{G}\text{H}$ with a symmetric, topologically non-trivial classical gauge field, embedded in an H-subgroup of the Yang-Mills group. These theories are known to be classically stabilized by gravity if G_YM = H, $\text{G}\text{H}$ is a sphere and d ≠ 3. We study classical instabilities caused by embedding H in a larger gauge group. The small fluctuation spectrum is completely calculable, and leads to a stability condition. For two-dimensional spheres this condition is precisely the Brandt-Neri stability condition for non-abelian monopole fields. For four-spheres we find stability for SU(2) instantons embedded in arbitrary gauge groups and we reproduce the fluctuation spectrum around instantons. For higher-dimensional spheres the stable solutions of this type are completely classified, and occur only for d = 5, 6, 8, 9, 10, 12 and 16. The results show a remarkable agreement with expected topological stability. We also give a few examples with other symmetric spaces, such as CP_n, where the stability criterion appears less restrictive.     

Symmetric space scalar field theory

Some quantum field theories, such as the chiral SU(2) ? SU(2) theory, can have a dynamics invariant under a group $\text{G}$ that is realized on a vacuum which is invariant only under a subgroup $\text{H}$ of $\text{G}$. These theories may be defined by scalar fields which are coordinates for the coset manifold $\text{G}$/$\text{H}$. They are thus non-polynomial theories on a symmetric space, with the group motions in this space described by a set of Killing vectors. We show how the Lagrange function may be constructed entirely from the Killing vectors. In particular, all physical quantities may be expressed in terms of the currents formed out of the Killing vectors. The current correlation functions do not exhibit the spurious wave function renormalizations which are encountered if ordinary Green's functions are computed. We illustrate the general method by calculating one-loop counter terms in a completely invariant fashion. An Appendix describes in simple terms the general theory of symmetric spaces, which should prove useful in other contexts.     

Dimensional reduction and flavor chirality

We show that phenomenologically realistic flavor-chiral Yang-Mills-Higgs theories in 4 dimensions can be derived by dimensional reduction of 10-dimensional vectorlike and gauge theories, where the extra 6 dimensions form a compact coset space with scale size r. The dimensional reduction often implies a symmetry breaking pattern like that of the electroweak theory, in which case it is natural to propose $\text{r ? G}{}_{\mathrm{<mtext>F</mtext>}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. Quantum effects then determine the short-distance behavior of the theory, including any additional symmetry breaking.     

Consistency requirements for 1N expandable field theories

It is demonstrated that $\text{1}\text{N}$ expansion in a class of field theories with an internal O(N) symmetry leads to pathologies. Therefore, these models are either not internally consistent, or their perturbation expansions (ordinary as well as $\text{1}\text{N}$ are unreliable.     

Manifest chiral symmetry preservation in the 1N-expanded SU(N) Thirring model

By utilizing manifestly chiral-invariant auxiliary field operators, it is demonstrated that continuous chiral symmetry is preserved explicitly in the $\text{1}\text{N}$ expansion of 2-dimensional theories whose fermions have a Gross-Neveu type of potential. The effective lagrangian derived in the $\text{1}\text{N}$ limit describes a massless scalar field whose derivative coupling to the fermions vanishes as N → ∞, and a decoupled massive scalar field.     

Foldy-Wouthuysen transformations for any spin

We give the generalized Foldy-Wouthuysen transformation for any 2(2J + 1)-component Poincaré-invariant Hamiltonian theory that describes free massive spin-J particles and that is subject to the conditions: (a) every observable $\text{O}$ is either Hermitian ($\text{O}$ = $\text{O}$⁺) or pseudo-Hermitian ($\text{O}$ = ?₃$\text{O}$⁺?₃) and (b) the theory is invariant under the discrete symmetries. The requirement that the Hamiltonian be defined in the rest frame specifies one and only one boost generator that is also defined at p = 0.     

A classification of compactifying solutions for d =11 supergravity

Supergravity in eleven dimensions is known to have classical solutions of the type (anti-de Sitter space-time) × (7-dimensional Einstein space). We give a list of all homogeneous 7-manifolds which admit an Einstein metric. Known solutions are reviewed, with some emphasis on the SU(3) × SU(2) × U(1) compactifications. Their topology is discussed in detail.The list includes three new solutions, with symmetry groups SU(3) × SU(2), SO(5) and SO(5) × U(1). The first solution has no supersymmetry, while the second and third yield respectively N = 1 and N = 2 supersymmetry in four dimensions. The last two solutions may be extended to solutions with nonzero internal photon curl, breaking all supersymmetry.The existence of a spin structure on homogeneous manifolds $\text{G}\text{H}$ is discussed and related to topological properties of $\text{G}\text{H}$. As an illustration, we treat the coset spaces SU(m + 1) × SU(n + 1)/SU(m) × SU(n) × U(1), which include the spaces with SU(3) × SU(2) × U(1) symmetry.     

Radial and Regge excitations in unified,grand unified and subconstituent models

Necessary group theoretic conditions for all elementary gauge bosons and fermions of an arbitrary renormalizable gauge theory to lie on Regge trajectories are reviewed. It is then argued that in properly unified gauge theories all particles of a given spin lie on Regge trajectories. This then implied that a properly unified gauge theory has no local U(1) factor groups, and no massive fermion singlets. A consideration of the general pattern of Regge and radial recurrences to be expected in quantum field theories suggests that the presence or absence of spin $\text{3}\text{2}$ quarks and/or leptons in the TeV region will provide crucial clues to enable one to distinguish between various classes of unified, grand unified, and subconstituent models. The correct interpretation of such excited fermions will require correlation with the Higgs boson mass and possible radial and Regge excitations of the weak vector bosons.     

Matter fields and metric deformations in multidimensional unified theories

This paper describes a first study of the effects due to including matter fields in generalized Kaluza-Klein (KK) theories with nonabelian compact gauge group G and nontrivial fibres $\text{V}$^K. The approach is based on the first-order Einstein-Cartan (EC) general relativity in (4 + K) dimensions. In the EC theory there are two basic mechanisms which can lead to a spontaneously compactified KK background geometry R⁴ × $\text{V}$^K: (A) a particular kind of energy-momentum density matter condensate in the quantized ground state, or (B) a particular kind of spin-density matter condensate. If (A) or (B) are operating, the inconsistencies usually found between the KK ansatz and the matter-free EC theory are avoided. Mechanism (B) works only when $\text{V}$^K is parallelizable. It is shown that the expansion of matter fields in normal modes on $\text{V}$^K implies that one must include deformations of the Yang-Mills (YM) potentials contained in the usual KK metrics. We discuss and characterize one class of such deformations. As a case study, we consider fibres $\text{V}$^K ～ G′, where G′ is a semisimple compact Lie group. We allow for the “maximal” YM gauge group G_L′ × G_R′. We carry out the harmonic analysis for spinor fields and study the mass spectrum and YM quantum numbers of the normal modes. We rely on mechanism (B) to provide a curvature-free connection (“parallelization”) on $\text{V}$^KG′ by means of a suitable vertical constant torsion. Minimal YM couplings are of size $\text{l}\text{L}\text{≡ g}$, where l is the Planck length and L is the length of the fibre; nonminimal YM couplings are of size L. Nonzero masses are of size L^?1. The massless modes are found and discussed. There would be no massless modes if the parallelizing vertical torsion were absent. This torsion also implies the vanishing of the cosmological constant. When the theory is restricted to massless modes, the YM deformations disappear and the dimensional reduction to four dimensions yields an effective YM theory, which is renormalizable at energies far below L^?1: the effective theory is obtained by letting L → 0 with g ? 1 fixed and by neglecting all masses of order L^?1; g corresponds to the bare YM coupling constant. The surviving effective YM gauge group is G_L′ and the matter fields are in a particular representation of G_L′ × G_R′, corresponding to the zero mass eigenvalue. Explicit examples are discussed for G′ = SU(2) and G′ = SU(3).     

Dynamical group quantization

The projective unitary irreducible representations U(G) of the space-time symmetry group G provides a unique quantization scheme for elementary particles. By extension a direct method of quantization for more general systems by the projective unitary representations U^$\text{G}$(G) induced from a dynamical group $\text{G}$ is outlined. Reducible relativistic composite systems are defined and the geometry of $\text{G}$ is discussed.     

A slice theorem for the space of solutions of Einstein's equations

A slice for the action of a group G on a manifold X at a point x ? X is, roughly speaking, a submanifold S_x which is transverse to the orbits of G near x. Ebin and Palais proved the existence of a slice for the diffeomorphism group of a compact manifold acting on the space of all Riemannian metrics. We prove a slice theorem for the group $\text{D}$ of diffeomorphisms of spacetime acting on the space $\text{E}$ of spatially compact, globally hyperbolic solutions of Einstein's equations. New difficulties beyond those encountered by Ebin and Palais arise because of the Lorentz signature of the spacetime metrics in $\text{E}$ and because $\text{E}$ is not a smooth manifold- it is known to have conical singularities at each spacetime metric with symmetries. These difficulties are overcome through the use of the dynamic formulation of general relativity as an infinite dimensional Hamiltonian system (ADM formalism) and through the use of constant mean curvature foliations of the spacetimes in $\text{E}$. (We devote considerable space to a review and extension of some special properties of constant mean curvature surfaces and foliations that we need.) The conical singularity structure of $\text{E}$, the sympletic aspects of the ADM formalism, and the uniqueness of constant mean curvature foliations play key roles in the proof of the slice theorem for the action of $\text{D}$ on $\text{E}$. As a consequence of this slice theorem, we find that the space $\text{D}$ = $\text{E}$/$\text{D}$ of gravitational degrees of freedom is a stratified manifold with each stratum being a sympletic manifold. The spaces for homogeneous cosmologies of particular Bianchi types give rise to special finite dimensional symplectic strata in this space $\text{G}$. Our results should extend to such coupled field theories as the Einstein-Yang-Mills equations, since the Yang-Mills system in a given background spacetime admits a slice theorem for the action of the gauge transformation group on the space of Yang-Mills solutions, since there is a satisfactory Hamiltonian treatment of the Einstein-Yang-Mills system, and since the singularity structure of the solution set is known.     

Splitting the degeneracy of harmonically-bound identical spinless bosons: General derivation

The Gartenhaus-Schwartz model of N particles interacting via pairwise harmonic potentials $\text{V}{}_{\mathrm{ij}}\text{=}\text{1}\text{2}\text{kR}{}_{\mathrm{ij}}{}^2$ is extended by addition of an arbitrary pairwise interaction small enough to be treated by degenerate perturbation theory. The theory is worked out for a rarefied fluid of N spinless bosons, to give the splittings of the degeneracies of the U(3N?3) symmetry inherent in the Gartenhaus-Schwartz model. Following Moshinsky and Kramer the external field is formally introduced in order to reduce the zeroth-order problem to the 3N-dimensional harmonic oscillator. For systems with more particles than quanta, a recipe is given to construct states in which all the excitation is in internal degrees of freedom and none is in translation. Analytical solutions are obtained for states with no more than two quanta in excitation of relative motion of any one pair of particles.     

Random walk on the homogeneous spaces of a group and generalized coherent states

The evolution of a quantum system characterized by a dynamical group G and subjected to random interactions can be reduced, through the use of the generalized coherent states associated with a group $\text{G}$, whose algebra is an ideal of the algebra of G, to a random walk on some homogeneous space S of $\text{G}$. G is the group of automorphisms of S. The connection between the symmetry properties of such a random walk and the structure of G are discussed, showing that it is always possible to map the original process into an equivalent process taking place in the manifold of G. The investigation of the general properties of this mapping quite naturally provides a deep characterization of the role played by G in determining the basic transient and stationary features of the evolution of the system.     

Relaxation of the 4G92 state of Nd3+ in LaCl3 and Nd3+ in La(Cl99.6Br0.4)3

The decay of the ⁴G_{$\text{9}\text{2}$} state of Nd³⁺ in LaCl₃ and La(Cl_99.6 Br_0.4)₃ was measured after pulsed laser excitation as a function of temperature. The decay rate is shown to depend besides the radiative transition on single-phonon relaxation between the states ⁴G_{$\text{9}\text{2}$} ($\text{μ =}\text{1}\text{2}$) and ⁴G_{$\text{9}\text{2}$} ($\text{μ =}\text{3}\text{2}$) and on multiphonon orbit-lattice relaxation from ⁴G_{$\text{9}\text{2}$} to ²G_{$\text{9}\text{2}$}. Partial substitution of Cl by Br only alters the radiative lifetime.     

A constraint on the distance dependence of the gravitational constant

Extended supergravity theories predict the existence of vector and scalar bosons, besides the graviton, which in the static limit couple to the mass. An example is the graviphoton, leading to antigravity. If these bosons have a small mass (?10^?4 eV), an observable Yukawa term would be present in the gravitational potential in the newtonian limit. This can be parametrized by a distance dependent effective gravitational “constant” G(r). Defining G₀ = G(10 cm) and G_c = G(10³ km), the comparison between theory and observations of the white dwarf Sirius B results in $\text{G}{}_{\mathrm{<mtext>c</mtext>}}\text{G}{}_0\text{= 0.98 ± 0.08}$.     

Massless composites with massive constituents

We construct model field theories in which a confining gauge interaction binds massive elementary fermions into massless composite particles. The massless composites are either Goldstone bosons or $\text{spin}\text{-}\text{1}\text{2}$ fermions. In these models, the manner in which exact chiral symmetries are realized changes at a critical value of the elementary fermion mass of order (e²/16π²)Λ, where Λ is the confinement scale and e is a weak gauge coupling.     

Exact solutions in λϕn theory

Families of exact solutions of the λ?ⁿ theory in d + 1-dimensional space—time are obtained. Certain finite-energy static solutions are found in the case when $\text{n =}\text{2d}\text{(d ? 2)}$.     

Membrabe models and generalized Z2 gauge theories

We consider models of (d?n)-dimensional membranes fluctuating in a d-dimensional space under the action of surface tension. We investigate the renormalization properties of these models perturbatively and in $\text{1}\text{n}$ expansion. The potential relationships of these models to generalized Z₂ gauge theories are indicated.