The Positive Action conjecture and asymptotically Euclidean metrics in quantum gravity

The Positive Action conjecture requires that the action of any asymptotically Euclidean 4-dimensional Riemannian metric be positive, vanishing if and only if the space is flat. Because any Ricci flat, asymptotically Euclidean metric has zero action and is local extremum of the action which is a local minimum at flat space, the conjecture requires that there are no Ricci flat asymptotically Euclidean metrics other than flat space, which would establish that flat space is the only local minimum. We prove this for metrics onR ⁴ and a large class of more complicated topologies and for self-dual metrics. We show that ifR 0 there are no bound states of the Dirac equation and discuss the relevance to possible baryon non-conserving processes mediated by gravitational instantons. We conclude that these are forbidden in the lowest stationary phase approximation. We give a detailed discussion of instantons invariant under anSU(2) orSO(3) isometry group. We find all regular solutions, none of which is asymptotically Euclidean and all of which possess a further Killing vector. In an appendix we construct an approximate self-dual metric onK3 — the only simply connected compact manifold which admits a self-dual metric.     

Existence of Resonances for Metric Scattering in Even Dimensions

Suppose _g is the (negative) Laplace–Beltrami operator of a Riemannian metric g on ⁿ which is Euclidean outside some compact set. It is known that the resolvent R()=(– _g –²)^–1, as the operator from L ² _comp( ⁿ ) to H ² _loc( ⁿ ), has a meromorphic extension from the lower half plane to the complex plane or the logarithmic plane when n is odd or even, respectively. Resonances are defined to be the poles of this meromorphic extension. We prove that when n is 4 or 6, there always exist infinitely many resonances provided that g is not flat. When n is greater than 6 and even, we prove the same result under the condition that the metric is conformally Euclidean or is close to the Euclidean metric.     

On some euclidean einstein metrics

We prove that the complex manifold of the superposition Eguchi-Hanson metric plus the pseudo-Fubini-Study metric is equal to the total space of the holomorphic line bundle of degree –n on the Riemann sphere. The apparent singularities of the metric can be resolved only if the Eguchi-Hanson parameter satisfies a ⁴=4(n–2)²(n+1)/3², n3. We give a geometrical explanation of the fact that we need n3. Finally, we generalize the metric of Gegenberg and Das to obtain a triaxial vacuum metric.     

On Existence of Static Metric Extensions in General Relativity

Motivated by problems related to quasi-local mass in general relativity, we study the static metric extension conjecture proposed by R. Bartnik [4]. We show that, for any metric on ¯B ₁ that is close enough to the Euclidean metric and has reflection invariant boundary data, there always exists an asymptotically flat and scalar flat static metric extension in M=³B ₁ such that it satisfies Bartnik's geometric boundary condition [4] on B ₁.     

The isoperimetric problem for pinwheel tilings

In aperiodic pinwheel tilings of the plane there exist unions of tiles with ratio (area)/(perimeter)² arbitrarily close to that of a circle. Such approximate circles can be constructed with arbitrary center and any sufficiently large radius. The existence of such circles follows from the metric on pinwheel space being almost Euclidean at large distances; ifP andQ are points separated by large Euclidean distanceR, then the shortest path along tile edges fromP toQ has lengthR+o(R).Research supported in part by NSF Grant No. DMS-9304269 and Texas ARP Grant 003658-113.Research supported in part by an NSF Mathematical Sciences Postdoctoral Fellowship and Texas ARP Grant 003658-037.     

Twistor spinors and gravitational instantons

We construct a complete Riemannian metric on the four-dimensional vector space ⁴ which carries a two-dimensional space of twistor spinor with common zero point. This metric is half-conformally flat but not conformally flat. The construction uses a conformal completion at infinity of theEguchi-Hanson metric on the exterior of a closed ball in ⁴.     

Hyper-Kähler metrics and a generalization of the Bogomolny equations

We generalize the Bogomolny equations to field equations on ^{3 n} and describe a twistor correspondence. We consider a general hyper-Kähler metric in dimension 4n with an action of the torusT ⁿ compatible with the hyper-Kähler structure. We prove that such a metric can be described in terms of theT ⁿ -solution of the field equations coming from the twistor space of the metric.     

On the Borel summability of planar perturbation series: Transition to Minkowski space

In recent years 't Hooft and Rivasseau proved the Borel summability of planar asymptotically free massive theories in Euclidean space. The corresponding Borel sums in Minkowski space are shown to exist as linear functionals if the Euclidean counterparts are bounded polynomially in momentum space and fulfill certain analyticity conditions. Both can be verified in massive planar wrong sign ₄ ⁴ using Rivasseau's approach. The functionals alternatively are densely defined and unbounded on anL ^p space or bounded on (the whole of) a Banach space with a more restrictive norm.Supported by a scholarship of the NATO scientific committee via the German academic exchange service (DAAD)After October 1, 1987: Max-Planck-Institut für Physik, Föhringer Ring 6, D-8000 München 40, Federal Republic of Germany     

The Riemannian geometry of the Yang-Mills moduli space

The moduli space of self-dual connections over a Riemannian 4-manifold has a natural Riemannian metric, inherited from theL ² metric on the space of connections. We give a formula for the curvature of this metric in terms of the relevant Green operators. We then examine in great detail the moduli space ₁ ofk=1 instantons on the 4-sphere, and obtain an explicit formula for the metric in this case. In particular, we prove that ₁ is rotationally symmetric and has finite geometry: it is an incomplete 5-manifold with finite diameter and finite volume.Partially supported by Horace Rackham Faculty Research Grant from the University of MichiganPartially supported by N.S.F. Grant DMS-8603461     

Geometroelectrodynamics

We present an elementary particle model that can be thought of as a unification of certain topological ideas abstracted from the string model and the standard Yang-Mills theory. The basic dynamical entity of the model is a spacelike 3-surfaceX ³ in some metric spaceH and is interpreted as a particle. The dynamics of the model is based on two ideas. First the model is formally a Yang-Mills theory on the surfaceX ⁴ representing the orbit(s) of the particle(s) inH. Secondly the Yang-Mills structure onX ⁴ is constructed using only the natural geometric structures of the space H by a process which we call induction. It is found that some rather general requirements highly fix the choice of the space H. In fact the minimal model, for which the space H is the product of Minkowski space and the 2-sphere, is obtained by requiring that the symmetry group of the theory is the product of the Poincaré group and the color groupSO(3). The unique feature of the minimal model is that it affords a purely topological mechanism for quark confinement.     

A theory of generally invariant lagrangians for the metric fields. I

The second-order generally invariant Lagrangians for the metric fields are studied within the framework of the Ehresmann theory of jets. Such a Lagrangian is a function on an appropriate fiber bundle whose structure group is the groupL _n ³ of invertible 3-jets with source and target at the origin 0 of the real,n-dimensional Euclidean spaceR ⁿ, and whose type fiber is the manifold T_n ²(R^n* R ^n*) of 2-jets with source at 0 R ⁿ and target in the symmetric tensor productR ^n* R^n*. Explicit formulas for the action ofL _n ³ onT _n ²(R^n* R ^n*) are considered, and a complete system of differential identities for the generally invariant Lagrangians is obtained.     

The Lipatov argument for φ 3 4 perturbation theory

We extend to ₃ ⁴ the work of S. Breen on the leading behavior at large order of ₂ ⁴ perturbation theory. Using a phase space expansion to obtain new estimates on the high energy behavior of ₃ ⁴ Feynman graphs, and a rigorous semiclassical expansion, we prove that the radius of convergence of the Borel transform of the pertubative series for ₃ ⁴ Euclidean field theory is the one computed by the Lipatov method.     

Topological aspects of Yang-Mills theory

The space of mapsS ³ G has components which give the topological quantum number of Yang-Mills theory for the groupG. Each component of the space has further topological invariants. WhenG=SU(2) we show that these invariants (the homology groups) are captured by the space of instantons. Using these invariants we show that potentials must exist for which the massless Dirac equation (in Euclidean 4-space) has arbitrarily many independent solutions (for fixed instanton number).     

Space Geometry of Rotating Platforms: An Operational Approach

We study the space geometry of a rotating disk both from a theoretical and operational approach; in particular we give a precise definition of the space of the disk, which is not clearly defined in the literature. To this end we define an extended 3-space, which we call relative space: it is recognized as the only space having an actual physical meaning from an operational point of view, and it is identified as the physical space of the rotating platform. Then, the geometry of the space of the disk turns out to be non Euclidean, according to the early Einstein's intuition; in particular the Born metric is recovered, in a clear and self consistent context. Furthermore, the relativistic kinematics reveals to be self consistent, and able to solve the Ehrenfest's paradox without any need of dynamical considerations or ad hoc assumptions.     

Minkowski space Yang-Mills fields from Euclidean self-dual fields

It is examined, if it is possible, to obtain solutions of the SU(2) Yang-Mills field equations in Minkowski space from Euclidean self-dual Yang-Mills fields by method proposed by Bernreuther. It is shown that the conditions, which are imposed on the Euclidean self-dual fields by this method, make every Euclidean self-dual field and the corresponding Minkowski space field obtained from it, equivalent to a pure gauge field, F _ab 0.     

Scattering theory,cross sections,and the lattice of propositions

The asymptotic conditions for the nonrelativistic quantum scattering of a particle by a center of force are derived in terms of a metric on the space of states on a complete orthocomplemented lattice. The flux of particles scattered into a coneC per unit incident flux, averaged over all displacements of the center of force at right angles to the axis of the incident beam, is expressed in terms of the differential cross sectiond/d when the motion is classical, and in terms of the scattering amplitudef when the motion is quantum mechanical. This enables the usual identificationd/d=|f|² to be made.     

Global conformal invariance in quantum field theory

Suppose that there is given a Wightman quantum field theory (QFT) whose Euclidean Green functions are invariant under the Euclidean conformal groupSO _e (5,1). We show that its Hilbert space of physical states carries then a unitary representation of the universal (-sheeted) covering group* of the Minkowskian conformal group SO _e (4, 2)₂. The Wightman functions can be analytically continued to a domain of holomorphy which has as a real boundary an -sheeted covering of Minkowski-spaceM ⁴. It is known that* can act on this space and that admits a globally*-invariant causal ordering; is thus the natural space on which a globally*-invariant local QFT could live. We discuss some of the properties of such a theory, in particular the spectrum of the conformal HamiltonianH=1/2(P ⁰+K ⁰).As a tool we use a generalized Hille-Yosida theorem for Lie semigroups. Such a theorem is stated and proven in Appendix C. It enables us to analytically continue contractive representations of a certain maximal subsemigroup of to unitary representations of*.     

Convergent perturbation expansions for Euclidean quantum field theory

Mayer perturbation theory is designed to provide computable convergent expansions which permit calculation of Greens functions in Euclidean quantum field theory to arbitrary accuracy, including nonper-turbative contributions from large field fluctuations. Here we describe the expansions at the example of 3-dimensional ⁴-theory (in continuous space). They are not essentially more complicated than standard perturbation theory. Then ^th order term is expressed in terms ofO(n)-dimensional integrals, and is of order ^k if 4k–3n4k.Dedicated to the memory of Kurt Symanzik     

Nonvacuum taub-type cosmological model

The Einstein universe is a simple model describing a static cosmological spacetime, having a constant radius and a constant curvature, and, as is well known, it does not describe our universe. We propose a model which is an extension of Einstein's. Our metric, havingR × S ³ topology, describes a nonisotropic homogeneous closed (finite) universe of Bianchi type IX. This metric is similar to that of Taub, but is simpler. Unlike the Taub solution (which is a cosmological extension of the NUT solution), however, the universe described by our metric contains matter. Like the Taub metric, our metric has two positive constants (, T). The gravitational red shift calculated from our metric is given. Similarly to the Schwarzschild metric, which has a singularity atr = 2m, this metric has the same kind of singularity att = 2. The maximal extension of the coordinates in our metric is fairly analogous to that of the Schwarzschild metric.