Conformal changes and geodesic completeness

Let (M, g) be a causal spacetime. ConditionN will be satisfied if for each compact subsetK ofM there is no future inextendible nonspacelike curve which is totally future imprisoned inK. IfM satisfies conditionN, then wheneverE is an open and relatively compact subset ofM the spacetimeE with the metricg restricted toE is stably causal. Furthermore, there is a conformal factor such that (M, ² g) is both null and timelike geodesically complete. IfM is an open subset of two dimensional Minkowskian space, thenM is conformal to a geodesically complete spacetime.     

Gleason's theorem and completeness criteria

We give some applications of Gleason's theorem to completeness criteria of inner product spaces using different families of subspaces, measures on them, and frame functions. Some open criteria problems are formulated.     

Poincaré transformations and Galilei transformations

Poincaré and Galilei transformations are seen to be contained in each other in one space dimension more.     

Interacting Bosons at Finite Temperature: How Bogolubov Visited a Black Hole and Came Home Again

The structure of the thermal equilibrium state of a weakly interacting Bose gas is of current interest. We calculate the density matrix of that state in two ways. The most effective method, in terms of yielding a simple, explicit answer, is to construct a generating function within the traditional framework of quantum statistical mechanics. The alternative method, arguably more interesting, is to construct the thermal state as a vector state in an artificial system with twice as many degrees of freedom. It is well known that this construction has an actual physical realization in the quantum thermodynamics of black holes, where the added degrees of freedom correspond to the second sheet of the Kruskal manifold and the thermal vector state is a state of the Unruh or the Hartle–Hawking type. What is unusual about the present work is that the Bogolubov transformation used to construct the thermal state combines in a rather symmetrical way with Bogolubov's original transformation of the same form, used to implement the interaction of the nonideal gas in linear approximation. In addition to providing a density matrix, the method makes it possible to calculate efficiently certain expectation values directly in terms of the thermal vector state of the doubled system.     

A signed measure completeness criterion

We show that an inner product space S is complete whenever its system E(S) of all splitting subspaces, i.e. of all subspaces M for which M+M =S holds, possesses at least one nonzero completely additive signed measure or, equivalently, iff S possesses at least one nonzero frame function. Moreover, we show a new and simple proof that S is complete iff E(S) contains the join of any sequence of orthogonal one-dimensional subspaces.     

Trapping and completeness of a spherical light-cone

This article concerns an outgoing spherically symmetric light-cone evolving according to the Einstein equations. The relevant equations are solved explicitly as integrals of the Mailing matter, which is unspecified apart from energy conditions and a consistency condition. A condition on the matter is derived which is necessary and sufficient for the cone to be trapped. If the cone is not trapped it is complete, and if it is trapped it will recollapse to a point. In either case, the asymptotic behaviour of the matter is consequently constrained.     

Asymptotic completeness forN-body Stark Hamiltonians

We prove asymptotic completeness for short- and long-rangeN-body Stark Hamiltonians with local singularities of at most Coulomb type. Our results include the usual models for atoms and molecules.Research partially supported by NSF grant DMS 9307147.     

Complex-scaled spectrum completeness for pedestrians

The Complex Scaling Method (CSM) is rephrased in terms of a Hamiltonian containing an optical potential. Traditional theorems concerning Jost solutions, the Jost function, residues of a suitable Green’s function at its poles, and the resolution of the identity are extended. Elementary derivations of the completeness of the CSM spectrum are detailed for the one and the two channel cases.     

Gleason's theorem and completeness of inner product spaces

We generalize the result of Hamhalter and Pták showing that an inner product space whose dimension is either a nonmeasurable cardinal or an arbitrary cardinal is complete iff its lattice of strongly closed subspaces possesses at least one state or one completely additive state, respectively. Moreover, we show that this lattice of any separable space possesses many-finite measures, and we give the Gleason analogue for them.     

Asymptotic completeness and multiparticle structure in field theories

The ideas developed in Part I (ref. [1]) are applied to the recently constructed massive Gross-Neveu model. We define in this case an irreducible kernel satisfying a regularized Bethe-Salpeter equation which is convenient to derive asymptotic completeness in the 2-particle region. As in Part I, the method allows direct graphical definition of general irreducible kernels and is well suited to the analysis of asymptotic completeness and related results in more general energy regions.A large part of the paper is devoted to a new self-contained construction (via phase space expansion) of the Gross-Neveu model. The presentation is somewhat simpler than previous ones, is more complete on some points and is best suited to our purposes.     

Growth conditions, Riemannian completeness and Lorentzian causality

The stationary-Randers correspondence (SRC) provides a deep connection between the property of standard stationary spacetimes being globally hyperbolic, and the completeness of certain Finsler metrics of Randers type defined on the fibres. In order to establish further results, we investigate pointwise conformal transformations of certain Riemannian metrics on the fibres and growth conditions on the corresponding conformal factors. In general, a conformal transformation may map a complete Riemannian metric onto a complete or incomplete metric. We prove a theorem for the growth of the conformal factor such that the conformally transformed Riemannian metric is also complete. As an application, we establish novel relations between the completeness of Riemannian metrics, growth conditions on conformal factors and the Cauchy hypersurface condition on the fibres of a standard stationary spacetime. These results also imply new conditions for the completeness of Randers-type metrics by the application of the SRC.     

Half-range completeness for the Fokker-Planck equation

Half-range completeness theorems are proved for eigenfunctions associated to the one-dimensional Fokker-Planck equation in a semi-infinite medium. Existence and uniqueness results for perfectly absorbing, partially absorbing, and purely specularly reflecting boundary conditions are deduced for the stationary and time-dependent problems. Similar results are obtained for a slab geometry.     

Note on the completeness of spacetimes

Three formally different definitions of completeness for normal hyperbolic manifolds are discussed and shown to be inequivalent.     

Quantum state preparation via asymptotic completeness

We demonstrate that any quantum state |chi> of a single mode radiation field can be prepared with arbitrarily high fidelity by interaction with a sequence of two-level atoms, prepared in a suitable initial state. No final state measurement of the atoms is needed.     

Gauge transformations and quasitriangularity

Natural conditions on a Poisson/quantum group G to implement Poisson/quantum gauge transformations on the lattice are investigated. In addition to our previous result that transformations on one lattice link require G to be coboundary, it is shown that for a sequence of links one needs a quasitriangular G.     

Quantum logics and completeness criteria of inner product spaces

We present the survey of measure-theoretic completeness criteria for inner product spaces using methods and notions important for quantum logics. Moreover, some new criteria and open problems are given.     

Contact transformations and conformal group. III. Finite nonrelativistic transformations

Contact conformal transformations (as defined in a previous work) are integrated giving the finite form of these transformations. Some properties of such a group of motions are pointed out.Research supported in part by the National Science Foundation under Grant No. MPS75-20427.On leave of absence from Departamento de Fisica Matematica, Universidad de Salamanca, Spain.     

Geometric phases and the informational completeness of quantum frames

The basic results on geometric phases are rederived by using infinite dimensional coordinate charts in line bundles, in Hopf bundles, and in projective Hilbert spaces. The determination of a quantum state can be then geometrically described as the measurement of Fubini-Study distances from that state to the elements of informationally complete quantum frames. The basic geometric features of such quantum frames are formulated, and their relationships to corresponding classical frames are analyzed.