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Biconformal gravity, based on gauging of the conformal group to 2n dimensions, reproduces n-dim scale-covariant general relativity on the co-tangent bundle in any dimension. We generalize this result to include Yang-Mills matter sources formulated as SU\left(N\right) gauge theories with a twisted action on the full 2n-dimensional biconformal space. We show that the coupling of the sources to gravity does not stop the reduction of effective dimension 2n\rightarrow n of the gravity theory, and instead forces the Yang-Mills source to reduce to ordinary n-dimensional Yang-Mills theory on the gravitating cotangent bundle, with the usual Yang-Mills energy tensor as gravitational source. The results apply as well to gravity with sources on K\"ahler manifolds and in double field theories., Comment: 67 pages including appendices and references Clarified discussion of Kahler and Killing forms; added discussion of alternative Weyl gravity sources

We consider the conformal group of a space of dim n=p+q, with SO(p,q) metric. The quotient of this group by its homogeneous Weyl subgroup gives a principal fiber bundle with 2n-dim base manifold and Weyl fibers. The Cartan generalization to a curved 2n-dim geometry admits an action functional linear in the curvatures. Because symmetry is maintained between the translations and the special conformal transformations in the construction, these spaces are called biconformal; this same symmetry gives biconformal spaces overlapping structures with double field theories, including manifest T duality. We establish that biconformal geometry is a form of double field theory, showing how general relativity with integrable local scale invariance arises from its field equations. While we discuss the relationship between biconformal geometries and the double field theories of T-dual string theories, our principal interest is the study of the gravity theory. We show that vanishing torsion and vanishing co-torsion solutions to the field equations overconstrain the system, implying a trivial biconformal space. Wih co-torsion unconstrained, we show that (1) the torsion-free solutions are foliated by copies of an n-dim Lie group, (2) torsion-free solutions generically describe locally scale-invariant general relativity with symmetric, divergence-free sources on either the co-tangent bundle of n-dim (p,q)-spacetime or the torus of double field theory, and (3) torsion-free solutions admit a subclass of spacetimes with n-dim non-abelian Lie symmetry. These latter cases include the possibility of a gravity-electroweak unification. It is notable that the field equations reduce all curvature components to dependence only on the solder form of an n-dim Lagrangian submanifold, despite the increased number of curvature components and doubled number of initial independent variables., 56 pages

We develop the properties of Weyl geometry, beginning with a review of the conformal properties of Riemannian spacetimes. Decomposition of the Riemann curvature into trace and traceless parts allows an easy proof that the Weyl curvature tensor is the conformally invariant part of the Riemann curvature, and shows the explicit change in the Ricci and Schouten tensors required to insure conformal invariance. We include a proof of the well-known condition for the existence of a conformal transformation to a Ricci-flat spacetime. We generalize this to a derivation of the condition for the existence of a conformal transformation to a spacetime satisfying the Einstein equation with matter sources. Then, enlarging the symmetry from Poincar\'e to Weyl, we develop the Cartan structure equations of Weyl geometry, the form of the curvature tensor and its relationship to the Riemann curvature of the corresponding Riemannian geometry. We present a simple theory of Weyl-covariant gravity based on a curvature-linear action, and show that it is conformally equivalent to general relativity. This theory is invariant under dilatations, but not the full conformal group., Comment: 27 pp Minor notational changes, additional example and conclusion added

According to the Coleman-Mandula theorem, any gauge theory of gravity combined with an internal symmetry based on a Lie group must take the form of a direct product in order to be consistent with basic assumptions of quantum field theory. However, we show that an alternative gauging of a simple group can lead dynamically to a spacetime with compact internal symmetry. The biconformal gauging of the conformal symmetry of n-dim Euclidean space doubles the dimension to give a symplectic manifold. Examining one of the Lagrangian submanifolds in the flat case, we find that in addition to the expected SO(n) connection and curvature, the solder form necessarily becomes Lorentzian. General coordinate invariance gives rise to an SO(n-1,1) connection on the spacetime. The principal fiber bundle character of the original SO(n) guarantees that the two symmetries enter as a direct product, in agreement with the Coleman-Mandula theorem., Comment: Added section to discuss full space as cotangent bundle

Of those gauge theories of gravity known to be equivalent to general relativity, only the biconformal gauging introduces new structures - the quotient of the conformal group of any pseudo-Euclidean space by its Weyl subgroup always has natural symplectic and metric structures. Using this metric and symplectic form, we show that there exist canonically conjugate, orthogonal, metric submanifolds if and only if the original gauged space is Euclidean or signature 0. In the Euclidean cases, the resultant configuration space must be Lorentzian. Therefore, in this context, time may be viewed as a derived property of general relativity., 21 pages (Reduced to clarify and focus on central argument; some calculations condensed; typos corrected)

A survey of topics of recent interest in Hamiltonian and Lagrangian dynamical systems, including accessible discussions of regularization of the central force problem; inequivalent Lagrangians and Hamiltonians; constants of central force motion; a general discussion of higher-order Lagrangians and Hamiltonians with examples from Bohmian quantum mechanics, the Korteweg-de Vries equation and the logistic equation; gauge theories of Newtonian mechanics; classical spin, Grassmann numbers, and pseudomechanics., Comment: Einstein Centennial Review Article, 48 pages

Biconformal supergravity models provide a new gauging of the superconformal group relevant to the Maldacena conjecture. Using the group quotient method to biconformally gauge SU(2,2|N), we generate a 16-dim superspace. We write the most general even- and odd-parity actions linear in the curvatures, the bosonic sector of which is known to descend to general relativity on a 4-dim manifold., 35 pages, adjusted group nomenclature, 1 reference and acknowledgements added

When the full connection of Weyl conformal gravity is varied instead of just the metric, the resulting vacuum field equations reduce to the vacuum Einstein equation, up to the choice of local units, if and only if the torsion vanishes. This result differs strongly from the usual fourth-order formulation of Weyl gravity., Final version, 11 pages. The role of torsion is clarified, two appendices - on structure equations and Bianchi identities and on the basis equation - have been added

Scale invariance provides a principled reason for the physical importance of Hilbert space, the Virasoro algebra, the string mode expansion, canonical commutators and Schrodinger evolution of states, independent of the assumptions of string theory and quantum theory. The usual properties of dimensionful fields imply an infinite, projective tower of conformal weights associated with the tangent space to scale-invariant spacetimes. Convergence and measurability on this tangent tower are guaranteed using a scale-invariant norm, restricted to conformally self-dual vectors. Maps on the resulting Hilbert space are correspondingly restricted to semi-definite conformal weight. We find the maximally- and minimally-commuting, complete Lie algebras of definite-weight operators. The projective symmetry of the tower gives these algebras central charges, giving the canonical commutator and quantum Virasoro algebras, respectively. Using a continuous, m-parameter representation for rank-m tower tensors, we show that the parallel transport equation for the momentum-vector of a particle is the Schrodinger equation, while the associated definite-weight operators obey canonical commutation relations. Generalizing to the set of integral curves of general timelike, self-dual vector-valued weight maps gives a lifting such that the action of the curves parallel transports arbitrary tower vectors. We prove that the full set of Schrodinger-lifted integral curves of a general self-dual map gives an immersion of its 2-dim parameter space into spacetime, inducing a Lorentzian metric on the parameter space. This immersion is shown to satisfy the full variational equations of open string.

The dimensional properties of fields in classical general relativity lead to a tangent tower structure which gives rise directly to quantum mechanical and quantum field theory structures without quantization. We derive all of the fundamental elements of quantum mechanics from the tangent tower structure, including fundamental commutation relations, a Hilbert space of pure and mixed states, measurable expectation values, Schroedinger time evolution, collapse of a state and the probability interpretation. The most central elements of string theory also follow, including an operator valued mode expansion like that in string theory as well as the Virasoro algebra with central charges., 8 pages, Latex, Honorable Mention 1997 GRG Essay

We make a thorough study of the regions near finite-order metric-singularity boundaries of static, spherically symmetric spacetimes. After distinguishing curvature singularities from other types of metric breakdown, we examine the eigenvalues of the energy tensor near the singularities for positivity and energy dominance, find the causal class of the t-translation (“static”) Killing field, and ascertain the presence or absence of timelike, null, and spacelike geodesic incompleteness for each spacetime. For a certain subclass of spacetimes, we also show the completeness of all timelike and spacelike curves despite the superficial failure of the metric.

We write a gravity theory with Yang-Mills type action using the biconformal gauging of the conformal group. We show that the resulting biconformal Yang-Mills gravity theories describe 4-dim, scale-invariant general relativity in the case of slowly changing fields. In addition, we systematically extend arbitrary 4-dim Yang-Mills theories to biconformal space, providing a new arena for studying flat space Yang-Mills theories. By applying the biconformal extension to a 4-dim pure Yang-Mills theory with conformal symmetry, we establish a 1-1, onto mapping between a set of gravitational gauge theories and 4-dim, flat space gauge theories., 27 pages; paper emphasis shifted to focus on gravity; references added

The biophysical origins of paramyotonia congenita and its exacerbation in cold temperatures were examined. Human skeletal muscle voltage-gated sodium channels were expressed in Xenopus oocytes and macroscopic currents were recorded from cell-attached patches. Wild-type (hNaV1.4) channels were compared to two mutant channel isoforms, T1313M and R1448C. The voltage dependence and temperature sensitivity of activation, fast-inactivation onset and recovery, and deactivation were studied. Although activation and the onset of fast-inactivation were temperature sensitive in all three isoforms, and although these properties in mutant channels differed from those in wild-type channels, they did not account for cold-exacerbation. Deactivation, however, was disproportionately slower in R1448C, but not in T1313M, than in hNaV1.4. These defects may, at least in part, account for the clinical symptoms of paramyotonia congenita and its exacerbation by cold, and provide a basis for studies into the therapeutic alleviation of these symptoms.

Biconformal spaces contain the essential elements of quantum mechanics, making the independent imposition of quantization unnecessary. Based on three postulates characterizing motion and measurement in biconformal geometry, we derive standard quantum mechanics, and show how the need for probability amplitudes arises from the use of a standard of measurement. Additionally, we show that a postulate for unique, classical motion yields Hamiltonian dynamics with no measurable size changes, while a postulate for probabilistic evolution leads to physical dilatations manifested as measurable phase changes. Our results lead to the Feynman path integral formulation, from which follows the Schroedinger equation. We discuss the Heisenberg uncertainty relation and fundamental canonical commutation relations., 34 pages; replaced discussion of multiparticle Hamiltonian dynamics

We derive both Lagrangian and Hamiltonian mechanics as gauge theories of Newtonian mechanics. Systematic development of the distinct symmetries of dynamics and measurement suggest that gauge theory may be motivated as a reconciliation of dynamics with measurement. Applying this principle to Newton's law with the simplest measurement theory leads to Lagrangian mechanics, while use of conformal measurement theory leads to Hamilton's equations., 44 pages, no figures, LaTeX

The quotient of the conformal group of Euclidean 4-space by its Weyl subgroup results in a geometry possessing many of the properties of relativistic phase space, including both a natural symplectic form and non-degenerate Killing metric. We show that the general solution posesses orthogonal Lagrangian submanifolds, with the induced metric and the spin connection on the submanifolds necessarily Lorentzian, despite the Euclidean starting pont. By examining the structure equations of the biconformal space in an orthonormal frame adapted to its phase space properties, we also find that two new tensor fields exist in this geometry, not present in Riemannian geometry. The first is a combination of the Weyl vector with the scale factor on the metric, and determines the timelike directions on the submanifolds. The second comes from the components of the spin connection, symmetric with respect to the new metric. Though this field comes from the spin connection it transforms homogeneously. Finally, we show that in the absence of conformal curvature or sources, the configuration space has geometric terms equivalent to a perfect fluid and a cosmological constant., Comment: 26 pages, no figures. Appreciable introductory material added. Results substantially strengthened and explained. New results concerning dark matter and dark energy candidates added to this version

Biconformal gauging of the conformal group has a scale-invariant volume form, permitting a single form of the action to be invariant in any dimension. We display several 2n-dim scale-invariant polynomial actions and a dual action. We solve the field equations for the most general action linear in the curvatures for a minimal torsion geometry. In any dimension n>2, the solution is foliated by equivalent n-dim Ricci-flat Riemannian spacetimes, and the full 2n-dim space is symplectic. Two fields defined entirely on the Riemannian submanifolds completely determine the solution: a metric, and a symmetric tensor., 35 pages, now includes comparisons with other theories; one added reference

A new 8-dim conformal gauging solves the auxiliary field problem and eliminates unphysical size change from Weyl's electromagnetic theory. We derive the Maurer-Cartan structure equations and find the zero curvature solutions for the conformal connection. By showing that every one-particle Hamiltonian generates the structure equations we establish a correspondence between phase space and the 8-dim base space, and between the action and the integral of the Weyl vector. Applying the correspondence to generic flat solutions yields the Lorentz force law, the form and gauge dependence of the electromagnetic vector potential and minimal coupling. The dynamics found for these flat solutions applies locally in generic spaces. We then provide necessary and sufficient curvature constraints for general curved 8-dimensional geometries to be in 1-1 correspondence with 4-dimensional Einstein-Maxwell spacetimes, based on a vector space isomorphism between the extra four dimensions and the Riemannian tangent space. Despite part of the Weyl vector serving as the electromagnetic vector potential, the entire class of geometries has vanishing dilation, thereby providing a consistent unified geometric theory of gravitation and electromagnetism. In concluding, we discuss the observability of the extra dimensions., 39 pages, plain TeX, no figures

General relativity may be formulated as a gauge theory more than one way using the quotient manifold approach. We contrast the structures that arise in four gravitational gauge theories, three of which give satisfactory gauge theoris of general relativity. Of particular interest is the quotient of the conformal group of a flat space by its Weyl subgroup, which always has natural symplectic and metric structures in addition to the requisite manifold. This quotient space admits canonically conjugate, orthogonal, metric submanifolds distinct from the original space if and only if the original flat space has signature n, -n or 0. In the Euclidean cases, the resultant configuration space must be Lorentzian. This gives a 1-1 mapping between Euclidean and Lorentzian submanifolds, with induced Euclidean gravity or general relativity, respectively.

General relativity can be constructed as a gauge theory using the quotient manifold strategy of [1, 2]. We consider a conformal gauging where the geometry is far richer than normal spacetime, including a symplectic form and the necessary emergence of Lorentzian signature. The resulting 2n-dim manifold constitutes a relativistic phase space, and general relativity is recovered when we demand that the momentum space is flat. However, the full geometry allows for curved phase space.

Among the solutions for anisotropic, time-dependent, maximally Gauss-Bonnet extended gravity, we find a class of curvature singularities for which the metric components remain finite. These new singularities therefore differ in type from the standard Kasner-like divergences expected for this class of theories. We study perturbative solutions near the singularity and show that there exist solutions with timelike paths that reach the singularity in finite proper time. Solving the equation of geodesic deviation in the same approximation, we show that the comoving coordinate system does not break down at the singularity. A brief classification of the corresponding singularity types in Robertson-Walker cosmologies is also provided.

When the full conformal algebra is gauged there arises a gauge field for inverse translations in addition to the usual translational gauge field. By considering every scale-invariant action constructible from the curvatures of the conformal group and the metric, we show that when the gauge field of the usual translations is identified as the vierbein, the gauge field of inverse translations may always be eliminated by its own field equation. After this elimination, every torsion-free, scale-invariant action reduces to a linear combination of the square of the conformal curvature tensor {ital C}{sub {alpha}{beta}{mu}{nu}} and the square of the Weyl field strength {ital C}{sub {alpha}{beta}}.

In an arbitrary number of dimensions, we find the full exact anisotropic, time-dependent, diagonal-metric solutions to maximally Gauss-Bonnet extended gravity theory. This class of theories, for which the lagrangian is an arbitrary linear combination of dimensionally extended Euler forms, is the most general gravitational theory in which the field equations contain no more than second derivatives of the metric. We show that the space-time exponentially approaches an asymptotic state of constant, anisotropic curvature and prove three theorems concerning two generic types of singularities. The first theorem gives conditions for the existence of Kasner-like curvature singularities. For these the metric diverges as t Pi where ∑ p i = 2 k max - 1 and k max is the highest power of the curvature in the lagrangian. Other critical point singularities can arise from the polynomial nature of the theory. The remaining theorems demonstrate that the generic solution is extendible at all of these other critical points and that the generic critical points occur at moments of external volume density of space-time. We give an explicit coordinate transformation which produces a smooth extension through the critical point. The space-time may therefore alternately expand and contract for many cycles before expanding forever or contracting to a singularity. Many particular cases are treated in detail including several power series solutions, the generalized Kasner solution to general relativity with or without cosmological constant, the perturbative solution for quadratic string gravity, and five-dimensional extended gravity.

We consider spacetime as having an a priori arbitrary, possibly fractional dimension p>0 and propose a new variational principle for actions defined on p-dimensional spaces. Demanding that the action be stationary with respect to variations in p leads to a constraint equation whose solution yields an explicit determination of the dimension at the classical level. We illustrate these concepts by analyzing a model which reduces to free scalar field theory when p is any positive integer.

Department of Physics and AstronomySwarthmore College, Swarthmore, Pennsylvania 19081AbstractA model for the interpretation of spacetime as a Weyl geometry is proposed, based on thehypothesis that a system moves on any given path with a probability which is inversely proportionalto the resulting change in length of the system. The results of physical measurements are calculatedas the product of Weyl-conjugate gauge dependent probabilities for the detection of conjugateobjects. Each probability, expressed as a Wiener integral, is the Green function for a diffusionequation. If the line integral of the Weyl field equals the action functional divided by +h , thisequation provides the stochastic equivalent of the Schrodinger equation, with expanding andcontracting states replacing the oscillating states of standard quantum theory. These dilatations arenot directly measurable within the theory, but the rates of expansion and contraction are. Weestablish the classical limit of the theory by proving that the most general form of the Weyl fieldwhich gives an extremum of the action is restricted to be of a special form. Particles obeying theequations of motion for these extremal Weyl fields experience no measurable dilatation. However,the Weyl field itself is classically detectable, and gives a correct description of the electromagneticinteraction of a point particle.Current address: Department of Physics, Utah State University, Logan, Utah 84322.

Motivated by the possibility of a finite theory of gravity provided by superstrings in ten space-time dimensions, we analyze the problem of space compactification in the context of string dynamics. Such an analysis is hampered by conceptual and technical problems, stemming from the existence of the quantum string's own graviton mode on the one hand, and from Witten's observation of anomalies in a not specially chosen curves space-time on the other hand. Still, in the context of a classical local field presentation of string theory a la Nambu and Hosotani, supplemented by gravitational and Kalb-Ramond interactions, we are able to find solutions with space compactification. It is the antisymmetric tensor zero modes that dictate this compactification towards three space-time dimensions for ordinary strings or towards four or five space-time dimensions for superstrings.

A study of 4-dimensional volume preserving covariant changes of the metric leads directly to string theory in 26 dimensions. In any conformally invariant theory of such metric distortions, a new principle of relativity is shown to imply that the maximum number of independent components of the distortion tensor is 26. Within the simplest such theory allowing metric compatible spaces as trivial solutions, we show that this maximum is realized by a solution equivalent to the 26-dimensional bosonic string.

The most general gravitational lagrangian which can be constructed from the curvature two-form, the vielbein one-form, and tensors invariant in the tangent space is a linear combination of dimensionally extended Euler characteristics. Several recent studies indicate that superstring lagrangians include such terms. In an arbitrary number of dimensions, with arbitrary torsion, we show that in the most general such extended theory the only static, spherically symmetric, massive solutions to the variational equations of motion contain gravitational singularities. The existence of an event horizon is proved for certain cases, and a bound on the location of singularities is found. A certain class of non-asymptotically flat solutions is found for which the metric and the torsion are not entirely determined by the field equations.

Normally, pure states of coherent light have equal uncertainties for pairs of conjugate variables. In recent years, however, it has become possible to produce and detect light in which fluctuations of one of the quadrature components are suppressed below the corresponding flutuations of a coherent state. Such radiation is said to be in a squeezed state. We explore the possibility that a strong gravitational field can produce a squeezed state of light. Such squeezing does in fact occur, and we derive an expression for the resulting uncertainties in a high frequency or long time limit. These results comprise a new, testable prediction of general relativity.

Low-energy limits of string theories suggest that gravity lagrangians should include quadratic and higheorder curvature terms, in the form of dimensionally continued Gauss-Bonnet densities. In an arbitrary number of dimensions, we consider the static, spherically symmetric solutions to the lowest order Gauss-Bonnet extended Einstein equations. We also find isotropic, homogeneous cosmological solutions with an ideal fluid source.