Scalar-tensor theories of gravitation: Foundations and prospects

A generalization of Einstein's theory is discussed in which the gravitation is described by a tensor and a scalar field. The theory is more consistent with Mach's principle and less reliant on absolute properties of space. The modification involves a violation of the “strong principle of equivalence” on which Einstein's theory is based. In the original version of this new theory, the “constant” of gravitationG is varying and particle masses are fixed. Later on another version of the theory was given in whichG is truly a constant and the particle masses vary. The two versions are related by a conformal transformation. The physical and mathematical foundations of this theory have been discussed and the field equations have been derived. The astrophysical and cosmological consequences of the theory have been elaborately reviewed.     

Euclidean D-branes and higher-dimensional gauge theory

We consider euclidean D-branes wrapping around manifolds of exceptional holonomy in dimensions seven and eight. The resulting theory on the D-brane—that is, the dimensional reduction of 10-dimensional supersymmetric Yang-Mills theory—is a cohomological field theory which describes the topology of the moduli space of instantons. The 7-dimensional theory is an N_T = 2 (or balanced) cohomological theory given by an action potential of Chern-Simons type. As a by-product of this method, we construct a related cohomological field theory which describes the monopole moduli space on a 7-manifold of G₂ holonomy.     

Divergence-free WKB theory

We present a divergence-free WKB theory, which is a new semiclassical theory modified by nonperturbative quantum corrections. Conventionally, the WKB theory is constructed upon a trajectory that obeys the bare classical dynamics expressed by a quadratic equation in momentum space. Contrary to this, the divergence-free WKB theory is based on a higher-order algebraic equation in momentum space, which represents a dressed classical dynamics. More precisely, this higher-order algebraic equation is obtained by including quantum corrections to the quadratic equation, which is the bare classical limit. An additional solution of the higher-order algebraic equation enables us to construct a uniformly converging perturbative expansion of the wavefunction. Namely, our theory removes the notorious divergence of wavefunction at a turning point from the WKB theory. Moreover, our theory is able to produce wavefunctions and eigenenergies more accurate than those given by the traditional WKB method. In addition, the divergence-free WKB theory that is based on the cubic equation allows us to construct a uniformly valid wavefunction for the nonlinear Schrödinger equation (NLSE). A recent short letter [T. Hyouguchi, S. Adachi, M. Ueda, Phys. Rev. Lett. 88 (2002) 170404] is the opening of the divergence-free WKB theory. This paper presents full formalism of this theory and its several applications concerning wavefunction and eigenenergy to show that our theory is a natural extension of the traditional WKB theory that incorporates nonperturbative quantum corrections.     

The Haldane-Rezayi quantum Hall state and conformal field theory

We propose field theories for the bulk and edge of a quantum Hall state in the universality class of the Haldane-Rezayi wavefunction. The bulk theory is associated with the c = −2 conformal field theory. The topological properties of the state, such as the quasiparticle braiding statistics and ground state degeneracy on a torus, may be deduced from this conformal field theory. The 10-fold degeneracy on a torus is explained by the existence of a logarithmic operator in the c = −2 theory; this operator corresponds to a novel bulk excitation in the quantum Hall state. We argue that the edge theory is the c = 1 chiral Dirac fermion, which is related in a simple way to the c = −2 theory of the bulk. This theory is reformulated as a truncated version of a doublet of Dirac fermions in which the SU(2) symmetry - which corresponds to the spin-rotational symmetry of the quantum Hall system - is manifest and non-local. We make predictions for the current-voltage characteristics for transport through point contacts.     

On the relation between Brueckner theory and Jastrow theory of nuclear matter

The two- and three-hole-line contributions to the ground state energy as calculated from Brueckner theory are derived from a cluster expansion followed by variation of the trial function. The implications of that derivation both for Brueckner theory and for Jastrow theory are worked out in detail. It is argued that the Jastrow theory is able to give simpler methods to calculate the ground state energy which may be of the same accuracy as current Brueckner calculations. It is shown that the single-particle potential of Brueckner theory is intimately related to a subsidiary condition used in the variation of the trial function. The main steps which have to be taken in a derivation of the general hole-line expansion from Jastrow theory are indicated. It is shown that the hole-line expansion is not a cluster expansion in the sense of Jastrow theory, and an interpretation is given of the “self-consistent choice” of the single-particle potential advocated in Brueckner theory.     

Infinite-component fields as a basis for a finite quantum field theory

A quantum field theory based on infinite-component fields is developed in which the spectrum of particles for all spins is composed of infinite sums of finite, non-unitary representations of the Lorentz group. This leads to a field theory free of causality problems. The problem of gauging away all unphysical modes in the infinite-component field theory is achieved by using infinite-parameter gauge fields which remove all unphysical modes, independently of the number of space-time dimensions. A model of an infinite-component quantum field theory is formulated, using perturbation theory, in which there are no ultraviolet divergences and the S-matrix is causal and unitary.     

Lagrangian formulation of a geometrical scalar-tensor theory of gravitation

The author's geometrical theory of the scalar-tensor gravitational field is extended by formulating it in terms of a Lagrangian. An exact solution of the coupled nonlinear field equations for a static point mass is also presented. This theory which is conformally equivalent to the empty spaceEinstein equations predicts the same results for experiments as the usual theory of Brans and Dicke which has a non-zero energy momentum tensor.     

An analytical theory for diffusion of fluids in crystalline nanoporous materials

An analytical theory for diffusion of fluids in zeolites and other nanoporous materials has been developed. The theory incorporates molecular level information about the nanoporous material, which is obtainable from an energy minimization and does not require molecular dynamics computer simulations. The theory is statistical mechanical in nature and assumes a lattice composed of adsorption sites. The theory yields a self-diffusion coefficient, which is a function of (i) temperature, (ii) adsorbate density, (iii) adsorbate size, (iv) adsorbate-adsorbate energetic interaction and (v) adsorbate-pore energetic interaction. The theory is generalized and is applicable to nanoporous materials with three-dimensional porous networks (e.g. faujusite) and one-dimensional porous networks (e.g. A1P0₄-5).

The theory is self-contained and incorporates no fitting parameters. The theory does not require computational effort beyond a few seconds on a standard personal computer.     

Construction of a covariant spinor field theory

A covariant theory is constructed of a spinor field in a space which is represented by the local topological product of a space X_n and a space of values of a geometrical object η. The covariant nonlinear spinor field theory constructed preserves the principles of the theory of the unified field and is compatible with the theory of gauge fields.     

What is Quantum Mechanics? A Minimal Formulation

This paper presents a minimal formulation of nonrelativistic quantum mechanics, by which is meant a formulation which describes the theory in a succinct, self-contained, clear, unambiguous and of course correct manner. The bulk of the presentation is the so-called “microscopic theory”, applicable to any closed system S of arbitrary size N, using concepts referring to S alone, without resort to external apparatus or external agents. An example of a similar minimal microscopic theory is the standard formulation of classical mechanics, which serves as the template for a minimal quantum theory. The only substantive assumption required is the replacement of the classical Euclidean phase space by Hilbert space in the quantum case, with the attendant all-important phenomenon of quantum incompatibility. Two fundamental theorems of Hilbert space, the Kochen–Specker–Bell theorem and Gleason’s theorem, then lead inevitably to the well-known Born probability rule. For both classical and quantum mechanics, questions of physical implementation and experimental verification of the predictions of the theories are the domain of the macroscopic theory, which is argued to be a special case or application of the more general microscopic theory.     

Physical foundations of the new theory of gravitation

It is shown that in generalizing simultaneously the special theory of relativity and the Newtonian theory of gravitation into a curved-space theory of gravity one needs to satisfy second-order correspondence conditions as well as the usual first-order ones. It is further shown that the imposition of these second-order conditions leads to a locally Lorentz invariant spin-2 field theory of gravity which is in agreement with all the known facts concerning gravitation. The second-order correspondence requirements endow the curved-space theory of gravitation with experimentally and theoretically novel features which will be discussed in some detail.     

Unified Field Theory From Enlarged Transformation Group. The Covariant Derivative for Conservative Coordinate Transformations and Local Frame Transformations

Pandres has developed a theory in which the geometrical structure of a real four-dimensional space-time is expressed by a real orthonormal tetrad, and the group of diffeomorphisms is replaced by a larger group called the conservation group. This paper extends the geometrical foundation for Pandres’ theory by developing an appropriate covariant derivative which is covariant under all local Lorentz (frame) transformations, including complex Lorentz transformations, as well as conservative transformations. After defining this extended covariant derivative, an appropriate Lagrangian and its resulting field equations are derived. As in Pandres’ theory, these field equations result in a stress-energy tensor that has terms which may automatically represent the electroweak field. Finally, the theory is extended to include 2-spinors and 4-spinors.     

Gravitation and source theory

Schwinger's source theory is applied to the problem of gravitation and its quantization. It is shown that within the framework of a flat-space the source theory implementation leads to a violation of probability. To avoid the difficulty one must introduce a curved space-time hence the source concept may be said to necessitate the transition to a curved-space theory of gravitation. It is further shown that the curved-space theory of gravitation implied by the source theory is not equivalent to the conventional Einstein theory. The source concept leads to a different theory where the gravitational field has a stress-energy tensor t_μ^ν, which contributes to geometric curvatures.     

Renormalization Proof for Spontaneously Broken Yang–Mills Theory with Flow Equations

In this paper we present a renormalizability proof for spontaneously broken SU(2) gauge theory. It is based on Flow Equations, i.e. on the Wilson renormalization group adapted to perturbation theory. The power counting part of the proof, which is conceptually and technically simple, follows the same lines as that for any other renormalizable theory. The main difficulty stems from the fact that the regularization violates gauge invariance. We prove that there exists a class of renormalization conditions such that the renormalized Green functions satisfy the Slavnov-Taylor identities of SU(2) Yang-Mills theory on which the gauge invariance of the renormalized theory is based.     

Regularization in quantum field theory from the causal point of view

The causal approach to perturbative quantum field theory is presented in detail, which goes back to a seminal work by Henri Epstein and Vladimir Jurko Glaser in 1973 [12]. Causal perturbation theory is a mathematically rigorous approach to renormalization theory, which makes it possible to put the theoretical setup of perturbative quantum field theory on a sound mathematical basis. Epstein and Glaser solved this problem for a special class of distributions, the time-ordered products, that fulfill a causality condition, which itself is a basic requirement in axiomatic quantum field theory. In their original work, Epstein and Glaser studied only theories involving scalar particles. In this review, the extension of the method to theories with higher spin, including gravity, is presented. Furthermore, specific examples are presented in order to highlight the technical differences between the causal method and other regularization methods, like, e.g. dimensional regularization.     

Understanding fields using strings: A review

In addition to being a prime candidate for a fundamental unified theory of all interactions in nature, string theory provides a natural setting to understand gauge field theories. This is linked to the concept of ‘D-branes’: extended, solitonic excitations of string theory which can be studied using techniques of string theory and which support gauge fields localized along their world-volumes. It follows that the techniques of string theory can be very useful even for those particle physicists who are not specifically interested in unification and/or quantum gravity. In this talk I attempt to review how strings help us to understand fields. The discussion is restricted to 3+1 spacetime dimensions.     

Quantization in the large

A model theory is constructed that exhibits quantization on a cosmic scale. A holistic rationale for the theory is discussed. The theory incorporates a fundamental length, of cosmic size, and preserves the weak, geometrical equivalence principle. The momentum operator is an integral, nonlocal, naturally contravariant operator, in contrast to the usual quantum case. In the limit of high quantum numbers the theory reduces to classical physics, giving rise to a world which is quantized both on the microscopic and cosmic scale, each of which passes over to the usual macroscopic, continuous, classical, world in the highn limit. The theory is applied to two experimental situations, absorption lines in high-z quasars and elliptical rings around normal galaxies, with suggestive but not definitive results.     

Topological aspect of vortex lines in two-dimensional Gross—Pitaevskii theory

Using the -mapping topological theory, we study the topological structure of vortex lines in a two-dimensional generalized Gross-Pitaevskii theory in (3+1)-dimensional space-time. We obtain the reduced dynamic equation in the framework of the two-dimensional Gross-Pitaevskii theory, from which a conserved dynamic quantity is derived on the stable vortex lines. Such equations can also be used to discuss Bose-Einstein condensates in heterogeneous and highly nonlinear systems. We obtain an exact dynamic equation with a topological term, which is ignored in traditional hydrodynamic equations. The explicit expression of vorticity as a function of the order parameter is derived, where the δ function indicates that the vortices can only be generated from the zero points of Φ and are quantized in terms of the Hopf indices and Brouwer degrees. The -mapping topological current theory also provides a reasonable way to study the bifurcation theory of vortex lines in the two-dimensional Gross-Pitaevskii theory.     

Einstein–Cartan gravity,Asymptotic Safety,and the running Immirzi parameter

In this paper we analyze the functional renormalization group flow of quantum gravity on the Einstein–Cartan theory space. The latter consists of all action functionals depending on the spin connection and the vielbein field (co-frame) which are invariant under both spacetime diffeomorphisms and local frame rotations. In the first part of the paper we develop a general methodology and corresponding calculational tools which can be used to analyze the flow equation for the pertinent effective average action for any truncation of this theory space. In the second part we apply it to a specific three-dimensional truncated theory space which is parametrized by Newton’s constant, the cosmological constant, and the Immirzi parameter. A comprehensive analysis of their scale dependences is performed, and the possibility of defining an asymptotically safe theory on this hitherto unexplored theory space is investigated. In principle Asymptotic Safety of metric gravity (at least at the level of the effective average action) is neither necessary nor sufficient for Asymptotic Safety on the Einstein–Cartan theory space which might accommodate different “universality classes” of microscopic quantum gravity theories. Nevertheless, we do find evidence for the existence of at least one non-Gaussian renormalization group fixed point which seems suitable for the Asymptotic Safety construction in a setting where the spin connection and the vielbein are the fundamental field variables.