The spherically symmetric Standard Model with gravity

Spherical reduction of generic four-dimensional theories is revisited. Three different notions of “spherical symmetry” are defined. The following sectors are investigated: Einstein-Cartan theory, spinors, (non-)abelian gauge fields and scalar fields. In each sector a different formalism seems to be most convenient: the Cartan formulation of gravity works best in the purely gravitational sector, the Einstein formulation is convenient for the Yang-Mills sector and for reducing scalar fields, and the Newman-Penrose formalism seems to be the most transparent one in the fermionic sector. Combining them the spherically reduced Standard Model of particle physics together with the usually omitted gravity part can be presented as a two-dimensional (dilaton gravity) theory.     

The auxiliary field structure in chirally extended supergravity

The auxiliary fields for Einstein supergravity with axial gauge coupling are those of Maxwell-Einstein supergravity. The gauge algebra is an irreducible extension of the gauge algebra of Einstein supergravity, so that the complete system is a gauge theory with an extra local chiral invariance, rather than a matter coupling.     

Modified strings in terms of zweibein fields

Possible modifications of the relativistic string which preserves conformal invariance in the conformal gauge is investigated using zweibein fields. A fully reparametrization-invariant action yielding Liouville's equation is then constructed without the introduction of auxiliary fields. This action breaks the local two-dimensional Lorentz invariance and the corresponding extra degree of freedom reduces in the conformal gauge to a free field. For open strings the variation of the action implies that the Liouville field and this free field are connected by a Bäcklund transformation at the boundary. In certain cases it is shown that this extends to hold everywhere. If the local Lorentz invariance is restored, then the reparametrization algebra acquires the anomalous term necessary for the quantization in subcritical dimensions.     

A complete action for the spinning string

We present an action for the Neveu-Schwarz-Ramond model from which follow both the field equations and the gauge and supergauge constraints. This is done by coupling the free-field action to two-dimensional supergravity in a geometrically clear way. The constraints arise as the supergravity field equations, the supergravity fields playing the role of Lagrange multipliers. The action is invariant under local supersymmetry transformations and, as a consequence, the field equations and the constraints are consistent. The commutator structure of the local supersymmetry algebra is exhibited. It is also shown that there exists a special gauge in which the action, the field equations and the constraints take the free-field from of the usual formulation of the Neveu-Schwarz-Ramond model.     

The Hamiltonian of Einstein affine-metric formulation of General Relativity

It is shown that the Hamiltonian of the Einstein affine-metric (first-order) formulation of General Relativity (GR) leads to a constraint structure that allows the restoration of its unique gauge invariance, four-diffeomorphism, without the need of any field dependent redefinition of gauge parameters as in the case of the second-order formulation. In the second-order formulation of ADM gravity the need for such a redefinition is the result of the non-canonical change of variables (arXiv:0809.0097). For the first-order formulation, the necessity of such a redefinition “to correspond to diffeomorphism invariance” (reported by Ghalati, arXiv:0901.3344) is just an artifact of using the Henneaux–Teitelboim–Zanelli ansatz (Nucl. Phys. B 332:169, 1990), which is sensitive to the choice of linear combination of tertiary constraints. This ansatz cannot be used as an algorithm for finding a gauge invariance, which is a unique property of a physical system, and it should not be affected by different choices of linear combinations of non-primary first class constraints. The algorithm of Castellani (Ann. Phys. 143:357, 1982) is free from such a deficiency and it leads directly to four-diffeomorphism invariance for first, as well as for second-order Hamiltonian formulations of GR. The distinct role of primary first class constraints, the effect of considering different linear combinations of constraints, the canonical transformations of phase-space variables, and their interplay are discussed in some detail for Hamiltonians of the second- and first-order formulations of metric GR. The first-order formulation of Einstein–Cartan theory, which is the classical background of Loop Quantum Gravity, is also discussed.     

Confronting Finsler space–time with experiment

Within all approaches to quantum gravity small violations of the Einstein Equivalence Principle are expected. This includes violations of Lorentz invariance. While usually violations of Lorentz invariance are introduced through the coupling to additional tensor fields, here a Finslerian approach is employed where violations of Lorentz invariance are incorporated as an integral part of the space–time metrics. Within such a Finslerian framework a modified dispersion relation is derived which is confronted with current high precision experiments. As a result, Finsler type deviations from the Minkowskian metric are excluded with an accuracy of 10⁻¹⁶.     

Gauge fields,space-time geometry and gravity

A general framework for the gauge theory of the affine group and its various subgroups in terms of connections on the bundle of affine frames and its subbundles is given, with emphasis on the correct gauging of groups including space-time translations. For consistency of interpretation, the appropriate objects to be identified with gravitational vierbeins in such theories are not the translational gauge fields themselves, but their pull backs,via appropriate bundle homomorphisms, to the bundle of frames. This automatically solves the problems usually encountered in constructing a gauge theory of the conventional sort for groups containing translations. We give a consistent formulation of the Poincare gauge theory and also of the theory based on translational gauge invariance which, in the absence of matter fields with intrinsic spin, gives a local Lorentz invariant theory equivalent to Einstein gravity.     

Lorentz invariant exact solution of the anomalous chiral Schwinger model

We present an exact solution of the anomalous chiral Schwinger model using Fermionic variables. We implement infrared regularization by considering the model on a spatial circleS ¹. Quantum effects modify the gauge constraints through the appearance of Schwinger terms in the gauge algebra. We perform a careful analysis of the resulting second class gauge constraints by implementing Dirac's method at the quantum level and obtain the spectrum of the theory. We get a consistent unitary Lorentz invariant theory for particular values of the counterterms. We find that when we regulate the fermionic sector of the model without reference to the gauge fields Lorentz invariance requires that we add both Lorentz variant and gauge variant counterterms.     

Curved Space-Times by Crystallization of Liquid Fiber Bundles

Motivated by the search for a Hamiltonian formulation of Einstein equations of gravity which depends in a minimal way on choices of coordinates, nor on a choice of gauge, we develop a multisymplectic formulation on the total space of the principal bundle of orthonormal frames on the 4-dimensional space-time. This leads quite naturally to a new theory which takes place on 10-dimensional manifolds. The fields are pairs of \(((\alpha ,\omega ),\varpi )\), where \((\alpha ,\omega )\) is a 1-form with coefficients in the Lie algebra of the Poincaré group and \(\varpi \) is an 8-form with coefficients in the dual of this Lie algebra. The dynamical equations derive from a simple variational principle and imply that the 10-dimensional manifold looks locally like the total space of a fiber bundle over a 4-dimensional base manifold. Moreover this base manifold inherits a metric and a connection which are solutions of a system of Einstein–Cartan equations.     

Gauge group of gravity,spinors, and anomalies

A discussion is given of the gravitational anomalies that arise from coupling Weyl spinors to gravity, treating the metric, the soldering form, and the connection as independent dynamical variables. This system is strictly analogous to Weyl spinors coupled to Yang-Mills fields and a nonlinear sigma model. The larger gauge group of this formulation is seen to lie at the root of the equivalence between Einstein and Lorentz anomalies.On leave of absence from SISSA, Trieste, Italy.     

Modified Faddeev–Jackiw quantization of massive non-Abelian Yang–Mills fields and Lagrange multiplier fields

Massive Yang–Mills fields and Lagrange multiplier fields are quantized by the modified Faddeev–Jackiw quantization method, and the method's comparisons with Dirac method and the usual Faddeev–Jackiw method are also given. We show that this method not only is equivalent to Dirac method, but also remains all the virtues of the usual Faddeev–Jackiw method. Moreover, the modified Faddeev–Jackiw quantization method is simpler than the usual one when obtaining new secondary constraints. Therefore, the modified Faddeev–Jackiw method is more economical and effective than Dirac method and the usual Faddeev–Jackiw method. Meanwhile, we find the new meanings of the Lagrange multipliers, and discover that the Lagrange multipliers and the zeroth components of gauge field are just a pair of canonical field variables except a constant factor in this system.     

The phase and critical point of quantum Einstein–Cartan gravity

By introducing diffeomorphism and local Lorentz gauge invariant holonomy fields, we study in the recent article [S.-S. Xue, Phys. Rev. D 82 (2010) 064039] the quantum Einstein–Cartan gravity in the framework of Regge calculus. On the basis of strong coupling expansion, mean-field approximation and dynamical equations satisfied by holonomy fields, we present in this Letter calculations and discussions to show the phase structure of the quantum Einstein–Cartan gravity, (i) the order phase: long-range condensations of holonomy fields in strong gauge couplings; (ii) the disorder phase: short-range fluctuations of holonomy fields in weak gauge couplings. According to the competition of the activation energy of holonomy fields and their entropy, we give a simple estimate of the possible ultra-violet critical point and correlation length for the second-order phase transition from the order phase to disorder one. At this critical point, we discuss whether the continuum field theory of quantum Einstein–Cartan gravity can be possibly approached when the macroscopic correlation length of holonomy field condensations is much larger than the Planck length.     

Realisation of a Lorentz algebra in Lorentz violating theory

A Lorentz non-invariant higher derivative effective action in flat spacetime, characterised by a constant vector, can be made invariant under infinitesimal Lorentz transformations by restricting the allowed field configurations. These restricted fields are defined as functions of the background vector in such a way that background dependence of the dynamics of the physical system is no longer manifest. We show here that they also provide a field basis for the realisation of a Lorentz algebra and allow the construction of a Poincaré invariant symplectic two-form on the covariant phase space of the theory.     

Cumulative author index B751–B754

The paper is devoted to the study of BRST charge in perturbed two-dimensional conformal field theory. The main goal is to write the operator equation expressing the conservation law of BRST charge in perturbed theory in terms of purely algebraic operations on the corresponding operator algebra, which are defined via the OPE. The corresponding equations are constructed and their symmetries are studied up to the second order in formal coupling constant. It appears that the obtained equations can be interpreted as generalized Maurer–Cartan ones. We study two concrete examples in detail: the bosonic nonlinear sigma model and perturbed first order theory. In particular, we show that the Einstein equations, which are the conformal invariance conditions for both these perturbed theories, expanded up to the second order, can be rewritten in such generalized Maurer–Cartan form.     

Nonlinear Connections and Nearly Autoparallel Maps in General Relativity

We apply the method of moving anholonomic frames with associated nonlinear connections to the (pseudo) Riemannian space geometry and examine the conditions when locally anisotropic structures (Finsler like and more general ones) could be modeled in the general relativity theory and/or Einstein–Cartan–Weyl extensions [1]. New classes of solutions of the Einstein equations with generic local anisotropy are constructed. We formulate the theory of nearly autoparallel (na) maps generalizing the conformal transforms and formulate the Einstein gravity theory on na–backgrounds provided with a set of na–map invariant conditions and local conservation laws. There are illustrated some examples when vacuum Einstein fields are generated by Finsler like metrics and chains of na–maps.     

Induced classical gravity

We review the induced-gravity approach according to which the Einstein gravity is a long-wavelength effect induced by underlying fundamental quantum fields due to the dynamical-scale symmetry breaking. It is shown that no ambiguities arise in the definition of the induced Newton and cosmological constants if one works with the path integral for fundamental fields in the low-scale region. The main accent is on a specification of the path integral which enables us to utilize the unitarity condition and thereby avoid ambiguities. Induced Einstein equations appear from the symmetry condition that the path integral of fundamental fields for a slowly varying metric is invariant under the local GL(4, R)-transformations of a tetrad, which contain the local Euclidean Lorentz, O(4)-rotations as a subgroup. The relationship to induced quantum gravity is briefly outlined.     

A New Formulation of the Equations of Dynamics

In a recent paper [1] the author introduced a new method for handling differentials and extrema in the presence of constraints, in which the constraints were represented by a projection matrix. For extrema this method can be used instead of Lagrange multipliers. Here the same idea is applied to the equations of motion in mechanics, leading to a new formulation in both Cartesian and generalized coordinates, equivalent to Lagrange's equations and applicable with both holonomic and nonholonomic constraints. The present formulation provides an alternative to a recent approach of Kalaba, Udwadia, and Xu [2,3], which too is based on linear algebra, and to Appell's classical method for the nonholonomic case     

Quantum scalar field in quantum gravity: the propagator and Lorentz invariance in the spherically symmetric case

We recently studied gravity coupled to a scalar field in spherical symmetry using loop quantum gravity techniques. Since there are local degrees of freedom one faces the “problem of dynamics”. We attack it using the “uniform discretization technique”. We find the quantum state that minimizes the value of the master constraint for the case of weak fields and curvatures. The state has the form of a direct product of Gaussians for the gravitational variables times a modified Fock state for the scalar field. In this paper we do three things. First, we verify that the previous state also yields a small value of the master constraint when one polymerizes the scalar field in addition to the gravitational variables. We then study the propagators for the polymerized scalar field in flat space-time using the previously considered ground state in the low energy limit. We discuss the issue of the Lorentz invariance of the whole approach. We note that if one uses real clocks to describe the system, Lorentz invariance violations are small. We discuss the implications of these results in the light of Hořava’s Gravity at the Lifshitz point and of the argument about potential large Lorentz violations in interacting field theories of Collins et al.     

Canonical structure of tetrad bimetric gravity

We perform the complete canonical analysis of the tetrad formulation of bimetric gravity and confirm that it is ghost-free describing the seven degrees of freedom of a massless and a massive gravitons. In particular, we find explicit expressions for secondary constraints, one of which is responsible for removing the ghost, whereas the other ensures the equivalence with the metric formulation. Both of them have a remarkably simple form and, being combined with conditions on Lagrange multipliers, can be written in a covariant way.