In this paper, we present a non-geometrodynamic quantum Yang-Mills theory of gravity based on the homogeneous Lorentz group within the general framework of the Poincare gauge theories. The obstacles of this treatment are that first, on the one hand, the gauge group that is available for this purpose is non-compact. On the other hand, Yang-Mills theories with non-compact groups are rarely healthy, and only a few instances exist in the literature. Second, it is not clear how the direct observations of space-time waves can be explained when space-time has no dynamics. We show that the theory is unitary and is renormalizable to the one-loop perturbation. Although in our proposal, gravity is not associated with any elementary particle analogous to the graviton, classical helicity-two space-time waves are explained. Five essential exact solutions to the field equations of our proposal are presented as well. We also discuss a few experimental tests that can falsify the presented Yang-Mills theory.
We study the dynamics of gauge theory and general relativity using fields of local observers, thus maintaining local Lorentz symmetry despite a space/time splitting of fields. We start with Yang--Mills theory, where observer fields are defined as normalized future-timelike vector fields. We then define observers without a fixed geometry, and find these play two related roles in general relativity: splitting fields into spatial and temporal parts, and "breaking" gauge symmetry, effectively reducing the spacetime SO(n,1) connection to an observer-dependent spatial SO(n) connection. In both gauge…
A new tetrad introduced within the framework of geometrodynamics for non-null electromagnetic fields allows for the geometrical analysis of the Lorentz force equation and its solutions in curved spacetimes. When expressed in terms of this new tetrad, the electromagnetic field displays explicitly maximum simplification, and the degrees of freedom are manifestly revealed. In our manuscript we are deducing the Lorentz force equation on purely Riemannian geometrical grounds. The equation arises on the basis of Frenet-Serret analysis through the use of our new tetrads and gauge invariance arguments…
In this thesis the Bohm-de Broglie interpretation of quantum mechanics is applied to canonical quantum gravity. It is shown that, irrespective of any regularization or choice of factor ordering of the Wheeler-DeWitt equation, the unique relevant quantum effect which does not break spacetime is the change of its signature from lorentzian to euclidean. The other quantum effects are either trivial or break the four-geometry of spacetime. A Bohm-de Broglie picture of quantum geometrodynamics is constructed, which allows the investigation of these latter structures. For instance, it is shown that a…
In four dimensions complexified General Relativity (GR) can be non-trivially deformed: There exists an (infinite-parameter) set of modifications all having the same count of degrees of freedom. It is trivial to impose reality conditions that give versions of the deformed theories corresponding to Riemannian and split metric signatures. We revisit the Lorentzian signature case. To make the problem tractable, we restrict our attention to a four-parameter set of deformations that are natural extensions of Ashtekar's Hamiltonian formalism for GR. The Hamiltonian of the later is a linear combinatio…
We consider quantum geometrodynamics and parametrized quantum field theories in the framework of the Bohm-de Broglie interpretation. In the first case, and following the lines of our previous work [1], where a hamiltonian formalism for the bohmian trajectories was constructed, we show the consistency of the theory for any quantum potential, completing the scenarios for canonical quantum cosmology presented there. In the latter case, we prove the consistency of scalar field theory in Minkowski spacetime for any quantum potential, and we show, using this alternative hamiltonian method, a concret…
A new set of tetrads is introduced within the framework of SU(2) X U(1) Yang-Mills field theories in four dimensional Lorentz curved spacetimes. Each one of these tetrads diagonalizes separately and explicitly each term of the Yang-Mills stress-energy tensor. Therefore, three pairs of planes also known as blades, can be defined, and make up the underlying geometrical structure, at each point. These tetrad vectors are gauge dependent on one hand, and also in their definition, there is an additional inherent freedom in the choice of two vector fields. In order to get rid of the gauge dependence,…
In the Ashtekar-Barbero formulation of canonical general relativity based on an SU(2) connection, Lorentz covariance is a subtle issue which has been the focus of some debate. Here we present a Lorentz covariant formulation generalising the notion of a foliation of spacetime to a field of local observers which specify a time direction only locally. This field spontaneously breaks the local SO(3,1) symmetry down to a subgroup SO(3); we show that the apparent symmetry breaking to SO(3) is not in conflict with Lorentz covariance. We give a geometric picture of our construction as Cartan geometrod…
Recently, a model for an emergent gravity based on $SO(5)$ Yang-Mills action in Euclidian 4-dimensional spacetime was proposed. In this work we provide some 1 and 2-loops computations and show that the model can accommodate suitable predicting values for the Newtonian constant. Moreover, it is shown that the typical scale of the expected transition between the quantum and the geometrodynamical theory is consistent with Planck scale. We also provide a discussion on the cosmological constant problem.
In Ashtekar's Hamiltonian formulation of general relativity, and in loop quantum gravity, Lorentz covariance is a subtle issue that has been strongly debated. Maintaining manifest Lorentz covariance seems to require introducing either complex-valued fields, presenting a significant obstacle to quantization, or additional (usually second class) constraints whose solution renders the resulting phase space variables harder to interpret in a spacetime picture. After reviewing the sources of difficulty, we present a Lorentz covariant, real formulation in which second class constraints never arise. …
From pure Yang-Mills action for the $SL(5,\mathbb{R})$ group in four Euclidean dimensions we obtain a gravity theory in the first order formalism. Besides the Einstein-Hilbert term, the effective gravity has a cosmological constant term, a curvature squared term, a torsion squared term and a matter sector. To obtain such geometrodynamical theory, asymptotic freedom and the Gribov parameter (soft BRST symmetry breaking) are crucial. Particularly, Newton and cosmological constant are related to these parameters and they also run as functions of the energy scale. One-loop computations are perform…
We construct a spin foam model of Yang-Mills theory coupled to gravity by using a discretized path integral of the BF theory with polynomial interactions and the Barret-Crane ansatz. In the Euclidian gravity case we obtain a vertex amplitude which is determined by a vertex operator acting on a simple spin network function. The Euclidian gravity results can be straightforwardly extended to the Lorentzian case, so that we propose a Lorentzian spin foam model of Yang-Mills theory coupled to gravity.
General relativity with fermions has two independent symmetries: general coordinate invariance and local Lorentz invariance. General coordinate invariance is implemented by the Levi-Civita connection and by Cartan's tetrads both of which have as their action the Einstein-Hilbert action. It is suggested here that local Lorentz invariance is implemented not by a combination of the Levi-Civita connection and Cartan's tetrads known as the spin connection, but by independent Lorentz bosons L_i that gauge the Lorentz group, that couple to fermions like Yang-Mills fields, and that have their own Yang…
The generalized Cauchy-Riemann equations (GCRE) in biquaternion algebra appear to be Lorentz-invariant. The Laplace equation is in this case replaced by a nonlinear (complexified) eikonal equation. GCRE contain the 2-spinor and the gauge structures, and their integrability conditions take the form of free-source Maxwell and Yang-Mills equations. For the value of electric charge from GCRE only the quantization rule follows, as well as the treatment of Coulomb law as a stereographic map. The equivalent geometrodynamics in a Weyl-Cartan affine space and the conjecture of a complex-quaternion stru…
Different black hole solutions of the coupled Einstein-Yang-Mills equations have been well known for a long time. They have attracted much attention from mathematicians and physicists since their discovery. In this work, we analyze black holes associated with the gauge Lorentz group. In particular, we study solutions which identify the gauge connection with the spin connection. This ansatz allows one to find exact solutions to the complete system of equations. By using this procedure, we show the equivalence between the Yang-Mills-Lorentz model in curved space-time and a particular set of exte…
The Yang-Mills theory associated with the restricted Lorentz group is revisited as a candidate for a theory of gravity. This is a natural idea because the principle of equivalence of gravitation and inertia suggests to introduce locally inertial coordinate systems with the gauge freedom of Lorentz transformations. Compared to previous implementations of the idea, we use a generalized expression for the vector potential of the gauge theory in terms of the metric, which involves a coupling constant. One can verify that, in the limit of small coupling constant, all the classical predictions of ge…
In the framework of Einstein-Yang-Mills theories, we study the gauge Lorentz group and establish a particular correspondence between this case and a certain class of theories with torsion within Riemann-Cartan space-times. This relation is specially useful in order to simplify the problem of finding exact solutions to the Einstein-Yang-Mills equations. The applicability of the method is divided into two approaches: one associated with the Lorentz group SO(1,n-1) of the space-time rotations and another one with its subgroup SO(n-2). Solutions for both cases are presented by the explicit use of …
We present arguments for the existence of a new type of solutions of the Euclidean Einstein-Yang-Mills-dilaton theory in $d=4$ dimensions. Possesing nonvanishing nonabelian charges, these nonselfdual configurations have no counterparts on the Lorentzian section. They provide, however, new saddle points in the Euclidean path integral.
For the Yang-Mills-type gauge-field theory with Lorentz symmetry group, we propose and verify an explicit expression for the conserved currents in terms of the energy-momentum tensor. A crucial ingredient is the assumption that the gauge symmetry arises from the decomposition of a metric in terms of tetrad variables. The currents exist under the weak condition that the energy-momentum tensor and the Ricci tensor commute. We show how the conserved currents can be used to obtain a composite theory of gravity and discuss the static isotropic field around a point mass at rest.
We develop the complete composite theory of gravity, in which the gauge vector fields of the Yang-Mills theory with Lorentz symmetry group are expressed in terms of the tetrad variables obtained from the decomposition of a metric. A key element of a compelling formulation of composite gravity are refined coordinate conditions that offer a natural coupling of the gravitational field to matter and ensure the closest relationship to general relativity. The composite theory of gravity is presented from three different perspectives highlighting its intuitive interpretation, its relationship to gene…
By regarding gravity as the convolution of left and right Yang-Mills theories together with a spectator scalar field in the bi-adjoint representation, we derive in linearised approximation the gravitational symmetries of general covariance, p-form gauge invariance, local Lorentz invariance and local supersymmetry from the flat space Yang-Mills symmetries of local gauge invariance and global super-Poincaré. As a concrete example we focus on the new-minimal (12+12) off-shell version of simple four-dimensional supergravity obtained by tensoring the off-shell Yang-Mills multiplets (4+4,N_L =1)and(…
We consider extension of some established techniques of study of tensor fields on Lorentzian manifolds of arbitrary dimension to non-Abelian gauge covariant fields. These are then applied to study of gauge fields with vanishing scalar curvature invariants and universal Yang-Mills fields, for which all possible higher-order corrections to the Yang-Mills equation identically vanish.
The Hamiltonian of Intrinsic Time Gravity is elucidated. The theory describes Schrodinger evolution of our universe with respect to the fractional change of the total spatial volume. Gravitational interactions are introduced by extending Klauder's momentric variable with similarity transformations, and explicit spatial diffeomorphism invariance is enforced via similarity transformation with exponentials of spatial integrals. In analogy with Yang-Mills theory, a Cotton-York term is obtained from the Chern-Simons functional of the affine connection. The essential difference is the fundamental va…
Recently, the {\it spacelike} part of the $SU(2)$ Yang--Mills equations has been identified with geometrical objects of a three--dimensional space of constant Riemann--Cartan curvature. We give a concise derivation of this Ashtekar type (``inverse Kaluza--Klein") {\it mapping} by employing a $(3+1)$--decomposition of {\it Clifford algebra}--valued torsion and curvature two--forms. In the subcase of a mapping to purely axial 3D torsion, the corresponding Lagrangian consists of the translational and Lorentz {\it Chern--Simons term} plus cosmological term and is therefore of purely topological or…
We describe the Lorentzian version of the Kapovitch-Millson phase space for polyhedra with $N$ faces. Starting with the Schwinger representation of the $\mathfrak{su}(1,1)$ Lie algebra in terms of a pair of complex variables (or spinor), we define the phase space for a space-like vectors in the three-dimensional Minkowski space $\mathbb{R}^{1,2}$. Considering $N$ copies of this space, quotiented by a closure constraint forcing the sum of those 3-vectors to vanish, we obtain the phase space for Lorentzian polyhedra with $N$ faces whose normal vectors are space-like, up to Lorentz transformation…
We introduce a gauge and diffeomorphism invariant theory on Yang-Mills phase space. The theory is well defined for an arbitrary gauge group with an invariant bilinear form, it contains only first class constraints, and the spacetime metric has a simple form in terms of the phase space variables. With gauge group $SO(3,C)$, the theory equals the Ashtekar formulation of gravity with a cosmological constant. For Lorentzian signature, the theory is complex, and we have not found any good reality conditions. In the Euclidean signature case, everything is real. In a weak field expansion around de Si…
Discrete versions of the Yang-Mills and Einstein actions are proposed for any finite group. These actions are invariant respectively under local gauge transformations, and under the analogues of Lorentz and general coordinate transformations. The case Z_n \times Z_n \times...\times Z_n is treated in some detail, recovering the Wilson action for Yang-Mills theories, and a new discretized action for gravity.
We derive a semiclassical equation of motion for a `composite' quark in strongly-coupled large-N_c N=4 super-Yang-Mills, making use of the AdS/CFT correspondence. The resulting non-linear equation incorporates radiation damping, and reduces to the standard Lorentz-Dirac equation for external forces that are small on the scale of the quark Compton wavelength, but has no self-accelerating or pre-accelerating solutions. From this equation one can read off a non-standard dispersion relation for the quark, as well as a Lorentz covariant formula for its radiation rate.
The dark matter and dark energy problem, that is now dominating the research in cosmology, makes the question of the origin of mass-energy content of the universe more urgent than ever. There are two philosophies regarding this question: according to Mach's principle it is matter that generates geometry of space-time, and according to Wheeler's geometrodynamics some configurations of space-time geometry are to be interpreted as its material content. Neither of these philosophies has led to success. In the present paper, we show that there exists an algebraic generalisation of geometry that rec…
In the Ashtekar and geometrodynamic formulations of vacuum general relativity, the Euclidean and Lorentzian sectors can be related by means of the generalized Wick transform discovered by Thiemann. For some vacuum gravitational systems in which there exists an intrinsic time variable which is not invariant under constant rescalings of the metric, we show that, after such a choice of time gauge and with a certain identification of parameters, the generalized Wick transform can be understood as an analytic continuation in the explicit time dependence. This result is rigorously proved for the Gow…
We reformulate an approach fist given by Barbour and Bertotti (BB) for implementing Mach's principle for nonrelativistic particles. This reformulation can deal with arbitrary symmetry groups and finite group elements. Applying these techniques to U(1) and SU(N) invariant scalar field theories, we show that BB's proposal is nearly equivalent to defining a covariant derivative using a dynamical connection. We then propose a modified version of the BB method which implements Mach's principle using gauge theory techniques and argue that this modified method is equivalent to the original. Given thi…
The local Lorentz group is introduced in flat space-time, where the resulting Dirac and Yang-Mills equations are found, and then generalized to curved space-time: if matter is neglected, the Lorentz connection is identified with the contortion field, while, if matter is taken into account, both the Lorentz connection and the spinor axial current are illustrated to contribute to the torsion of space-time.
The canonical formulation of general relativity is based on decomposition space--time manifold $M$ into $ R\times Σ$, this decomposition has to preserve the invariance of general relativity, invariance under general coordinates, and local Lorentz transformations. These symmetries associate with conserved currents that are coupled to gravity. In this paper, we try to solve the equations of motion of general relativity in self-dual formalism using only the spin currents(Lorentz currents), in static case, and without needing using the Einstein's equation, that makes the general relativity similar…
We propose a modified gravitational action containing besides the Einstein-Cartan term some quadratic contributions resembling the Yang-Mills lagrangian for the Lorentz spin connections. We outline how a propagating torsion arises and we solve explicitly the linearised equations of motion on a Minkowski background. We identify among torsion components six degrees of freedom: one is carried by a pseudo-scalar particle, five by a tachyon field. By adding spinor fields and neglecting backreaction on the geometry, we point out how only the pseudo-scalar particle couples directly with fermions, but…
I suggest in this letter a new strategy to attack the problem of the reality conditions in the Ashtekar approach to classical and quantum general relativity. By writing a modified Hamiltonian constraint in the usual $SO(3)$ Yang-Mills phase space I show that it is possible to describe space-times with Lorentzian signature without the introduction of complex variables. All the features of the Ashtekar formalism related to the geometrical nature of the new variables are retained; in particular, it is still possible, in principle, to use the loop variables approach in the passage to the quantum t…
A four dimensional generally covariant modified Yang-Mills action, which depends on the lorentzian complex structure of spacetime and not its metric, is presented. The extended Weyl symmetry, implied by the effective metric independence, makes the lagrangian model renormalizable. The modified Yang-Mills action generates a linear potential, instead of the Coulomb-like (1/r) potential of the ordinary action. Therefore the Yang-Mills excitations must be perturbatively confined. The metric, which admits an integrable lorentzian complex structure, can be extended to a Kaehler metric and the spaceti…
In abstract Yang-Mills theory the standard instanton construction relies on the Hodge star having real eigenvalues which makes it inapplicable in the Lorentzian case. We show that for the affine connection an instanton-type construction can be carried out in the Lorentzian setting. The Lorentzian analogue of an instanton is a spacetime whose connection is metric compatible and Riemann curvature irreducible ("pseudoinstanton"). We suggest a metric-affine action which is a natural generalization of the Yang-Mills action and for which pseudoinstantons are stationary points. We show that a spaceti…
Special relativity asserts that physical phenomena appear the same for all inertially moving observers. This symmetry, called Lorentz symmetry, relates long wavelengths to short ones: if the symmetry is exact it implies that spacetime must look the same at all length scales. Several approaches to quantum gravity, however, suggest that there may be a Lorentz violating microscopic structure of spacetime, for example discreteness, non-commutativity, or extra dimensions. Here we determine a very strong constraint on a type of Lorentz violation that produces a maximum electron speed less than the s…
In the low energy domain of four-dimensional SU(2) Yang-Mills theory the spin and the charge of the gauge field can become separated from each other. The ensuing field variables describe the interacting dynamics between a version of the O(3) nonlinear $σ$-model and a nonlinear Grassmannian $σ$-model, both of which may support closed knotted strings as stable solitons. Lorentz transformations act projectively in the O(3) model which breaks global internal rotation symmetry and removes massless Goldstone bosons from the particle spectrum. The entire Yang-Mills Lagrangian can be recast into a gen…
This is an introductory chapter in a series in which we take a systematic study of the Yang-Mills equations on curved space-times. In this first, we provide standard material that consists in writing the proof of the global existence of Yang-Mills fields on arbitrary curved space-times using the Klainerman-Rodnianski parametrix combined with suitable Grönwall type inequalities. While the Chruściel-Shatah argument requires a simultaneous control of the $L^{\infty}_{loc}$ and the $H^{2}_{loc}$ norms of the Yang-Mills curvature, we can get away by controlling only the $H^{1}_{loc}$ norm instead, …
The natural constraints for the weak-field approximation to composite gravity, which is obtained by expressing the gauge vector fields of the Yang-Mills theory based on the Lorentz group in terms of tetrad variables and their derivatives, are analyzed in detail within a canonical Hamiltonian approach. Although this higher derivative theory involves a large number of fields, only few degrees of freedom are left, which are recognized as selected stable solutions of the underlying Yang-Mills theory. The constraint structure suggests a consistent double coupling of matter to both Yang-Mills and te…
