Canonical decomposition of the general relativistic initial value problem

A canonical transformation is employed to implement a conformal transformation of the configuration variables of general relativity. The transformation is so chosen that the spatial constraints become algebraic in the trace of the momentum density. The temporal constraint is then found to have the form of York and O'Murchadha. The role played by the York coordinate condition in decoupling the constraint equations is examined, and a procedure to solve the constraint equations without employing such a coordinate condition is sketched.     

Complex Canonical Gravity and Reality Constraints

Ashtekar canonical variables for generalrelativity yielding low degree polynomial constraintsare complex and describe complex canonical gravity. Topick the real sector, a la Dirac, one must introduce reality constraints; they turn out to besecond-class. It is shown here that this holds not onlyfor pure gravity but also for a scalar fieldnon-minimally coupled to gravity. The originalsimplicity produced by the complex variables is spoiled if one getsrid of the second-class constraints via Dirac brackets,however. To circumvent such an undesirable feature,alternative possibilities are pointed out.     

The Hamiltonian formulation of general relativity: myths and reality

A conventional wisdom often perpetuated in the literature states that: (i) a 3 + 1 decomposition of spacetime into space and time is synonymous with the canonical treatment and this decomposition is essential for any Hamiltonian formulation of General Relativity (GR); (ii) the canonical treatment unavoidably breaks the symmetry between space and time in GR and the resulting algebra of constraints is not the algebra of four-dimensional diffeomorphism; (iii) according to some authors this algebra allows one to derive only spatial diffeomorphism or, according to others, a specific field-dependent and non-covariant four-dimensional diffeomorphism; (iv) the analyses of Dirac [21] and of ADM [22] of the canonical structure of GR are equivalent. We provide some general reasons why these statements should be questioned. Points (i–iii) have been shown to be incorrect in [45] and now we thoroughly re-examine all steps of the Dirac Hamiltonian formulation of GR. By direct calculation we show that Dirac’s references to space-like surfaces are inessential and that such surfaces do not enter his calculations. In addition, we show that his assumption g _0k = 0, used to simplify his calculation of different contributions to the secondary constraints, is unwarranted; yet, remarkably his total Hamiltonian is equivalent to the one computed without the assumption g _0k = 0. The secondary constraints resulting from the conservation of the primary constraints of Dirac are in fact different from the original constraints that Dirac called secondary (also known as the “Hamiltonian” and “diffeomorphism” constraints). The Dirac constraints are instead particular combinations of the constraints which follow directly from the primary constraints. Taking this difference into account we found, using two standard methods, that the generator of the gauge transformation gives diffeomorphism invariance in four-dimensional space-time; and this shows that points (i–iii) above cannot be attributed to the Dirac Hamiltonian formulation of GR. We also demonstrate that ADM and Dirac formulations are related by a transformation of phase-space variables from the metric _{g μν} to lapse and shift functions and the three-metric g _km , which is not canonical. This proves that point (iv) is incorrect. Points (i–iii) are mere consequences of using a non-canonical change of variables and are not an intrinsic property of either the Hamilton-Dirac approach to constrained systems or Einstein’s theory itself.     

Representations of spacetime diffeomorphisms. II. Canonical geometrodynamics

Our method of constructing representations of the spacetime diffeomorphism group DiffMin parametrized field theories is generalized to canonical geometrodynamics. The gravitational configuration space Riem Σ is extended by the space of embeddings of the spatial manifold Σ in the spacetimeM. Spacetime metrics are limited by Gaussian conditions with respect to an auxiliary foliation structure. As a result of these conditions, the super-Hamiltonian and supermomentum constraints are temporarily suspended. There are, however, new constraints in the theory associated with the canonical pair of the embedding variables and their conjugate momenta. By smearing the new constraint functions by vector fields V ∈ LDiffMrestricted to the embeddings, we construct a homomorphism from the Lie algebra of the spacetime diffeomorphism group into the Poisson bracket algebra of the dynamical variables on the extended geometrodynamical phase space. The dynamical evolution generated by such dynamical variables automatically preserves both the new and the old constraints and builds a Ricci-flat spacetime. The implications of the scheme for the canonical quantization of gravity are discussed.     

On Dirac's Conjecture

The extended canonical Noether identities and canonical first Noether theorem derived from an extended action in phase space for a system with a singular Lagrangian are formulated. Using these canonical Noether identities, it can be shown that the constraint multipliers connected with the first-class constraints may not be independent, so a query to a conjecture of Dirac is presented. Based on the symmetry properties of the constrained Hamiltonian system in phase space, a counterexample to a conjecture of Dirac is given to show that Dirac's conjecture fails in such a system. We present here a different way rather than Cawley's examples and other's ones in that there is no linearization of constraints in the problem. This example has a feature that neither the primary first-class constraints nor secondary first-class constraints are generators of the gauge transformation.     

Symmetries in constrained Hamiltonian systems

We derive an algorithm for the construction of all the gauge generators of a constrained hamiltonian theory. Dirac's conjecture that all secondary first-class constraints generate symmetries is revisited and replaced by a theorem. The algorithm is applied to Yang-Mills theories and metric gravity, and we find generators which operate on the complete set of canonical variables, thus producing the correct transformation laws also for the unphysical coordinates. Finally we discuss the general structure of the Hamiltonian for constrained theories. We show how in most cases one can read off the first-class constraints directly from the Hamiltonian.     

Generalized Canonical Noether Theorem and Poincare—Cartan Integral Invariant ofr a System with a Singular High—Order Lagrangian and an Application

Based on the canonical action,a generalized canonical first Noether theorem and Poicare-Cartan integralinvariant for a system with a singular high-order Lagrangian are derived.It is worth while to point out that the constraints are invariant under the total variation of canonical variables including time.We can also deduce the result,which differs from the previous work to reuire that the constraints are invariant under the simultaneous variations of canonical variables.A counter example to a conjecture of the Dirac for a system with a singular high-order Lagrangian is given,in which there is no linearization of constraint.     

On the two-dimensional classical motion of a charged particle in an electromagnetic field with an additional quadratic integral of motion

The problem of quadratic Hamiltonians with an electromagnetic field commuting in the sense of the standard Poisson brackets has been considered. It has been shown that, as in the quantum case, any such pair can be reduced to the canonical form, which makes it possible to construct the complete classification of the solutions in the class of meromorphic solutions for the main function of one variable. The transformation to the canonical form is performed through the change of variables to the Kovalevskaya-type variables, which is similar to that in the theory of integrable tops. This transformation has been considered for the two-dimensional Hamiltonian of a charged particle with an additional quadratic integral of motion.     

How commutators of constraints reflect the spacetime structure

The structure constants of the “algebra” of constraints of a parametrized field theory are derived by a simple geometrical argument based exclusively on the path independence of the dynamical evolution; the change in the canonical variables during the evolution from a given initial surface to a given final surface must be independent of the particular sequence of intermediate surface used in the actual evaluation of this change. The requirement of path independence also implies that the theory will propagate consistently only initial data such that the Hamiltonian vanishes. The vanishing of the Hamiltonian arises because the metric of the surface is a canonical variable rather than a c-number. It is not assumed the constraints can be solved to express four of the momenta in terms of the remaining canonical variables. It is shown that the signature of spacetime can be read off from the commutator of two Hamiltonian constraints at different points. The analysis applies equally well irrespective of whether the spacetime is a prescribed Riemannian background or whether it is determined by the theory itself as in general relativity. In the former case the structure of the commutators imposes consistency conditions for a theory in which states are defined on arbitrary spacelike surfaces; whereas, in the later case it provides the conditions for the existence of spacetime— “embeddability” conditions which ensure that the evolution of a three-geometry can be viewed as the “motion” of a three-dimensional cut in a four-dimensional spacetime of hyperbolic signature.     

Algebraic programming of Hamiltonian formalism in general relativity: Application to inhomogeneous space-times

We describe the structure and the use of a program written in the algebraic programming languagereduce 2, giving the super-Hamiltonian and supermomenta constraints, as well as Hamilton's canonical equations in terms of the canonical variables, for vacuum relativistic space-times. The program uses as input the components of the spatial metric tensor and of the corresponding canonically conjugate momenta in a coordinate or in a spatial Cartan basis. The results of the application of the program to a series of inhomogeneous (cosmological as well as noncosmological) space-times are given: in particular, the constraints, the Dirac Hamiltonian and the canonical equations are explicitly written for axisymmetric space-times, constituting the starting point for the study of the dynamics and of the canonical quantization of these configurations.     

On a two-dimensional Schrödinger equation with a magnetic field with an additional quadratic integral of motion

The problem of commuting quadratic quantum operators with a magnetic field has been considered. It has been shown that any such pair can be reduced to the canonical form, which makes it possible to construct an almost complete classification of the solutions of equations that are necessary and sufficient for a pair of operators to commute with each other. The transformation to the canonical form is performed through the change of variables to the Kovalevskaya-type variables; this change is similar to that in the theory of integrable tops. As an example, this procedure has been considered for the two-dimensional Schrödinger equation with the magnetic field; this equation has an additional quantum integral of motion.     

On the space-time interpretation of classical canonical systems II: Relativistic canonical systems

Relativistic canonical systems and their symmetries are defined and classified within the class of canonical systems treated in a previous paper. Their algebra of variables contains a subset of monotone variables which satisfy a certain uniqueness condition and are later shown to increase strictly in the course of the dynamical evolution of the system on all physically acceptable states. This leads to a unique space-time interpretation of relativistic canonical systems. Finally we study space-time factorizations of such systems and introduce the appropriate notion of states. For a certain simple class of states the theory is shown to describe the motion of relativistic matter in some external gravitational and electromagnetic field.     

Review: Dirac's Observables for the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge

We define the rest-frame instant form of tetrad gravity restricted to Christodoulou-Klainermann spacetimes. After a study of the Hamiltonian group of gauge transformations generated by the 14 first class constraints of the theory, we define and solve the multitemporal equations associated with the rotation and space diffeomorphism constraints, finding how the cotriads and their momenta depend on the corresponding gauge variables. This allows to find a quasi-Shanmugadhasan canonical transformation to the class of 3-orthogonal gauges and to find the Dirac observables for superspace in these gauges. The construction of the explicit form of the transformation and of the solution of the rotation and supermomentum constraints is reduced to solve a system of elliptic linear and quasi-linear partial differential equations. We then show that the superhamiltonian constraint becomes the Lichnerowicz equation for the conformal factor of the 3-metric and that the last gauge variable is the momentum conjugated to the conformal factor. The gauge transformations generated by the superhamiltonian constraint perform the transitions among the allowed foliations of spacetime, so that the theory is independent from its 3+1 splittings. In the special 3-orthogonal gauge defined by the vanishing of the conformal factor momentum we determine the final Dirac observables for the gravitational field even if we are not able to solve the Lichnerowicz equation. The final Hamiltonian is the weak ADM energy restricted to this completely fixed gauge.     

On the Hamiltonian formalism of the tetrad-gravity with fermions

We extend the analysis of the Hamiltonian formalism of the d-dimensional tetrad-connection gravity to the fermionic field by fixing the non-dynamic part of the spatial connection to zero (Lagraa et al. in Class Quantum Gravity 34:115010, 2017). Although the reduced phase space is equipped with complicated Dirac brackets, the first-class constraints which generate the diffeomorphisms and the Lorentz transformations satisfy a closed algebra with structural constants analogous to that of the pure gravity. We also show the existence of a canonical transformation leading to a new reduced phase space equipped with Dirac brackets having a canonical form leading to the same algebra of the first-class constraints.     

A New Parametrization for Tetrad Gravity

A new version of tetrad gravity in globally hyperbolic, asymptotically flat at spatial infinity spacetimes with Cauchy surfaces diffeomorphic to R ³ is obtained by using a new parametrization of arbitrary cotetrads to define a set of configurational variables to be used in the ADM metric action. Seven of the fourteen first class constraints have the form of the vanishing of canonical momenta. A comparison is made with other models of tetrad gravity and with the ADM canonical formalism for metric gravity.     

Separation of Variables for a Lattice Integrable System and the Inverse Problem

We investigate the relation between the local variables of a discrete integrable lattice system and the corresponding separation variables, derived from the associated spectral curve. In particular, we have shown how the inverse transformation from the separation variables to the discrete lattice variables may be factorized as a sequence of canonical transformations.