119290899
Darboux Coordinates for the Hamiltonian of First Order Einstein-Cartan Gravity
Based on our preliminary analysis of the Hamiltonian formulation of the first order Einstein-Cartan action (arXiv:0902.0856 [gr-qc] and arXiv:0907.1553 [gr-qc]) we derive the Darboux coordinates, which are a unique and uniform change of variables that preserve equivalence with the original action in all spacetime dimensions higher than two. Considerable simplification of the Hamiltonian formulation using the Darboux coordinates, as compared with the direct analysis, is explicitly demonstrated. Even an incomplete Hamiltonian analysis in combination with known symmetries of the Einstein-Cartan action and the equivalence of Hamiltonian and Lagrangian formulations allows us to unambiguously conclude that the unique gauge invariances generated by the first class constraints are translation and rotation in the tangent space. Diffeomorphism invariance, though a manifest invariance of the action, is not generated by the first class constraints of the theory.
122626470
Remarks on 'Note about Hamiltonian formalism of healthy extended Horava-Lifshitz gravity' by J. Kluson
We reassess the conclusion by Klusoyn (J. High Energy Phys. 1007 (2010) 038) that the Hamiltonian formulation of the healthy extended Hoyrava-Lifshitz gravity does not present any problem.
117314216
Comment on `Hamiltonian formulation for the theory of gravity and canonical transformations in extended phase space' by T P Shestakova
We argue that the conclusion, `we cannot consider the Dirac approach as fundamental and undoubted', made in the paper by Shestakova (Class. Quantum Grav. 28 055009, 2011), is based upon an incomplete and flawed analysis of the simple model presented in section 3 of the article. We re-examine the analysis of this model and find that it does not support the author's conclusion. For the theory of gravity neither the equivalence of the effective action nor its Hamiltonian formulation is given by the author, therefore, we only provide a brief commentary.
118512255
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 g0k = 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 g0k = 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 gμν to lapse and shift functions and the three-metric gkm, 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.
119222199
The Hamiltonian formulation of tetrad gravity: Three-dimensional case
The Hamiltonian formulation of tetrad gravity in any dimension higher than two, using its first-order form where tetrads and spin connections are treated as independent variables, is discussed, and the complete solution of the three-dimensional case is given. For the first time, applying the methods of constrained dynamics, the Hamiltonian and constraints are explicitly derived and the algebra of Poisson brackets among all constraints is calculated. The algebra of Poisson brackets, among first-class secondary constraints, locally coincides with Lie algebra of the ISO(2,1) Poincaré group. All the first-class constraints of this formulation, according to the Dirac conjecture and using the Castellani procedure, allow us to unambiguously derive the generator of gauge transformations and find the gauge transformations of the tetrads and spin connections which turn out to be the same as found by Witten without recourse to the Hamiltonian methods [Nucl. Phys. B 311, 46 (1988)]. The gauge symmetry of the tetrad gravity generated by the Lie algebra of constraints is compared with another invariance, diffeomorphism. Some conclusions about the Hamiltonian formulation in higher dimensions are briefly discussed; in particular, that diffeomorphism invariance is not derivable as a gauge symmetry from the Hamiltonian formulation of tetrad gravity in any dimension where tetrads and spin connections are used as independent variables.
119213240
Diffeomorphism invariance in the Hamiltonian formulation of General Relativity
Abstract It is shown that when the Einstein–Hilbert Lagrangian is considered without any non-covariant modifications or change of variables, its Hamiltonian formulation leads to results consistent with principles of General Relativity. The first-class constraints of such a Hamiltonian formulation, with the metric tensor taken as a canonical variable, allow one to derive the generator of gauge transformations, which directly leads to diffeomorphism invariance. The given Hamiltonian formulation preserves general covariance of the transformations derivable from it. This characteristic should be used as the crucial consistency requirement that must be met by any Hamiltonian formulation of General Relativity.
100638900
On Separation of Gagolinium's Isotopes by Means of Isotopically Selective ICRH