Hamiltonian formulation for the theory of gravity and canonical transformations in extended phase space

Shestakova, T. P.

doi:10.1088/0264-9381/28/5/055009

Cited by 19 publications

(102 citation statements)

References 17 publications

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“…Consequently, Π ∂ a Ψ = 0 in Eq. (48). In such a case, the shift-vector constraint (21) reduces to that of standard GRT and π ab can be readily calculated by existing methods.…”

Section: Shift-covector Constraint Equationmentioning

confidence: 99%

Canonical reduction for dilatonic gravity in3+1dimensions

et al. 2016

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We generalize the 1 + 1-dimensional gravity formalism of Ohta and Mann to 3 + 1 dimensions by developing the canonical reduction of a proposed formalism applied to a system coupled with a set of point particles. This is done via the Arnowitt-Deser-Misner method and by eliminating the resulting constraints and imposing coordinate conditions. The reduced Hamiltonian is completely determined in terms of the particles' canonical variables (coordinates, dilaton field and momenta). It is found that the equation governing the dilaton field under suitable gauge and coordinate conditions, including the absence of transverse-traceless metric components, is a logarithmic Schrödinger equation. Thus, although different, the 3 + 1 formalism retains some essential features of the earlier 1 + 1 formalism, in particular the means of obtaining a quantum theory for dilatonic gravity.

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“…Consequently, Π ∂ a Ψ = 0 in Eq. (48). In such a case, the shift-vector constraint (21) reduces to that of standard GRT and π ab can be readily calculated by existing methods.…”

Section: Shift-covector Constraint Equationmentioning

confidence: 99%

Canonical reduction for dilatonic gravity in3+1dimensions

et al. 2016

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“…The action (1.4) includes second derivatives, and to construct the BRST charge one should used the Noether theorem generalized for theories with high order derivatives. In our case we have 5) ϕ a stands for field variables and ghosts. It gives the expression…”

Section: Introductionmentioning

confidence: 99%

The role of BRST charge as a generator of gauge transformations in quantization of gauge theories and Gravity

Shestakova¹

2016

Proceedings of Frontiers of Fundamental Physics 14 — PoS(FFP14)

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In the Batalin -Fradkin -Vilkovisky approach to quantization of gauge theories a principal role is given to the BRST charge which can be constructed as a series in Grassmannian (ghost) variables with coefficients given by generalized structure functions of constraints algebra. Alternatively, the BRST charge can be derived making use of the Noether theorem and global BRST invariance of the effective action. In the case of Yang -Mills fields both methods lead to the same expression for the BRST charge, but it is not valid in the case of General Relativity. It is illustrated by examples of an isotropic cosmological model as well as by spherically-symmetric gravitational model which imitates the full theory of gravity much better. The consideration is based on Hamiltonian formulation of General Relativity in extended phase space. At the quantum level the structure of the BRST charge is of great importance since BRST invariant quantum states are believed to be physical states. Thus, the definition of the BRST charge at the classical level is inseparably related to our attempts to find a true way to quantize gravity.

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“…However, in their approach gauge variables were still considered as non-physical, secondary degrees of freedom playing just an auxiliary role in the theory. While the BFV approach aimed at reproducing the results of Dirac's canonical quantization on a path integral level, our construction of extended phase space guarantees equivalence of Lagrangian and Hamiltonian dynamics of a constrained system for a wide enough class of parametrizations, and constraints and gauge condition get a status of Hamiltonian equations in extended phase space [11]. The algebra of Poisson brackets turns out to be invariant under reparametrizations from this class.…”

mentioning

confidence: 97%

“…Then, we can write the effective action including gauge and ghost sectors as it appears in the path integral approach to gauge field theories, It has been shown [11] for the full gravitational theory that the transformation…”

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confidence: 99%

A view on the problems of Quantum Gravity

Shestakova¹

2012

J. Phys.: Conf. Ser.

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The existing approaches to quantization of gravity aim at giving quantum description of 3-geometry following to the ideas of the Wheeler -DeWitt geometrodynamics. In this description the role of gauge gravitational degrees of freedom is missed. A probable alternative is to consider gravitational dynamics in extended phase space, taking into account the distinctions between General Relativity and other field theories. The formulation in extended phase space leads to some consequences at classical and quantum levels. At the classical level, it ensures that Hamiltonian dynamics is fully equivalent to Lagrangian dynamics, and the algebra of Poisson brackets is invariant under reparametrizations in a wide enough class including reparametrizations of gauge variables, meantime in the canonical Dirac approach the constraints' algebra is not invariant that creates problems with quantization. At the quantum level, the approach come to the description in which the observer can see various but complementary quantum gravitational phenomena in different reference frames that answers the spirit of General Relativity and Quantum Theory. Though until now the approach was applied to General Relativity in its original formulations, its implementation in different trends, including Quantum Loop Gravity or some other representations of gravitational variables, would also be of interest.I am grateful to the Organizers of the Conference for the opportunity to present my point of view on the problems of Quantum Gravity. In our attempts to quantize gravity we inevitably rely on approaches which work satisfactory for ordinary field theories, often without a careful analysis of applicability one or another approach to gravitation. An attractive point of Loop Quantum Gravity is that it aims at searching for its own way of description of Quantum Geometry and the Hilbert space of quantum states.

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Hamiltonian formulation for the theory of gravity and canonical transformations in extended phase space

Cited by 19 publications

References 17 publications

Canonical reduction for dilatonic gravity in3+1dimensions

Canonical reduction for dilatonic gravity in3+1dimensions

The role of BRST charge as a generator of gauge transformations in quantization of gauge theories and Gravity

A view on the problems of Quantum Gravity

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