Space-time covariant form of Ashtekar’s constraints

Abstract

Summary. -The Lagrangian formulation of classical field theories and in particular general relativity leads to a coordinate-free, fully covariant analysis of these constrained systems. This paper applies multisymplectic techniques to obtain the analysis of Palatini and self-dual gravity theories as constrained systems, which have been studied so far in the Hamiltonian formalism. The constraint equations are derived while paying attention to boundary terms, and the Hamiltonian constraint turns out to be linear in the multimomenta. The equivalence with Ashtekar's formalism is also established. The whole constraint analysis, however, remains covariant in that the multimomentum map is evaluated on any spacelike hypersurface. This study is motivated by the non-perturbative quantization program of general relativity. More recently, the work by Ashtekar, Rovelli, Smolin and their collaborators on connection dynamics and loop variables has made it possible to cast the constraint equations of general relativity in polynomial form, and then find a large class of solutions to the quantum version of constraints [7][8][9][10][11][12][13][14]. However, the quantum theory via the Rovelli-Smolin transform still suffers from severe mathematical problems in 3+1 space-time dimensions [15], and there appear to be reasons for studying non-perturbative quantum gravity also from a Lagrangian, rather than Hamiltonian, point of view (see below). The aim of this paper is therefore to provide a multisymplectic, Lagrangian framework for general relativity [16][17][18], to complement the present attempts to quantize general relativity in a non-perturbative way. The motivations of our analysis are as follows.(i) In the case of field theories, there is not a unique prescription for taking duals, on passing to the Hamiltonian formalism. For example, algebraic and topological duals are different. In turn, this may lead to inequivalent quantum theories. SPACE-TIME COVARIANT FORM OF ASHTEKAR'S CONSTRAINTS(ii) The 3+1 split of the Lorentzian space-time geometry, with the corresponding Σ×R topology, appears to violate the manifestly covariant nature of general relativity, as well as rely on a very restrictive assumption on the topology [19].(iii) In the Lagrangian formalism, explicit covariance is instead recovered. The first constraints one actually evaluates correspond to the secondary first-class constraints of the Hamiltonian formalism. At least at a classical level, the Lagrangian theory of constrained systems is by now a rich branch of modern mathematical physics [4,20,21], although the majority of general relativists are more familiar with the Hamiltonian framework.(iv) In the ADM formalism [5,6], the invariance group is not the whole diffeomorphism group, but a subgroup given by the Cartesian product of diffeomorphisms on the real line with the diffeomorphism group on spacelike three-surfaces. By contrast, in the Lagrangian approach, the invariance group of the theory is the full diffeomorphism group of fourdimensional Lorentzian space-time...

Covariant canonical quantization of fields and Bohmian mechanics

“…However, if this 4-diffeomorhism invariance is explicitly realized, then one expects that ultraviolet divergences should be absent [3]. Perhaps this could be realized explicitly by introducing a time-space symmetric version of loop quantum gravity based on the spacetime covariant Ashtekar variables [37].…”

Hamiltonian approach to GR – Part 2: covariant theory of quantum gravity

“…Nevertheless, it must be mentioned that in the case of continuum fields, the appropriate formalism is actually well established, being provided by the Weyl-De Donder Lagrangian and Hamiltonian treatments [13][14][15]. The need to adopt an analogous approach also in the context of classical GR, and in particular for the Einstein equation itself or its possible modifications, has been recognized before [29][30][31][32][33]. The fulfillment of the physical prerequisites indicated above in the context of a classical treatment of SF-GR and the definition of the related conceptual framework for GR has been provided recently by Part 1 and Refs.…”

Peierls Brackets in Field Theory

“…The last term in (20) requires new calculations because it contains functional derivatives of G ij . These can be dealt with after taking infinitesimal variations of Eq.…”

“…(26) give vanishing contribution to (20). One is therefore left with the contributions involving third functional derivatives of the action.…”