Revisiting observables in generally covariant theories in the light of gauge fixing methods

Abstract

We derive for generally covariant theories the generic dependency of observables on the original fields, corresponding to coordinate-dependent gauge fixings. This gauge choice is equivalent to a choice of intrinsically defined coordinates accomplished with the aid of spacetime scalar fields. With our approach we make full contact with, and give a new perspective to, the "evolving constants of motion" program. We are able to directly derive generic properties of observables, especially their dynamics and their Poisson algebra in terms of Dirac brackets, extending earlier results in the literature. We also give a new interpretation of the observables as limits of canonical maps.

Effective approach to the problem of time: General features and examples

“…The effective framework is, thus, amenable to techniques, usually aimed at a solution to the classical problem of observables, such as [9,11] and the perturbative expansions of [10]. Moreover, concrete evaluations of the constrained systems are usually done by employing gauge fixing, for which classical methods such as those of [25] are useful.…”

“…In particular, the effective approach avoids the Hilbert space problem altogether since no use of representations or physical inner products has been made at any point of the algebraic construction. The tedious problem of constructing physical states and inner products, which is often even practically impossible, 25 is replaced by evaluating an (infinite) coupled set of quantum variables which can be consistently truncated to a finite solvable system, for instance, at semiclassical order; necessary physicality conditions for observables are ultimately imposed just by reality conditions. At this stage, the effective framework can be implemented numerically and its physical properties can be studied in detail.…”

“…The first class constraint with vanishing flow on these variables is now given by C q . Solving this constraint then implies ∆(qp) = − i 2 and, together with (25), the saturation of the uncertainty relation betweenq andp. The "Hamiltonian constraint" of the q-gauge reads…”

“…We recall that the original gauge orbit for the truncated system of constraints (14) is, in general, four-dimensional. The three gauge-fixing equations of either (15) or (25) restrict us to a one-dimensional flow on this gauge orbit generated by the remaining first-class constraint (17) or (26), respectively. In order to ensure that the two sets of variables lie on the same four-dimensional gauge orbit we need to find a gauge transformation which takes us from the surface defined by (15) to the one defined by (25) and vice versa.…”

“…From the point of view of the Poisson manifold of the effective framework no variables or gauges are preferred over others and we could, in principle, choose a q-gauge like (25) and still use t as our clock for relational evolution. However, as we will see in the second model in Sec.…”