Observables in classical canonical gravity: Folklore demystified

Abstract

We give an overview of some conceptual difficulties, sometimes called paradoxes, that have puzzled for years the physical interpetation of classical canonical gravity and, by extension, the canonical formulation of generally covariant theories. We identify these difficulties as stemming form some terminological misunderstandings as to what is meant by "gauge invariance", or what is understood classically by a "physical state". We make a thorough analysis of the issue and show that all purported paradoxes disappear when the right terminology is in place. Since this issue is connected with the search of observables -gauge invariant quantities -for these theories, we formally show that time evolving observables can be constructed for every observer. This construction relies on the fixation of the gauge freedom of diffeomorphism invariance by means of a scalar coordinatization. We stress the condition that the coordinatization must be made with scalars. As an example of our method for obtaining observables we discuss the case of the massive particle in AdS spacetime.

Three denials of time in the interpretation of canonical gravity

“…6 Geometrically we can characterise such a generic constrained 5 It is important to note that this key aspect to our analysis represents a departure from both the received and dissenting view on this matter (although it is close to the spirit of Pons et al (2010)). Whereas, the received view is that the Hamiltonian constraints purely generate unphysical transformations (e.g.…”

“…2 As pointed out by Pons, Salisbury, and Sundermeyer (2010) general relativity actually admits the larger symmetry group of field-dependent infinitesimal co-ordinate transformations and so Dif f (M) is properly a sub-group of the fundamental symmetry group. This difference will not be important for our purposes.…”

“…This sub-set can be shown (e.g. Pons et al (2010)) to be infinitesimal coordinate transformations x µ → x µ − µ (x) such that:…”

“…Thus, at a classical level at least there is no detectable difference. We can in fact explicitly construct a canonical symmetry generator acting upon the space of solutions that maps between the same diffeomorphically related spacetimes that we take to be symmetry related in the covariant formalism (see Pons et al (2010) and references therein for more details). Thus, so long as we are concerned with globally hyperbolic solutions, the two formalisms admit identical symmetry relations.…”

Equivalent Theories and Changing Hamiltonian Observables in General Relativity

“…Relatedly, Pons, Salisbury and Sundermeyer have proposed a reformed definition of observables using the Anderson-Bergmann-Castellani gauge generator G, a tuned sum of all first-class constraints including the primaries [2,11,39].…”