General relativity histories theory

Abstract

The canonical description is based on the prior choice of a spacelike foliation, hence making a reference to a spacetime metric. However, the metric is expected to be a dynamical, fluctuating quantity in quantum gravity. After presenting the developments in the History Projection Operator histories theory in the last seven yearsgiving special emphasis on the novel temporal structure of the formalism -we show how this problem can be solved in the histories formulation of general relativity. We implement the 3 + 1 decomposition using metricdependent foliations which remain spacelike with respect to all possible Lorentzian metrics. This allows us to find an explicit relation of covariant and canonical quantities which preserves the spacetime character of the canonical description. In this new construction we have a coexistence of the spacetime diffeomorphisms group Diff(M) and the Dirac algebra of constraints.

The role of phase space geometry in Heisenberg’s uncertainty relation

“…Since the decoherence functional is constructed from the metric and the phase space action, our construction provides a stepping stone for a geometric quantisation algorithm for histories, which can be naturally generalised to include parameterised systems, such as the relativistic particle we considered in section 5. For this reason it might prove relevant in the histories quantisation of general relativity [44]. In fact, the search for a geometric procedure for the quantisation of histories has been one of the main motivations of this work.…”

“…However, the distinction is natural in the histories framework and may be essential in general relativity [11,44] as any attempt to quantise this theory needs to deal with the "problem of time"…”

“…In classical mechanics and -to a large extent-in quantum theory this distinction is not essential: the timeparameter of Hamilton's or Heisenberg's time evolution is usually thought of as incorporating both notions of time. However, the distinction is natural in the histories framework and may be essential in general relativity [11,44] as any attempt to quantise this theory needs to deal with the "problem of time" 2 (see [14,15] for a review).…”

Quantum probabilities for time-extended alternatives

“…Therefore the form of the smearing function determines the physical interpretation of the observable we consider [18].…”

“…It will be convenient to represent the proposition about the time-averaged value of  by the operator D(a) := e −i ĤT Ĉ(a), (18) or else…”