Switching Internal Times and a New Perspective on the ‘Wave Function of the Universe’

Abstract

Despite its importance in general relativity, a quantum notion of general covariance has not yet been established in quantum gravity and cosmology, where, given the a priori absence of coordinates, it is necessary to replace classical frames with dynamical quantum reference systems. As such, quantum general covariance bears on the ability to consistently switch between the descriptions of the same physics relative to arbitrary choices of quantum reference system. Recently, a systematic approach for such switches has been developed [1-3]. It links the descriptions relative to different choices of quantum reference system, identified as the correspondingly reduced quantum theories, via the reference-system-neutral Dirac quantization, in analogy to coordinate changes on a manifold. In this work, we apply this method to a simple cosmological model to demonstrate how to consistently switch between different internal time choices in quantum cosmology. We substantiate the argument that the conjunction of Dirac and reduced quantized versions of the theory defines a complete relational quantum theory that not only admits a quantum general covariance, but, we argue, also suggests a new perspective on the 'wave function of the universe'. It assumes the role of a perspective-neutral global state, without immediate physical interpretation, that, however, encodes all the descriptions of the universe relative to all possible choices of reference system at once and constitutes the crucial link between these internal perspectives. While, for simplicity, we use the Wheeler-DeWitt formulation, the method and arguments might be also adaptable to loop quantum cosmology.

Construction of quantum Dirac observables and the emergence of WKB time

“…Although there is currently no consensus about the correct way to quantise the gravitational field, promising approaches can be developed if the gauge symmetry of the theory is genuinely understood. Following [15,[27][28][29][30][31][32][33][34][35][36][37][38], we have taken the view that the dynamical content of a generally covariant theory is relational and can be comprehended through gauge-invariant extensions of the dynamical variables. Classically, such extensions can be constructed by the "Faddev-Popov procedure", which expresses gauge-fixed variables in an arbitrary gauge by means of integral formulas.…”

“…Indeed, for the case of the relativistic particle (which we will analyse in section 3), the operator definition chosen in [73] yieldsÔ[1|q 0 = cs] = sgn(p 0 ) =1. We believe that (38) should be the correct requirement. In fact, eq.…”

“…This result, although expected, clarifies the relation between the "semiclassical approach to the problem of time" and the more complete quantum theory based on the induced inner product * . Moreover, we discuss the dynamics of quantum Dirac observables also in the non-relativistic limit to compare the formalism here presented with the results of [37,38].…”

“…The dynamics is not "frozen", but rather encoded in the relational evolution of Dirac observables. Variations of this argument have been explored in the literature [27][28][29][30][31][32][33][34][35][36][37][38], but a systematic way to construct Dirac observables is lacking. Moreover, the explicit connection between this relational view and other approaches to the "problem of time" has remained unclear.…”

A change of perspective: switching quantum reference frames via a perspective-neutral framework

“…However, in generic systems, globally valid perspectives of quantum reference frames will become impossible (this is related to the Gribov problem in gauge theories) and, accordingly, such transformations between them will likewise not be of global validity. This is illustrated in the companion paper [2], where we extend our discussion to the three-dimensional N -body problem, and in [3,4], where it will be shown how to change relational quantum clocks, using our new method. In particular, in contrast to [22][23][24], these clock changes will also be valid beyond a semiclassical regime.…”