Gauge Fixing and the Semiclassical Interpretation of Quantum Cosmology

Abstract

We make a critical review of the semiclassical interpretation of quantum cosmology and emphasise that it is not necessary to consider that a concept of time emerges only when the gravitational field is (semi)classical. We show that the usual results of the semiclassical interpretation, and its generalisation known as the Born-Oppenheimer approach to quantum cosmology, can be obtained by gauge fixing, both at the classical and quantum levels. By 'gauge fixing' we mean a particular choice of the time coordinate, which determines the arbitrary Lagrange multiplier that appears in Hamilton's equations. In the quantum theory, we adopt a tentative definition of the (Klein-Gordon) inner product, which is positive definite for solutions of the quantum constraint equation found via an iterative procedure that corresponds to a weak coupling expansion in powers of the inverse Planck mass. We conclude that the wave function should be interpreted as a state vector for both gravitational and matter degrees of freedom, the dynamics of which is unitary with respect to the chosen inner product and time variable.

Construction of quantum Dirac observables and the emergence of WKB time

“…(56) together with (48) implies that sgn(ṫ) = −sgn(p t ) = σ = const., which leads to |t(b) − t(a)| = σ(t(b) − t(a)). Using (56) and (57), we can now insert the relational solution (50) in the integrand of (47) to obtain the on-shell action:…”

“…When higher orders in 1 c 2 are included, S σ solves corrected Newtonian constraints, where the corrections come from the expansion of the square-root in (56). The same expansion procedure can be performed for any minisuperspace model of cosmology with a non-vanishing potential and, formally, for the field-theoretical case [50]. The corrected Newtonain constraint leads to a corrected Schrödinger equation in the quantum theory.…”

“…For this reason, this emergent semiclassical time is often referred to as "WKB time" [39,40,49]. Recently [50], the author has argued that such a "semiclassical approach to the problem of time" coincides with a particular choice of gauge (i.e., time coordinate) and can be extended beyond the semiclassical level.…”

“…The second objetive is a continuation of [50], in which it was extensively argued how the results of the usual "semiclassical approach to the problem of time" can be recovered from a complete quantum theory where the notion of "gauge fixing" was paramount. The work of [50] was, however, limited by the use of the indefinite Klein-Gordon inner product in the Hilbert space of physical states.…”

“…The second objetive is a continuation of [50], in which it was extensively argued how the results of the usual "semiclassical approach to the problem of time" can be recovered from a complete quantum theory where the notion of "gauge fixing" was paramount. The work of [50] was, however, limited by the use of the indefinite Klein-Gordon inner product in the Hilbert space of physical states. In the present article, we adopt the positive-definite Rieffel induced inner product [53][54][55][56][57][58][59] (see also [60][61][62][63]), and we show how the emergence of "WKB time" occurs in the simple example of a relativistic particle, which is sufficient to illustrate the connection between the "semiclassical time" and the exact relational dynamics at the fully quantum level.…”