We describe a method of construction of gauge-invariant operators (Dirac observables or "evolving constants of motion") from the knowledge of the eigenstates of the gauge generator of time-reparametrisation invariant mechanical systems. These invariant operators evolve unitarily with respect to an arbitrarily chosen time variable. We emphasise that the dynamics is relational, both in the classical and quantum theories. In this framework, we show how the "emergent WKB time" often employed in quantum cosmology arises from a weak-coupling expansion of invariant transition amplitudes, and we illustrate an example of singularity avoidance in a vacuum Bianchi I (Kasner) model. * Gauge conditions of the form given in (5) are sufficient for our purposes, although more general gauge conditions are possible [12].† This corresponds to the statement that invariant extensions are obtained by writing gauge-fixed quantities in an arbitrary gauge.

We present a new method for renormalisation group improvement of the effective potential of a quantum field theory with an arbitrary number of scalar fields. The method amounts to solving the renormalisation group equation for the effective potential with the boundary conditions chosen on the hypersurface where quantum corrections vanish. This hypersurface is defined through a suitable choice of a field-dependent value for the renormalisation scale. The method can be applied to any order in perturbation theory and it is a generalisation of the standard procedure valid for the one-field case. In our method, however, the choice of the renormalisation scale does not eliminate individual logarithmic terms but rather the entire loop corrections to the effective potential. It allows us to evaluate the improved effective potential for arbitrary values of the scalar fields using the tree-level potential with running coupling constants as long as they remain perturbative. This opens the possibility of studying various applications which require an analysis of multi-field effective potentials across different energy scales. In particular, the issue of stability of the scalar potential can be easily studied beyond tree level.

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.

Radiative symmetry breaking (RSB) is a theoretically appealing framework for the generation of mass scales through quantum effects. It can be successfully implemented in models with extended scalar and gauge sectors. We provide a systematic analysis of RSB in such models: we review the common approximative methods of studying RSB, emphasising their limits of applicability and discuss the relevance of the relative magnitudes of tree-level and loop contributions as well as the dependence of the results on the renormalisation scale. The general considerations are exemplified within the context of the conformal Standard Model extended with a scalar doublet of a new SU(2) X gauge group, the so-called SU(2)cSM. We show that various perturbative methods of studying RSB may yield significantly different results due to renormalisation-scale dependence. Implementing the renormalisation-group (RG) improvement method recently developed in ref.[1], which is well-suited for multi-scale models, we argue that the use of the RG improved effective potential can alleviate this scale dependence providing more reliable results.

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We show that the usual Born–Oppenheimer type of approximation used in quantum gravity, in which a semiclassical time parameter emerges from a weak-coupling expansion of the Wheeler–DeWitt constraint, leads to a unitary theory at least up to the next-to-leading order in minisuperspace models. As there are no unitarity-violating terms, this settles the issue of unitarity at this order, which has been much debated in the literature. Furthermore, we also show that the conserved inner product is gauge-fixed in the sense that the measure is related to the Faddeev–Popov determinant associated with the choice of semiclassical time as a reparametrization gauge. This implies that the Born–Oppenheimer approach to the problem of time is, in fact, an instance of a relational quantum theory, in which transition amplitudes can be related to conditional probabilities.