Restricted Weyl symmetry and spontaneous symmetry breakdown of conformal symmetry

In this paper, we elucidate the relation between the restricted Weyl symmetry and spontaneous symmetry breakdown of conformal symmetry. Using a scalar–tensor gravity, we show that the restricted Weyl symmetry leads to spontaneous symmetry breakdown of a global scale symmetry when the vacuum expectation value of a scalar field takes a nonzero value. It is then shown that this spontaneous symmetry breakdown induces spontaneous symmetry breakdown of special conformal symmetry in a flat Minkowski spacetime, but the resultant Nambu–Goldstone boson is not an independent physical mode but expressed in terms of the derivative of the dilaton which is the Nambu–Goldstone boson of the global scale symmetry. In other words, the theories which are invariant under the general coordinate transformation and the restricted Weyl transformation exhibit a Nambu–Goldstone phase where both special conformal transformation and dilatation are spontaneously broken while preserving the Poincaré symmetry.

Quadratic gravity and restricted Weyl symmetry

In this paper, we investigate the relationship between quadratic gravity and a restricted Weyl symmetry where a gauge parameter [Formula: see text] of Weyl transformation satisfies a constraint [Formula: see text] in a curved spacetime. First, we briefly review a model with a restricted gauge symmetry on the basis of QED, where a [Formula: see text] gauge parameter [Formula: see text] obeys a similar constraint [Formula: see text] in a flat Minkowski spacetime, and explain that the restricted gauge symmetry removes one on-shell mode of gauge field, which together with the Feynman gauge leaves only two transverse polarizations as physical states. Next, it is shown that the restricted Weyl symmetry also eliminates one component of a dipole field in quadratic gravity around a flat Minkowski background, leaving only a single scalar state. Finally, we show that the restricted Weyl symmetry cannot remove any dynamical degrees of freedom in static background metrics by using the zero-energy theorem of quadratic gravity. This fact also holds for the Euclidean background metrics without imposing the static condition.

SU(2) Symmetry of Qubit States and Heisenberg–Weyl Symmetry of Systems with Continuous Variables in the Probability Representation of Quantum Mechanics

In view of the probabilistic quantizer–dequantizer operators introduced, the qubit states (spin-1/2 particle states, two-level atom states) realizing the irreducible representation of the S U ( 2 ) symmetry group are identified with probability distributions (including the conditional ones) of classical-like dichotomic random variables. The dichotomic random variables are spin-1/2 particle projections m = ± 1 / 2 onto three perpendicular directions in the space. The invertible maps of qubit density operators onto fair probability distributions are constructed. In the suggested probability representation of quantum states, the Schrödinger and von Neumann equations for the state vectors and density operators are presented in explicit forms of the linear classical-like kinetic equations for the probability distributions of random variables. The star-product and quantizer–dequantizer formalisms are used to study the qubit properties; such formalisms are discussed for photon tomographic probability distribution and its correspondence to the Heisenberg–Weyl symmetry properties.

Selected topics in scalar–tensor theories and beyond

Scalar fields have played an important role in the development of the fundamental theories of physics as well as in other branches of physics such as gravitation and cosmology. For a long time, these escaped detection until 2012 when the Higgs boson was observed for the first time. Since then, alternatives to the general theory of relativity like the Brans–Dicke (BD) theory, scalar–tensor theories of gravity and their higher derivative generalizations — collectively known as Horndeski theories — have acquired renewed interest. In the present review, we discuss several selected topics regarding these theories, mainly from the theoretical perspective but with due mention of the observational aspect. Among the topics covered in this review, we pay special attention to the following: (1) the asymptotic dynamics of cosmological models based on the BD, scalar–tensor and Horndeski theories, (2) inflationary models, extended quintessence and the Galileons, with emphasis in causality and stability issues, (3) the chameleon and Vainshtein screening mechanisms that may allow the elusive scalar field to evade the tight observational constraints implied by the solar system experiments, (4) the conformal frames conundrum with a brief discussion on the disformal transformations and (5) the role of Weyl symmetry and scale invariance in the gravitation theories. The review is aimed at specialists as well as at nonspecialists in the subject, including postgraduate students.