Classical versus quantum gravity

Is Einstein's metric theory of gravitation to be quantized to yield a complete and logically consistent picture of the geometry of the real world in the presence of quantized material sources? To answer this question, we give arguments that there is a consistent way to extend general relativity to small distances by incorporating further geometric quantities at the level of the connection into the theory and introducing corresponding field equations for their determination, allowing thereby the metric and the Levi-Civita connection to remain classical quantities. The dualism between matter and geometry is extended to quantized fields with the help of a Hibert bundle ? raised over a Riemann-Cartan spacetime. Quantized subnuclear matter fields (generalized quantum mechanical wave functions) are sections on ? which determine generalized bilinear currents acting as sourc currents for the bundle geometry at small distances. The established dualism between matter and the underlying bundle geometry contains general relativity as a classical part.     

Dark energy versus modified gravity

There is now strong observational evidence that the expansion of the Universe is accelerating. The standard explanation invokes an unknown "dark energy" component. But such scenarios are faced with serious theoretical problems, which has led to increased interest in models where instead general relativity is modified in a way that leads to the observed accelerated expansion. The question then arises whether the two scenarios can be distinguished. Here we show that this may not be so easy, demonstrating explicitly that a generalized dark energy model can match the growth rate of the Dvali-Gabadadze-Porrati model and reproduce the 3+1 dimensional metric perturbations. Cosmological observations are then unable to distinguish the two cases.     

Galilei invariance and superfluidity II

Continuing the (heuristic) analysis of the mathematical structure of the Landau excitations, we find that inone dimension they may be described by a vector bundle over the base space of the boosts. The total space is a direct integral of all irreducible representations (of a given class) of the Galilei group. The existence of an energy-momentum spectrum requires the action of the boosts to be non-linear. This action can also be formulated as a superselection rule.     

Quantum gravitational bremsstrahlung: massless versus massive gravity

The massive spin-2 quantum gauge theory previously developed is applied to calculate gravitational bremsstrahlung. It is shown that this theory is unique and free from defects. In particular, there is no strong coupling if the graviton mass becomes small. The cross sections go over smoothly into the ones of the massless theory in the limit of vanishing graviton mass. The massless cross sections are calculated for the full tensor theory.     

Poincaré transformations and Galilei transformations

Poincaré and Galilei transformations are seen to be contained in each other in one space dimension more.     

On the representations of deformed Galilei group

The projective representations of k-Galilei group G _k are found by contracting the relevant representations of –Poincare group. The projective multiplier is found. It is shown that it is not possible to replace the projective representations of G _k by vector representations of some its extension.     

An extended Galilei group and its application

The Galilei group is combined with two one-dimensional groups, to form a twelve-dimensional extended Galilei group. Irreducible representations of this group depend upon two indicesm, s that can, respectively, be interpreted as the mass and spin of a non-relativistic particle. It turns out that the true irreducible representations of the ordinary Galilei group correspond tom=0, and this explains why these representations have no physical interpretation.     

Integral versus local conservation laws in metric theories of gravity

Previously we considered extensions of metric theories of gravitation with non-zero divergence of the energy-momentum tensor. We confirm here that lack of local conservation laws does not exclude, a priori, existence of integral conservation laws.     

Induced Representations of the One-Dimensional Quantum Galilei Group

We study the representations of the quantum Galilei group by a suitable generalization of the Kirillov method on spaces of noncommutative functions. On these spaces, we determine a quasi-invariant measure with respect to the action of the quantum group by which we discuss unitary and irreducible representations. The latter are equivalent to representations on ², i.e. on the space of square summable functions on a one-dimensional lattice.     

Galilei Covariance Does Not Imply Minimal Electromagnetic Coupling

The postulate of Galilei covariance in one-particle classical and quantum mechanics is reinvestigated, with particular intent to correct some current misconceptions concerning the rǒle of minimal electromagnetic coupling in Galilei covariant theories.     

Orthosymplectic unification of the Galilei group and BRST

The inclusion of BRST generators into the Poincarè group in D dimensions is known to lead IOsp[D,2|2]. Similarly, conformal symmetry gets extended into Osp[D×1,3|2]. For the non-relativistic case we find that the Galilei symmetry gets extended, by inclusion of the BRST generators, into an orthosymplectic symmetry possessing Osp[D,1|2] as a subgroup. All such extensions express the possibility of formulating the classical theories in reparametrization invariant ways. They include besides the generators of the initial kinematical symmetry (Poincarè, or conformal, or Galilei), the generators of Parisi-Sourlas transformations. The extended symmetries follow directly through BRST quantization.     

On dynamical realizations of l-conformal Galilei groups

We consider the dynamics invariant under the action of l-conformal Galilei group using the method of nonlinear realizations. We find that by an appropriate choice of the coset space parametrization one can achieve the complete decoupling of the equations of motion. The Lagrangian and Hamiltonian are constructed. The results are compared with those obtained by Galajinsky and Masterov (2013) [14].     

Dark energy versus modified gravity: Impacts on measuring neutrino mass

In this paper,we make a comparison for the impacts of smooth dynamical dark energy,modified gravity,and interacting dark energy on the cosmological constraints on the total mass of active neutrinos.For definiteness,we consider theΛCDM model,the w CDM model,the f(R)model,and two typical interacting vacuum energy models,i.e.,the IΛCDM1 model with Q=βHρc and the IΛCDM2 model with Q=βHρΛ.In the cosmological fits,we use the Planck 2015 temperature and polarization data,in combination with other low-redshift observations including the baryon acoustic oscillations,the type Ia supernovae,the Hubble constant measurement,and the large-scale structure observations,such as the weak lensing as well as the redshift-space distortions.Besides,the Planck lensing measurement is also employed in this work.We find that,the w CDM model favors a higher upper limit on the neutrino mass compared to theΛCDM model,while the upper limit in the f(R)model is similar with that in theΛCDM model.For the interacting vacuum energy models,the IΛCDM1 model favors a higher upper limit on neutrino mass,while the IΛCDM2 model favors an identical neutrino mass with the case ofΛCDM.     

Conformal Galilei algebras,symmetric polynomials and singular vectors

We classify and explicitly describe homomorphisms of Verma modules for conformal Galilei algebras \({\mathfrak {cga}}_\ell (d,\mathbb {C})\) with \(d=1\) for any integer value \(\ell \in {\mathbb {N}}\). The homomorphisms are uniquely determined by singular vectors as solutions of certain differential operators of flag type and identified with specific polynomials arising as coefficients in the expansion of a parametric family of symmetric polynomials into power sum symmetric polynomials.     

On a condition characterizing the Lorentz and Galilei groups

We prove by topological and Lie group theoretical methods that, in the class of the subgroups of $\text{GL}\text{(n,}\text{R}\text{)(n ≥ 3)}$, the n-dimensional Lorentz and Galilei groups are characterized by a condition which is the n-dimensional generalization of an “isotropy of space axiom” previously introduced. The inclusion of the n = 3 case implies a complete solution for the hitherto unsolved problem of determining the subgroups of GL(3, R) compatible with the three-dimensional isotropy condition.     

Convenient versus unique effective action formalism in 2d dilaton-Maxwell quantum gravity

The structure of one-loop divergences of two-dimensional dilaton-Maxwell quantum gravity is investigated in two formalisms: one using a convenient effective action and the other a unique effective action. The one-loop divergences (including surface divergences) of the convenient effetive action are calculated in three different covariant gauges: (i) De Witt, (ii) Ω-degenerate De Witt, and (iii) simplest covariant. The on-shell effective action is given by surface divergences only (finiteness of theS-matrix), which yet depend upon the gauge condition choice. Off-shell renormalizability is discussed and classes of renormalizable dilaton and Maxwell potentials are found which coincide in the cases of convenient and unique effective actions. A detailed comparison of both situations, i.e. convenient vs. unique effective action, is given. As an extension of the procedure, the one-loop effective action in two-dimensional dilaton-Yang-Mills gravity is calculated.