Can quantum fluctuations differentiate between standard and unimodular gravity?

We formally prove the existence of a quantization procedure that makes the path integral of a general diffeomorphism-invariant theory of gravity, with fixed total spacetime volume, equivalent to that of its unimodular version. This is achieved by means of a partial gauge fixing of diffeomorphisms together with a careful definition of the unimodular measure. The statement holds also in the presence of matter. As an explicit example, we consider scalar-tensor theories and compute the corresponding logarithmic divergences in both settings. In spite of significant differences in the coupling of the scalar field to gravity, the results are equivalent for all couplings, including non-minimal ones.

Dual Numbers and Invariant Theory of the Euclidean Group with Applications to Robotics

<p>In this thesis we study the special Euclidean group SE(3) from two points of view, algebraic and geometric. From the algebraic point of view we introduce a dualisation procedure for SO(3;ℝ) invariants and obtain vector invariants of the adjoint action of SE(3) acting on multiple screws. In the case of three screws there are 14 basic vector invariants related by two basic syzygies. Moreover, we prove that any invariant of the same group under the same action can be expressed as a rational function evaluated on those 14 vector invariants. From the geometric point of view, we study the Denavit-Hartenberg parameters used in robotics, and calculate formulae for link lengths and offsets in terms of vector invariants of the adjoint action of SE(3). Moreover, we obtain a geometrical duality between the offsets and the link lengths, where the geometrical dual of an offset is a link length and vice versa.</p>

Dual Numbers and Invariant Theory of the Euclidean Group with Applications to Robotics

<p>In this thesis we study the special Euclidean group SE(3) from two points of view, algebraic and geometric. From the algebraic point of view we introduce a dualisation procedure for SO(3;ℝ) invariants and obtain vector invariants of the adjoint action of SE(3) acting on multiple screws. In the case of three screws there are 14 basic vector invariants related by two basic syzygies. Moreover, we prove that any invariant of the same group under the same action can be expressed as a rational function evaluated on those 14 vector invariants. From the geometric point of view, we study the Denavit-Hartenberg parameters used in robotics, and calculate formulae for link lengths and offsets in terms of vector invariants of the adjoint action of SE(3). Moreover, we obtain a geometrical duality between the offsets and the link lengths, where the geometrical dual of an offset is a link length and vice versa.</p>

Modular symmetry at level 6 and a new route towards finite modular groups

We propose to construct the finite modular groups from the quotient of two principal congruence subgroups as Γ(N′)/Γ(N″), and the modular group SL(2, ℤ) is ex- tended to a principal congruence subgroup Γ(N′). The original modular invariant theory is reproduced when N′ = 1. We perform a comprehensive study of $$ {\Gamma}_6^{\prime } $$ Γ 6 ′ modular symmetry corresponding to N′ = 1 and N″ = 6, five types of models for lepton masses and mixing with $$ {\Gamma}_6^{\prime } $$ Γ 6 ′ modular symmetry are discussed and some example models are studied numerically. The case of N′ = 2 and N″ = 6 is considered, the finite modular group is Γ(2)/Γ(6) ≅ T′, and a benchmark model is constructed.

Non-local probes of entanglement in the scale-invariant gravity

In this paper, we study the generic action for the scale-invariant theory of gravity and then by making use of the holographic methods, we compute some specific holographic measures of entanglement. Precisely, we calculate the entanglement entropy, mutual and tripartite information and show that the mutual information is always positive while the tripartite information becomes negative. This indeed recovers the monogamy property of mutual information in this context.