arXiv subjects

Giuseppe Sellaroli

Publications and source records attributed to Giuseppe Sellaroli.

7 recordsLinked to original sources

arXiv2026

A Combinatorial Origin of Locality

Locality and unitarity are fundamental principles of quantum field theory. For tree-level scattering amplitudes, locality determines which propagator poles may occur together, while unitarity fixes their residues through factorization. Previous uniqueness theorems have repeatedly shown that unitarity can emerge from locality combined with other physical principles. In this Letter we give the first complete all-multiplicity proof for the emergence of locality in this setting. We focus on Tr($ϕ^3$) theory and phrase locality as a problem in geometric combinatorics. Planar propagators are chords of a polygon, and a denominator is local exactly when its chords triangulate the polygon, and non-local otherwise. We show that hidden zeros---regular kinematic conditions on which the amplitude vanishes---enforce pole compatibility and therefore imply a local singularity structure. The proof rests on two ingredients: a graphical \emph{star test} determines when a chosen zero eliminates a pole product, and a \emph{smoothing} procedure finds such a zero. Thus, once the planar pole alphabet and denominator bound are specified, locality and unitarity emerge together from a single on-shell principle: hidden zeros. Equivalently, our proof provides a novel characterization of a familiar concept: polygon triangulations are precisely the chord configurations that evade every star test.

hep-th↗

arXiv2017

An algorithm to reconstruct convex polyhedra from their face normals and areas

A well-known result in the study of convex polyhedra, due to Minkowski, is that a convex polyhedron is uniquely determined (up to translation) by the directions and areas of its faces. The theorem guarantees existence of the polyhedron associated to given face normals and areas, but does not provide a constructive way to find it explicitly. This article provides an algorithm to reconstruct 3D convex polyhedra from their face normals and areas, based on an method by Lasserre to compute the volume of a convex polyhedron in $\mathbb{R}^n$. A Python implementation of the algorithm is available at https://github.com/gsellaroli/polyhedrec.

cs.CG↗

arXiv2017

SO*(2N) coherent states for loop quantum gravity

A SU(2) intertwiner with N legs can be interpreted as the quantum state of a convex polyhedron with N faces (when working in 3d). We show that the intertwiner Hilbert space carries a representation of the non-compact group SO*(2N). This group can be viewed as the subgroup of the symplectic group Sp(4N,R) which preserves the SU(2) invariance. We construct the associated Perelomov coherent states and discuss the notion of semi-classical limit, which is more subtle that we could expect. Our work completes the work by Freidel and Livine which focused on the U(N) subgroup of SO*(2N).

math-ph↗

arXiv2016

Non-compact groups, tensor operators and applications to quantum gravity

This work focuses on non-compact groups and their applications to quantum gravity, mainly through the use of tensor operators. First, the mathematical theory of tensor operators for a Lie group is recast in a new way which is used to generalise the Wigner-Eckart theorem to non-compact groups. The result relies on the knowledge of the recoupling theory between finite-dimensional and infinite-dimensional irreducible representations of the group; here the previously unconsidered cases of the 3D and 4D Lorentz groups are investigated in detail. As an application, the Wigner-Eckart theorem is used to generalise the Jordan-Schwinger representation of SU(2) to both groups, for all representation classes. Next, the results obtained for the 3D Lorentz group are applied to (2+1) Lorentzian loop quantum gravity to develop an analogue of the well-known spinorial approach used in the Euclidean case. Tensor operators are used to construct observables and to generalise the Hamiltonian constraint introduced by Bonzom and Livine (2012) for 3D gravity to the Lorentzian case. The Ponzano-Regge amplitude is shown to be a solution of this constraint by recovering the (opportunely generalised) Biedenharn-Elliott relations. Finally, the focus is shifted on the intertwiner space based on SU(2) representations, widely used in loop quantum gravity. When working in the spinorial formalism, it has been shown that the Hilbert space of n-valent intertwiners with fixed total area is a representation of U(n). Here it is shown that the full space of all n-valent intertwiners forms an irreducible representation of the non-compact group SO*(2n). This fact is used to construct a new kind of coherent intertwiner state (in the sense of Perelomov). Hints of how these coherent states can be interpreted in the semi-classical limit as convex polyhedra are provided.

math-ph↗

arXiv2015

3d Lorentzian loop quantum gravity and the spinor approach

We consider the generalization of the "spinor approach" to the Lorentzian case, in the context of 3d loop quantum gravity with cosmological constant $Λ=0$. The key technical tool that allows this generalization is the recoupling theory between unitary infinite-dimensional representations and non-unitary finite-dimensional ones, obtained in the process of generalizing the Wigner-Eckart theorem to SU(1,1). We use SU(1,1) tensor operators to build observables and a solvable quantum Hamiltonian constraint, analogue of the one introduced by V. Bonzom and his collaborators in the Euclidean case (with both $Λ=0$ and $Λ\neq0$). We show that the Lorentzian Ponzano-Regge amplitude is solution of the quantum Hamiltonian constraint by recovering the Biedenharn-Elliott relation (generalized to the case where unitary and non-unitary SU(1,1) representations are coupled to each other). Our formalism is sufficiently general that both the Lorentzian and the Euclidean case can be recovered (with $Λ=0$).

gr-qc↗

arXiv2015

Wigner-Eckart theorem and Jordan-Schwinger representation for infinite-dimensional representations of the Lorentz group

The Wigner-Eckart theorem is a well known result for tensor operators of SU(2) and, more generally, any compact Lie group. This paper generalises it to arbitrary Lie groups, possibly non-compact. The result relies on knowledge of recoupling theory between finite-dimensional and arbitrary admissible representations, which may be infinite-dimensional; the particular case of the Lorentz group will be studied in detail. As an application, the Wigner-Eckart theorem will be used to construct an analogue of the Jordan-Schwinger representation, previously known only for finite-dimensional representations of the Lorentz group, valid for infinite-dimensional ones.

math-ph↗

arXiv2015

Wigner-Eckart theorem for the non-compact algebra sl(2,R)

The Wigner-Eckart theorem is a well known result for tensor operators of su(2) and, more generally, any compact Lie algebra. In this paper the theorem will be generalized to the particular non-compact case of sl(2,R). In order to do so, recoupling theory between representations that are not necessarily unitary will be studied, namely between finite-dimensional and infinite-dimensional representations. As an application, the Wigner-Eckart theorem will be used to construct an analogue of the Jordan-Schwinger representation, previously known only for representations in the discrete class, which also covers the continuous class.

math-ph↗