Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1609.07795

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

Giuseppe Sellaroli
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math-phgr-qchep-thmath.MPmath.RT

Keep this discovery

BibTeXRIS

Giuseppe Sellaroli. 2016-09-25. Non-compact groups, tensor operators and applications to quantum gravity. https://arxiv.org/abs/1609.07795

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Why we should condition denoising diffusion generative models on windows of past observations

Data assimilation (DA) is, traditionally, a cycling process that relies on time-dependent priors to propagate information from past observations to future cycles. Using denoising diffusion generative modeling for DA is challenging because standard approaches use a fixed training data set, which in turn leads to a static prior that ignores information from past observations. Because past observations are ignored, DA systems with static priors lead to larger posterior errors than cycling DA systems. Incorporating time-dependent priors into generative models, however, requires expensive and frequent retraining. Motivated by linear systems theory - where the dependence of a prediction of a Kalman filter on past observations decays exponentially - we condition diffusion models on short windows of past observations. Specifically, we describe training procedures for two frameworks: a diffusion DA system predicting the current state given a set of past observations, and a diffusion ``direct observation prediction'' (DOP) system, predicting future observations given a set of past observations. Using a canonical linear system, we show that both systems can achieve the minimal posterior error characteristic of a fully-cycled DA/DOP system, without re-training, provided the time windows are long enough. The linear setup ensures analytical tractability, avoids confounding neural network training errors, and confirms that conditioning on windows of past observations is required for efficient and accurate diffusion-based DA or DOP.

math-ph↗
arXiv2026

The kinematic structures and the inertial geometry of a moving charge

We ask how much of the geometry a charged particle moves in is fixed by its motion, and how much a particle must bring. We ask of a symplectic structure only that it relate velocity to momentum as Hamilton's equations do, and we ask it of every energy at once. In particular, we show that the structures meeting that demand are the canonical one and its twists by a closed two-form of the base. A field provides the two-form, a particle the multiplier before it, which we identify constitutively with its charge. Thus, a single energy governs a family of structures, and each particle takes the one its charge fixes. We then ask what a particle must bring to be given a momentum, and we show that the degree of that map settles the degree at which a field enters Newton's Second Law. An antisymmetric bilinear form returns no Lorentz force, whilst a Randers metric returns one --- a length whose difference from a Riemannian one is linear in the velocity. Moreover, we find that metric already within the twisted structure, as its primitive over a level of the free energy, and its law of transport to be nonlinear, no affine connection being known to serve. Under an indefinite signature the length parts from the dynamics, and the extremals turn from shortest to longest. On the round sphere a monopole flux leaves no such metric, whilst the transport remains and prequantisation, given a unit of action, restricts the charge to a lattice. In this manner, we conclude that each charge-to-mass ratio receives a geometry of its own, so that by a functionalist criterion none of them is the spacetime of a charged particle.

math-ph↗