The Noncommutative Integral

On a spectral geometry approach to Hořava–Lifshitz gravity: a spectral dimension

“…The task of the spectral geometry is extracting geometrical information about a manifold from the spectral properties of some differential operators. There are many references on spectral (or non-commutative) geometry [6,10,13,14]. Here, we do not need the whole machinery, so we just briefly review the main idea.…”

“…Toward this end, one tries to obtain geometric information from the spectrum of some relevant operator (Laplacian or Dirac). For example, it is well known (see, e.g., [6]) that the Riemannian geometry of a closed manifold, M, can be completely recovered from the so-called spectral triple, (A, H, D), where A = C ∞ (M), H = L 2 (M, S)-the Hilbert space of spinors on M, and D = D / = iγ μ (∂ μ + ω μ )-a standard Dirac operator on M. Using more general spectral triples, (A, H, D) (subject to some natural conditions), allows us to construct generalized geometries in purely algebraic way, including cases when the geometric construction either does not exist or obscure. Moreover, because the Dirac operator is also very important from the physics point of view, this shows that choosing different Dirac operators, i.e.…”

“…The strongest motivation to use (11) as the definition of dimension comes from the Dixmier trace [16], which is a natural generalization of the integral over manifold. In the construction of this integral the role of the volume form is played by |D /| −n , where n is given by (11) (see [6] for details and also some further comments in the next section).…”

“…From the geometrical perspective, the key difference between Einstein's general relativity and the HL model is that we now have a foliated manifold instead of just a manifold. It is well known that the geometrical information, in the case of the usual differential and Riemannian geometry, can be completely recovered from the knowledge of a standard Dirac operator (plus an algebra of functions and a Hilbert space, but these are needed already on the level of topology) [6]. When some additional structure, as foliation, is present the corresponding Dirac operator should respect it.…”

Phase transitions of a GUP-corrected Schwarzschild black hole within isothermal cavities

“…The most interesting one of all quantum gravity candidates is the existence of a minimal observable length of the order of the Planck length. The idea of a minimal length can be modeled in terms of a quantized spacetime and goes back to the early days of quantum field theory [2] (see also [3][4][5]). An alternative approach consists in promoting the Heisenberg uncertainty principle (HUP) to the generalized uncertainty principle (GUP) [6,7].…”