Abstract
Different approaches to quantum gravity generally predict that the dimension of spacetime at the fundamental level is not 4. The principal tool to measure how the dimension changes between the IR and UV scales of the theory is the spectral dimension. On the other hand, the noncommutative-geometric perspective suggests that quantum spacetimes ought to be characterised by a discrete complex set -- the dimension spectrum. Here we show that these two notions complement each other and the dimension spectrum is very useful in unravelling the UV behaviour of the spectral dimension. We perform an extended analysis highlighting the trouble spots and illustrate the general results with two concrete examples: the quantum sphere and the $\kappa$-Minkowski spacetime, for a few different Laplacians. In particular, we find out that the spectral dimensions of the former exhibit log-periodic oscillations, the amplitude of which decays rapidly as the deformation parameter tends to the classical value. In contrast, no such oscillations occur for either of the three considered Laplacians on the $\kappa$-Minkowski spacetime.
Highlights
The concept of spacetime, understood as a differentiable manifold, has proven to be extremely fruitful in modeling gravitational phenomena
We show that these two notions complement each other and the dimension spectrum is very useful in unraveling the UV behavior of the spectral dimension
The purpose of this work is to revisit the concept of the spectral dimension from the perspective of the dimension spectrum. We show that the latter is a valuable rigorous tool to study the UV behavior of the spectral dimension
Summary
The concept of spacetime, understood as a differentiable manifold, has proven to be extremely fruitful in modeling gravitational phenomena. Complex dimensions (or complex critical exponents) and the corresponding log-periodic oscillations can arise in the systems with discrete scale invariance, which is observed in many contexts, including some particular cases of holography [28], as well as condensed matter physics, earthquakes and financial markets, see, e.g., [29] It has already been recognized by Connes and Moscovici in 1995 [30] that quantum spaces—understood as spaces determined by noncommutative algebras of observables—ought to be characterized by a discrete subset of the complex plane—the dimension spectrum, rather than a single number. V and discuss their consequences for model building in quantum gravity
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