Spectral dimensions from the spectral action

Abstract

The generalised spectral dimension $D_S(T)$ provides a powerful tool for comparing different approaches to quantum gravity. In this work, we apply this formalism to the classical spectral actions obtained within the framework of almost-commutative geometry. Analysing the propagation of spin-0, spin-1 and spin-2 fields, we show that a non-trivial spectral dimension arises already at the classical level. The effective field theory interpretation of the spectral action yields plateau-structures interpolating between a fixed spin-independent $D_S(T)=d_S$ for short and $D_S(T)=4$ for long diffusion times $T$. Going beyond effective field theory the spectral dimension is completely dominated by the high-momentum properties of the spectral action, yielding $D_S(T)=0$ for all spins. Our results support earlier claims that high-energy bosons do not propagate.

arXiv comment9 pages, 3 figures; version to be published in Physical Review D

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