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High Energy Physics - Theory, hep-th,General Relativity and Quantum Cosmology, gr-qc,Mathematical Physics, math-ph,Mathematics, Mathematical Physics, math.MP
Abstract:
We construct a theory of fields living on continuous geometries with
fractional Hausdorff and spectral dimensions, focussing on a flat background
analogous to Minkowski spacetime. After reviewing the properties of fractional
spaces with fixed dimension, presented in a companion paper, we generalize to a
multi-fractional scenario inspired by multi-fractal geometry, where the
dimension changes with the scale. This is related to the renormalization group
properties of fractional field theories, illustrated by the example of a scalar
field. Depending on the symmetries of the Lagrangian, one can define two
models. In one, the scalar has a continuum of massive modes, while in the other
it only has a mass pole. If the effective dimension flows from 2 in the
ultraviolet (UV), geometry constrains the infrared limit to be
four-dimensional. At the UV critical value, the model is rendered
power-counting renormalizable. However, this is not the most fundamental
regime. Compelling arguments of fractal geometry require an extension of the
fractional action measure to complex order. In doing so, we obtain a hierarchy
of scales characterizing different geometric regimes. At very small scales,
discrete symmetries emerge and the notion of a continuous spacetime begins to
blur, until one reaches a fundamental scale and an ultra-microscopic fractal
structure. This fine hierarchy of geometries has implications for
non-commutative theories and discrete quantum gravity. In the latter case, the
present model can be viewed as a top-down realization of a quantum-discrete to
classical-continuum transition.
