Abstract
We explicitly construct fractals of dimension 4-epsilon on which dimensional regularization approximates scalar-field-only quantum-field-theory amplitudes. The construction does not require fractals to be Lorentz-invariant in any sense, and we argue that there probably is no Lorentz-invariant fractal of dimension greater than 2. We derive dimensional regularization's power-law screening first for fractals obtained by removing voids from 3-dimensional Euclidean space. The derivation applies techniques from elementary dielectric theory. Surprisingly, fractal geometry by itself does not guarantee the appropriate power-law behavior; boundary conditions at fractal voids also play an important role. We then extend the derivation to 4-dimensional Minkowski space. We comment on generalization to nonscalar fields, and speculate about implications for quantum gravity.
Highlights
Is “dimension deficit” really the correct physical meaning of the parameter ε in dimensional regularization? The only way to prove it is by explicitly constructing a fractal spacetime on which dimensional regularization approximates quantumfield amplitudes
Introducing such a construction for scalaronly quantum-field theories is the purpose of this paper. This is the first step in a longer research program aimed at extending this construction to non-scalar fields
Even if that does not materialize, the scalar construction should be of interest in its own right, as it casts a fresh light on the foundations of dimensional regularization, one of the cornerstones of modern quantum-field theory
Summary
The only way to prove it is by explicitly constructing a fractal spacetime on which dimensional regularization approximates quantumfield amplitudes. Introducing such a construction for scalaronly quantum-field theories is the purpose of this paper. This is the first step in a longer research program aimed at extending this construction to non-scalar fields.
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