Physical model of dimensional regularization II: complex dimensions, non-scalar fields and the cosmological constant

Jonathan Schonfeld


In an earlier paper on the foundations of dimensional regularization, I formulated a model of a scalar quantum field whose propagator exhibits short-distance power-law screening with real positive exponent.  In this paper, I heuristically generalize the model so that the propagator exhibits power-law screening at short distance with complex exponent.  I further extend the model to Abelian gauge fields and Dirac spinors.  As an unexpected byproduct, the spinor case leads to interesting extensions of the “bag” boundary conditions for the Dirac and Weyl equations.  If the world really had complex dimension, it might explain in a natural way why the preponderance of observed fundamental interactions are renormalizable; and why the non-renormalizability of quantum gravity, which balances dimensional-regularization poles against a very weak coupling constant, is both acceptable and too small to be observed under ordinary circumstances.  It might also motivate the famous factor of 10-120 between the observed cosmological constant and naïve dimensional analysis.


Dimensional regularization; complex dimension; fractal; quantum gravity; boundary conditions; cosmological constant

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