Fractal propagators in QED and QCD and implications for the problem of confinement

Abstract

We show that QED radiative corrections change the propagator of a charged Dirac particle so that it acquires a fractional anomalous exponent connected with the fine structure constant. The result is a nonlocal object which represents a particle with a roughened trajectory whose fractal dimension can be calculated. This represents a significant shift from the traditional Wigner notions of asymptotic states with sharp well-defined masses. Nonabelian long-range fields are more difficult to handle, but we are able to calculate the effects due to Newtonian gravitational corrections. We suggest a new approach to confinement in QCD based on a particle trajectory acquiring a fractal dimension which goes to zero in the infrared as a consequence of self-interaction, representing a particle which, in the infrared limit, cannot propagate.

Particles, Fields, Pomerons and Beyond

“…So much for free particles! As discussed in [12], the corresponding corrections due gravity can also be calculated in the Newtonian limit and one finds a similar, though much smaller correction. Nevertheless, as a matter of principle, any particle coupled to an infinite-range field cannot reasonably be expected to have a propagator with simple poles.…”

“…An extremely interesting point raised in [12] is that if one considers a "red" (say) quark, even propagating in empty space (probably a bad approximation, as we will see later) will get corrections with a γ of opposite sign which could be large enough to force the exponent to which the propagator is raised to zero and drop the path dimension also to zero -the quark would be unable to propagate in the IR limit due to interactions with its own glue -it would be confined! It has not been possible to make this argument rigorous yet, but it certainly is an interesting way to think about confinement and how a particle could be effectively unable to propagate in the IR limit.…”

Spin-statistics violations in superstring theory

“…Regardless of the details of the instantonlike solution, we see the propagator will necessarily be modified into something nonlocal, requiring an infinite number of derivatives. It is probably not coincidence that such solutions likely require coupling to a gauge field via (3.6), as happens in field theory [22], allowing one to evade the spin-statistics theorem [23]:…”

Casimir Effect Associated with Fractional Klein–Gordon Field

“…There also exists work on axiomatic approach to fractional Klein-Gordon field, where properties of the n-point Schwinger or Euclidean Green functions and their analytic continuation to the corresponding n-point Wightman functions were given 55,56 . Fractal fermion propagator has been obtained as a consequence of the QED radiative corrections, which requires the incorporation into the propagator with a fractional exponent connected with the fine structure constant 57 . More recent work is related to the Casimir effect for the massless and massive fractional fields at zero and positive temperature; and work on Casimir effect due to fractional Klein-Gordon field subject to fractional Neumann boundary conditions which interpolate between the usual Dirichlet and Neumann conditions have been carried out 58 .…”