@ Victor,

Why don't you submit your work to a decent peer-reviewed journal. Most of the equations (ones with fractional-order derivatives) in your work are dimensionally inconsistent and therefore wrong. Check it yourself!

One has to be careful with fractional order equations. Please write only equations which fulfil the physical nessities. Principal I think it should be possible.  The Standard model is for me the only right model. All others have prionciple weaknesses. 

It is difficult enough to comprehend physics without the application of fractals. Why do you need fractals?

This question is natural  and should be answered before thinking on such kind of generalizations. Otherwise there exist enough models  in biomath, mathematical physics where fractional order differential operators makes sense. 

The foundations of physical reality are not accessible for observations. Investigation of these foundations can only be achieved by mathematical test models. The foundation must be simple and therefore it must be easily comprehensible by skilled scientists, but it is not sensible to start that investigation by applying exotic methods. Some test models exist. A well known example is the orthomodular lattice, which is better known as quantum logic and which was already discovered by John von Neumann and Garret Birkhoff. See the link below,

http://arxiv.org/abs/1101.5690

Why would anybody be interested in QED in a fractal medium? Which medium will be fractal; what is it supposed to describe?

A fractal medium would break global Poincaré invariance explicitly-and gauge transformations, that rely on the definition of a derivative, are problematic-so whatever the theory is, it isn't quantum electrodynamics. Of course, using a lattice approximation to a fractal, it is possible to assign U(1) valued variables to links  and construct U(1) invariant plaquette variables; and assign correspondingly  charged degrees of freedom to the vertices; however, such a construction is but an approximation to a fractal and, whether the theory, defined on such a lattice, possesses a Poincaré invariant scaling limit,  is doubtful. The reason is  that, in that case, the fractal properties wouldn't survive and be but artifacts of the regularization. If the scaling limit isn't Poincaré invariant, then the theory isn't electrodynamics.

The confusion, also,  results from forgetting that space and spacetime are different things. 

This discussion is depressing! I sign out.

Stam Nicolis' response above is correct.

However, the only instance where all consistency requirements mandated by QFT (including compliance with Poincaré and gauge symmetries) are asymptotically met, is a spacetime manifold endowed with arbitrarily small deviations from four dimensions (epsilon = 4 - D

Once more: if the ``good set'' of correlation functions can be defined with the symmetries of Minkowski spacetime, then the fractal properties have been eliminated, therefore they don't play any role. Minkowski spacetime isn't fractal.

Not all quantum field theories are invariant under Lorentz transformations-nor are all Lorentz invariant classical theories the limits of quantum field theories. Classical electrodynamics, however, is invariant under Lorentz transformations and is the limit of quantum electrodynamics that's, also, Lorentz invariant, therefore fractal properties of spacetime are not relevant for it. 

Once more: Electrodynamics has certain properties: U(1) gauge invariance and Poincaré invariance. A theory that doesn't have one of them isn't electrodynamics-it's something else. A fractal worldvolume is not invariant under either translations or Lorentz transformations, but under other transformations. These transformations induce deviations from Lorentz covariance of the fields. And the difficulty of defining translations means that gauge transformations have problems. There do exist lattice approximations to fractal worldvolumes-but these depend on cutoffs and the lattice structure used. Electrodynamics, whether classical or quantum, does not. How the fine structure constant depends on energy is well known and, at the energies where experiments and calculations are possible, no evidence of deviations from the Standard Model has been found. There isn't any particular value of the fine structure constant-that statement is known to be wrong.  The fine structure constant is the coupling constant of the electromagnetic interactions and its value depends on the energy scale, as predicted by the Standard Model-and this dependence can be calculated and measured. Its only particular value would be a zero of its β function-but that isn't known, since the theory isn't asymptotically free and doesn't make sense by itself at high energies, where weak interaction effects become relevant. It's a fact that its value at atomic scales is small enough that perturbation theory works very well.

Disordered materials can have fractal structure and it is possible to ask about their electromagnetic properties-and these have been extensively studied. However, in these materials, the fractal structure is embedded in a Minkowski spacetime and there are cutoffs, defined by the material. And the energy scales are such that the shot noise of the photons is decoupled from the shot noise of the charges. The threshold for pair production is much beyond the endurance of the materials. That's why it doesn't make sense to worry about a régime where the two kinds of shot noise might be coupled-which would be what could be called quantum electrodynamics in fractal media.

Stam Nicolis,

You are indeed right in your assessment. There is no doubt that any theory targeting physics beyond the Standard Model (BSM) must comply with all consistency requirements mandated by relativistic QFT in the perturbative régime of low-energies, in particular local gauge and Poincaré symmetries.

Having said that, the daunting task is to convincingly explain the vast array of open challenges facing both the Standard Model and QFT beyond the well-behaved approximations of effective Lagrangian formulations. To give only a handful of examples, the problems related to triviality/Landau poles, the absence of non-perturbative regulators in chiral gauge theories, the naturalness problems linked to the Higgs sector, the correct non-perturbative descriptions of the Renormalization Group and strongly-coupled sector (and so on).  

This is where the science of nonlinear dynamics and complexity may bring useful insights that should not be easily discarded. There is a lot of work under development in these areas and only time will tell if these lines of research will pan out. In particular, the asymptotic approach to self-similarity and the role of fractal dimensions in the transition from smooth to non-differentiable spacetime manifolds are currently topics pursued by several research groups.

The claimed challenges don't have anything to do with the fractal properties of spacetime.  The putative Landau pole doesn't have anything to do with the subject and, while there is  certain expectation that the φ4 theory, by itself,  is, in fact, a free theory, also, in four spacetime dimensions, there isn't any expectation that, within the Standard Model, it remains so and there is a wealth of evidence, theoretical and experimental, that it doesn't. 

It is known how to formulate gauge theories, coupled to fermions in a complex representation of the gauge group, in a non-perturbative way, for more than twenty years-the issues aren't of principle, but of practice, how to perform the calculations efficiently. 

The discovery of the Higgs implies, inevitably, the existence of new degrees of freedom. However it hasn't led to the deviations that a fractal structure of spacetime  at LHC energies would imply. These are relevant for describing gravitational effects, which are completely irrelevant at the scales probed by the LHC, from the measurements available to date. 

Sam Nicolis,

You are certainly entitled to your opinion, but I disagree with your viewpoint. The onset of spacetime having arbitrarily small deviations from four-dimensions begins to develop above the electroweak scale, without impacting the Standard Model predictions. Analysis shows that the onset of these deviations provides valuable clues on a number of unsolved puzzles associated with the Standard Model. Numerical results are promising because they show good agreement with all experimental observations recorded to date.  

Unfortunately, these topics are too complex to be debated here. Let's agree to disagree.

The statements about Lorentz violations-this is what a deviation from four spacetime dimensions *means*-aren't a matter of personal opinion, but of impersonal calculation and measurement. If they don't impact the Standard Model predictions, this *means* they're consistent with the statement that there aren't any such deviations at the scale probed. Any such deviations would, also, require an explanation for how come they hadn't affected any other calculations or measurements. There's been a lot of work in describing Lorentz violating effects in the Standard Model, e.g. http://www.physics.indiana.edu/~kostelec/faq.html

Stam,

All consistency requirements mandated by the Standard Model (including Lorentz invariance) are met, if (and only if) the deviations from four dimensions are continuous and arbitrarily small (

This is my last reply to Stam Nicolis.

"If they don't impact the Standard Model predictions, this *means* they're consistent with the statement that there aren't any such deviations at the scale probed. Any such deviations would, also, require an explanation for how come they hadn't affected any other calculations or measurements.”

As shown in my recent book, the reason why the arbitrarily small dimensional deviations don’t impact the SM predictions is that their magnitudes are on the order of magnitude of (electroweak scale / Planck scale) ^2 and are thus strongly suppressed in the scattering amplitude expansion.

The very existence of such deviations stems from the behavior of the RG flow near fixed points and from the intrinsic scale-invariant properties of the flow near its fixed points. Numerical results derived from this approach are in good agreement with observational data and clarify some of the open questions left unanswered by the SM.       

The short list of electric charges and color charges that occur in the standard model are a direct corollary of the symmetry flavors that quaternionic number systems can represent. Why these symmetry flavors produce these charges can be derived from the application of the generalized Stokes theorem to a quaternionic model. Multidimensional integration appears to be sensitive to the symmetries of coexisting parameter spaces.