The title of this paper is misleading. Fortunately, its contents 
is interesting and it is not difficult to re-arrange the message 
in comprehensible form. One simply has to start - and continue - 
reading from the end. The last Appendix B introduces the first 
idea by reminding us that it is easy to generalize exponentials 
exp(x) to Mittag-Lefler functions E(x) defined by Taylor series 
with a new free parameter $\nu$. The second idea may be found in 
the Appendix A. There, one imagines that the relationship between 
the exponentials exp(x) and certain elementary differential 
equations may be also paralleled by a similar relationship 
between the Mittag-Lefler functions E(x) and certain less 
elementary integro-differential equations called, for the sake of 
brevity (though at an expense of clarity) fractional-differential 
equations. Then, the third input idea appears, in the 
rationalized ordering of my own re-presentation, in section VI 
where one imagines that the Schroedinger and Klein-Gordon 
description of the time-evolution of the system (i.e., the 
respective first- and second-order differential equations with 
respect to time) may be interpreted as the two specific 
integer-parameter special cases of the generalized, 
``fractional-differential" equations of the above-mentioned type. 
Now the reader is prepared to start experimenting with the latter 
generalized equations in special cases, being assisted by the two 
most elementary examples as outlined in sections IV and V. 
Finally, being already firmly aware that all that nice game has 
definitely nothing in common with physics, the kind reader may 
continue reading section III where the fractional derivative is 
decomposed into integral derivative plus a remainder. For the 
sake of brevity, we are offered just the 
Schroedinger-like-looking option with the order of the separated 
differentiation equal to one, and we are informed that the 
residual term has almost as many nasty properties as many of us 
have already expected. Still, being pretty sure now that all the 
objects with pretty names (like ``probability density" etc) have 
really nothing in common with the standard physics (while no 
alternative physics can be offered within 14 pages of the text of 
course), we feel no objections reading, finally, sections II 
(clarifying the ambiguities and the correct usage of units) and 
Introduction (where we are painfully guided towards a few further 
speculative ideas about possible connections of what we just read 
to several areas of the real physics ranging from the problems of 
diffusion (in condensed matter etc) and of random walks till 
non-Markovian processes and, perhaps, also quantum mechanics. 
Feeling satisfied by what we were told, consulting a few 
recommended related references could be our next step.

MR2077514 Naber, Mark . Time fractional Schrödinger equation. J. 
Math. Phys. 45 (2004), no. 8, 3339--3352.