[400.2.1.1] What does it mean to replace an ordinary time derivative
with a fractional derivative ?
[400.2.1.2] Are fractional time derivatives the infinitesimal
generators of translations or other symmetry
transformations, and, if yes, what is their nature ?
[400.2.1.3] Which fractional derivative should be used ?

[400.2.2.1] These questions have been generally neglected by all
workers in the field, and were only recently addressed and
answered in Refs. [21, 22, 4, 5, 3].
[400.2.2.2] It was found that generalized fractional time
evolutions Tα, whose infinitesimal generators are
fractional time derivatives of order α, arise very
generally in the transition between microscopic and macroscopic
time scales.
[400.2.2.3] The fractional time evolution Tαt for duration
t is defined through its action on an observable ft0
depending on the time instants t0 by
[21, 22, 4, 5, 3]

Tαtft0=∫0∞ft0-shαstdst | | (1) |

where t≥0 and 0<α≤1.
[400.2.2.4] The kernel function hαx is the one sided stable
probability density with stable index α
[21, 22, 4, 5, 3].
[400.2.2.5] Its Mellin transform is known to be [23]

Mhαxt0=1αΓ1-s/αΓ1-s. | | (2) |

[400.2.2.6] This allows to identify its density function as
[24, 22, 4, 5, 25]

hα(x)=1αxH1110(1x|0,10,1/α) | | (3) |

in terms of H-functions [26, 27].
[400.2.2.7] Its well known Laplace transform reads

[400.2.2.8] The operators Tαt form a semi-group and
obey the basic semi-group relation
[page 401, §0]

[401.1.0.1] The infinitesimal generator Aα of the fractional semi-group Tα

Aαft=-Dαft=-1Γ-α∫0∞ft-s-ftsα+1ds | | (6) |

is the fractional Marchaud-Hadamard derivative [28].
[401.1.0.2] For α=1 the fractional semi-group T1t becomes the
semi-group Ttft0=ft-t0 of simple translations.
[401.1.0.3] Because of this and because of the properties (1)
and (5) the fractional semi-group Tαt
will also be called “fractional translation” for short.

[401.1.1.1] The fractional time evolution/translation Tαt seems
to have been first introduced into physics in connection with
the discovery of a new class of phase transitions [21].
[401.1.1.2] It was later derived for dynamical systems from ergodic theory
in [22, 4, 5].
[401.1.1.3] Based on these results it
was argued that fractional time evolutions and fractional
dynamics actually exist in nature.
[401.1.1.4] Recently the physical basis for formula (1)
was generalized further using the idea of coarse graining [3].
[401.1.1.5] Formula (1) was previously known in pure mathematics
where it has close connections with the theory of semi-groups
and subordination [29, 30].
[401.1.1.6] It did not find direct applications in physics until the
present author used it as the foundation for the theory
of fractional time evolutions in physics.
[401.1.1.7] Formula (1) was recently rediscovered
in physics in the more restricted context of fractional
diffusion [31].