^{1}

^{2}

^{1}

^{2}

^{3}.

We here propose to extend the concept of helicity to include it in a fractional scenario and we write down the left- and the right-handed Weyl equations from first principles in this extended framework. Next, by coupling the different fractional
Weyl sectors by means of a mass parameter, we arrive at the fractional version of Dirac's equation which, whenever coupled
to an external electromagnetic field and reduced to the nonrelativistic regime, yields a fractional Pauli-type equation.
From the latter, we are able to present an explicit expression for the gyromagnetic ratio of charged fermions in terms of
the fractionality parameter. We then focus our efforts to relate the coarse-grained property of space-time to fractionality
and to the (

The electron’s gyromagnetic ratio,

It is now known that the anomalous magnetic moment (AMM) of the muon is considered to be one of the most promising observables that can unveil effects of some new physics beyond the Standard Model [

Nonrelativistic quantum mechanics predicts the remarkable result,

In this paper, we investigate the fractional coarse-grained aspects of the electron’s anomalous magnetic

FC is one of the possible generalizations of classical calculus. It provides a redefinition of a number of mathematical concepts and it seems very useful to deal with anomalous and frictional systems [

Fractional systems are described as being dissipative [

It is worthy to point out here the significant work by Svozil and Zeilinger [

Here, we claim that the use of an approach of FC based on a sequential form of the modified Riemann-Liouville (MRL) fractional calculus [

It seems that a reasonable way to probe the classical framework of physics is to highlight that, in our real world, the generic point is not infinitely small (or thin); it rather has a thickness. In a coarse-grained space, a point is not infinitely thin, and, here, this feature is modeled by means of a space in which the generic differential is not

The great majority of actual classical systems are nonconservative but, in spite of that, the most advanced formalisms of classical mechanics deal only with conservative systems [

Field theory aspects of nonlinear dynamics are today an important subject of study in different subareas of physics and mathematics, but the real success and radically new understanding of nonlinear processes have acquired body over the past 40 years. This understanding has been inspired by the discovery and insights of chaotic dynamics, where the randomness of physical processes is considered and more precisely when particle trajectories are indistinguishable for random process [

In [

Here, to achieve our goals, we carefully build up a fractional Dirac equation from underlying fractional Weyl equations written down in terms of the helicity operator. We begin by discussing the fractional angular momentum algebra in a coarse-grained scenario in order to understand the fractional spin operator used to set up the helicity operator. Also, a gauge-covariant fractional derivative is proposed so that, from the gauge transformation of charged matter fields, the gauge transformation of the fractional vector potential can be read off. Then, minimal coupling to the electromagnetic field is naturally achieved by writing down fractional Lagrangian densities in terms of fractional gauge-covariant derivatives of the matter fields. It is important to stress that we are not proposing the matter transformation from the very start [

Our paper is outlined as follows. In Section

In the sequel, we adopt an alternative approach by considering a fractional coarse-grained space-time instead of fractional space functions, meaning that neither the space nor the time is infinitely thin but has instead some “thickness”. As the use of certain calculation rules is essential to our approach, we briefly comment on this point, before presenting these rules.

The Riemann-Liouville and Caputo approaches for FC are well known and have their rules rigorously proved, as the reader may find in the standard textbooks [

But, by strictly referring to the context of modified Riemann-Liouville (MRL) formalism, it seems to us worthy to notice that the chain rule, as well as the Leibniz product rule, has had their validity mathematically proven only recently [

Following the the MRL definition, we find that the fractional derivative of a constant is zero; and, next, we can use it for both classes of differentiable as nondifferentiable functions. They are cast as follows [

simple rules:

simple chain rules:

For further details on the MRL formalism, we suggest the readers to follow [

Here, another comment is pertinent and concerns certain definitions called local fractional derivatives, as the ones introduced by Kolwankar and Gangal [

Now that we have set up these fundamental expressions, we are ready to carry out the calculations of our main concern.

Here, we will derive the commutation algebra for spin-1/2 particles in a coarse-grained medium.

Since in the MRL approach the chain and Leibniz rules hold, it is not difficult to obtain the commutation relation between position and momentum operators and to define the fractional angular momentum components [

Again, one can obtain the commutation relations using the definitions given above and the chain (

Let us now draw the reader’s attention to the different character of the

The square of fractional angular moment operator is defined as in [

The commutation relations then follow as below:

The commutation relations written above indicate that the ordinary integer angular momentum algebra does not change. This implies that the raising (lowering) operator, acting on an eigenstate

Another important conclusion is that, for the ordinary Pauli spin matrices, the representation for the basis of

This section sets out to show our proposal for building fractional versions of Weyl and Dirac equations on the basis of fundamental principles. Rather than adopting a fractionalized version for these equations from the very beginning, we propose to extend the concept of helicity to account for fractionality and, once the suitable helicity operators are written down, we adopt them to propose the fractional Weyl equation for left- and the right-handed fermions. Next, by coupling the different fractional Weyl equations by means of a mass parameter, we will end up with what we propose to be the fractional Dirac equation.

Weyl’s equation is a relativistic wave equation to describe massless spin-1/2 particles. Recalling that helicity is the projection of the spin onto the direction of momentum, we proceed to write down Weyl’s equations as statements on the helicity of the left- and right-handed fermions, since, in the case of massless fermions, chirality and helicity are equivalent.

The projection of spin onto the linear momentum is written as follows:

For a massless particle, the relativistic relation between energy and momentum reads as given below [

Let us consider a

To get a quantum wave equation, we invoke the correspondence principle and we propose the fractional operators that represent energy and momentum to be given by the equations that follow:

By virtue of the correspondence principle, (

The Weyl equation above can be cast in a covariant form as it follows below:

Here,

Also, defining the conjugated Pauli-spin matrices as

Inserting

The last equation can be rewritten, with the help of the properties of

Now, the spinor field is noted as

In compact notation, the fractional derivative is given by

The two Weyl equations can be written in a more compact form in terms of a four-dimensional spinorial field

We go ahead now by introducing a mass parameter that mixes up the two chiral components to obtain an equation that describes the dynamics of a charged massive fermion. To achieve that, we present the two fractional Weyl equations as

From the second of (

This equation indicates, upon comparison with a fractional Klein-Gordon [

We start off with the conjugated Dirac’s equation:

Now, multiplying (

The quantity in the bracket can be identified as a conserved current and this equation is interpreted as a fractional continuity equation. From standard electromagnetism coupling with matter, we now take that

For the fractional case, we can think of a minimal coupling term as

Following the Lagrangian for the model with integer dimensions, we can write down the Lagrangian with electromagnetic coupling in the fractional case as given by

In order that the theory remains covariant, we take the spinor field transformation as follows:

The fractional covariant derivative of the field

The solution to the equation above reads

The fractional Dirac equation, in a coarse-grained scenario, may now be written with the minimal electromagnetic coupling as

Splitting the space and time components, multiplying by

Using the correspondence principle, we have

Now, we write the Dirac spinor as a column-vector in the coarse-grained scenario as

In order to work out the nonrelativistic limit of the fractional Dirac’s Equation, we have to consider that the dominant term in the Hamiltonian is the rest energy, given by

Inserting this ansatz into the fractional equation (

We now consider that the mass term is dominant over the electrostatic interaction energy, that is,

The expression above readily gives that the weak component is, within the approximations carried out above, suppressed with respect to the strong bispinor,

Now, inserting this result into the first equation (

We define the fractional momentum operator as

Equation (

Recalling that the spin term is

The factor into the bracket can be identified as the fractional gyromagnetic ratio or

Notice that (

We can map a fractional parameter with the CODATA [

Since we expect the

We believe that, for the sake of completeness, it would be instructive to present the Gordon decomposition for the electronic current in its fractional form. The Gordon decomposition readily manifests the magnetic dipole piece of the relativistic current. So, after we have studied the effect of the fractionality on the

From (

The above equation may be rewritten as

From the Dirac equation (

Now, inserting these results in the first and second terms of (

For the gamma-matrices, we can write that

Using the definitions above and inserting them into (

Defining

Therefore, we are here giving the Gordon decomposition carried out in configuration space rather than presenting it in Fourier space. We choose to do so because we are not dealing with generalized functions, but with nondifferentiable functions.

With the help of (

For the electron (CODATA), we have that

Analogously, for the muon (CODATA),

Notice the hierarchy for the

We can interpret these results as follows: the higher is the

Now, in terms of mean life, muon decays in

From QED calculations [

Comparing the parameters above from QED with those from the measured anomalies [

We can see the opposite behavior for muon:

We can interpret these results as follows: for the electron, the results may be indicating that the complexity of the interactions taken into account in the QED calculation may be lower than it would be in the reality seen by the experiment. That is, the electron, as a pseudoparticle, keeps hidding other interactions or even structures that are not well described or are incomplete in the SM description. Thus, this may justify the deviation from the QED calculations as compared to the experimental results. In other words, in the experimental reality for electrons, we may think that the complexity is larger than that considered in the QED calculations based on the SM and the fractionality should indicate that the SM, although very good, may not be providing all information necessary to describe the interactions in a more complete view, either at high energies or in granular or fractal space-time. Thus, the FC may give evidences that the SM could require corrections (or higher order calculations by QED).

For the muon, an opposite behavior in terms of fractional parameter could be observed and indicates that the QED calculations might be taken into account with more interactions and a consequent complexity than the particle really experience, suggesting that the SM may not be complete to describe the whole interaction scenario.

In summary, in the present work, we have built up a fractional Dirac equation in a coarse-grained scenario by taking into account a fractional Weyl equation, a fractional angular momentum algebra, by introducing a mass parameter and imposing that the equations be compatible with the fractional energy-momentum relation. Considering then a fractional gauge-covariant derivative, we could minimally couple the charged fermion to an external electromagnetic field. The step further consisted in working out the nonrelativistic limit of the Dirac’s equation to obtain a fractional version of the Pauli’s equation. We have then investigated the anomalous magnetic moment for the charged leptons: electrons, muons, and taus. With the fractional approach to a coarse-grained scenario, we were able to get the fractionality associated to each leptonic species.

We have shown that a mapping of the anomalous magnetic

Defining a fractional current density, we have also performed a fractional Gordon decomposition and identified the spin contribution to the fractional electromagnetic current density.

We also suggested that the understanding of the results comparatively for electrons, muons and taus may be thought in the realm of complexity, mean life, coarse-grainedness, and pseudoparticle concepts.

We have finally shown that the fractionalities,

The authors declare that there is no conflict of interests regarding the publication of this paper.

Cresus F. L. Godinho is acknowledged for the discussions at an early stage of this work. The authors wish to express their gratitude to FAPERJ-Rio de Janeiro and CNPq-Brazil for the partial financial support.