The first consequence of MEFs is related to a static electric field, that is, under the static condition of, Equation (2.14) reduces to

For a point charge, it yields a Yukawa typical potential

with an exponential decay range of. The exponential deviation from Coulomb’s law will provide many approaches to test for the photon mass in laboratory experiments.

It is important that, the electromagnetic induction described by Equation (2.7) can make MEFs spread in vacuum as free wave

The most typical aspect of massive photon is its frequent dependence (corresponding to Equation (2.2))

is the angular wave number(i.e. spin quantum number) of photon, the angular wave length. Notice that, in general the two possess the quantized values, namely,. Following Equation (3.4) is the group velocity differing from the phase velocity

both tend to c together, only as.

By generalized wave vector of, we also obtain a 4-velocity

where, is the generalized wave length in space, the total wave number, by which the generalized group and phase velocities can be define as

They have the same value of c. However, when projecting to the real space, the two occur immediately differentiation, since one is displayed as a direct projection, the other represents the velocity of the intersection point of wave surface and z-axis moving along the axis (see Figure 2).

From the geometric relation above, we find

In the case of, and. This means, at present, the photon is rest in real space, but in

generalized space moving with velocity c. Correspondingly, the two velocities along f-axis can be given by

Then we have

For nonzero mass particle of spin

with a total wave number

denotes the Compton wave length, the Compton wave number. Now, by introducing the Compton group and phase velocities

we get

Clearly, for the photon of zero Compton wave number, it reduces to (3.10).

Maxwell’s theory points out two polarized directions, both of which are orthogonal to the propagating direction of photon. However, MEWs described by PEs would result in a third state of polarization, in which the electric field points along the line of motion, corresponding to longitudinal photon [21] . Here, we still approve the third polarized state, but emphasize that, there will be three types of electromagnetic oscillations propagating along z-axis, those are pure transverse wave (PTW), pure longitudinal wave (PLW) and longitudinal-transverse mixed wave (LTMW). These waves possess a general solution

a) Pure transverse wave. For PTW with generalized potential of, we have the following fields by Equation (2.7)

The determined energy flows can be written as the energy density w multiplied by its traveling velocity, namely

in agreement with Equation (2.9). When the polarized fields (, b) are neglected, the presented will naturally reduce to the usual form. The field structure of PTW is shown as Figure 3(a).

b) Pure longitudinal wave. Correspondingly, the potential of PLW reads. The potential combining with Lorentz condition

gives the following nonzero field components

The energy flows come mainly from the polarized fields, namely

It tells us that, as a special radiation involving the polarized fields (, b), PLW (only about of usual

radiation) has no classical correspondence, but represents a natural induction process: Varying polarized vector field generates the polarized scalar and electric fields, in turn, when the latter two change, the former is induced. Figure 3(b) presents the structure of PLW.

c) Longitudinal-transverse mixed wave. With regard to LTMW of potential, it is easy to find the nonzero components of MEFs

followed by the flows just along the travelling direction of photon (see Figure 4).

However, in Proca theory, the situation is completely different, since it only gives three field components

with an energy flow defined by [21] [22]

not along the direction of wave vector. This means, with continuous transmission, the wave energy would break away from its travelling direction, and thus leads to a physically unacceptable result. Notice that, serving as a substantial basis of electromagnetic radiation, the energy flow must represent the motion of MEWs.

A well-known topological interference effect is called AB effect, which concerns a phase shift for moving elec-

trons (mass) diffracting around a tube of magnetic flux [23] , and it arises from the presence of generalized vector potential in the Lagrangian

where the vector potential for tube system of magnetic flux, can written as. When an electron beam is split and then recombined, there will be a phase shift from the interference effect. Now, consider (Lorentz condition), we have

, (3.27)

denotes the flow of through any surface bounded by the closed curve l, R the distance from the solenoid to the observational location. The result contains the standard AB effect in the limit of vanishing photon mass, and an additional effect predicted from, is a correction for massive photon.

An extension of the AB effect was presented by Aharonov and Casher [24] , called AC effect, which emphasizes that, a neutral particle possessing a magnetic dipole moment should experience an analogous phase shift when diffracted around a line of electric charge. Consider a magnetic dipole with mass and moment diffracted around an infinitely long line of linear charge density, its generalized Lagrangian can be given by

The corresponding phase shift of the split beam at the recombination point reads

The shift for this case is shown to reduce smoothly to that of the standard AC effect in the limit of vanishing photon mass, as was observed in the neutron interferometry experiment [25] . An additional effect is predicted from

where, the scalar potential of line charge system. The AC effect in massive electrodynamics was demonstrated by Fuchs [26] , neither the AB nor AC effects would provide a practical approach for bounding the photon mass in technology until now.

Vacuum polarization field not only can delay movement of photon, but require a generalized form of flow conservation, that is,. So that, for the plane MEWs, there should be an average steady equation

with a damping solution

is the mean number density of photon. With the help of this solution, we can write the spectral shift of photon in unit distance as

Such the effect was first discovered by Hubble in astronomical observation [27] , that is formulated as Hubble law: the recession velocity of celestial body determined by spectral redshift is always proportional to its distance r i.e. with a ratio H called Hubble constant. Taking the observational value [28] of, we get, which has been introduced into GMEs as a natural constant like speed c.

Because of having cosmological meaning, we can use the astronomical observation (i.e. Hubble constant H) to affirm the effective mass of photon

just equal to its ultimate upper limit estimated by uncertainty principle. This mass is determined by spin, and so called the spin mass of photon rather than the usual one.

The modified postulate inspires us to discuss motion in 5-dimensional Minkowski space, and write its invariant interval

Here, the invariance means every inertial observer would obtain the same value for this particular combination. The interval is directly related to the time interval in the rest frame of particle with no spatial displacement, that is, or

The rest-frame time coordinate is called the proper time. Since there is only one rest-frame, its time interval must be unique: all observers should agree on the value, namely

(4.3)

gives

where called generalized Lorentz factor. This is the physical basis for the invariance of the motion equation under generalized coordinate transformation.

In order to contain completely such that, GMEs should keep the same form, and the modified transformation for coordinates, must necessarily bring about the change of observers from frame S to moving with a generalized relative velocity along x-axis (due to the frame of reference having no spin), that is

GMEs have Lorentz symmetry because they are covariant under such the transformation (called generalized Lorentz transformation (GLT)), instead of the usual one.

A further 5-vector is the 5-velocity

with an invariant length of

Its components transform into each other under GLT in the same manner as transform into each other, namely

With the help of GLT, we can get the basic formula for velocity addition as follows:

In particular, if generalized speed and, we find

This is just a restatement of the fact that, if a particle (or light) has in one frame of reference, then it has the same result of in all frames of reference. To understand physically why this is the case, it is necessary to turn to consideration of relativistic dynamics.

In order for relativistic mechanics to be Lorentz symmetric, we need to generalize the familiar 4-momentum to the 5-dimensional form of. For massive photon, it reads

with an invariant length of, implying the conventional mass of photon is still zero. Correspondingly, for a particle with mass and spin , we have

is the spin to mass ratio. Obviously, for meson, a should be zero (due to), and for photon, infinity (due to). Thus we see that, the linear momentum, spin and energy E of moving particle can naturally form a covariant 5-vector, whose components transform in a definite way under GLT: observers in relative motion will see different generalized momentum and energy E, or, as we say, the generalized momentum and energy can transform into each other.

Moreover, we also have another 4th component of velocity

The first result guarantees (consistent with our modified postulate), the second implies a

very small velocity for electron. Repeatedly for the earth, it reads

Now, by the invariant length of, we can also lead to the generalized energy-momentum relation

which naturally contains particle spin. The presented allows us to formally take the limit of, and in this limit it gives

meaning only zero mass particle can travel at speed c in generalized space, i.e., and only zero mass particle with zero spin can travel at the limit speed of in real space.

In 5-dimensional relativity, all the related concepts can be given by analogy with their corresponding relativistic versions. Thus, we define the generalized force by

In the case of, it reduces to the usual form. The generalized relativistic work done by during a small displacement, can be written analogy as

The rate at which the generalized force does work is then

is the generalized relativistic kinetic energy. So that, integrating with respect to t gives

By requiring that for, we find

As should be the case, the generalized energy tends to the classical form for moving particle with a small velocity.

When particle is force-free, its generalized momentum and energy conserve

of which, the 4th component indeed represents the spin conservation. To see the physical meaning of the above, we consider the breakup of a body (such as a radioactive nucleus) of rest mass and spin into two pieces of rest masses, and spins,. If suppose the initial body is stationary, and the debris flies apart with velocities and, then some of the rest mass of the original body would be converted into the kinetic energy of the two masses produced. So by the generalized conservation laws, we have

In particularly, for meson decay, , , it gives

Nevertheless, the truly remarkable aspect of the above conclusions is that it has its fundamental origin in the fact that there exists a universal maximum possible speed c and a characteristic length, which are built into the structure of 5-dimensional Minkowski space. This structure ultimately exerts an effect on the properties of matter occupying space, phase and time, those are the linear momentum, angular momentum (spin) and energy of moving particle.

With in hand, it is natural to suppose that the dynamic equation for a particle with charge q and mass in MEFs is

By potential, the equation can be transformed into

and further manipulated into the Lagrange’s form

with the Lagrangian

For zero spin particle of, it reduces to the usual form

To derive the Hamiltonian H for charged particle in MEFs, we write the canonical momentum

Then, applying Legendre transform to gives

Consider the energy-momentum relation and, we have the finally Hamiltonian form

followed by Hamilton’s equations:

These equations are first-order in time in contrast to the second-order Lagrange’s.

The clue we need to construct a transparently covariant comes from GMEs, whose structure guarantees that the equations are form-invariant to translations in generalized space. Therefore, to show that massive electrodynamics is covariant, it is sufficient to show that the fundamental equations can be written entirely in terms of Lorentz tensors, whose components change under a Lorentz boost. The path to writing GMEs in covariant form begins with the introduction of the MEF tensor

Of which, the components transform according to the rule of. Tensor (5.1) can help us to express the homogeneous and non-homogeneous equations in Equation (2.7) respectively as

including continuity equation and Lorenz condition. If gauge invariance held, the Lagrangian density can be given by

The variation of the density with respect to yields.

Now, it is easy to confirm that, the tensor transformation rule applied to reproduces the field transformation formulae

Also noteworthy is the Lorentz invariant scalar function

The presented results reflect the transform properties of MEFs. The approach can provide us all the knowledge of generalized electrodynamics.

In order to organize the conservation laws of electromagnetism, we write the stress of 5-current as

and then get

denotes the generalized electromagnetic stress-energy tensor. An immediate consequence of Equation (5.7) is that a free MEF has a divergence-free stress-energy tensor

To examine the elements of, we find

The latter just simplifies to the negative of electromagnetic energy density w. The off-diagonal elements are proportional to the Cartesian components of the corresponding flow density defined in Equation (2.9), namely

A bit of algebra confirms that the space-phase components are the mixed flow density

Furthermore, the space-space components can be given by

Putting all the presented results together gives the matrix of as

With the representation (5.13) in hand, it is straightforward to confirm that Equation (5.7) contains two conservation laws in differential form. The first is a statement of the conservation of generalized momentum

The second gives Poynting’s theorem of energy conservation

Now, with the help of Equation (5.13), it is easy to write out the stress-energy tensor of plane MEWs discussed in Section 3, that is

So, we have the following conservation equations

corresponding to a steady form

due to. The equation implies a constant velocity, namely

Treating H as the Hubble constant gives, and then

with representing the cosmological time. The small value of (e.g. for a photon) determines that the spin phase is also a very small quantity. Notice that, is nothing but the full velocity of light defined explicitly in ref [29] , which can naturally lead to Hubble redshift, in agreement with the result of Equation (3.33). Importantly, such the agreement enables us to investigate motion in full velocity space, and further introduce a modified relativistic postulate:

The generalized full speed of light is a constant c.

By the postulate, we can incorporate the presented theoretical form into a more generalized unified framework of spatial relativity [29] , the unified framework could give a satisfactory account of the relativistic phenomena.

Now, by force (5.6) we define the generalized Lorentz torque density tensor as

The structure of this anti-symmetric tensor is

Analogous to Equation (5.7), it is possible to write the second-rank torque density as the five divergence of a generalized third-rank Lorentz tensor:

The anti-symmetry of with respect to its indices implies that only 50 of its components are independent. In detail, we use Equations (5.7), (5.23), and the symmetry of the stress-energy tensor to write (5.21) in the form

or given by

We focus on the 30 components , and then find that the twenty-four with are exactly the components of the second rank tensor of generalized angular momentum current density,. The six components are similarly the components of angular momentum flow density. These six independent components can be collected into the continuity-like equation for angular momentum flow, i.e. the first of Equation (5.25). The remainders (with) have been collected into the second.

Now, let us investigate the similarity between an electromagnetic pulse and a relativistic particle. When no source exists, Equation (5.24) becomes

followed by the conservation equations

In which, the conserved quantities are defined as the angular momentum flow densities of massive electromagnetic radiations, and. For a single photon of generalized momentum, we have

with,. By dispersion relation (3.4), we can get the Hamiltonian function of moving photon, that is

It can help us to write Equation (5.27) in term of Poisson formulation (due to,)

with Poisson brackets defined respectively by

It suggests that, when the brackets with the Hamiltonian vanish, the generalized angular momentums of moving photon will be constant.

To combine relativistic invariance with quantum mechanics, let us to write out the generalized Dirac Hamiltonian of and the generalized Dirac equation (GDE)

with an explicit representation of Hermitian matrices given by

in term of the unit I and Pauli matrices, and, are proposed to be anticommuting matrices of square equal to one:

To study the interaction of a Dirac particle with an external MEF characterized by potential, we need to write the covariant from of GDE, that is

where the matrices are related to and through

with Hermitian, Antihermitian. The relevant coupling can be obtained from the free GDE through the coupling prescription, that is

Multiplying it by the operator yields

with a generalized spin tensor defined by

whose components read

The 10 matrices of plusing the 5 components of and scalar I just constitute a set of 16 linear independent matrices. Therefore, the spin coupling term can be written as

When the polarized fields neglected (i.e.,), it will tend to the usual result [30] .

It is easy to find the similarity between Hamiltonian (5.29) and the Dirac formation. This similarity would provide us a very useful analytical device to study the angular momentum of photon from standpoint of Dirac theory. Thus, based on the fact that, the 3rd component of photon spin is always parallel to its momentum (i.e.), and no zero spin photon existing, we introduce the following hermitian matrices

and write the generalized Hamiltonian operator of photon as

where, , and thus only, are required to be anticommuting, namely

Now, it is easy to find that, the generalized angular momentum of photon is not commuting with

To meet the requirement, we introduce the matrices

and define the generalized spin tensor of photon by, with

The definition allows us to treat the total angular momentum as a conserved quantity rather than. This conversed condition requires

implying five conserved components, namely

Accordingly, MEWs behave like relativistic particles in the sense that their angular momentum transform like the energy-momentum vector of a particle. The situation suggests the conversation law of angular momentum should be modified as: the nature of MEWs is no longer to keep the generalized, but the total (including photon spin) constant. This means, only the quantities of commuting with Hamiltonian, will be the electrodynamic constants.

To study the physical implication of the Hamiltonian (6.12), we introduce a 5-dimensional bispinor to describe the massive electromagnetic potential, and propose the following Dirac typical equation (together with a modified Lorentz condition)

We here emphasize: 1) The components of must satisfy the Klein-Gordon equation, meaning a plane

MEW with (i.e.) is a solution. 2) There exists a 5-current density which

is conserved and whose firth component is a positive density of photon. So that, in the 5-dimensional representation (6.15), may be written as

in terms of two-component spinors and, which satisfy

Solving the equation gives its eigensolutions and the corresponding eigenvalues (shown as Table 1).

where, the spin states and the helicity operator read respectively.

the parameter equal to

From the above, we find a 4D mixed representation of consisting of the energy symbol (±) and the spin chirality, that is

Then by Equation (6.19) and its conjugate form

we get conservation equation

The number density of photon is positive, and the density current transforms as a generalized Lorentz 5-vector.

It is interesting to notice the equivalence between the covariant form of Equation (6.19)

and the homogeneous d’Alembert’s,. This equivalence allows us to express formally the MEF tensor as

including the stress-energy one

For plane MEWs of propagating along z-axis, we have Lorentz condition

and then get the nonzero field components (corresponding to Equation (3.22))

followed by the energy density

identical to expressions (3.23) and (6.26). The transverse and longitudinal fields in (6.31) represent respectively PTW and PLW, and coexistence of the both will bring us LTMW.

Up to now, we have transformed GMEs into Dirac form, which practically identical but conceptually different with the usual electrodynamics, looks upon MEWs from a viewpoint of quantum physics: MEWs are nothings but a collection of a large number of massive photon with unit spin, which can be described by a Dirac typical equation. Importantly, we find a new route that can be followed to study the motion of photon in the mathematical clothes of Dirac theory.