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Based on the mechanism of vacuum polarization, we here establish a set of new electromagnetic field equations (EFEs) in 5-dimensional Minkowski coordinate system, which can be used to consider some physical implications, such as the dispersion, the polarized states and the Hubble redshift of massive photon. It shows that, the effective mass of photon is related to the Hubble constant H , and finally determined by its unit spin h . Importantly, these obtained equations, working as a generalization of Maxwell’s equations (MEs), enable us to develop the special relativity into 5-dimensional form. In developed relativity, the particle spin will voluntarily go into the motion equation, since it plus the linear momentum and energy can just form a 5-dimensional covariant vector. Moreover, by reorganizing the conservation laws of generalized electrodynamics, we find that the Hamiltonian of massive photon is similar to the Dirac formation. This similarity allows us to construct a new Dirac typical equation to study the motion of massive photon from a standpoint of Dirac theory.

A basic implication of Maxwell’s theory is that, all electromagnetic radiations propagate in vacuum at a constant velocity c. This conclusion was further raised to the postulate of special relativity, and soon after that described successfully as the moving behavior of massless photon by quantum theory. Despite all these, a substantial experimental effort [

Now, it is considered to be almost certainly impossible to do any experiment to confirm the value of^{10} years [

In quantum field theory [

In Maxwell’s theory, electromagnetic phenomena are always characterized by the electric and magnetic fields (E, B), which are thought of as the quantum of light in term of photon. If photon is massive instead of massless, its motion equations would become Proca form (in the Heaviside-Lorentz system of units) [

where,

The quantum theory can provide a basis for massive electromagnetic theory, since according to the theory, vacuum is not empty, but filled with a large number of virtual particle-antiparticle pairs flashing in and out of existence [

These equations can be further expressed in 5-dimensional Minkowski space with an extra-dimension

Now, we introduce a polarized vector field

and note that, the positive and negative charge elements

where

It shows that, the charge is conserved in space

called generalized Maxwell’s equations (GMEs). The performance of massive electromagnetic induction can be summarized as follows:

1) Varying magnetic and polarized scalar fields generate respectively an electric and a polarized vector fields, described by Equations (b) and (e).

2) Varying electric field generates a magnetic field and a polarized vector field, by Equation (d).

3) Varying polarized vector field generates a polarized scalar and an electric fields, by Equation (g).

GMEs can provide a complete and self-consistent description of electromagnetic phenomena, and help us to calculate the stress of current

with

Now, by the 5-dimensional potential

but need to supply a generalized Lorentz condition

It is easy to verify that, MEFs still have gauge invariance under the generalized transformation of

which has the following retarded solution

for current

The first consequence of MEFs is related to a static electric field, that is, under the static condition of

For a point charge

with an exponential decay range of

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))

both tend to c together, only as

By generalized wave vector of

where,

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

From the geometric relation above, we find

In the case of

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

Then we have

For nonzero mass

with a total wave number

we get

Clearly, for the photon of zero Compton wave number

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 [

a) Pure transverse wave. For PTW with generalized potential of

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

in agreement with Equation (2.9). When the polarized fields (

b) Pure longitudinal wave. Correspondingly, the potential of PLW reads

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 (

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.

c) Longitudinal-transverse mixed wave. With regard to LTMW of potential

followed by the flows just along the travelling direction of photon (see

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

with an energy flow defined by [

not along the direction of wave vector

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

trons (mass

where the vector potential

, (3.27)

An extension of the AB effect was presented by Aharonov and Casher [

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 [

where,

Vacuum polarization field not only can delay movement of photon, but require a generalized form of flow conservation, that is

with a damping solution

Such the effect was first discovered by Hubble in astronomical observation [

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

Special relativity is the theory of how different observers, moving at constant velocity with respect to one another, report their experience of the same physical event. And all the descriptions are completely based on the following two postulates:

I. The laws of physics take the same form in every inertial frame;

II. The speed of light in vacuum is the same in every inertial frame.

However, in massive electromagnetic theory, the speed of light is dependent of frequency rather than a unique constant. Thus, there needs a new postulate to be proposed to restore the features of special relativity, and the proposed should be aimed at the existence of a unique limiting speed c, to which speeds of all bodies tend when their energy becomes much larger than their mass [

The generalized speed of light is a constant c.

The modified postulate inspires us to discuss motion in 5-dimensional Minkowski space

Here, the invariance means every inertial observer would obtain the same value for this particular combination. The interval

The rest-frame time coordinate

(4.3)

gives

where

In order to contain completely such that, GMEs should keep the same form, and the modified transformation for coordinates

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

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

In particular, if generalized speed

This is just a restatement of the fact that, if a particle (or light) has

In order for relativistic mechanics to be Lorentz symmetric, we need to generalize the familiar 4-momentum

with an invariant length of

Moreover, we also have another 4th component of velocity

The first result guarantees

very small velocity

Now, by the invariant length of

which naturally contains particle spin

meaning only zero mass particle can travel at speed c in generalized space, i.e.

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

The rate at which the generalized force does work is then

By requiring that

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

of which, the 4th component

In particularly, for

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

With

By potential

and further manipulated into the Lagrange’s form

with the Lagrangian

For zero spin particle of

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

Then, applying Legendre transform to

Consider the energy-momentum relation

followed by Hamilton’s equations:

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

It is natural that, the electrodynamics of moving bodies could be in agreement with the developed relativistic principles, under which all the related problems could be discussed. In particularly, when we say GMEs are covariant, we eventually must specify the transform properties of that, it is not only the space and time coordinates that will change, but also MEFs.

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

including continuity equation

The variation of the density with respect to

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

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

and then get

To examine the elements of

The latter just simplifies to the negative of electromagnetic energy density w. The off-diagonal elements

A bit of algebra confirms that the space-phase components

Furthermore, the space-space components can be given by

Putting all the presented results together gives the matrix of

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

Treating H as the Hubble constant gives

with

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 [

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

or given by

We focus on the 30 components

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

followed by the conservation equations

In which, the conserved quantities are defined as the angular momentum flow densities of massive electromagnetic radiations,

with

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.

In this section, our aim will be try to construct a Dirac typical equation to describe the motion of photon. Along the way, we shall encounter some challenges, which ultimately will force us to a recasting of Dirac equation.

To combine relativistic invariance with quantum mechanics, let us to write out the generalized Dirac Hamiltonian of

with an explicit representation of Hermitian

in term of the

To study the interaction of a Dirac particle with an external MEF characterized by potential

where the

with

Multiplying it by the operator

with a generalized spin tensor defined by

whose components read

The 10 matrices of

When the polarized fields neglected (i.e.

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

and write the generalized Hamiltonian operator of photon as

where,

Now, it is easy to find that, the generalized angular momentum of photon

To meet the requirement, we introduce the

and define the generalized spin tensor of photon by

The definition allows us to treat the total angular momentum

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

To study the physical implication of the Hamiltonian (6.12), we introduce a 5-dimensional bispinor

We here emphasize: 1) The components of

MEW with

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

in terms of two-component spinors

Solving the equation gives its eigensolutions and the corresponding eigenvalues (shown as

Eigenvalues | Eigensolutions | |||
---|---|---|---|---|

where, the spin states

the parameter

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

Then by Equation (6.19) and its conjugate form

we get conservation equation

The number density of photon

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

and the homogeneous d’Alembert’s,

including the stress-energy one

For plane MEWs of

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.

In this paper, we have made a special effort to illustrate the physical consequences of vacuum polarization. This practice could provide a direct pathway to develop Maxwell’s theory, and the development has leaded to many surprising results. To sum up, these results are:

1) The starting point of our work is to establish a set of new EFEs by the mechanism of vacuum polarization, which is expressed in generalized space with an added dimension identified with the spin phase, hence possessing gauge invariance. These equations could give us a complete and self-consistent description of electromagnetic phenomena.

2) The effects of massive photon were incorporated into electromagnetism straightforwardly through GMEs, which can be used to consider some physical implications, such as deviations in the behavior of static electromagnetic fields, the dispersion of light, the polarized states of MEWs and the Hubble redshift. In particularly, we emphatically discuss the field structures of three typical MEWs, i.e. the pure transverse, pure longitudinal and longitudinal-transverse mixed waves.

3) By a modified relativistic postulate of that: The generalized speed of light is a constant c, we develop the special relativity into a 5-dimensional Lorentz symmetric form; this form contains two natural constants: a velocity constant c and a length constant

4) To guarantee electrodynamics to be covariant, we have written entirely the fundamental equations of the subject in terms of Lorentz tensors, whose components change under a Lorentz boost, but whose essential tensor character does not change. The covariant notation provides a powerful way to organize the conservation laws of electromagnetism for the linear momentum, angular momentum and energy, including a further revisiting by Lagrangian method.

5) By reevaluating the similarity between the energy form of photon and the Hamiltonian of Dirac particle, we constructed a Dirac typical equation for free massive photon. The plane wave solutions of the equation are presented, which could bring a significant change to electrodynamics.

Finally, let us review the developed electrodynamics. The original equations (i.e. MEs) were not accurate and had to be reformulated. The modified equations (i.e. GMEs) were involved in a generalized field formulation. This field formulation always gives the same results when applied to the different frameworks, and thus the Lorentz covariance is guaranteed.

Qiankai Yao, (2015) The Unified Theoretical Form of Massive Electrodynamics. Open Access Library Journal,02,1-23. doi: 10.4236/oalib.1101732