Dirac type relativistic quantum mechanics for massive photons

Constructing a relativistic quantum mechanics (RQM) for a massive photon, without appealing to the quantum field theoretical approach such as the massive Proca model, we find a new theoretical particle solution which allows the massive photon having either positive or negative energy as solutions. In particular, we predict the existence of the so-called anti-photon corresponding to the negative energy solution, similar to the positron in the Dirac RQM for an electron. Note that the anti-photons could be an intense radiation flare of the gamma ray burst. In this RQM for the massive photon, we construct a positive definite probability density and a nontrivial diagonal Hamiltonian, and also discuss a massless photon. Moreover we confirm the covariances of the relativistic equation of motion for the massive photon and the corresponding probability continuity equation under the Lorentz transformation.


I. INTRODUCTION
Generalizing the Hawking-Penrose singularity theorem (HPST) [1], the stringy cosmology in a higher dimensional total spacetime has been investigated [2,3] with a success that one can describe precisely the motion types of stringy congruence in terms of the universe expansion rate after the Big Bang. Moreover, in the stringy HPST described in the higher dimensions, we can have an advantage that the degrees of freedom (DOF) of the rotation and shear of the stringy congruence are introduced naturally in the early universe. In the stringy HPST with the higher dimensional spacetime, assuming that the smallest particle is the photon, one can find that in the universe there exists a massive photon possessing the finite size which is filled with mass. Note that the higher dimensional theories have been exploited in the higher dimensional de Sitter cosmology [4].
Next, the gamma ray burst (GRB) in the universe has been detected in 1967, and the GRB discovery has been published in 1973 [5]. Since then, to explain the GRB, there have been extremely lots of theoretical models. For a recent detection of the GRB associated with the magnetar, see Ref. [6] for instance. If the GRB sources were from within the Milky Way, they would be strongly concentrated near the galactic plane. The absence of any such pattern in the GRB has suggested a strong evidence that the GRB can come from beyond our own galaxy. The intense radiation flare of most detected GRB has been then assumed to emerge from a supernova, during a high-mass star implodes to construct a neutron star or a black hole.
On the other hand, Dirac has formulated the relativistic quantum mechanics (RQM) for an electron [7]. In his theory, a new particle solution has been proposed to allow the electron having either positive or negative energy solutions. In particular, the positron corresponding to the negative energy solution has been theoretically predicted and later has been experimentally confirmed [8]. Note that the Dirac equation in condensed matter systems has been investigated in terms of bound state in continuum like solutions, in addition to discrete energy bound state solutions [9].
Recently making use of an open string which performs both rotational and pulsating motions, we have predicted the intrinsic frequency ω γ = 9.00 × 10 23 sec −1 [10] for the bosonic photon with spin one which is comparable to the intrinsic frequencies ω N = 0.87 × 10 23 sec −1 and ω ∆ = 1.74 × 10 23 sec −1 [11] of the nucleon and delta baryon with spin 1/2, respectively. Next we have calculated the finite photon size r 2 1/2 (photon) = 0.17 fm in the phenomenological stringy photon model [10].
In this paper, we will propose a new RQM for a massive photon with finite size, without appealing to the quantum field theoretical approach such as the massive Proca model. In this RQM for the massive photon, we will theoretically predict a new particle solution which allows the massive photon having either positive or negative energy as solutions.
In the massless limit we will also recover the RQM for the photon with transverse polarizations only. Moreover, we will investigate the intense radiation flare of the GRB in terms of the negative energy solution of the massive photon.
Next we will discuss the covariances of the relativistic equation of motion for the massive photon and its corresponding probability continuity equation under the Lorentz transformation.
In Sec. II, we will construct the Hamiltonian for a massive photon in our model. In Sec. III, we will investigate the phenomenology of the RQM for the massive photon. Explicitly we will show that the probability density for the massive photon is positive definite in our model. The negative energy solution of the massive photon will be also discussed together with the GRB. In Sec. IV, we will study the massless limit of the photon. We will also compare the RQM for the massive photon, with the Proca model. In Sec. V, we will investigate the Lorentz transformation for the massive photon to discuss the covariances of its relativistic equation of motion and probability continuity equation. Sec. VI includes conclusions.

II. HAMILTONIAN CONSTRUCTION IN RQM FOR A MASSIVE PHOTON
In this section, we will construct the Hamiltonian in the RQM for a massive photon. To do this, we assume that the photon trajectory is a straight line along the z direction, for simplicity. We then have a relativistic relation E 2 = m 2 + p 2 where p = p z = | p|. The RQM equation of motion for the massive photon is then given by Here φ a denotes the wave function for the massive photon, explicitly given by where the superscript t stands for the transpose of the wave function components. Here the spin index a (a = 0, 1, 2, 3) denotes the spin DOF for the massive photon with spin one. The component index A (A = 1, 2) stands for the two DOF which have the same DOF of the positive and negative energy solutions with the energy index ± in (3.2), since the positive and negative energy solutions are given by linear combinations of the two wave functions with the component indices. Note that the wave function φ a A is described in terms of a 1 × (2 energy ⊗ 4 spin ) = 1 × 8 column vector. From now on we will drop the index A in the wave functions except (3.9) and (3.24) below, for simplicity.
Note that, in the quantum field theoretical Proca model for the massive photon, we need a 1 × 1 matrix Hamiltonian as in (4.5) below. Moreover, in this model, we cannot have a negative energy solution. Now we include a possibility of a negative energy solution for the massive photon in our model, by following Dirac idea for the positron. To do this, for the corresponding Hamiltonian for the bosonic massive photon, we introduce a minimal 2 × 2 matrix associated with the positive and negative energy solutions. 1 Next, complying with the Dirac algorithm for the RQM for the positron, we proceed to find H in (2.1) for the case of the RQM for the massive photon. The Hamiltonian H is then given by a 2 × 2 matrix acting on the component index A only where A i (i = 1, 2, 3) and B are 2 × 2 matrices. Using the relation E 2 = m 2 + p 2 , we obtain the algebra among A i and B as follows where I is a 2 × 2 unit matrix. Note that eigenvalues of A i or B are ±1 and trA i = trB = 0. In our construction, the photon spin DOF is included in the wave function in (2.2), as in the Proca model wave function in (4.5). As in the Dirac relativistic formalism for the positron, exploiting the above relations in (2.4) together with the massive photon Hamiltonian in (2.3), we obtain the representations for A i and B given by A 1 = A 2 = 0, A 3 = σ 1 and B = σ 3 , with σ i being the Pauli matrices. Inserting the above representations for A i and B into (2.3), we arrive at the desired 2 × 2 Hamiltonian of the form Making use of (2.1) and (2.5), we find the relativistic equation of motion for the massive photon as follows where φ a (x) is a function of x µ and Γ µ is given by

III. ANTI-PHOTON IN RQM FOR A MASSIVE PHOTON
In this section, we will investigate the phenomenological aspects of the RQM for the massive photon. To do this, we start with the equation in (2.6), which describes the motion of the massive photon. Since, for the relativistic massive photon satisfying the relation E 2 = m 2 +p 2 , we have two kinds of solutions corresponding to E = ±(m 2 +p 2 ) 1/2 ≡ ±p 0 , we introduce an ansatz for the wave function φ a as follows for a positive (negative) solution with an upper (lower) sign. Next we find the positive energy solution φ a where u a (p µ ) and v a (p µ ) are given by the nontrivial forms, Here ǫ a is a unit polarization vector possessing the spacetime index a (a = 0, 1, 2, 3) which is the same as the spin index and is needed to incorporate minimally the spin DOF for the massive photon. Here we have considered the Lorentz frame where ǫ a is purely space-like so that we can readily find that ǫ a ǫ a = ǫ · ǫ = 1, or ǫ a ǫ a = − ǫ · ǫ = −1.
Moreover we have the relation ǫ a p a = 0, since for the massive photon we have longitudinal component in addition to transverse ones, similar to the phonon associated with massive particle lattice vibrations [12]. Next we have the normalization relations: 2) into (2.6), we obtain the equations of motion in the momentum space where p = Γ µ p µ . It is well known that, in the Dirac RQM for the positron, we have the electron and positron corresponding to the positive and negative energy solutions, respectively. It is now interesting to note that, similar to the Dirac theory for the positron, there exists the massive photon with negative energy solution.
Reshuffling the equation of motion in (2.6) we obtain Taking Hermitian conjugate of the equation in (3.5), we next construct Exploiting (3.5) and (3.6), we find the probability continuity equation where the probability density ρ and probability current J, respectively, are given by Here we have used the notation φ a ≡ ǫ a (φ 1 , φ 2 ) t . Note that the probability density ρ is positive definite in our model. We emphasize that the quantities ρ and J in (3.9) are physically well defined to yield a good quantization, similar to the Dirac RQM for the electron where the corresponding probability density ρ D is also positive definite. However, in the Klein-Gordon model, the probability density ρ KG for a relativistic spinless boson is not positive definite [13]. Note that the probability continuity equation in (3.8) can be rewritten in the covariant form as follows where we have used the four probability current J µ = (ρ, J ). For the positive energy solution with E > 0, inserting φ a + (x) in (3.2) into ρ and J in (3.9), we obtain Next, for the negative energy solution with It seems appropriate to comment on the four probability current J µ D = (ρ D , J D ) in the Dirac RQM for an electron with mass m e . The relativistic equation of motion for the electron is given by [13] (iγ µ ∂ µ − m e )ψ(x) = 0, (3.13) with γ µ (µ = 0, 1, 2, 3) being given by 4 × 4 matrices γ 0 = β and γ i = βα i where The electron wave equation is then given by for a positive (negative) solution with an upper (lower) sign. Now we find the positive energy solution ψ + (x) with E e > 0 and the negative energy solution ψ − (x) with E e = −|E e | < 0, respectively. For a given index I (I = 1, 2) corresponding to spin ±1/2 states we construct where, for a given relativistic four momentum variable p µ e = (E e , p e ), u I (p µ ) and v I (p µ ) are given by where χ 1 = (1, 0) t and χ 2 = (0, 1) t . In the Dirac RQM for the electron, the four probability current J µ D = (ρ D , J D ) is given by [13] Making use of (3.15)-(3.17), we find for the positive energy solution with E e > 0, and for the negative energy solution with E e = −|E e | < 0. Here we notice that the four probability current J µ D satisfies the probability continuity equation both for the positive and negative solutions [13] ∂ µ J µ D = 0. (3.21) Note that there exists the similarity such as J µ in (3.11) and (3.12), and J µ D in (3.19) and (3.20) between the RQM for the massive photon and the Dirac RQM for the electron. Now we have some comments to address on the negative solution φ a − in the RQM for the massive photon. First, for the negative energy solution of the massive photon, since we have ρ = |φ 1 | 2 + |φ 2 | 2 > 0, ρ = |E| m implies that m is positive. Even in this negative energy solution case, the positive mass m moves with the probability current J along the z direction. From now on, we will name the photon possessing the characteristic that the particle has the positive mass m and positive energy |E| and is associated with the negative energy solution, an anti-photon. To be more specific, we find that the anti-photon possessing the positive mass m has a positive definite probability density ρ and propagates with a probability current J along the direction of p.
Second, we propose that the anti-photon related with the negative energy solution is defined to interact repulsively with the ordinary massive photon, oppositely to the ordinary massive photon-photon attractive gravitational interaction pattern. 2 Next, the positron and electron can annihilate each other via the particle and anti-particle pair annihilation mechanism. However, the uncharged anti-photon scatters away from the ordinary massive particle including the photon, in the repulsive gravitational interaction between the anti-photon and the ordinary massive particle.
Third, since the anti-photon is repulsive against charged or uncharged ordinary massive matters, the anti-photon does not adhere to the ordinary massive matters, so that the anti-photons can yield an intense radiation flare. Now we propose that the anti-photons with positive masses could be the intense radiation flare of the GRB released from a supernova, during a high-mass star implodes and then forms a neutron star or a black hole.
Next we diagonalize the Hamiltonian in (2.5) by introducing a unitary operator matrix and its inverse one, respectively, Even though the equation of motion in (3.24) seems to be simpler than that in (2.6), in investigating the phenomenology of the massive photon, the wave function φ a A (x) is not so physically meaningful since it is described in terms of the rather non-physical component index A, instead of the physical energy index ± in (3.2). The equation of motion in (3.24) will be discussed in the next section, to investigate the differences among the RQM for the massless photon, the massive Proca model and the RQM for the massive photon.

IV. RQM FOR MASSLESS PHOTONS
In this section, we will formulate wave functions for massless photons. To do this, we revisit the RQM equation of motion in (2.6) and the solutions in (3.2) and (3.3) from which, in the massless limit, we obtain the following positive and negative energy solutions where a normalization factor N will be fixed later. Note that the above solutions in (4.1) satisfy the following equation of motion (x) as shown in (4.1). Moreover, since the neutral massless photon is equal to its massless anti-particle, we construct the massless photon wave functions in terms of the wave functions possessing the spin index only. In other words, by exploiting φ a(m=0) ± (x) in (4.1) and the unit polarization vector ǫ a , we find φ a(m=0) (x) corresponding to the massless photon wave function possessing the spin index a only without the component index A Here the normalization factor N ≡ 1 √ 2p0V associated with p 0 and space volume V is now fixed so that the massless photon energy ω in the electromagnetic wave can be given by ω = p 0 = | p| [13]. Note that φ a(m=0) (x) ≡ A a (x) is the four-vector potential in the elecromagnetism for the massless photon.
It seems appropriate to comment on the normalization factor N . In the Bose-Einstein statistics [14], we can find the massive photon statistics where the total number of the photons is countable and fixed so that we can have the constraint of the form r n r = N . Here n r is the number of photons in quantum state r and N is the total photon number. However in treating the massless photon we have a puzzle that, the total number of the massless photons is not fixed so that we cannot have the above constraint on the total number of the photons. Now, this puzzle can be solved by the intrinsic property that the massless photon is a point-like particle without any size, differently from the massive photon with finite size. After quantization of the light we thus cannot count the number of quantized massless photons. Without resorting to the above constraint on the total number of the photons, the quantum statistics for the massless photon is then known to yield the Planck distribution [14]n s = 1 e βEs −1 , where β = 1/kT with k and T being the Boltzmann constant and temperature, respectively, and E s (s = 1, 2, ...) is the energy in state s. For the massless point-like photons, we thus have the normalization factor in (4.3), which is different from those in (3.3) for the massive photon with finite size.
Note that in constructing φ a(m=0) (x), we have reduced the eight components of the wave function for the massive photon into the four components of the wave function for the massless photon. Here we have included only the DOF originated from the spin index a in φ a(m=0) (x) for the massless photon. In other words, we do not have the DOF associated with the positive and negative energy solutions, and thus φ a(m=0) (x) do not have the energy index. This construction is consistent with the traditional photon relativistic representation. Note also that the polarization vector ǫ a satisfies the transversality condition ǫ a p a = 0, which is needed since the massless photon has the transverse components only. Moreover we can readily check that the above wavefunction in (4.3) fulfills the equation of motion for the massless photon φ a(m=0) (x) = 0. (4.4) The above transversality condition ǫ a p a = 0 then yields ǫ · p = 0, so that we can define two space-like polarization vectors ǫ I (I = 1, 2) satisfying ǫ I · ǫ J = δ IJ . Note that ǫ I and p form a three dimensional orthogonal basis system as desired [13]. Next, in the massive Proca model described by a wave function ϕ a (x) with spin index a only, we find the relativistic equation of motion for a massive photon ( + m 2 )ϕ a (x) = 0, (4.5) which is different from (3.24), since ϕ a (x) in (4.5) does not possess the component index A. Here the Hamiltonian is given by a 1 × 1 matrix. Note that the wave function ϕ a (x) in (4.5) describes only a positive energy solution for the massive photon, and thus it cannot explain the anti-photon aspects discussed in the Dirac type RQM for the massive photon.

V. LORENTZ TRANSFORMATION FOR A MASSIVE PHOTON
In this section we will investigate the Lorentz transformation for a massive photon. To do this, we first consider the infinitesimal Lorentz transformation given by where ǫ µ ν are the full Lorentz group transformation parameters. Note that for the positive and negative energy solutions for a given relativistic four momentum variable p µ = (E, p) in the RQM for the massive photon, we obtain the four probability current J µ for finding the photon in (3.11) and (3.12). Next the probability density and current in (3.8) must form a four vector under the Lorentz transformation in order to confirm the covariance of the continuity equation and of the probability interpretation associated with the Born's rule in a space-plus-time split of spacetime manifold, as in the Dirac RQM for the electron. Moreover the equation for the massive photon in (2.6) should be shown to be Lorentz covariant. 3 Exploiting (5.1), we will show the Lorentz covariance of the relativistic equation of motion for the massive photon in (2.6). Now we introduce an equation which takes the form of (2.6) in the primed system Similar to the scheme exploited in the Dirac theory for the electron [13], we now make an ansatz for φ a′ (x ′ ) as follows Here S(a) is a function of a µν and a 2 × 2 matrix acting on the 2 energy -component colume vector φ a (x) for a given spin index a. In order to find S(a) satisfying ( Expanding S(a) in powers of ǫ µν given by a µν in (5.1) and keeping only the linear term in the infinitesimal generators, we make an ansatz where σ µν (= −σ νµ ) are 2 × 2 matrices in the RQM for the massive photon. Inserting (5.1) and (5.6) into (5.5), we are left with from which we obtain the covariance condition of the form The problem of finding the Lorentz covariance of the relativistic equation of motion for the massive photon in (2.6) under the full Lorentz group transformation is now reduced to that of constructing matrices σ ρσ satisfying (5.8). Similar to the algorithm used in the Dirac theory for the electron [13], the simplest guess to make is an anti-symmetric product of two matrices, and we find that is the desired matrix which satisfies the covariance condition in (5.8). Here we have used (2.7) and the ensuing results for σ ρσ in the RQM for the massive photon, given by We thus prove that (5.2) also holds covariantly in the primed system, and we finally show the covariance of the relativistic equation of motion for the massive photon in (2.6) under the full Lorentz group transformation. Note that, in the RQM for the massive photon, the generators of the full Lorentz group K i = 1 2 σ 0i and N i = 1 4 ǫ ijk σ jk (i = 1, 2, 3) satisfy the commutation relations 11) similar to the corresponding relations in the RQM for the electron [15]. Next we consider explicitly the full Lorentz group transformation for a massive photon whose trajectory is a straight line along the z axis. To do this, we introduce a rotation around z axis and a boost along z axis, which are physically of interest in the RQM for the massive photon where θ and ω are the rotation and boost parameters, respectively. Exploiting (5.1) and (5.12) we obtain the nonvanishing Lorentz transformation parameters ǫ µν rot and ǫ µν boost for the rotation and boost with the conditions θ ≪ 1 and ω ≪ 1 given by Now, by making use of σ µν in (5.10), we explicitly find the corresponding matrix S in (5.6) for the rotation and boost associated with ǫ µν in (5.13). First, for the case of the rotation around z axis, we readily obtain S z rot = I, (5.14) regardless of the non-vanishing rotation parameters ǫ 12 rot = −ǫ 21 rot = −θ in (5.13), since we have σ 12 = σ 21 = 0 in (5.10). In particular, for the case of a 2π radian rotation, we still find S z rot = I implying a bosonic photon property that it takes a rotation of 2π radian to return φ a (x) to its original value. Note that for the fermionic electron case it takes a 4π radian rotation to return ψ(x) to its original value [13] and this characteristic is also discussed in terms of the Möbius strip structure of the hypersphere manifold in the hypersphere soliton model [16]. Second, for the case of the boost along z axis related with the non-vanishing boost parameters ǫ 03 boost = −ǫ 30 boost = ω, we find Next we investigate the covariance of the probability continuity equation in (3.10) under the full Lorentz group transformation. To do this, we first rewrite the four probability current J µ in (3.9) in terms of Γ µ J µ =φ a Γ µ φ a . For the rotation around z axis associated with S z rot = I, we readily find to produce For the boost along z axis related with S z boost = I − 1 2 ωσ 1 , keeping only the linear term in the infinitesimal generators we obtain the non-vanishing components of J µ′ boost J 0′ boost =φ a Γ 0 (I − ωσ 1 )φ a , J 3′ boost =φ a Γ 3 (I − ωσ 1 )φ a .

VI. CONCLUSIONS
In summary, making use of the RQM for the electron, Dirac predicted the existence of the positron before the quantum field theory had not been developed. Similarly, exploiting the RQM for the massive photon, we have predicted the existence of the anti-photon without resorting to the quantum field theory. To do this, following the Dirac formalism for the RQM for the electron, we have developed a physics algorithm for the RQM for a massive photon, to formulate a 2×2 Hamiltonian matrix for the massive photon. Exploiting the Hamiltonian, we have found a new theoretical particle solution which allows the massive photon possessing either positive or negative energy solutions. In particular, we have proposed theoretically the anti-photon corresponding to the negative energy solution, similar to the positron in the Dirac RQM for an electron. In our model, the massive photon has been shown to possess the longitudinal polarization in addition to the transverse ones as in the case of a phonon. Moreover we have formulated a nontrivial diagonal Hamiltonian for the massive photon. We also have investigated the Lorentz transformation associated with the RQM for the massive photon, to ensure the covariances of the relativistic equation of motion and the corresponding probability continuity equation. One of the main points of this paper is that, the anti-photons could be the candidate for the intense radiation flare of the mysterious GRB released from a supernova which is located far from the Earth. Note that the RQM for the anti-photon possessing positive mass is a new phenomenology, which could be consistent with the well-established Dirac positron theory and related with a fundamental prediction of the GRB.