**6. Conclusions**

**5.8 Unresolved issues regarding the generalized spacetime matrix equation**

1. For the case when *κ* ¼ *moc=*ℏ, we found there were four orthonormal

The following are the key points found in this analysis:

traditional Dirac equation.

*Progress in Relativity*

mentioned in Case 1.

Maxwell equations.

**34**

The eigenvectors and eigenvalues associated with the generalized spacetime matrix equation, for the special case of a time-harmonic plane-wave propagating in free space in the +*z* direction, have been determined for both *κ* ¼ *moc=*ℏ and *κ* ¼ 0.

eigenvectors (two having positive energy eigenvalues þ*γEo* and two having negative energy eigenvalues �*γEo*) describing waves having transverse properties. From **Table 2**, each of these four eigenvectors have components Δ<sup>3</sup> ¼ Δ<sup>4</sup> ¼ Ω<sup>3</sup> ¼ Ω<sup>4</sup> ¼ 0*:* Using the linear transformation equation (71), these eigenvectors map nicely into four orthonormal eigenvectors satisfying the

2. For the case when *κ* ¼ *moc=*ℏ, we found there were also four orthonormal eigenvectors (again two having positive energy eigenvalues þ*γEo* and two having negative energy eigenvalues �*γEo*) describing waves having nontransverse properties. From **Table 2**, each of these four eigenvectors have components Δ<sup>1</sup> ¼ Δ<sup>2</sup> ¼ Ω<sup>1</sup> ¼ Ω<sup>2</sup> ¼ 0*:* Again, using the linear transformation equation (71), these four eigenvectors map nicely into the same four orthonormal eigenvectors satisfying the traditional Dirac equation as

3. Therefore, for the case when *κ* ¼ *moc=*ℏ, the generalized spacetime matrix equation (49) for free space provides eight orthonormal eigenvector solutions (both transverse and non-transverse) which map into four orthonormal eigenvector solutions satisfying the traditional Dirac equation (65).

eigenvectors (two associated with waves propagating in free space with speed +*c* and two associated with waves propagating in free space with speed -*c*) describing waves having transverse properties. From **Table 4**, each of these four eigenvectors have components Δ<sup>3</sup> ¼ Δ<sup>4</sup> ¼ Ω<sup>3</sup> ¼ Ω<sup>4</sup> ¼ 0*:* For the case of transverse waves propagating with +*c*, these eigenvectors are associated with real electromagnetic waves predicted by the traditional

4.For the case when *κ* ¼ 0, we found there were four orthonormal

5. For the case when *κ* ¼ 0, we found there were also four orthonormal

these four eigenvectors has components Δ<sup>1</sup> ¼ Δ<sup>2</sup> ¼ Ω<sup>1</sup> ¼ Ω<sup>2</sup> ¼ 0*:*

is simply the Maxwell spacetime matrix equation for free space. The

Maxwell vector equations (11) and (12) for free space.

eigenvectors (two associated with waves propagating in free space with speed +*c* and two associated with waves propagating in free space with speed -*c*) describing waves having non-transverse properties. From **Table 4**, each of

6.The generalized spacetime matrix equation for *κ* ¼ 0 when Δ<sup>4</sup> � 0 and Ω<sup>4</sup> � 0

generalized spacetime matrix equation for *κ* ¼ *moc=*ℏ when Δ<sup>4</sup> � 0 and Ω<sup>4</sup> � 0 is simply the Dirac spacetime matrix equation for free space. In addition, the Dirac spacetime matrix equation for free space is equivalent to the four Dirac spacetime vector equations (37) and (38) for free space resembling the four

