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

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. The following are the key points found in this analysis:

In the de Broglie-Bohm picture of quantum mechanics, Hardy [16] and Bell [17]

suggest empty waves represented by wave functions propagating in spacetime, but not carrying energy or momentum, can exist. This same concept was called ghost waves or ghost fields by Albert Einstein (see [18]). The controversy as to whether matter waves correspond to real waves or ghost waves has been and is

In Section 5.1, we mentioned that the number of unanswered questions and mysteries regarding the universe from the smallest to the largest, in the fields of physics and astronomy, is unimaginable. Allowing the elements Δ<sup>4</sup> and Ω<sup>4</sup> to have nonzero values in the generalized spacetime matrix equation certainly raises a number of unanswered questions. The following is the author's list of 12 unanswered questions and mysteries regarding our analysis of the generalized spacetime

still a subject of debate and controversy.

*DOI: http://dx.doi.org/10.5772/intechopen.86982*

For relativistic quantum mechanics—matter waves:

*Eight-by-Eight Spacetime Matrix Operator and Its Applications*

For classical electrodynamics—electromagnetic waves:

What class of particles do the transverse eigenvectors represent? Do the transverse eigenvectors represent real or ghost waves?

What can be said about those waves propagating with speed -*c*? Do these represent a new type of electromagnetic wave?

What class of particles do the non-transverse eigenvectors represent? Do non-transverse eigenvectors represent real or ghost waves?

What can be said about those waves having a longitudinal component? What can be said about those waves having a fourth component? Could these be associated with undiscovered electromagnetic waves?

Why do the *Dirac* and *Maxwell* vector equations resemble each other? Does the spacetime matrix operator *M*^ have more surprises in store?

1. The four classical electromagnetic microscopic Maxwell field equations have been rewritten as a single matrix equation, referred to as the Maxwell spacetime matrix equation, using the spacetime matrix operator *M*^ . The Maxwell spacetime matrix equation is relativistic invariant under a Lorenz

summarized next. Other fundamental equations of electromagnetic theory have also been expressed as single matrix equations using the spacetime matrix

3. The traditional relativistic Dirac equation for free space has been expressed as a new matrix equation, referred to as the Dirac spacetime matrix equation for free space, using the same spacetime matrix operator *M*^ . The Dirac spacetime matrix equation is also relativistic invariant under a Lorenz transformation.

2. The square eight-by-eight matrix operator *M*^ has several benefits as

electromagnetic potential wave equations.

operator *M*^ , namely, the electromagnetic wave and charge continuity equations, the Lorentz conditions and electromagnetic potentials, and the

Are the transverse and non-transverse eigenvectors equivalent in some way?

matrix equation for free space:

And two last questions:

**6. Conclusions**

**35**

transformation.


In the de Broglie-Bohm picture of quantum mechanics, Hardy [16] and Bell [17] suggest empty waves represented by wave functions propagating in spacetime, but not carrying energy or momentum, can exist. This same concept was called ghost waves or ghost fields by Albert Einstein (see [18]). The controversy as to whether matter waves correspond to real waves or ghost waves has been and is still a subject of debate and controversy.

In Section 5.1, we mentioned that the number of unanswered questions and mysteries regarding the universe from the smallest to the largest, in the fields of physics and astronomy, is unimaginable. Allowing the elements Δ<sup>4</sup> and Ω<sup>4</sup> to have nonzero values in the generalized spacetime matrix equation certainly raises a number of unanswered questions. The following is the author's list of 12 unanswered questions and mysteries regarding our analysis of the generalized spacetime matrix equation for free space:

For relativistic quantum mechanics—matter waves: What class of particles do the transverse eigenvectors represent? Do the transverse eigenvectors represent real or ghost waves? What class of particles do the non-transverse eigenvectors represent? Do non-transverse eigenvectors represent real or ghost waves? Are the transverse and non-transverse eigenvectors equivalent in some way?

For classical electrodynamics—electromagnetic waves: What can be said about those waves propagating with speed -*c*? Do these represent a new type of electromagnetic wave? What can be said about those waves having a longitudinal component? What can be said about those waves having a fourth component? Could these be associated with undiscovered electromagnetic waves?

### And two last questions:

Why do the *Dirac* and *Maxwell* vector equations resemble each other? Does the spacetime matrix operator *M*^ have more surprises in store?
