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Quantum electrodynamics (QED) is built on the original Dirac equation, an equation that exhibits perfect symmetry in that it is symmetric under charge conjugation (C), space (P) and time (T) reversal and any combination of these discrete symmetries. We demonstrate herein that while the proposed Lorentz invariant Curved Spacetime Dirac Equations ( CSTD-equations) obey C, PT and CPT- symmetries, these equations readily violate P, T, CP and CT- symmetries. Realising this violation, namely the T and CT- violation, we take this opportunity to suggest that the Curved Spacetime Dirac Equations may help in solving the long standing riddle and mystery of the preponderance of matter over antimatter. We come to the tentative conclusion that if these CSTD-equations are to explain the preponderance of matter over antimatter; then, photons are to be thought of as described by the spherically curved version of this set of equations, while ordinary matter is to be explained by the parabolically and hyperbolically curved spacetime versions of this same set of equations.

The Dirac equation is a relativistic quantum mechanical wave equation serendipitously discovered by the eminent British physicist, Professor Paul Adrien Maurice Dirac [

The Dirac equation was discovered as part of an effort (by Dirac) to overcome the criticism levelled against the Klein-Gordon equation [

The Dirac equation applies to a flat Minkowski spacetime. Thus, it was born without the corresponding curved spacetime version. Realising this gap to be filled, several researchers proposed their own versions of the curved spacetime versions of the Dirac equation [

As is well known, Dirac [

Along the same line as that Dirac used in his derivation, it is a fundamental mathematical fact that a two-rank tensor (such as the metric tensor

where the matrices ^{1} are defined such that:

where

The index “a” is not an active index as are the Greek indices―its an index which labels a particular representation of the metric―it labels a particular curvature of spacetime i.e. whether spacetime is spherically^{2}, parabolically or hyperbolically curved. Written in full, the three metric tensors

Especially for a scientist and/or mathematician, there is little if anything they can do but accept facts as they

stand and present them-self thus the writing of

mate mathematical fact for as long as

written as

It is not a difficult exercise to show that multiplication of (5) from the left hand-side by the conjugate operator

gauge condition

As it stands, Equation (5) would be a horrible equation insofar as its solutions are concerned because the vector

where now the matrices

where

Now, on the main business of the day: we shall work-out the symmetries of the curved spacetime Dirac equations for the case

Under a T-transformation, the charges and positions will remain unchanged, whereas the currents will flow in opposite direction, in which case we will get:

Using similar arguments as above, we will get for the C-transformation, the following:

Finally under the combined CPT-transformation the charges and currents change sign and the electric and magnetic fields will retain their signs. These properties can be summarised in terms of the four vector potential

Of particular importance here is the transformations (11) of the four vector potential

To demonstrate the symmetries of the CSTD-equations under charge conjugation, we proceed as usual, that is, we bring the curved spacetime Dirac particle

Equation (12) represents the curved spacetime Dirac particle

If the CSTD-equations is symmetric under charge conjugation, then, there must exist some mathematical transformation, which if applied to (13) would lead us back to an equation that is equivalent to (12).

Starting from (13), in-order to revert back to (12), the first mathematical operation to be applied to (13) the complex conjugate operation on the entire equation. So doing, we will have:

If (12) is invariant under charge conjugation, then, there must exist a matrix

If such a matrix

where

We shall prove that the matrix

Taking the complex conjugate on both-sides of (17) and knowing that

By some legitimate mathematical operation, we need to remove the complex conjugate on

Multiplying both-sides of (19) by

so that:

and from this, it follows that we must have

therefore:

as desired in (15).

A parity transformation requires that we reverse the space coordinates i.e.

If (6) is invariant under a parity transformation, then, there must exist a matrix

There does not exist such a matrix

A time reversal transformation requires that we reverse the time coordinate i.e.

If (6) is invariant under a time reversal transformation, then, there must exist a matrix

There does not exist such a matrix

A simultaneous charge conjugation and parity transformation requires that we reverse the particle’s electromagnetic field and that of the ambient electromagnetic magnetic field i.e.

If (12) is invariant under a simultaneous charge conjugation and parity transformation, then, there must exist a matrix

This matrix

A simultaneous charge conjugation and time reversal requires that we reverse the ambient electromagnetic magnetic field i.e.

If (12) is invariant under a simultaneous charge conjugation and time transformation, then, there must exist a matrix

This matrix

If we are to reverse the spacetime coordinates, that is

If we are to reverse the ambient electromagnetic magnetic field together with the spacetime coordinates i.e.

In

Now, without simultaneously acting on the CSTD-equations with the parity operator (P), the effect of acting exclusively on the CSTD-equations with the spin operator (S) is the same as exclusively acting on this same equation with the P-operator. What this effectively means is that the spherically curved CSTD-equations i.e., the case

Symmetry | Case | ||
---|---|---|---|

C | Yes | Yes | Yes |

P | Yes | No | No |

T | Yes | No | No |

CP | Yes | No | No |

CT | Yes | No | No |

PT | Yes | Yes | Yes |

CPT | Yes | Yes | Yes |

Lorentz | Yes | Yes | Yes |

demonstrate how the violation of the P, T, CP and S-symmetries can be used harmoniously to explain Universe that we live in. These three CSTD-equations together with these symmetry violations neatly explain the existence of radiation and the preponderance of matter over antimatter.

The symmetries of the Lorentz invariant CSTD-equations have here been worked out and we have shown that the parabolic

In our view―insofar as the preponderance of matter over antimatter is concerned, one of the problems with the original Dirac equation is that it was born solo, as an equation explaining a Minkowski flat spacetime particle with no curved spacetime version of it. Realising the clear evident gap, over the years, researches proposed curved spacetime versions of the Dirac equation (cf. Refs.: [

If the predictions of the original Dirac equations together with its descendants [

In 1967, Professor Andrei Dimitriev Sakharov described three minimum properties of Nature which are required for any baryogenesis to occur, regardless of the exact mechanism leading to the excess of baryonic matter. In his seminal paper, Sakharov [

1) At least one B-number violating process.

2) C and CP-violating processes.

3) Interactions outside of thermal equilibrium.

These conditions must be met by any explanation in which

Therefore, the current thrust in research especially at CERN^{3} is to search for physical processes in Nature that violate CP-symmetry. In 2011 during high-energy Proton collisions in the LHCb experiment [

The first violations of CP-symmetry was first documented in Brookhaven Laboratory in the US in the 1960s in the decay of neutral Kaon particles. Since then, Japanese and US labs forty years later found similar behaviour in B^{0}-mesons systems where they detected similar CP-symmetry violations. LHCb-Collaboration [

This is not the case with the CSTD-equations which clearly predict T, CT and CP-violation as a permissible Laws of Nature. That is to say, in as much as the Dirac equation is taken as a Law of Nature, here we have (if we accept these equations) these CSTD-equations standing as candidate Laws of Nature in which case they predict T, CT and CP-violation. If we accept them as legitimate equations of physics as is the case with the Dirac equation, then, we can use them to explain the apparent preponderance of matter without the need for the Sakhorov conditions.

The Sakhorov conditions assume that the Laws of Nature are symmetric with respect to matter and antimatter. According to these pre-conditions, the preponderance of matter will arise in a Universe whose laws are perfectly symmetric with respect to matter and antimatter if there exists physical mechanisms and processes satisfying these conditions. If however the Laws of Nature are asymmetric with respect to matter and antimatter, there is no need for the Sakhorov conditions to explain the preponderance of matter over antimatter.

If all the three CSTD-equations are to operate simultaneously in the same Universe (there is nothing stopping this occurrence), then, the spherically curved spacetime version of the CSTD-equations, i.e., the case

First we must realise that each of the four discrete symmetries C, P, T and S have two states. If all of them where obeyed, then, a shown in

1) Clearly, T-violation implies that only Electrons moving forward in time will be allowed to exist. This means the eight Electrons [

T | |||||
---|---|---|---|---|---|

C | C | ||||

S | P | ||||

S | |||||

P | |||||

2) S-violation implies that exclusively positive or negative spin Electrons will be allowed in the Universe. This means of the eight candidate Electrons [

3) P-violation implies that exclusively left or right handed Electrons will be allowed in the Universe. This means of the remaining four candidate Electrons [

4) Finally, we must realise that of the two remaining Electrons

Clearly, there here is no need for the Sakhorov conditions in-order for there to be a preponderance of matter over antimatter. We do not say nor make the claim that this is “The Solution” to the long-standing problem of the preponderance of matter over antimatter, but that, it is (perhaps) a viable solution worthy of consideration.

The present work is to be taken as work in progress toward a Unified Field Theory [

1) The parabolic

2) If the Lorentz invariant CSTD-equations are to explain the prepondarance of matter over antimatter, then, photons are to be thought of as obeying the flat CSTD-equations i.e., the CSTD-equations for which

We are grateful to the National University of Science & Technology (NUST)’s Research & Innovation Department and Research Board for their unremitting support rendered toward our research endeavours; of particular mention, Prof. Dr. Mundy, Dr. P. Makoni and Prof. Dr. Y. S. Naiks unwavering support. This paper is dedicated to my mother Setmore Nyambuya and to the memory of departed father Nicholas Nyambuya (1947-1999).

Golden GadzirayiNyambuya, (2015) On the Preponderance of Matter over Antimatter. Journal of Modern Physics,06,1441-1451. doi: 10.4236/jmp.2015.611148