General Spin Dirac Equation (II)

In an earlier reading [1], we did demonstrate that one can write down a general spin Dirac equation by modifying the usual Einstein energy-momentum equation via the insertion of the quantity “ s ” which is identified with the spin of the particle. That is to say, a Dirac equation that describes a particle of spin 1 2 s  S σ where is the normalised Planck constant, are the Pauli matrices and  σ 2 2    1, 2, 3, ,etc. s      . What is not clear in the reading [1] is how such a modified energy-momentum relation would arise in Nature. At the end of the day, the insertion by the sleight of hand of the quantity “ s ” into the usual Einstein energy-momentum equation, would then appear to be nothing more than an idea belonging to the domains of speculation. In the present reading—by making use of the curved spacetime Dirac equations proposed in the work [2], we move the exercise of [1] from the realm of speculation to that of plausibility.


Introduction
In an earlier reading [1], it is argued without a proper physical basis but more out of mathematical curiosity that the modified dispersion relation or the modified Einstein energy-momentum relation: grid.In the dispersion relation (1.1), is the total energy of the particle, is its momentum, 0 its rest mass and is the speed of light in vacuum.What is not clear in this reading [i.e. in Ref. 1] is how such an energy-momentum relation would arise in Nature in a manner that can be justified without making ad hoc and hand-waving arguments.At the end of the day, the insertion by the sleight of hand of the quantity " E m p s " into the usual Einstein energy-momentum equation: would then appear to be nothing more than a product of agile mathematical curiosity, speculation and chicanery, without anything to do with physical and natural reality as we know it.Herein, by making use of the three curved spacetime Dirac equations proposed in [2], we move the exercise of [1] from the realm of curiosity, speculation and chicanery to that of plausibility.As already stated, in (1), it is not clear why the quantity " s " has to take integral values   1, 2, 3, , etc. s      .Because spin has to take integral and half integral values, it was assumed without proof that this quantity " s " has to take integral values.This off cause is a hole in the theory that needs to be filled.This reading will furnish this missing part in the "General Spin Dirac Equation" proposed in [1].We not only demonstrate how " s " comes to be part of the dispersion relation , but how and why this quantity comes to take only integer values.Now, in-closing this section, let us give a brief synopsis of the present reading.It is as follows.In the next section, we are going to give a brief exposition of the curved spacetime Dirac equation first presented in [2].In the successive section, we are going to dwell on the main thrust of the present reading by demonstrating how " s " comes to be part of the dispersion relation and as-well how and why " s " comes to take only integer values.Thereafter, we give a general discussion and the conclusions drawn thereof.Lastly, we are of the very strong view that any reader that wants or seeks to make sense of the present reading must first go through the readings [1,2] as these are minimum prerequisites.Otherwise, if they [the reader] do not do so, they will miss the main content and morass substance of the present reading.

Curved Spacetime Dirac Equations
As is well known, the Dirac equation is derived from the fundamental equation , where We know that its equivalent in curved spacetime is given by: where the four momentum p  is given by p and g  is the metric of spacetime.In order to aid the reader in visualizing (3) in a way that conforms to the end that we seek, we have to write this equation in its equivalent matrix form, i.e.: Above in (4), the " T " in the superscript of the column vector denotes the transpose operation on that column vector.Now, in writing down the curved spacetime version of the Dirac equation [in the reading 2], we made a novel suggestion of writing down the spacetime metric tensor g  as: where A  is some four vector and .In general, the metric , there are no off-diagonal terms in the metric, while for the cases , we have off diagonal terms [see 2].As shown there in [2], the resulting three curved spacetime Dirac equations are given by: where 2 : In the above (and hereafter), 2 I is the 2 identity matrix, , should be taken as a gauge condition restricting this four vector.In the next section, we are going to demonstrate the Lorentz invariance of the curved spacetime Dirac equation (7).

Lorentz Invariance
To prove Lorentz invariance 3 , two conditions must be satisfied, these two conditions are: 1) Given any two inertial observers O and O anywhere in spacetime, if in the frame we have is the equation describing the same state but in the frame 2 In Equation (1.8) above, the term  must be treated as a single object with one index  .This is what this object is.One can set In-order that we are on the same level of understanding with the general reader, we do not have to deviate from the standard terminology.
2) Given that   The meaning of the above is that the matrices   In Case (II), we can have this transform under a multiplication of  by some constant matrix .If , then this matrix will have to be such that in-order for Lorentz invariance to hold.
The present exercise to re-demonstrate the Lorentz invariance of (7) has been conducted so as to demonstrate the all-important difference that we must always take note of, that is, in the bare Dirac theory, the Additionally, we have shown here that Equation ( 7) is not Lorentz covariant but Lorentz invariant.The orginal Dirac equation is not Lorentz invariant but Lorentz convariant-this is something to be noted as it distinguishes the present effort from that of [3,4].

General Magnitude of a Four Vector
In this section, we are going to look into the issue of the magnitude of a four vector.For example, the square of the magnitude of the four momentum p  is such that . If we take a general four vector  , is a constant, it has the same value everywhere all the time; so that in general we can assume that the , is a constant aswell.We ask, "In general, does  have to be a constant?"The answer to this question is a bold no!It only has to be a scalar since the quantity g V V    is a scalar.A constant is a special kind of a scalar, it is a scalar that takes the same value everywhere all the times.If  is a general scalar, then

Given the above thesis i.e.
, what we seek here is a function that gives the value of is itself a scalar, we propose that, in general, the magnitude of all four vectors in spacetime be such that g   , so that: where *  is a constant which takes the same value everywhere all the times for-all observers.The quantity *  has the dimensions as that of V  .
One may very well be tempted to ask the good question "What is the motivation for (10)?" Well-as will be seen in the next section; the motivation for the proposal ( 10) is that if we do not have such a setting, then contrary to experience, the rest mass of a particle in a curved spacetime will have to depend on where the particle is, and when it is at that place where it issimple, To avoid this, we have no choice but to impose (10).

Energy Solutions
The energy-momentum equation for the particles described by Equation ( 7) is: where in line with (10), we will have A , we will have: which goes against experience.It is for this reason that we afore-proposed the condition (10).Now, setting ; and inserting these settings into the above, we will have: the subject of the formula, we will have: (13) From this, it is clear that we will have three negative en 4) to ju ergy particles and three positive energy particles.Now, in the next section, we are going to use (1 stify the insertion of " s " into the Einstein equation Demonstrating how the " s " comes to be p , also p ves for the other cases art of ro 1    .

Justification e case
Let us consider th 0   is ass . Space is usually assumed to be isotropic.Th umption finds solid justification form experience since observations reveal no directional properties of space, the deeper meaning of which is that space must have no preferential direction or directional properties.In the case of the metric (5), isotropy would mean that the space parts of the four vector A  must all be equal or identical to each other, that is space e above mentioned proof, let us write down the general spin dispersion relationship for a particle whose spacetime is isotropic.This we are going to do so that, we supply, not only the proof of why and how s   for the case 0   , but for the other two cases ll i.e. 1 as-we    .T general dispersion relationship of a particle e spacetime is isotropic is given by: he whos Now, (7) can be written in the general Schrödinger formulation as ˆ     where  and  are the Hamiltonian and rators res ctively o doing, i.e. writing (7) in the said form, we will have: From this, it follows that the new General Spin Dirac Hamiltonian is given by: This General Dirac Hamiltonian commutes with the total angu entum operator . The proof of this assertion is supplied in the Appendix.This fact th is the total an .The operator gular momentum of the particle since it commutes with the Hamiltonian , 1 where, and , 2 and as-well: and .
The 's are k  4 4  matrices such that:   Now, we propose the following eigenvalue equation: where   the th k -component of the phase of the particle.That is, if  is the four momentum of a particle and x  is its four position in spacetime, then, the phase of this particle S is such th S p x at  


. This phase can be s  .From this, it follows that we can rewrite (1.17) as: Acting on this equation from the left by    (27   z S , one can ea  sily show by using the fact (24), namely ˆˆ, and aswell the fact that , one e resu arrives at th lting equation: which then acts on (28).That is, acting from the left on (28) using this new operator   ˆk    , and thereafter perform g th necessary algebraic operations, the resulting equation is: is an integer.From the foregoing, it thus follows that " k s " will take only integral values i.e.
  . We have not only proved that " k s " is an integer, but in so doing, we have also proved spin is a physical quantity.

Metric of a General Spin Dirac Particle
From the above fin the general spacetime metric of a l spin Di e.We have argued that the four vector why quantised dings, we can compute genera rac particl . From this, we can write down a four spin quantum number s  .To do this, we note that the four vector A  can be written with its components as Further, this can be written as . The quantity   .From this, the four vector A  can now be written as   into (5), we will have: ju plification-from ten potentials to just one potential!At this point, the reader m legitimately want to ask if g  has the same meaning as in Einstein's General Theory of Relativity (GTR)?To answer this question, one has to visit the reading [5].It shown there in [5] that the vector is A  gives raise to the nuclear force non- abelian gauge field.The details of the Unified Field Theory presented in [5] are still being worked out.What the reader can do for now simple take A  as a four vector and nothing else.As to whether this vector represents a gravitational, electric or any force field for that matter is of no consequence here since we are not concerned with the force field which this four vector represents.

Discussion and Conclusion
We strongly believe that this reading stifies the assertion made in [1], namely that the modified Einstein dispersion relation .Not only have we justified this, we have also argued that " s " must take integral values.This means that, the work presented in [1] has be re acceptable pedestal.The reason we say this is because we believe that despited the fact that the true meaning and significance the curved spacetime Dirac equation derived in [2] has not been found yet, these curved spacetime Dirac equations are credible, mathematically and physically legitimate equations.Ac emonstrated that these curved spacetime Dirac equation a e key to the attainment of a general spin Dirac equation.
Insofar as the unification programme of physics is concerned, we believe that the writing down of an acceptable general spin Dirac equation is a step in the right direction.If discovered, the final unified theory is expected to be such that a "single equation/principle will explain about every observable phenomenon.Amongst others, it is expected that a single equation must be able to explain all particles from a simple unifying principle.In the light of the aforestated, it i put on a mu tually, it has been d r s so ch mo en mewhat sad to say that the current state of physics vis the equations purporting to explain particles-is very "ugly".For example, the Schrödinger equation describes spin-0 atoms and molecules [6], the Klein-Gordon equation describes spin-0 particles (that is carriers of forces), while the Dirac equation describes spin-1 2 particles, and the Rarita-Schwinger equation describes spin-3/2 particles [7].From this rather "ugly" trend, does it mean we have to look for another equation to describe spin-2 particles, and then another for spin-5 2 particles etc?This does not look beautiful, simple, or at the very least suggest at the far and deeper end, a unification of the Natural Laws.It is on this note that we feel the present endeavour are worthwhile.
Another interesting outcome is that ( 7) is no lon r restricted to the description of Fermions, but Bosons aswel If this equation proves successful as h ppened with Dirac's original equation, then, it will perhaps be the first equat n in physics to describe both Fermi and Bosons from a single unified principle or standpoint.Further, this equation shares some common ground with super-symmetry theories-that is, theories that try and unify quantum mechanics and gravitation is.For all we know is that from an abstract mathematical standpoint, this is what one must do.Our hope is that these and other seemingly strange concepts and operations will become clear as horizons of our insight deepens.
In-closing, we would like to point out something of note that we have not made mention of, namely that, the writing down of the general spin Dirac Equations (30) has bro im c Equ ught about a great simplification of the three curved spacet e Dira ations (7).When these equations were first written down [in 2], we wondered if they would be soluble at all.To dramatise and express this feeling, this reading [2] was started with a quote from Paul Dirac, namely: when measured in different random directions.
4) It has been shown that the curved spacetime Dirac equation leads to a Dirac wavefunction that can take a scalar nature, i.e., the resulting four component wavefunction  , together with the

Acknowledgements
I am grateful to the various anonymous Reviewers for their effort that g "The underlying Physical Laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these Laws leads to equations much too complicated to be soluble".
The apparent insolubility is because of the presence of four vector A  in Equation ( 7).Our guess then was that (7) would need to be solved numerically in-order to solve for  , but the present ef arguments presented herein.National University of Scie fort has unequivocally shown that this is not the case since

C
e Dirac particles naturally quantised i.e. it comes in integral fundamental basic unit of spin.This spi fact that the spin of a particle is measured to be th that space is is onclusion Assuming the acceptability (correctness) of the ideas propagated herein, we hereby make the following conclusions: 1) We have demonstrated that the curved spacetime Dirac equations [presented in Ref. 2] naturally lead to a general spin Dirac equation.
2) The spin of these curved spacetim quantization strongly appears to be wholly a part and parcel of the fabric of spacetime itself.
[6] E. Schrödinger, Physical Review, Vol. 28, 1 e same independent of the orientation; this fact suggests very strongly that spacetime must be isotropic on a quantum scale.If this were not the case 1049-1070.doi:10.1103/PhysRev.28.1049 [7] W. Ratita and J. Schwinger, Physical Review, Vol. 60, 1941, p. 61. doi:10.1103/Photropic on the quantum scale, then, according to the ideas propagated herein, a particles' spin will be different ysRev.60.61Appendix e are going to prove the crucial assertion that we stated page (2022) without any proof, namely that: . To begin, we know that: from this and as-well from the fact that: We also know that: . We only have to prove this for , , j x y z  , this prove is sufficient as prove for the rem We shall prove this for the case aining two cases.j x  .We know that: . From this, it follows that: , e case (A.1) implies that for th   2 a  , we will have: In this way, our task is now much easier, if we can show t , 0 Clearly, upon correct algebraic operations, one can verify that: is such that: which is equal to:  .11)which invariably implies that: ence we arrive at our desired result, namely, .Hence, according to our earlier arguments, it follows that the for-all and for-all


a General Spin Dirac Equation.That is to say, the resulting Dirac equation describes a particle of spin Pauli matrices and i, j, k are the three orthonormal basis on the xyz c and the Dirac four component  is represented in Case (I) where it is a scalar.The Dirac four component  is not constrained to only be a scalar.
-well the four component function  , do transform under a Lorentz transformation.This is not the case here;   a   is a constant matrix and the Dirac four component function  is scalar.In the reading [2], this very important fact that   a   is a constant matrix and that the Dirac four component function  can be scalar, was missed altogether, hence the need to make this clear at the present moment in the further development of the curved spacetime Dirac equation.
of the particle in question.Now, dividing (11) throughout by   2 0 equation.When this assertion was made in [1], it was not clear then, as to how such a it allows for the transmutation of a Fermion to a Boson and vice-versa.We believe this equation might very well be of interest to physicists working in this field.To transform a Fermion to a Boson and vice-versa, one simple acts on the wavefunction  with the operator   ˆk    .In physical terms, we have no idea what an operation on  with   ˆk    to take integer values thus literally eliminating what appeared to be a sure and impending mathematical nightmare of a numerical solution of the  .search & Innovation Department and Research Board for their unremitting support rendered toward my research endeavours; of particular mention, Dr. P. Makoni and Prof. Y. S. Naik's unwavering support.This publication proudly acknowledges a GRANT from the National University of Science and Technology's Research Board.