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Concerning the Dirac [gamma]-Matrices Under a Lorentz Transformation of the Dirac Equation.

"We have found it of paramount importance that in order to progress we must recognize our ignorance and leave room for doubt."

--Richard Phillips Feynman (1918-1988)

1 Introduction

As taught to physics students through the plethora of textbooks available on our planet e.g., refs. [1-5], the Dirac 4 x 4 [gamma]-matrices ([[gamma].sup.[mu]]) are usually presented as objects that undergo a transformation during a Lorentz transformation of the Dirac [6.7] equation. This issue of the transformation of these [gamma]-matrices is not well represented in the literature [8]. There, thus, is a need to clear the air around this issue regarding the proper transformation properties of these matrices. To that end, we here argue in favour of these matrices as physical four-vectors and as such, they must under a Lorentz transformation transform as four-vectors. In-fact, it is well known that the [[gamma].sup.i]-matrices (i = 1,2,3) represent spin (i.e., [mathematical expression not reproducible]) because, together with the angular momentum operator ([??]), their sum total of the orbital angular momentum and spin [mathematical expression not reproducible] commutes with the Dirac Hamiltonian ([H.sub.D]), i.e. ([??], [H.sub.D]] = 0), implying that [??] is a constant of motion.

For a particle whose rest-mass and Dirac [6, 7] wavefunction are [m.sub.0] and [psi] respectively, the corresponding Dirac [6.7] equation is given by:

i[??][[gamma].sup.[mu]][[partial derivative].sub.[mu]][psi] = [m.sub.0]c[psi], (1)

where:

[mathematical expression not reproducible], (2)

are the 4 x 4 Dirac [gamma]-matrices where [I.sub.2] and 0 are the 2 x 2 identity and null matrices respectively, and |[psi]> is the four component Dirac [6,7] wave-function, h is the normalized Planck constant, c is the speed of light in vacuum, i = [square root of -1], and:

[mathematical expression not reproducible], (3)

is the 4 x 1 Dirac [6,7] four component wavefunction and [[psi].sub.L] and [[psi].sub.R] are the Dirac [6,7] bispinors that are defined such that:

[mathematical expression not reproducible]. (4)

Throughout this reading--unless otherwise specified; the Greek indices will here-and-after be understood to mean ([mu], v, ... = 0,1,2,3) and the lower case English alphabet indices (i, j, k... = 1,2,3).

2 Lorentz Transformation of the Dirac as usually presented

To prove Lorentz Invariance (Covariance) two conditions must be satisfied:

1. The first condition is that: given any two inertial observers O and O' anywhere in spacetime, if in the frame O we have:

[i[??][[gamma].sup.[mu]][[partial derivative].sub.[mu]] - [m.sub.0]c][psi](x) = 0, (5)

as the Dirac equation for the particle f, then:

[i[??][[gamma].sup.[mu]'] [[partial derivative].sub.[mu]'] - [m.sub.0]c][psi]'(x') = 0 (6)

is the equation describing the same state but in the frame O'.

2. The second condition is that', given that [psi](x) is the wavefunction as measured by observer O, there must be a prescription for observer O' to compute [psi](x') from [psi](x) where [psi]'(x') describes to O' the same physical state as that measured by O. The conserve must be true as-well, that is: there must exist a prescription such that starting from equation (6), one can arrive at (5).

In simpler mathematical terms, the above two requirements are saying that: starting from equation (5), there must exist some physically legitimate transformations within the framework of Lorentz transformations that can take (map) us from this equation (5) to equation (6) and vice-versa. If we can find these, then, the Dirac equation is said to be Lorentz Invariant (Covariant).

Now, since the Lorentz transformations are linear, it is to be required or expected of the transformations between f (x) and [psi]'(x') to be linear too, i.e.:

[psi]'(x') = [psi]'([LAMBDA]x) = S ([LAMBDA])[psi](x) = S([LAMBDA])[psi]([[LAMBDA].sup.-1] x'), (7)

where S([LAMBDA]) is a 4 x 4 matrix which depends only on the relative velocities of O and O' and [LAMBDA] is the Lorentz transformation matrix. S ([LAMBDA]) has an inverse if O [right arrow] O' and also O' [right arrow] O. The inverse is:

[psi](x) = [S.sup.-1]([LAMBDA])[psi]'(x') = [S.sup.-1]([LAMBDA])[psi]'([LAMBDA]x), (8)

or we could write:

[psi](x) = S ([[LAMBDA].sup.-1])[psi]'([LAMBDA]x) [right arrow] S ([[LAMBDA].sup.-1]) = [S.sup.-1]([LAMBDA]). (9)

We can now write equation (5), as:

[mathematical expression not reproducible], (10)

and multiplying this from the left by S ([LAMBDA]), we have:

[mathematical expression not reproducible], (11)

and hence:

[mathematical expression not reproducible]. (12)

Therefore, for the above equation to be identical to equation (6) (hence Lorentz Invariant), the requirement is that:

[[gamma].sup.[mu]'] = S([LAMBDA])[[gamma].sup.[mu]] [partial derivative][x.sup.[mu]']/[partial derivative][x.sup.[mu]] [S.sup.-1]([LAMBDA]), (13)

hence, we have shown that--for as long as [S.sup.-1]([LAMBDA]) exists, equation (5) is Lorentz Invariant.

3 Dirac [mathematical expression not reproducible]-matrices as a four-vector

The Dirac equation (1) can be re-written in the traditional Schrodinger formulation as (H[psi] = E[psi]) where H and E are the energy and Hamiltonian operators respectively. In this Schrodinger formulation, H, will be such that it is given by:

H = [[gamma].sup.0][m.sub.0][c.sup.2] - i[??]c[[gamma].sup.0] [[gamma].sup.j][[partial derivative].sub.j], (14)

and (E = i[??][partial derivative]/[partial derivative]t).

Now, according to the quantum mechanical equation governing the evolution of any quantum operator Q, we know that:

i[??]Q/[partial derivative]t = QH - HQ = [Q, H], (15)

hence, if:

[Q, H] [equivalent to] 0, (16)

then, the quantum mechanical observable corresponding to the operator Q is a conserved physical quantity.

With this [equation (15)] in mind, Dirac asked himself the natural question--what the "strange" new [gamma]-matrices appearing in his equation really represent. What are they? In-order to answer this question, he decided to have a "look" at or make a closer "inspection" of the quantum mechanical orbital angular momentum operator [L.sub.i] which we all know to be defined:

[mathematical expression not reproducible], (17)

where, [[epsilon].sub.ijk] is the completely-antisymmetric three dimensional Levi-Civita tensor. In the above definition of [L.sub.i], the momentum operator [??] is the usual quantum mechanical operator, i.e.:

[mathematical expression not reproducible]. (18)

From this definition of Li given in equation (17), it follows from equation (15) that i[??][partial derivative][L.sub.i]/[partial derivative]t = [[L.sub.i], H], will be such that:

[mathematical expression not reproducible]. (19)

Now, because the term [[gamma].sup.0][m.sub.0][c.sup.2] is a constant containing no terms in pt, it follows from this very fact that ([[epsilon].sub.ijk][[x.sub.j][[partial derivative].sub.k], [[gamma].sup.0]] [equivalent to] 0), hence equation (19) will reduce to:

[mathematical expression not reproducible]. (20)

From the commutation relation of position ([x.sub.i]) and momentum (-i[??][[partial derivative].sub.j]) due to the Heisenberg uncertainty principle [9], namely (-i[??][[x.sub.i], [[partial derivative].sub.j]] = -i[??][[delta].sub.ij]) where [[delta].sub.ij] is the usual Kronecker-delta function, it follows that if in equation (20), we substitute ([[partial derivative].sub.l][x.sub.j] = [x.sub.j][[partial derivative].sub.l] + [[delta].sub.lj]), this equation is going to reduce to:

[mathematical expression not reproducible]. (21)

The term with the under-brace vanishes identically, that is to say: ([x.sub.j][[partial derivative].sub.k][[partial derivative].sub.l] - [x.sub.j][[partial derivative].sub.l][[partial derivative].sub.k] [equivalent to] 0); and ([[epsilon].sub.ijk][[gamma].sup.0][[gamma].sup.l][[delta].sub.ij] = [[epsilon].sub.ilk][[gamma].sup.0] [[gamma].sup.l]), it follows from this that equation (21), will reduce to:

i[??] [partial derivative][L.sub.i]/[partial derivative]t = [[??].sup.2]c[[epsilon].sub.ilk][[gamma].sup.0] [[gamma].sup.l][[partial derivative].sub.k]. (22)

Since this result [i.e., equation (22) above] is non-zero, it follows from the dynamical evolution theorem [i.e., equation (16)] of Quantum Mechanics (QM) that none of the angular momentum components [L.sub.i] are--for the Dirac particle-going to be constants of motion. This result obviously bothered the great and agile mind of Paul Dirac. For example, a non-conserved angular momentum would mean spiral orbits i.e., Dirac particles do not move in fixed and well defined orbits as happens with electrons of the Hydrogen atom for example; at the very least, this is very disturbing because it does not tally with observations. The miniature beauty that Dirac had--had the rare privilege to discover and, the first human being to "see" with his beautiful and great mind--this-had to be salvaged * somehow.

Now--enter spin! Dirac figured that "Subtle Nature" must conserve something redolent with orbital angular momentum, and he considered adding something to [L.sub.i] that would satisfy the desired conservation criterion, i.e.: call this unknown, mysterious and arcane quantity [S.sub.i] and demand that:

i[??] [partial derivative]([L.sub.i] + [S.sub.i])/[partial derivative]t [equivalent to] 0. (23)

This means that this strange quantity Si must be such that:

i[??] [partial derivative][S.sub.i]/[partial derivative]t = [[S.sub.i], H] = -[[??].sup.2]c[[epsilon].sub.ilk] [[gamma].sup.0] [[gamma].sup.l][[partial derivative].sub.k]. (24)

Solving equation (24) for Si, Dirac arrived at:

[mathematical expression not reproducible], (25)

where ([[gamma].sup.5] = i[[gamma].sup.0][[gamma].sup.1][[gamma].sup.2][[gamma].sup.3]), is the usual Dirac gamma-5 matrix.

Now, realising that:

1. The matrices [[sigma].sub.i] are Pauli matrices and they had been ad hocly introduced in 1925 into physics to account for the spin of the Electron by the Dutch-American theoretical physicists, George Eugene Uhlenbeck (1900-1988) and his colleague, Samuel Abraham Goudsmit (1902-1978) [10].

2. His equation--when taken in the non-relativistic limit, it would account for the then unexplained gyromagnetic ratio (g = 2) of the Electron and this same equation emerged with explaining the Electron's spin.

The agile Paul Dirac seized the golden moment and forthwith identified [S.sub.i] with the [psi]-particle's spin. The factor 1/2 [??] in [S.sub.i] implies that the Dirac particle carries spin 1/2, hence, the Dirac equation (1) is an equation for a particle with spin 1/2!

While in this esoteric way (i.e., as demonstrated above) Dirac was able to explain and "demystify" Wolfgang Pauli (1900-1958)'s strange spin concept which at the time had only been inserted into physics by "the sleight of hand" out of an unavoidable necessity, what bothers us (i.e., myself) the most is:
   How it comes about that we (physicists) have had
   issues to do with the transformational properties
   of the [gamma]-matrices? Why? Really--why? The
   fact that orbital angular momentum [??] is a vector
   invariably leads to the indelible fact that [??] is a
   vector as-well, because we can only add vectors
   to vectors.


If [??] is a vector, then the matrices [[gamma].sup.l] must be components of a 3-vector, so must the matrix [[gamma].sup.0] be the component of the time-vector in the usual four-vector formalism, hence [[gamma].sup.[mu]] must be a four-vector. So, right from the word go--with little or no resistance whatsoever, it must have been pristine clear that the [gamma]-matrices must be four-vectors.

4 Dirac equation with the [gamma]-matrices as a four-vector

With [gamma]-matrices now taken as a four-vector, the object [[gamma].sup.[mu]][[partial derivative].sub.[mu]] is a scalar, the meaning of which is that the Dirac equation will now accommodate two types of spinors "the usual Dirac bispinor" and a new "scalar-bispinor", i.e.:

1. A spinor that is a scalar. Let us here call this a scalar-bispinor and let us denote it with the symbol f and because of its scalar nature--under a Lorentz transformation, we will have ([phi]' = [phi]). Just like the ordinary Dirac wavefunction [psi] is a 4 x 1 component object, f is also a 4 x 1 object, i.e.:

[mathematical expression not reproducible], (26)

where [[phi].sub.L] and [[phi].sub.R] are the scalar-spinors--which are like the ordinary left and right handed Dirac spinors ([[psi].sub.L], [[psi].sub.R]); [[psi].sub.L] and [[phi].sub.R] are defined:

[mathematical expression not reproducible]. (27)

Consideration of the scalar-bispinor has been made in the past by others e.g., [11].

2. The ordinary Dirac bispinor [phi]: that transforms linearly under a Lorentz transformation i.e. ([phi]' = S[phi]), where, a usual, Lorentz Invariance (Covariance) requires that the function S = S ([x.sup.[mu]], [[??].sup.[mu]]) be such that:

[[gamma].sup.[mu]'] [[partial derivative].sub.[mu]']S = [[gamma].sup.[mu]] [[partial derivative].sub.[mu]], S = 0, (28)

and:

[[gamma].sup.[mu]] = [S.sup.-1] [[gamma].sup.[mu]]S, (29)

which implies:

[S, [[gamma].sup.[mu]]] = 0. (30)

Now, we certainly must ask "What does this all mean". That is to say, the fact that the Dirac equation allows for the existence of the usual Dirac bispinor [psi] and in addition to that --a scalar-bispinor [phi]? Taken at the same level of understanding that the Dirac equation's prediction of the existence of antimatter is premised on the Dirac equation being symmetric under charge conjugation--on that very same level of understanding, this fact that the Dirac equation in its most natural and un-tempered state as presented herein--it, allows for the existence of the usual Dirac bispinor [psi] and scalar-bispinor [phi]; in the same vein of logic, this naturally implies that for every Dirac bispinor [psi], there must exist a corresponding scalar-bispinor [phi]. That is, the Dirac bispinor [psi] and the scalar-bispinor [phi] must come in pairs. There is no escape from this train of logic.

If we are thinking of Leptons and Neutrinos, the above pair-picture of ([psi], [phi]) makes perfect sense. Based on this picture, we can write the Dirac equation for this pair ([psi], [phi]) as:

[mathematical expression not reproducible], (31)

where [eta] is a scalar-constant that we have introduced so as to accommodate the possibility that the particle-pair ([psi], [phi]), may have different masses. In this way, one can begin to entertain ideas on how to explain the Lepton-Neutrino pairing [([e.sup.[+ or -]], [v.sub.e]), ([[mu].sup.[+ or -]], [v.sub.[mu]]), ([[tau].sup.[+ or -]], [v.sub.[tau]])]. We have no intention of doing this or going any deeper on this matter but merely to point out--as we have just done--that, this idea may prove a viable avenue of research to those seeking an explanation of why this mysterious pairing occurs in nature.

5 General discussion

We must categorically state that--what we have presented herein is not new at all. All we have endeavoured is to make bold the point that the [gamma]-matrices constitute a four-vector. Perhaps the only novelty there is--in the present contribution --is the suggestion that we have made--namely that, the resulting scalar-bispinor (0) and the usual Dirac bispinor ([psi]) can be used as a starting point to explain the currently open problem of the three generation Lepton-Neutrino pairing ([e.sup.[+ or -]], [v.sub.e]), ([[mu].sup.[+ or -]], [v.sub.[mu]]) and ([[mu].sup.[+ or -]], [v.sub.[tau]]); where the scalar-bispinor can be assumed to be the Neutrino while the usual Dirac bispinor can be thought of the Lepton. In the sequatial reading [12], we will demonstrate how this formulation of the Dirac equation can be used to explain how massless neutrinos can oscillate.

Acknowledgements

We are grateful for the assistance rendered unto us by the National University of Science and Technology's Research Board toward our research endeavours.

Submitted on March 16, 2018

References

[1.] Itzykson C. and Zuber J.-B. Quantum Mechanics II. McGraw-Hill, New York, USA, 1980.

[2.] Messiah A. Quantum Mechanics II. North-Holland Publishing Company, Amsterdam, Netherland, second edition, 1962.

[3.] Sakurai J. J. Advanced Quantum Mechanics. Addison-Wesley Publishing Company, Massachusetts, USA, 1967.

[4.] Schweber S. S. An Introduction to Relativistic Quantum Field Theory. Row, Peterson and Company, New York, USA, 1961.

[5.] Zee A. Quantum Field Theory in a Nutshell. Princeton University Press, USA, second edition, 2010.

[6.] Dirac P. A. M. The Quantum Theory of the Electron. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 1928, v.117(778), 610-624.

[7.] Dirac P. A. M. The Quantum Theory of the Electron. Part II. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 1928, v. 118(779), 351-361.

[8.] Nikolic H. How (Not) To Teach Lorentz Covariance of the Dirac Equation. European Journal of Physics, 2014, v.35(3), 035003.

[9.] Heisenberg W. Ueber den anschaulichen Inhalt der quantentheoretischen Kinematik and Mechanik. Zeitschriftfur Physik, 1927, v.43, 172198. Engilish Translation: Wheeler J. A. and Zurek W. H. (eds) Quantum Theory and Measurement. Princeton (NJ), Princeton University Press, 1983, 62-84.

[10.] Uhlenbeck G. E. and Goudsmit S. Ersetzung der Hypothese vom unmechanischen Zwang durch eine Forderung beziiglich des inneren Verhaltens jedes einzelnen Elektrons. Die Naturwissenschaften, 1925, v.13(47), 953-954.

[11.] Chapman T. C. and Leiter D. J. On the Generally Covariant Dirac Equation. American Journal of Physics, 1976, v.44(9), 858-862.

[12.] Nyambuya G. G. Oscillating Massless Neutrinos. Progress in Physics, 2018, v. 14, 94-98.

G. G. Nyambuya

National University of Science and Technology, Faculty of Applied Sciences--Department of Applied Physics, Fundamental Theoretical and Astrophysics Group, P. O. Box 939, Ascot, Bulawayo, Republic of Zimbabwe

E-mail: physicist.ggn@gmail.com

* Such is the indispensable attitude of the greatest theoretical physicists that ever graced the face of planet Earth--beauty must and is to be preserved; this is an ideal for which they will live for, and if needs be, it is an ideal for which they will give-up their life by taking a gamble to find that unknown quantity that restores the beauty glimpsed!
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Author:Nyambuya, G.G.
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Date:Apr 1, 2018
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