Volume 14 (2018) PROGRESS IN PHYSICS Issue 2 (April) Concerning the Dirac γ-Matrices Under a Lorentz Transformation of the Dirac Equation 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: [email protected] We embolden the idea that the Dirac 4 × 4 γ-matrices are four-vectors where the space components (γi) represent spin and the forth component (γ0) should likewise represent the time component of spin in the usual four-vector formalism of the Special Theory of Relativity. With the γ-matrices as four-vectors, it is seen that the Dirac equation admits two kinds of wavefunctions — (1) the usual four component Dirac bispinor and (2) a scalar four component bispinor φ. Realizing this, and knowing forehand of the existing mystery as to why Leptons and Neutrinos come in pairs, we seize the moment and make the suggestion that the pair ( ; φ) can be used as a starting point to explain mystery of ± ± ± why in their three generations [(e ; νe); (µ ; νµ); (τ ; ντ)], Leptons and Neutrinos come in doublets. In this suggestion, the scalar-bispinor φ can be thought of as the Neutrino while the usual Dirac bispinor can be thought of as the Lepton. “We have found it of paramount importance that in are the 4 × 4 Dirac γ-matrices where I2 and 0 are the 2×2 order to progress we must recognize our ignorance identity and null matrices respectively, and j i is the four and leave room for doubt.” component Dirac [6, 7] wave-function, ~ is the normalizedp — Richard Phillips Feynman (1918-1988) Planck constant, c is the speed of light in vacuum, { = −1, and: 0 1 1 Introduction B 0 C 0 1 B C B L C B 1 C B C As taught to physics students through the plethora of text- = B C = B C ; (3) B 2 C @B AC books available on our planet e.g., refs. [1–5], the Dirac 4 × 4 @B AC R 3 γ-matrices (γµ) are usually presented as objects that undergo is the 4 × 1 Dirac [6,7] four component wavefunction and a transformation during a Lorentz transformation of the Dirac L and are the Dirac [6,7] bispinors that are defined such that: [6, 7] equation. This issue of the transformation of these γ- R 0 1 0 1 matrices is not well represented in the literature [8]. There, B 0 C B 2 C B C B C thus, is a need to clear the air around this issue regarding the L = B C and R = B C : (4) @B AC @B AC proper transformation properties of these matrices. To that 1 3 end, we here argue in favour of these matrices as physical Throughout this reading — unless otherwise specified; four-vectors and as such, they must under a Lorentz transfor- the Greek indices will here-and-after be understood to mean mation transform as four-vectors. In-fact, it is well known (µ, ν; ::: = 0; 1; 2; 3) and the lower case English alphabet in- that the γi-matrices (i = 1; 2; 3) represent spin (i.e., S~ = dices (i; j; k ::: = 1; 2; 3). 1 1~ˆ 1 2~ˆ 1 3~ˆ 2 ~γ i + 2 ~γ j + 2 ~γ k) because, together with the angular momentum operator (L~), their sum total of the orbital angu- 2 Lorentz Transformation of the Dirac as usually pre- lar momentum and spin (J~ = L~ + S~) commutes with the sented Dirac Hamiltonian (HD), i.e. ([J~; HD] = 0), implying that J~ To prove Lorentz Invariance (Covariance) two conditions is a constant of motion. must be satisfied: For a particle whose rest-mass and Dirac [6, 7] wave- 1. The first condition is that: given any two inertial ob- 0 function are m0 and respectively, the corresponding Dirac servers O and O anywhere in spacetime, if in the frame [6, 7] equation is given by: O we have: µ ~ µ − {~γ @µ = m0c ; (1) [i γ @µ m0c] (x) = 0; (5) as the Dirac equation for the particle , then: where: µ0 0 0 [i~γ @ 0 − m c] (x ) = 0 (6) 0 I 1 0 i 1 µ 0 B 2 0 C B 0 σ C 0 B C i B C is the equation describing the same state but in the fra- γ = B C ; γ = B C ; (2) @B AC @B i AC 0 0 −I2 −σ 0 me O . 90 G. G. Nyambuya. Transformation of the Dirac Matrices Issue 2 (April) PROGRESS IN PHYSICS Volume 14 (2018) 2. The second condition is that: given that (x) is the 3 Dirac 4 ×~ 4 ~γ-matrices as a four-vector wavefunction as measured by observer O, there must be The Dirac equation (1) can be re-written in the traditional a prescription for observer O0 to compute 0(x0) from 0 0 0 Schrodinger¨ formulation as (H = E ) where H and E are (x) where (x ) describes to O the same physical the energy and Hamiltonian operators respectively. In this state as that measured by O. The conserve must be true Schrodinger¨ formulation, H , will be such that it is given by: as-well, that is: there must exist a prescription such that 0 2 0 j starting from equation (6), one can arrive at (5). H = γ m0c − {~cγ γ @ j; (14) In simpler mathematical terms, the above two require- and (E = i~@=@t). ments are saying that: starting from equation (5), there must Now, according to the quantum mechanical equation gov- exist some physically legitimate transformations within the erning the evolution of any quantum operator Q , we know framework of Lorentz transformations that can take (map) us that: from this equation (5) to equation (6) and vice-versa. If we @Q {~ = QH − HQ = [Q ; H ] ; (15) can find these, then, the Dirac equation is said to be Lorentz @t Invariant (Covariant). hence, if: Now, since the Lorentz transformations are linear, it is to [Q ; H ] ≡ 0; (16) be required or expected of the transformations between (x) then, the quantum mechanical observable corresponding to 0 0 and (x ) to be linear too, i.e.: the operator Q is a conserved physical quantity. 0 0 0 −1 0 With this [equation (15)] in mind, Dirac asked himself (x ) = (Λx) = S (Λ) (x) = S (Λ) (Λ x ); (7) the natural question — what the “strange” new γ-matrices appearing in his equation really represent. What are they? where S (Λ) is a 4 × 4 matrix which depends only on the rel- 0 In-order to answer this question, he decided to have a “look” ative velocities of O and O and Λ is the Lorentz transforma- at or make a closer “inspection” of the quantum mechanical tion matrix. S (Λ) has an inverse if O ! O0 and also O0 ! O. orbital angular momentum operator Li which we all know to The inverse is: be defined: Li = (~r × ~p)i = −{~i jk x j@k; (17) (x) = S −1(Λ) 0(x0) = S −1(Λ) 0(Λx); (8) where, i jk is the completely-antisymmetric three dimensional or we could write: Levi-Civita tensor. In the above definition of Li, the momen- tum operator ~p is the usual quantum mechanical operator, i.e.: (x) = S (Λ−1) 0(Λx) =) S (Λ−1) = S −1(Λ): (9) ~p = −{~r~ ) pi = {~@i: (18) We can now write equation (5), as: From this definition of Li given in equation (17), it fol- lows from equation (15) that {~@Li=@t = [Li; H ], will be such " µ0 # µ @x −1 0 0 that: i~γ @µ0 − m0c S (Λ) (x ) = 0; (10) @xµ @L h i h i {~ i = −{~m c2 x @ ; γ0 + ~2c x @ ; γ0γl@ : (19) @t 0 i jk j k i jk j k l and multiplying this from the left by S (Λ), we have: 0 2 Now, because the term γ m0c is a constant containing no " µ0 # h 0i µ @x −1 0 0 terms in pi, it follows from this very fact that (i jk x j@k; γ ≡ S (Λ) i~γ @ 0 − m c S (Λ) (x ) = 0; (11) @xµ µ 0 0), hence equation (19) will reduce to: @L h i and hence: {~ i = ~2c γ0γl x @ ;@ @t i jk j k l " µ0 # (20) µ @x −1 0 0 i~S (Λ)γ S (Λ)@µ0 − m0c (x ) = 0: (12) ~2c γ0γl x @ @ − @ x @ : @xµ = i jk j k l l j k From the commutation relation of position (xi) and mo- Therefore, for the above equation to be identical to equa- mentum (−{~@ j) due to the Heisenberg uncertainty princi- tion (6) (hence Lorentz Invariant), the requirement is that: h i ple [9], namely (−{~ xi;@ j = −{~δi j) where δi j is the usual µ0 Kronecker-delta function, it follows that if in equation (20), 0 @x µ µ −1 we substitute (@ x = x @ + δ ), this equation is going to γ = S (Λ)γ µ S (Λ); (13) l j j l l j @x reduce to: −1 hence, we have shown that — for as long as S (Λ) exists, @Li 2 0 l 2 0 l {~ =~ ci jkγ γ x j@k@l−x j@l@k +~ ci jkγ γ δl j@k: (21) equation (5) is Lorentz Invariant. @t | {z } G. G. Nyambuya. Transformation of the Dirac Matrices 91 Volume 14 (2018) PROGRESS IN PHYSICS Issue 2 (April) The term with the under-brace vanishes identically, that is The agile Paul Dirac seized the golden moment and forth- 0 l 0 l 1 to say: (x j@k@l − x j@l@k ≡ 0); and (i jkγ γ δl j = ilkγ γ ), it with identified Si with the -particle’s spin.