Foundations of Physics manuscript No. (will be inserted by the editor)
Dirac Equation for General Spin Particles Including Bosons G. G. Nyambuya
Received: date / Accepted: date
Abstract We demonstrate (show) that the Dirac equation – which is universally assumed to represent only spin 1/2 particles; can be manipulated using legal mathematical operations – starting from the Dirac equation – so that it describes any general spin particle. If our approach is acceptable and is what Nature employs, then, as currently obtaining, one will not need a unique and separate equation to describe particles of different spins, but only one equation is what is needed – the General Spin Dirac Equation. This approach is more economic and very much in the spirit of unification – i.e., the tie-ing together into a single unified garment – a number of phenomenon (or facets of physical and natural reality) using a single principle, which, in the present case is the bunching together into one theory (equation), all spin particles into the General Spin Dirac Equation.
1 Introduction In the wider (research) literature and in most – if not all – textbooks on the planet that deal with the Dirac (1928a,b) equation, this equation (Dirac equation) is said to describe only spin 1/2 particles and nothing else. Dirac (1936) was the first to make an attempt at extending his equation so that it can explain particles with spin other than spin 1/2. Dirac (1936)’s efforts where followed up by Fierz (1939) and latter by Fierz and Pauli (1939). Also, American theoretical physicists, William Rarit`a (1907 − 1999) and Julian Seymour Schwinger (1918 − 1994) constructed the Rarit`aSchwinger equation which is a relativistic field equation believe to explain spin 3/2 Fermions (Rarit`a and Schwinger 1941). There exists other attempts at a general spin equation (e.g. Dowker 1967; Baisya 1995). What all these aforeG. G. Nyambuya National University of Science & Technology, Faculty of Applied Sciences – Department of Applied Physics, Fundamental Theoretical Astrophysics Group, P. O. Box AC 939, Ascot, Bulawayo, Republic of Zimbabwe. Tel.: +263-77-7960-452 E-mail:
[email protected]
cited attempts including those that we have not mentioned is that – the resulting equation is fundamentally different from the original Dirac (1928a,b). With better clarity and insight in the present than before (Nyambuya 2009, 2013), we write down a general spin Dirac equation. The equation that we write down is not fundamental different from Dirac (1928a,b)’s original equation. The Dirac (1928a,b) equation is given by: ı¯hγ µ ∂µ ψ = m0 cψ ,
(1)
where m0 is the rest-mass of the particle, c is the speed of light in vacuum, ∂µ the four partial derivative, ψ is the Dirac four component wavefunction (i.e. 4 × 1 “vector” field) and:
γ0 =
I 0 0 σi , , γi = −σ i 0 0 −I
(2)
are the 4 × 4 Dirac gamma matrices with I and 0 being the 2 × 2 identity and null matrices respectively. Throughout this reading, the Greek indices will be understood to mean (µ , ν , ... = 0, 1, 2, 3) and lower case English alphabet indices (i, j, k... = 1, 2, 3). In the next section, we shall that one can easily write down a general spin equation from the original Dirac equation and this equation represents a particle of spin s/2 where (s = ±1, ±2, ±3 . . . etc). This implies that this equation applies to both Boson and Fermions. In-order to make this future task much easer, it is perhaps wise for us to askand-answer here-and-now, the question: Why is the Dirac equation said to represent a particle of spin 1/2? To do this, we shall follow Dr. William O. Straub’s presentation1. This 1 Straub, W. O.; Paper Title : “Weyl’s Spinor and Dirac’s Equation”, http://www.weylmann.com/ . Visited on this day: Nov. 2, 2015@12h38GMT + 2
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G. G. Nyambuya
approach is the standard approach that is used in-order to demonstrate the fact that the Dirac equation – indeed – is, an equation that represents spin 1/2 particles.
2 Spin of the Dirac Particle The Dirac equation (1) can be re-written in the Schr¨odinger formulation as (H ψ = E ψ ) where H and E are the energy and Hamiltonian operators respectively. In this Schr¨odinger formulation, H , will be such that it is given by: H = γ 0 m0 c2 − ı¯hcγ 0 γ j ∂ j ,
(3)
and (E = i¯h∂ /∂ t). Now, according to the quantum mechanical equation governing the evolution of any quantum operator Q, we have:
ı¯h
∂Q = QH − H Q = [Q, H ] . ∂t
(4)
If ([Q, H ] ≡ 0), then, the quantum mechanical observable corresponding to the operator H is a conserved physical quantity. With this [equation (4)] in mind, Dirac asked himself the natural question of what the “strange” new γ -matrices appearing in his equation really represent. What are they? In-order to answer this question, he decided to have a look at the quantum mechanical angular momentum operator: Li = (r × p)i = −ı¯hεi jk x j ∂k ,
(5)
where, εi jk is the completely-antisymmetric three dimensional Levi-Civita tensor. In the above definition of Li the momentum operator p is the usual quantum mechanical operator, i.e.:
From the commutation relation of position (xi ) and momentum (−ı¯h∂ j ) due to the Heisenberg (1927) uncertainty principle, namely (−ı¯h [xi , ∂ j ] = −ı¯hδi j ) where δi j is the usual Kronecker-delta function, it follows that (8) if we substitute (∂l x j = x j ∂l + δl j ) into (8), this equation is going to reduce to:
ı¯h
∂ Li = h¯ 2 cεi jk γ l (x j ∂k ∂l − x j ∂l ∂k ) +¯h2 cεi jk γ l δl j ∂k . {z } | ∂t
(9)
The term with the under-brace2 vanishes identically, that is to say: (x j ∂k ∂l − x j ∂l ∂k ≡ 0); and (εi jk γ l δl j = εilk γ l ), it follows that (9) will reduce to:
ı¯h
∂ Li = h¯ 2 cεilk γ l ∂k . ∂t
(10)
Since this result is non-zero, it follows from the dynamical evolution theorem of quantum mechanics that none of the angular momentum components Li are – for the Dirac particle – going to be constants of motion. This result greatly bothered the great and agile mind of Paul Dirac. For example, a non-conserved angular momentum would mean spiral orbits – 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 salvaged3 somehow. Enter spin! Dirac figured that Subtle Nature must conserve something redolent with orbital angular momentum, and he considered adding something to Li that would satisfy the desired conservation criterion, i.e.: call this unknown, mysterious and arcane quantity Si and demand that:
ı¯h
∂ (Li + Si ) ≡ 0. ∂t
(11)
Solving (11) for Si , Dirac arrived at: p = −ı¯h∇
⇒
pi = ı¯h∂i .
(6)
From this definition of Li given in (5), it follows from (4) that ı¯h∂ Li /∂ t = [Li , H ], will be such that:
ı¯h
i h ∂ Li = −ı¯hεi jk x j ∂k , γ 0 m0 c2 + −ı¯hεi jk x j ∂k , −ı¯hcγ l ∂l . ∂t (7)
Because the term γ 0 m0 c2 is aconstant, it follows from this fact that −ı¯hεi jk x j ∂k , γ 0 m0 c2 ≡ 0, hence (7) will reduce to: ı¯h
∂ Li = h¯ 2 cεi jk γ l [x j ∂k , ∂l ] = h¯ 2 cεi jk γ l (x j ∂k ∂l − ∂l x j ∂k ) . ∂t (8)
1 1 σi 0 = h¯ γ 5 γ i . Si = h¯ 0 σ 2 2 i
(12)
Realising that: (1) the matrices σi are Pauli matrices and they had been ad hocly introduced into physics to account for the spin of the Electron (Uhlenbeck and Goudsmit 1925); (2) and that, his equation when taken in the non-relativistic 2 When we get to §(4), the momentum p and position xi will i be required to submit to a new type of Lie-Algebra. This new LieAlgebra will be such that the term in the under-brace will be such that (x j ∂k ∂l − x j ∂l ∂k = −∂k ), so that ı¯h∂ Li /∂ t = 0. 3 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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Dirac Equation for General Spin Particles Including Bosons
limit, it would account for the then unexplained gyromagnetic ratio (g = 2) of the Electron and this same equation emerged with σi explaining the Electron’s spin, Dirac seized the golden moment and forthwith identified Si with the ψ particle’s spin. The factor 21 h¯ in Si 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 Dirac was able to explain and “demystify” spin in this way (i.e., as demonstrated above), we are of the strong view that the non-independent conservation of spin and orbital angular momentum is problematic insofar as the stability of the Dirac particle. We shall elucidate on this matter and proffer a solution in §(4). In the subsequent section, we shall demonstrate how one can modify the Dirac equation so that it represents a general spin particle other than spin-zero particles. 3 Modification to General Spin Equation What we would like to do now is to demonstrate that – starting from an acceptable premise and from there-on applying logic, the Dirac equation can be modified so that it represents a general spin particle. To do this, we need first to ask ourself how the spin of any general particle can be represented. We are certain that the reader will – without any qualms – agree that if Si as given in (12) represents the spin of a spin 1/2 particle, then, the spin Si (s) of any general particle will be such that: 1 Si (s) = si h¯ 2
σi 0 0 σi
for (i = 1, 2, 3),
(13)
where (si = ±1, ±2, ±3, . . . etc). If (si = s; ∀i = 1, 2, 3), then, when s is even, we have a Boson and when is s odd, we have a Fermion. If one agrees to the above said that Si (s) is the spin of any general particle, then, they will invariably agree that if Li (s) is the corresponding orbital angular momentum operator such that the total orbital angular momentum operator [J (s) = Li (s)+ Si (s)] commutes with the Hamiltonian H (s), then, the equation:
∂ψ (14) = H (s)ψ , ∂t should be the Dirac equation of a general spin particle. If we define Li (s) and H (s), such that: ı¯h
Li (s) = −ı¯hεi jk x j γ k sk ∂k ,
(15)
and: H (s) = γ 0 m0 c2 − ı¯hcγ 0 γ j s j ∂ j ,
(16)
then, one can – as has been done in the previous – demonstrate that:
[Ji (s), H (s)] = 0.
(17)
Written explicitly, it follows from the foregoing that the Dirac equation for a general spin particle is such that:
ı¯hγ 0
∂ψ ∂ψ + ı¯hs j γ j j = m0 cψ . ∂ ct ∂x
(18)
Accordingly, this equation must apply to both Bosons and Fermions whose spin is non-zero. Using the usual quantum mechanical spin ladder operators, one can show that if, si , where to change from one value to the next, it will do so by increasing or decreasing by one integral unit. Since the Dirac state is one such that (si = +1), it follows from this that (si = ±1, ±2, ±3, . . . etc).
4 Conserved Spin and Orbital Angular Momentum Quantum Mechanics (QM), as taught to university sophomores – demands that all operators associated with physical observables must be Hermitian. The reason for this – is so that the resulting eigenvalues (observables) associated with these operators will be real-valued physical quantities. This is taught as one of the six fundamental postulates of QM. Despite this, the usual energy (E = ı¯h∂ /∂ t) and momentum (pi = −ı¯h∂ /∂ xi ) operators used in QM are not Hermitian. Given the reason for the Hermiticity of QM operators and the aforesaid non-Hermitian nature of the energy and momentum operators of QM, one will be right and spot-on to wonder if the energy and momentum eigenvalues of quantum mechanical operators are real. The answer is yes, they are real physical quantities. How can this be so if QM requires that only the eigenvalues associated with those operators that are Hermitian are those that will be real. How does this come about that non-Hermitian operators [(E = ı¯h∂ /∂ t) and (pi = −ı¯h∂ /∂ xi )] have real eigenvalues? The reason why these operators [(pi = −ı¯h∂ /∂ xi ) and (E = ı¯h∂ /∂ t)] result in real eigenvalues is that – quantum mechanical wavefunctions are typically of the generic form, Ψ (r,t) = NeiS(r,t) , where N is a (normalization) constant and S(r,t) is a real-valued function. If N where not a constant [i.e., N = N(r,t) : real or complex], then, the corresponding eigenvalues of these operators would be complex. In the present case – with regard to the non-Hermiticity of the energy and momentum operators; we shall concern ourself with the momentum operator. In-line with the demand that quantum mechanical operators must be Hermitian, we can define the momentum op-
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G. G. Nyambuya
erator so that it is no-longer given as it is given in (6), but is now given by:
pi (s) = −ısi h¯ γi
∂ . ∂ xi
(19)
This operator (19), is Hermitian and as said above, this is what quantum mechanics requires of every operator associated with a physical observable. Physically, the Hermitian property is necessary in order for the measured values (eigenvalues/ physical observables) to be constrained such that they are real and not complex quantities. As far as physical and natural reality dictates to us insofar as what Nature is, this seems to be the state of affairs – that all physical observables are real and not complex quantities. Defined the way it is defined in (19), the momentum operator pi (s) is a 4 × 4 matrix. Defined [pi (s)] in this way, it follows that the orbital angular momentum (5) is now such that: Li (s) = −ı¯hεi jk x j si γ i ∂i .
(20)
If – as the spin and orbital angular momentum – the momentum operator pk (s) and position xk obey the following Lie-Algebras: pi (s)p j (s) = ı¯h pk (s)/xk
...
(a)
p j (s)pk (s) = ı¯h pi (s)/xi
...
(b) ,
pk (s)pi (s) = ı¯h p j (s)/x j
...
(c)
(21)
then, the following anti-commutation and commutations relations will result there-from:
pi (s), p j (s) = 0
It is not difficult to demonstrate that if Si (s) is defined such that: 1 1 0 σi Si (s) = si h¯ = si h¯ γi , −σ i σ i 2 2
(25)
then, equation (24) will hold true. We have shown that it is possible to obtain a Dirac particle system which is such that the orbital angular momentum ans spin are constants. The motivation for this search has been necessitated by the fact that we realised that the Dirac particle system as currently understood, requires the preservation of the combination of spin and orbital angular angular momentum. Under such a setting, the orbital angular momentum and spin will not be constants of motion but differ from one moment to the next. If the spin and orbital angular angular momentum where to vary from one moment to the next, this invariably implies that the spin of particles must not be sharply defined as is obtained from experiments where – for example – it is known that the Electron is a spin 1/2 particle and that its spin never ever changes; from the moment of its “creation”, it [Electron spin] has and will always be a spin 1/2 particle and nothing else. Actually, ever since it became manifest to humanity that particles possess this “mysterious” property of spin, there has never ever been observed in the laboratory a particle changing its spin. Except in the realm of speculative theories such as Supersymmetry (and related theories), there is not even the slightest hint whatsoever that the spin of a particle is susceptible to change. In a nutshell, the spin of any known particle always stays fixed at the known value for the given particle. The only-and-only way for this to be so, is if we have ([S , H ] = 0), and this constraint ([S , H ] = 0) – via the conservation of total angular momentum ([J , H ] = 0); implies ([L , H ] = 0).
. . . (a) .
(22)
5 General Discussion
[pi (s), p j (s)] = 2ı¯h pk (s)/xk . . . (b) From the re-definition (20) of the orbital angular momentum, and the commutation relation (22 b), it follows from (4) that:
ı¯h
∂ Li (s) = 0. ∂t
(23)
If this is the case that the orbital angular momentum is a conserved quantity, then, the conservation of total angular momentum requires that the spin be conserved too, i.e.:
ı¯h
∂ Si (s) = 0. ∂t
(24)
We have herein demonstrated that the Dirac equation – which is universally assumed to represent only spin 1/2 particles; can be manipulated using legal mathematical operations – starting from the Dirac equation – so that it describes any general spin particle. Most if not all approaches (e.g. Dirac 1936; Fierz and Pauli 1939; Rarit`a and Schwinger 1941, etc) at achieving a general spin particle involve a fundamental change and meaning in the Dirac (1928a,b). The present approach is unique in that regard in that there is no fundamental change and meaning in the Dirac (1928a,b). The very same Dirac (1928a,b) equation in the same form – except the introduction of the number, si ; explains describes both Bosons and Fermions. From a vantage point of unity, this is an appealing aspect of the present approach.
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Dirac Equation for General Spin Particles Including Bosons
Unification requires the explanation of a wide range of phenomenon using a minimal number principles. An explain of a diverse of phenomenon using a single principle (equation) as is being proposed herein is the most desired. If our approach is acceptable and is what Nature employs, then, as currently obtaining, one will not need a unique and separate equation to describe particles of different spins, but only one equation is what is needed – the General Spin Dirac Equation. This approach is more economic and as pointed out above, it is very much in the spirit of unification – i.e., the tie-ing together into a single unified garment – a number of phenomenon (or facets of physical and natural reality) using a single principle, which, in the present case is the bunching together into one theory (equation), all spin particles into the General Spin Dirac Equation. In the present scheme, in general, the spin of a particle is given by: 1 1 1 S = s1 h¯ σ 1 + s2 h¯ σ 2 + s3 h¯ σ 3 . 2 2 2
(26)
The associated spin quantum numbers (s1 , s2 , s3 : si ) are in general not equal but are different i.e.: (s1 6= s2 ), (s1 6= s3 ) and (s2 6= s3 ). If this is so, then, for the Electron and all other particles whose spin we known, we have found that the spin is a fixed physical quantity. Each time we measure the spin of particle, we always measure the spin for along any one of the three axises (x, y, z). If the spin where different along these these axises, then, we would find that spin is not a fixed quantity. The fact that experiments reveal that the spin is a fixed physical quantity, this – for the present paradigm – suggests that the spin quantum number si is to be equal for the three axises, i.e. (si = s); so that equation (18) may – for practical purposes be written as: ı¯hγ 0
∂ψ ∂ψ + ı¯hsγ j j = m0 cψ . ∂ ct ∂x
(27)
this equation (27) represents – in the present paradigm – a particle of spin s/2.
References Baisya, H. L. (1995), ‘An Equation for Particles with Spin 5/2’, Progress of Theoretical Physics 94(2), 271–292. Dirac, P. A. M. (1928a), ‘The Quantum Theory of the Electron’, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 117(778), 610–624. Dirac, P. A. M. (1928b), ‘The Quantum Theory of the Electron. Part II’, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 118(779), 351–361. Dirac, P. A. M. (1936), ‘Relativistic Wave Equations’, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 155(886), 447–459. Dowker, J. S. (1967), ‘A Wave Equation for Massive Particles of Arbitrary Spin’, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 297(1450), 351–364. ¨ Fierz, M. (1939), ‘Uber die Relativistische Theorie Kr¨aftefreier Teilchen mit Beliebigem Spin’, Helv. Phys. Acta. 12(1), 3–37. Fierz, M. and Pauli, W. (1939), ‘On Relativistic Wave Equations for Particles of Arbitrary Spin in an Electromagnetic Field’, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 173(953), 211–232. Heisenberg, W. (1927), ‘Ueber den anschaulichen Inhalt der quantentheoretischen Kinematik and Mechanik’, Zeitschrift f¨ur Physik 43, 172–198. Engilish Translation: Wheeler, J. A. and Zurek, W. H. (eds) (1983) Quantum Theory and Measurement (Princeton NJ: Princeton University Press), pp. 62-84. Nyambuya, G. G. (2009), ‘General Spin Dirac Equation’, Apeiron 16(4), 516–531. Nyambuya, G. G. (2013), ‘General Spin Dirac Equation (II)’, Journal of Modern Physics 4(8), 1050–1058. Rarit`a, W. and Schwinger, J. (1941), ‘On a Theory of Particles with Half-Integral Spin’, Phys. Rev. 60, 61–61. Uhlenbeck, G. E. and Goudsmit, S. (1925), Naturwissenschaften 33(47), 953.
