Dirac Equation for General Spin Particles Including

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.

exists other attempts at a general spin equation (e.g. Dowker 1967; Baisya 1995). What all these aforecited 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 fundamentally different from Dirac (1928a,b)’s original equation. Written down in its usual covariant form, the original Dirac (1928a,b) equation is given by:

Keywords: Boson – Dirac equation – Fermion – general spin.

ı¯hγ µ ∂µ ψ = m0 cψ , 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 – theoretical physicists, William Rarità (1907−1999) and Julian Seymour Schwinger (1918 − 1994) constructed the Rarità-Schwinger equation which is a relativistic field equation which is assumed to explain spin 3/2 Fermions (Rarità and Schwinger 1941). There G. 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]

(1)

where m0 is the rest-mass of the particle, c is the speed of light in vacuum, ∂µ are the four partial derivatives of space and time, ψ is the Dirac four component wavefunction (i.e. 4 × 1 “vector” field) and:

γ0

=

I 0 0 σi i , γ = , −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 “see” 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 single equation applies to both Bosons and Fermions. In-order to make this future task (of writing down the general spin Dirac equation) much easier – especially, the understanding of how to determined what spin a particular equation explains, it is perhaps wise and instructive for us to ask-and-answer the question:

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G. G. Nyambuya

Why is the Dirac equation said to represent (describe) a particle of spin 1/2? To do this, we shall – in the subsequent section; follow Dr. William O. Straub’s presentation1. This approach is the standard approach that is used to demonstrate the fact that the Dirac equation – indeed – is, an equation that represents (describes) spin 1/2 particles. In §(3), we will motivate the need for Dirac particle that – separately – has a conserved spin and conserved orbital angular momentum. Lastly, in §(5), we will give a general discussion.

From this definition of Li given in (6), it follows from (4) that ı¯h∂ Li /∂ t = [Li , H ], will be such that:

ı¯h

no term Because – the term γ 0 m0 c2 is a constant containing 0 in pi , it follows from this fact that (εi jk x j ∂k , γ ≡ 0), hence (8) will reduce to:

2 Spin of the Dirac Particle ı¯h The Dirac equation (1) can be re-written in the Schrödinger formulation as (H ψ = E ψ ) where H and E are the energy and Hamiltonian operators respectively. In this Schrödinger 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:

∂Q = QH − H Q = [Q, H ] . ∂t

(4)

∂ Li = h¯ 2 cεi jk γ 0 γ l (x j ∂k ∂l − x j ∂l ∂k ) +¯h2 cεi jk γ 0 γ l δl j ∂k . {z } | ∂t

(10)

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 0 γ l δl j = εilk γ 0 γ l ), it follows that (10) will reduce to:

If: [Q, H ] ≡ 0,

(5)

then, the quantum mechanical observable corresponding to the operator Q is a conserved physical quantity. With this [equation (4)] in mind, Dirac asked himself the natural question – 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 orbital angular momentum operator: Li = (r × p)i = −ı¯hεi jk x j ∂k ,

(6)

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.: p = −ı¯h∇

∂ Li = h¯ 2 cεi jk γ 0 γ l [x j ∂k , ∂l ] = h¯ 2 cεi jk γ 0 γ l (x j ∂k ∂l − ∂l x j ∂k ) . ∂t (9)

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 (9) if we substitute (∂l x j = x j ∂l + δl j ) into (9), this equation is going to reduce to:

ı¯h ı¯h

h i ∂ Li = −ı¯hm0 c2 εi jk x j ∂k , γ 0 + h¯ 2 cεi jk x j ∂k , γ 0 γ l ∂l . ∂t (8)

⇒

pi = ı¯h∂i .

(7)

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

ı¯h

∂ Li = h¯ 2 cεilk γ 0 γ l ∂k . ∂t

(11)

Since this result (11) is non-zero, it follows from the dynamical evolution theorem (5) of Quantum Mechanics (QM) 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. 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

Now - 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

(12) ı¯h

Solving (12) for Si , Dirac arrived at:

1 σi 0 Si = h¯ 0 σi 2

1 = h¯ γ 5 γ i . 2

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 as given in (13), Si duly represents a spin 1/2 particle, then, the spin Si (s) of any general particle may as-well be represented by Si (s), such that:

σi 0 0 σi

for (i = 1, 2, 3),

(15)

(13)

3 Modification to General Spin Equation

∂ψ = H (s)ψ , ∂t

should be the Dirac equation of a general spin particle. If we define Li (s) and H (s), such that:

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 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 is concerned. 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.

1 Si (s) = si h¯ 2

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:

(14)

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.

Li (s) = −ı¯hεi jk x j γk sk ∂k ,

(16)

and: H (s) = γ 0 m0 c2 − ı¯hcγ 0 γ j s j ∂ j ,

(17)

then, one can – as has been done in the previous section – demonstrate that: [Ji (s), H (s)] = 0.

(18)

This fact (18) has “manually” (but in a less unclear manner) been demonstrated in the reading Nyambuya (2009). 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

(19)

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 sacrosanct and 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 i.e. (E † = −E) and (p†i = −pi ): the operation † is the usual combination of the complex conjugate

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G. G. Nyambuya

followed by the transpose operation. One can easily write these energy and momentum operators as 4 × 4 matrices as (E = ı¯hI4 ∂ /∂ t) and (pi = −ı¯hI4 ∂ /∂ xi ) where I4 is the 4 × 4 identity matrix. 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 inviolable demand that quantum mechanical operators must be Hermitian, we can define the momentum operator so that it is no-longer given as it is given in (7), but is now given by:

pi (s) = −ısi h¯ γi

∂ . ∂ xi

(20)

This operator (20), is Hermitian and as said above, this is what quantum mechanics requires of every operator associated with a physical observable. As already said – 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. Therefore, it is most logical to demand that all quantum mechanical operators be Hermitian and this must include the energy and momentum operator. Defined the way it is defined in (20), the momentum operator pi (s) is a 4 × 4 matrix. Defined [pi (s)] in this way, it follows that the orbital angular momentum (6) is now such that: Li (s) = −ı¯hεi jk x j sk γk ∂k .

(21)

Under the new definition (20) of momentum [and hence of orbital angular momentum: in (21)], one can show that apart

from satisfying the Dirac equation, if the Dirac wavefunction satisfied the following equation:

ıs j γk xk

1 ∂ψ ∂ 2ψ =− ∂ x j ∂ xi 2 ∂ xi

(22)

then:

∂ Li (s) = 0. (23) ∂t That is, the orbital angular momentum is a conserved quantity as per desideratum. 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

ı¯h

∂ Si (s) = 0. ∂t

(24)

It is not difficult to demonstrate that if Si (s) is defined such that: 1 1 0 σi = si h¯ γi , Si (s) = si h¯ −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 and 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 are not 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 of 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).

5

Dirac Equation for General Spin Particles Including Bosons

5 General Discussion We have herein demonstrated that the Dirac equation – which is universally assumed to describe 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; Rarità and Schwinger 1941, etc) aimed at achieving a general spin particle involve a fundamental change in the Dirac equation. The present approach is unique in that regard in that there is no fundamental change and meaning in the original Dirac equation. The very same Dirac equation in the same form – except the introduction of the numbers, si ; here describes both Bosons and Fermions. From a vantage point of unity, this is an appealing aspect of the present approach. Unification requires the explanation of a wide range of phenomenon using a minimal number principles. An explanation of a diverse of phenomenon using a single principle (equation), as has been proposed herein – this is the desideratum of the purest soul of the searching theoretical physicist. 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 (19). In the present scheme – in general, the spin of a particle is here given by: S = 12 s1 h¯ σ 1 + 12 s2 h¯ σ 2 + 12 s3 h¯ σ 3 . 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 arbitrarily measure the spin 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 set equal for the three axises, i.e. (si = s). So doing, it follows that equation (19) may – be written as: ı¯hγ 0

∂ψ ∂ψ + ı¯hsγ j j = m0 cψ . ∂ ct ∂x

(26)

this equation (26) represents – in the present paradigm – a particle of spin s/2. The corresponding Einstein energymomentum equation for the particle described by the relativistic equation (26), is such that: E 2 = s2 p2 c2 + m20 c4 .

(27)

This energy equation (27) was first written down in the reading Nyambuya (2009). In-closing, we should say that, while we have written down [in a much more lucid manner than before (Nyambuya 2009, 2013)] an equation for a general spin particle, there is nothing in the theory that tells us how to achieve such higher spin particles in the laboratory or how they occur in Nature. Therefore, the present work may be relevant in the future if discoveries of higher spin particles are ever made. It is off-cause always good to have forehand a theory ready to explain these eventualities if they ever “visit our world”.

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