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On the Extra Anomalous Gyromagnetic
Ratio of the Electron and Other General
Spin-1/2 Particles
Nyambuya, G. G.
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On the Extra Anomalous Gyromagnetic Ratio of
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Nyambuya, G.G., On the Extra Anomalous
Gyromagnetic Ratio of the Electron and Other General
Spin-1/2 Particles., pp.1-6.
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On the Extra Anomalous Gyromagnetic Ratio of the
Electron and Other General Spin-1/2 Particles
G. G. N YAMBUYA1 ∗
1
National University of Science & Technology, Faculty of Applied Sciences,
Department of Applied Physics, P. O. Box 939, Ascot, Bulawayo, Republic of Zimbabwe.
Abstract. As is well known, amongst the greatest successes of the Dirac equation at its birth
where its ability to naturally account for spin as a relativistic phenomenon and its almost arcane prediction of the correct gyromagnetic ratio of the Electron. The Dirac equation forms
the most fundamental basis of Quantum Field Theory (QFT). QFT is one of the most successful fields of human endeavour. The Dirac equation gives an accurate account of the Electron
so much that since its birth, it has always has been thought of as an equation for the Electron.
However, in its bare and natural form, it is widely believed that the same can not be said of
the Dirac equation when it comes to the Proton and Neutron, which – as the Electron; are
both spin-1/2 particles. For example, the Dirac equation has never in its natural form been
used to explain the gyromagnetic ratio of the Proton and Neutron. Much to the contrary to
this prevalent wisdom, here we show that one might account for the gyromagnetic ratio of any
general spin-1/2 particle such as that of the Proton, Neutron, together with the Electron’s extra anomalous gyromagnetic ratio as second order effects within the minimally coupled Dirac
theory that i.e., from the same theory that unveiled spin as a relativistic phenomenon and at the
same time predicted the correct gyromagnetic ratio of the Electron.
Keywords: Dirac equation, anomalous: gyromagnetic ratio, proton, Electron.
“I . . . have a secret fear that new generations
may not necessarily have the opportunity to become familiar with dissident ideas1 .”
– Julian Seymour Schwinger (1918 − 1994)
1 Introduction
T
HE Dirac (1928a,b) equation is a relativistic quantum mechanical wave equation discovered by the
pre-eminent British physicist – Prof. Paul Adrian Maurice (1902 − 1984). This equation was originally designed by Dirac to overcome the criticism of the Klein-Gordon equation. The Klein-Gordon
equation gave negative probabilities and this is considered to be physically meaningless. Despite this
fact, this equation accounts well for Bosons, that is spin zero particles. This criticism levelled against the
Klein-Gordon equation, motivated Dirac to successfully seek an equation devoid of negative probabilities.
However, recently, it has been shown (in Nyambuya 2013) that one can – in the framework of the KleinGordon, overcome the issue of negative probabilities without taking the root taken by Dirac, but by making
a proper choice of the Klein-Gordon probability current density.
The Dirac equation is consistent with Quantum Mechanics (QM) and fully consistent with the Special
Theory of Relativity (STR). This equation accounts in a natural way for the nature of particle spin as
a relativistic phenomenon and amongst its prophetic achievements was its successful prediction of the
∗ Email:
[email protected]
quoted by Sameer Shah in “If you can’t join ’em, beat ’em”: Julian Schwinger’s Conflicts in Physics. Directions in Cultural
History, The UCLA Historical Journal, Vol. 21, 2005 − 2006, p.50
1 As
1
existence of anti-particles. In its bare form, the Dirac equation provided us with an impressive and accurate
description of the Electron hence it being referred in most of the literature as the “Dirac Equation for
the Electron”. It also accounts very well for quarks and other spin-1/2 particles although in some of the
cases, there is need for modifications while in others is fails - for example, one needs the Proca equation to
describe the neutron which is a spin-1/2 particle as the Electron.
The first taste of glory of the Dirac equation was it being able to account for the gyromagnetic ratio of
the Electron, that is g = 2, which can not be accounted for using non-relativistic QM. For several years
after it’s discovery, most physicists believed that it described the Proton and the Neutron as-well, which
are both spin-1/2 particles. In simple terms, it was thought or presumed that the Dirac equation was a
universal equation for spin-1/2 particles.
However, beginning with the experiments of Stern and Frisch in 1933, the magnetic moments of these
particles were found to disagree significantly with the predictions of the Dirac equation. The Proton was
found to have a gyromagnetic ratio2 gp = 5.585694713(46) which is 2.79 times larger than that predicted
by the Dirac equation. The Neutron, which is electrically neutral spin-1/2 particle was found to have a
gyromagnetic ratio3 gn = −3.82608545(90).
These “anomalous magnetic moments” of the Neutron and Proton which are clearly not confirmatory
to the Dirac Theory have been taken to be experimental indication that these partices are not fundamental
particles. In the case of the Neutron, yes it is clearly not a fundamental particle since it does decay into a
Proton, Electron and Neutrino, that is (n −→ p + e− + ν̄e ). If the Dirac equation is a universal equation
for fundamental fermion particles, then any fundamental fermion particle must conform to this equation.
Simple, any spin-1/2 particle that can not be described by it, must therefore not be a fundamental particle
of nature. By definition a fundamental particle is a particle known to have no sub-structure, that is, it can
not be broken down into smaller particles thus will not decay into anything else.
From the Standard Model, we know that the Proton and Neutron are composed of quarks thus are
not fundamental particles. The question is, is this the reason why these particle’s gyromagnetic ratio is
different from that predicted by the bare Dirac equation? Prevailing wisdom (cf. Brodsky & Drell 1980,
Dehmelt 1989) suggests that anomalous gyromagnetic ratio arise because the particles under question are
not fundamental particles. From the theory laid down here, the answer to this is a clear no. All a particle
needs to have a non-zero g-factor is spin. We show herein that the same unmodified Dirac Theory that
predicted the gyromagnetic ratio of the Electron can in principle be used to explain the gyromagnetic ratio
of the Proton and Neutron and perhaps any general spin-1/2 particle.
2 Dirac Equation
For a particle whose rest-mass and wave-function are m0 and ψ respectively, its Dirac equation is given by:
[i~γ µ ∂µ − m0 c] ψ = 0,
(1)
where:
0
γ =
I
0
0
−I
i
, γ =
0
−σ i
σi
0
,
(2)
are the 4 × 4 Dirac gamma matrices (I and 0 are the 2×2 identity and null matrices respectively) and ψ
is the four component
Dirac wave-function, ~ is the normalized Planck constant, c is the speed of light in
√
vacuum, i = −1 and:


ψ0
 ψ1 

ψ=
(3)
 ψ2  ,
ψ3
2 http://physics.nist.gov/cgi-bin/cuu/Value?gp|search_for=all!
3 http://physics.nist.gov/cgi-bin/cuu/Value?gnn|search_for=all!
2
is the Dirac 4 × 1 four component wavefunction. 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.
3 Dirac Gyromagnetic Ratio
For latter instructive purposes, we show here how the Dirac equation accounts very well for the gyromagnetic ratio of the Electron. As already said, amongst others, it is for this reason that the Dirac equation is
said to account very well for the Electron. The present discussion follows closely that of Prof. Zee (2003,
pp.177 − 179).
ex
In the presence of an ambient magnetic field Aex
µ , the derivatives transform as ∂µ 7−→ Dµ = ∂µ +iκAµ
ex
where κ is a constant that has been inserted to ensure dimensional consistency. The field Aµ is to be
2
assumed to be a real normalized function the meaning of which is that Aµex Aex
µ = a∗ where a∗ is a unit
µ
magnitude of the four vector Aex . The unit magnitude a∗ is the same for any given normalized four
electromagnetic vector potential thus a∗ is a universal constant. Making this replacement (i.e. ∂µ 7−→
Dµ = ∂µ + iκAex
µ ) results in equation (1) reducing to:
µ
[i~γ µ Dµ − m0 c] ψ = 0.
µ ν
(4)
m20 c2 /~2
Acting on this equation with (i~γ Dµ + m0 c), we obtain γ γ Dµ Dν +
ψ = 0. We have
γ µ γ ν Dµ Dν = 21 ({γ µ , γ ν } + [γ µ , γ ν ]) Dµ Dν = Dµ Dµ − iσ µν Dµ Dν and for iσ µν Dµ Dν we have
ex
ex
iσ µν Dµ Dν = (i/2)σ µν [Dµ , Dν ] = (1/2)κσ µν Fµν
where Fµν
is the electromagnetic field tensor of
the applied external field. The above calculations reduce to:
1 µν ex m20 c2
µ
ψ = 0.
(5)
Dµ D − κσ Fµν + 2
2
~
Now consider a weak constant magnetic field in the z-axis such that A = (1/2)r × B where B =
(0, 0, B) so that F12 = B. Neglecting second order terms we have
Di D i
=
=
=
ex
2
2
(∂i )2 − κ(∂i Aex
i + Ai ∂i ) + κ O(Aex,i )
2
1
2
2
2
(∂i ) − κB(x ∂2 − x ∂1 ) + κ O(Aex,i ) ,
∇2 − κB · L + κ2 O(A2ex,i )
(6)
where L = r×p is the orbital angular momentum operator which means that the orbital angular momentum
generates orbital magnetic moment that interacts with the magnetic field. Φ
Now, if we write the Dirac four component wave-function as ψ =
, we find that in the
χ
non-relativistic limit the component χ dominates. Thus, σ µν Fµν /2 acting on Φ is effectively equal to
1
3
−im0 t
Ψ where Ψ oscillates much more
2 κσ (F12 − F21 ) = 2κB · S since S = (σ/2). Writing Φ = e
im0 t
2
2 2
2 −im0 c2 t/~
−im0 c2 t/~
slowly than e
so that (∂0 + m0 c /~ )e
Ψ≃ e
[−(2im0 c/~)∂0 Ψ]. Putting all the
bits and pieces together, we have:
2
~
∂Ψ
∇2 + κµB B · (L + 2S) Ψ = −i~
,
(7)
2m0
∂t
and this equation above and below embodies the historic fit of the Dirac equation in that it automatically
tells us that the gyromagnetic ratio of the Electron is 2. However as already explained, precise measurements the lastest of which are those by Hanneke et al. (2011) give (ge − 2)/2 = 0.0115965218073(28).
This value is slightly above 2 and this discrepancy in observations and theory serenaded the theorist back
to the drawing board to seek harmony with observations, the result of which was the creation of QED.
With the emergence of QED, this discrepancy was solved by the consideration of particle-particle interactions through the so-called Feynman diagrams/method. This approach has yielded the best ever agreement
for any theory ever conceived by the human mind. The agreement between theory and observation is so
impressive that QED has be dubbed “the best theory we have”. These impressive calculations where first
performed by Julian Seymour Schwinger (1918 − 1994) in 1948.
3
4 New Approach
Perhaps, we must begin this section by making a confession about Feynman diagrams/method. Despite
the impressive computational power of the Feynman method (via the famous Feynman diagrams), we
have never been comfortable with this approach because it appears at best as an artificial way to arrive
at results that agree with experience. This feeling is shared by a number of notable physicists who have
always wondered if “God uses Feynman diagrams too” or whether or not “God integrates empirically as is
required by the Feynman method”.
The fact that the gyromagnetic ratio of the Electron differs slightly from the predicted Dirac value of
(ge = 2) implies that there is an extra unaccounted for interaction of the spin with the ambient magnetic
field. To take this into account, naturally, we would modify (7) so that it reads:
2
∂Ψ
~
∇2 + κµB B · (L + 2(1 + ∆g )S) Ψ = −i~
.
(8)
2m0
∂t
where ∆g is the extra anomalous gyromagnetic ratio. Having done so – i.e. writing (7) as we have done
in (8) above, we would then follow up this by making the most logical hypothesis namely that the second
order terms that we neglected in our initial calculation (6) are what is responsible for the extra anomalous
gyromagnetic ratio of the Electron. This second order term can be written as:
#
"
κO(A2ex,i )
2
κµB B · S,
(9)
O(Aex,i ) =
µB B · S
thus in-cooperating this term into the new calculation, one easily deduces that:
∆g =
κ O(A2ex,i )
.
2µB B · S
(10)
a2∗
κ
.
2µB B · S
(11)
a2∗ κ/~µB
.
B cos ϑ
(12)
0
ex
i
ex
2
0
ex
Now, given that O(A2ex,i ) = Aex,i Aiex and that Aµex Aex
µ = Aex A0 + Aex Ai = a∗ , Aex = A0 = 0, it
follows that O(A2ex,i ) = Aex,i Aiex = a2∗ , hence:
∆g =
Now, B · S = 21 ~B cos ϑ where ϑ is the angle between the orientation of the Electron and the ambient
magnetic field; with this given, we will now have:
∆g =
From the above, it follows that:
B cos ϑ =
a2∗ κ
.
~µB ∆g
(13)
What the above implies is that the extra anomalous gyromagnetic ratio of the Electron can be thought of as
being second order perturbation effect between the interaction of the the Electron and the ambient magnetic
field. This extra anomalous gyromagnetic effect depends only on the component of the magnetic field along
the orientation of the spin since B cos ϑ is the component of the magnetic field along the orientation of the
spin. Further, since a2∗ κ/~µB ∆g is a constant, it follows that the spin will configure its self in the ambient
magnetic field in a such a manner that the component of the magnetic field along the orientation of the
spin has the same value always. Clearly, without the use of Feynman diagrams or elaborate mathematical
formulae as is the case with QED calculations by Schwinger (1948), we have shown that one can account
for the extra anomalous gyromagnetic ratio of the Electron with relative ease using the same theory that for
the first time successfully explained Electron gyromagnetic ratio.
Since κ, ~, µB and ∆g are all constants and | cos ϑ| ≤ 1, it follows from (13) that there is a minimum
magnetic field (Bmin ) necessary to measure this effect because from this equation (13) and the condition
| cos ϑ| ≤ 1, it follows that:
4
|B| ≥
1 |a2∗ κ|
= |Bmin |.
~µB |∆g |
(14)
In-closing, we shall substitute the parameters κ and µB into (12), i.e. κ = e/~ and µB = e~/2m, and
from this substitution, it follows that:
ma2∗
.
(15)
2~3 B cos ϑ
Thus, it is now clear that ∆g depends on the mass of the particle and because of this, one would expect that
this quantity will vary for particles of different masses.
∆g =
5 Proton, Neutron and Nuclear Gyromagnetic Ratios
The calculation of the previous section is not limited to the Electron but extends to any spin-1/2 particle
since the Dirac equation is an equation describing spin-1/2 particles. This means we can extend this to
include the Proton, Neutron and nuclei with spin-1/2 such as 57 Fe. Thus, the Dirac equation can now be
thought of as an equation describing any general spin-1/2 particle – and in its description, it is able – in
principle – to account for both spin and g-factor of these particles in the same manner as it successfully
explains the Electron. The only setback now is that it no longer predicts forehand the value to be excepted
upon measurement of the extra anomalous gyromagnetic ratio of the Electron, but merely tells us that the
spin of a particular particle (Electron in this case) will always align itself with the ambient magnetic field
in such a manner that the component of the magnetic field along the orientation of the spin always has
the same magnitude. Why this is so, the theory can not say. If the spin did align itself randomly to the
ambient magnetic field, the Dirac theory here is saying the extra anomalous gyromagnetic ratio ∆g would
be a variable, its value would differ depending on the orientation of the spin to the magnetic field at the
particular time of measurement.
6 General Discussion
In his celebrated landmarking calculation spanning about 72 pages and published in a very condensed
form in the paper Schwinger (1948); the brilliant American physicist – Julian Schwinger, demonstrated
that the extra anomalous gyromagnetic ratio of the Electron could be computed and shown to be equal to
(α/2π = 0.00116171491308), where (α = e2 /4πε0 ~c ∼ 1/137) is the famous fine structure constant.
Because of its amazing agreement with experience, this calculation has stood unchallenged ever since and
it is one of the great sources of pride for any quantum field theorist. Its (calculation) almost magical nature
i.e., out of a zoo of symbols emerges the simple result such as α/2π and given that this result is in-complete
accord with experiments – this alone led to the almost universal acceptance of Schwinger’s method of doing
physics. Be that it may, the problem that has always been known is that Schwinger (1948)’s calculation
when applied to the Proton, Neutron and atomic nuclei, it - sadly; does not yield the right numbers that are
in tandem with experience. The reason often cited (cf. Brodsky & Drell 1980, Dehmelt 1989) is that the
Proton, Neutron and nuclei are composite particles, they are not fundamental in their nature as is assumed
of the Electron.
One advantage of the present calculation is that, it applies to any general spin-1/2 particle – albeit, unlike the Dirac prediction, it (present calculation) does not give the values to be expected from experiments.
All one can deduce is that particles of the same kind will have their spins always aligned in such a manner
that the component of the magnetic field along the orientation of the spin always has the same magnitude.
The theory has two unknown parameters namely, ϑ and a∗ . It is in principle possible that for a given magnetic field B one can measure ϑ. If this is possible, then a∗ can be found and this value is expected to be
the same for all particles independent of their masses. Thus the measurement of this constant is the real
test of the present ideas.
Besides, the present ideas are not in any way a modification of any existing theories. All we have
done is to take into account the omitted second order term in the Dirac calculation leading to the historic
5
prediction of the the gyromagnetic ratio of the Electron (ge ∼ 2). Perhaps, one thing that we must mention
is that the original Dirac calculation leading to the historic (ge ∼ 2), this calculation is posted in most if
not all physics textbooks as if it is an exact calculation and the emergence of the the gyromagnetic ratio
from this “exact and perfect Dirac value (gD = 2)” is that the Electron is not a point particle, for if it is was
a point particle, it would yield the exact Dirac value. The truth is that the Dirac value (gD = 2) is not an
exact prediction of the Dirac Theory, but a first order approximation result. The present calculation is just
one step further and more exact than the celebrated Dirac calculation because we merely have taken-up the
second order terms and asked what their contribution to the gyromagnetic ratio would be and the answer is
something we never expected at all.
In our modest view, the present result is surely a profound result for it – in principle, now allows us
to say that the Dirac equation does – to a reasonable extent, explain not only the Proton, but the Neutron
and any general spin-1/2 particle. The very fact that one would not account for the Proton and Neutron’s
gyromagnetic ratio, this led physicists to think that in its bare form, the Dirac equation is not an equation
for the Electron and because this, some physicists have sought for a Dirac equation for the Proton and
Neutron (cf. Chirgwin & Flint 1945). Certainly, there is need to ponder on this result further than has been
conducted herein.
7 Conclusion
Assuming the acceptability of what has been presented herein, we hereby make the following conclusion:
1. The Dirac equation can be considered to account for the extra anomalous gyromagnetic ratio of all spin-1/2
particles and this extra anomalous gyromagnetic ratio arise as second order effect so to the spin’s alignment
with the ambient magnetic field.
2. The fact that the extra anomalous gyromagnetic ratio of a particular particle is the same for all particles of that
kind, this implies that the spin of a particular particle always aligns itself in such a manner that the component
of the magnetic field along the orientation of the spin always has the same magnitude.
Acknowledgments: We are heftily grateful to the National University of Science & Technology (NUST)’s
Research & Innovation Department and Research Board for their unremitting support rendered toward our
research endeavours; of particular mention, Prof. Dr. P. Mundy, Dr. P. Makoni, Dr. D. J. Hlatywayo and
Prof. Dr. Y. S. Naik’s unwavering support. This reading is dedicated to my mother Setmore Nyambuya and
to the memory of my dearly missed departed father Nicholas Nyambuya (27.10.1947 − 23.09.1999).
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