Moment of Inertia of Fundamental Particles based on Analogy

Abstract

Analogy has been a powerful technique in physics to teach and derive useful results. At this point, in the relativistic free particle Hamiltonian H=cα∙p+βmc^2 proposed by Dirac, the first term cα∙p describes the motion of the particle, while the second term βmc^2 is concerned with the properties at rest. This Hamiltonian is linear in the momentum vector p and rest mass m terms. Dirac had introduced two operators α and β and he was successful in associating the first operator α with the spin of the particle. On the basis of analogy, it will be shown in the present note that the second operator β is associated with the moment of inertia of the particle.

Full text

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Moment of inertia of fundamental particles based on analogy
Dulli Chandra Agrawal
Department of Farm Engineering
Institute of Agricultural Sciences
Banaras Hindu University, Varanasi 221005, India
Email:dca_bhu@yahoo.com
Abstract
Analogy has been a powerful technique in physics to teach and derive useful
results. At this point, in the relativistic free particle Hamiltonian
H=c α ∙ p +βm c2
proposed by Dirac, the first term
c α ∙ p
describes the motion
of the particle, while the second term
βm c2
is concerned with the properties
at rest.
The Hamiltonian
H
is linear in the momentum vector
p
and rest mass
m
terms. Dirac had introduced two operators
α
and
β
,
then he was successful in
associating the first operator
α
with the spin of the particle. On the basis of
analogy, it will be shown in the present note that the second operator
β
is
associated with the moment of inertia of the particle.
PACS number: 14.60.Cd Electrons (including positrons)
Keywords: Relativistic free particle Dirac Hamiltonian, spin, analogy, moment
of inertia, mass, electron, positron.
Analogy has been a powerful instrument to understand even apparently hard

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physics concepts amongst researchers, teachers, and students. Understanding
how these analogies work is a rich area of physics education and research. For
example, Coulomb’s law is frequently taught as analogous to Newton’s law of
gravitation, electric current can be linked to water flowing through a pipe, and
so on. Now, on the basis of analogy, an attempt has been made to propose some
important observations for fundamental particles such as the electron, proton,
positron, and so on. For this purpose, we start with the relativistic free particle
Hamiltonian proposed by Dirac [1].
H=c α ∙ p +βm c2
(1)
This Hamiltonian is linear in the momentum vector
p
and rest mass
m
terms.
Dirac had introduced two operators
α
and
β
and he was successful in
associating the first operator
α
with the spin of the particle. On the basis of
analogy, it will be shown in the present note that the second operator
β
is
associated with the moment of inertia of the particle.
Since the above Hamiltonian describes a free particle, the four quantities
αx, α y, α z
and
β
must not depend on the space coordinates or the time. That is
α
and
β
are independent of the space coordinates
r
, time
t
, momentum
p
and
energy
E
and therefore they commute with all of them. This does not
necessarily mean that
α
and
β
are numbers because they need not commute
with each other [1]. In fact, they anti-commute rather than commute with each
other; that is why they are matrices.
In this Hamiltonian, the first term
c α ∙ p
takes care of the motion of the particle
while the second term
βm c2
is concerned with the properties at rest. The first
term has two momentums
−¿
linear momentum and angular momentum; in the
first term, the linear momentum
p
is dotted with the operator
σ
(because
α=ρ σ
), which is essentially the inherent spin of the particle having value
1
2ℏ
or
−1
2ℏ
and whose classical analog is rotational angular momentum; also, the sum of
linear momentum
(
L
)
and angular momentum
(
S
)
is well known to be conserved.
A similar analogy should also hold for the second term
βm c2
:
the mass
m
,
which
is inertia for the linear motion, should have been multiplied by a term having
two characteristics; firstly, it has to be an inherent property, and secondly, it
should be a rotational analog of inertia, that is, the moment of inertia for
rotational motion. Hence it may be proposed [2] that
β
essentially measures the
moment of inertia
I0
of the particle, and the suggested expression [3] could be
I0=β
(
ℏ2/m c2
)
. This is plausible because both
β
and
I0
should be Hermitian
operators with positive and negative eigen values corresponding to particle
(electron) and antiparticle (positron), respectively. Starting with this idea, the
following speculations can be made: As
β
commutes with
α
, and therefore the

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moment of inertia
I0
should be independent of the spinning of the particle or the
antiparticle. Also, just as conservation law holds for the sum of linear and
angular momentums
(
L+S
)
in the case of the first term in (1), we expect
a n o th e r c o n s e r v a t i o n l a w c o m p r i s i n g th e m o m e n t of i n e r t i a
I0
and inertia
m
of the particle. Also, the relation
I0ω=ℏS
would define the
classical analog of angular frequency. These discussions will be fruitful in
understanding the excited states of these particles as well.
It may be added that the estimates of moment of inertia
(
I0
)
for electron, positron,
and proton would be
5.43 ∙10−55 ,−5.43 ∙10−55 ,∧2.96 ∙10−58 kg . m2
, respectively,
whereas their corresponding spinning frequencies
(
ω
)
would be
9.71∙1019 ,9.71∙1019 ,∧1.79 ∙1023 per second
through
I0ω=Sℏ=5.27 ∙10−35 J ∙ s
, [4].
References
1. Leonard I. Schiff, Quantum Mechanics (McGraw-Hill, Tokyo, 1955) 323.
2. D.C. Agrawal, “Notes”, Lat. Am. J. Phys. Educ., 2 (May 2008) p.220. This
was the initial draft, which was published without a title and so could not
get due attention.
3. Johan K. Fremerey (September 2017), Triple-gyro model for deduction
of proton radius and magnetic moment
https://www.researchgate.net/publication/319525987_Triple-
gyro_model_for_deduction_of_proton_radius_and_magnetic_moment
4. David Halliday, Robert Resnick and Jearl Walker, Fundamentals of
Physics (JohnWiley, New York, 4th edition, 1993) 921