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Progress In Electromagnetics Research Symposium Proceedings, Marrakesh, Morocco, Mar. 20–23, 2011 723
The Equivalent Rest-mass of Photon
Antonio Puccini
Department of Neurophysiology, Order of Malta, Naples, Italy
Abstract— The photon (P), one of the elementary particles of the Standard Model, is considered
massless. The Universal Gravitation Law states that the gravity force (GF) takes place only
between bodies having a mass. Einstein’s prediction, based on the Theory of General Relativity,
provides that the light is conditioned by the action of the GF. However observations showed that
the P feels GF. Lord Eddington, with his famous observation in 1919, confirmed completely what
Einstein had said. During a sun eclipse, indeed, it was possible to observe that the light coming
from far away stars, when passing next to the sun rather than propagate in straight line, followed
a trajectory bended towards the sun.
We know that two massless Ps, meeting, materialise in a couple of quark-antiquark, or even in
a couple muon-antimuon which, summed up, have a mass of 400 electrons.
The energetic value (E) of the P is represented by the Planck’s constant (h) — equal to 6.625 ·
10−27 [erg · sec] — times its frequency (f ): E = hf .
The value of f is not defined, but extremely variable, according to the EM spectrum. Thus we
should indicate f with 10n [1/s], where n can start from 0, that is one cycle per second (we cannot
exclude it), to a very high frequency γP. Thus it is possible to represent the E of a P in this way:
E = h · 10n [1/s].
If we apply the Principle of Equivalence Mass-Energy (E = mc2 ) to the P, our calculations show
that the P has a minimum equivalent rest-mass (Zero Point Mass) equal to 10−48 [g].
1. INTRODUCTION
The P is the light quantum, the electromagnetic radiation (EMR) quantum, which energetic value
is represented by the Planck’s constant (h) — equal to 6.625 · 10−27 [erg· sec] — times the number
of oscillations the P makes in a second, that is times its frequency (f ):
E = hf.
(1)
As we learn from the electromagnetic spectrum the number of oscillations (10n ) of a EMR quantum
can be considerably different. It starts from 1027 c.s., or more, of a highly energetic γP, to an
extremely low frequency P, which may have few oscillations per second, or just one cycle per second
(we cannot exclude it). Thus the value of f in Eq. (1) is not defined, but extremely variable,
according to the EM band we consider. Thus we should indicate f with 10n [1/s], where n can start
from 0, that is one oscillation per second (since 100 = 1), to 27 [c/s] of a very high frequency γP,
or even a bit further. Thus it is possible to represent the energy of a P as follows:
E = h · 10n [1/s],
(2)
where n, of course, gives the number of oscillation per second. What cannot be accepted is f = 0,
since the P is always emitted by the EM source along with an oscillating propagation motion,
besides f = 0 would cancel the energy of the P itself, as we infer from (1). Moreover it is the
Quantum Mechanics (QM) itself, through Heisenberg Uncertainty Principle (HUP), to avoid a P
with f = 0, since a P which does not oscillate is a motionless P, in this case we would know
simultaneously 2 complementary parameters of the same particle: its position and momentum.
2. DISCUSSION
Along with the Eq. (1), Feynman said: “the energy of a P is Planck’s constant times the frequency.
That is true, but since the frequency of light can be anything, there is no law saying that energy has
to be a certain definite amount. Unlike Dennis’ blocks, there can be any amount of energy, at least
as presently understood. So we do not understand this energy as counting something at the moment,
but just as a mathematical quantity, which is an abstract and rather peculiar circumstance” [1].
However if we consider the minimum oscillation as a point of reference (hypothetically possible)
which a P can make in a second, that is one oscillation, f = 100 [1/s], we have:
E = h · 100 [1/s] = h · 1 [1/s].
(3)
PIERS Proceedings, Marrakesh, MOROCCO, March 20–23, 2011
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In this way the (3) can represent the possible Minimum Energy State of a P. At this concern
Chandrasekhar said: “a fundamental consequence of the quantum nature of the matter is that the
lowest possible energy for a system cannot be zero, it must have a value 6= 10, it is called Zero
Point Energy” [2]. It is clear that the Eq. (3) gives a limit case, the minimum value hypothetically
possible. Besides Einstein Principle of Equivalence Mass-Energy, represented by Eq. (4):
E = mc2 ,
(4)
refers first of all to values at rest of the particle considered. This is how Einstein himself defined this
Principle: “it represents the connection between inertial mass and energy” [3]. Thus, according to
this Law, if we wish to study the inertial mass of a particle, it seems important to consider first
the inertial energy of the particle itself, which should correspond, be equivalent, just to its possible,
hypothetical value of minimum energy which, in the case of P, is represented by the Eq. (3). Thus,
if we substitute the values of (3) with those of Eq. (4), we have:
m = E/c2 = h(100 [1/s])/c2 = (6.6252 · 10−27 [erg · s]) · (1 [1/s])/(2.9979 · 1010 [cm/s]).
(5)
If we write the (5) according to the cgs system we have:
m = (6.6252 · 10−27 )/(2.9979)2 · 1020 [erg]/[cm/s]2 .
(6)
Since the erg value is expressed in [g · cm/s2 · cm], that is in [g · cm2 /s2 ], we have:
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m = 6.6252/(2.9979)2 · 10(−27−20) g · cm2 /s2 / cm2 /s2
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= 6.6252/8.9874 · 10−47 g · cm2 /s2 · s2/cm2 .
Then we have:
m = 0.7372 · 10−47 [g],
(7)
(8)
and:
m = 7.372 · 10−48 [g].
(9)
This is the hypothetical minimum inertial mass (rest mass) of the P, the smallest possible, since
we considered the minimum possible oscillation for the P, that is one oscillation per second! It is
an extremely low value, yet 6= 0. We may call it: Zero Point Equivalent-Mass of the P.
3. CONCLUSIONS
In order to simplify the calculations we considered the smallest inertial energy of the P. Eq. (9)
takes into account the minimum possible frequency for the P, that is n = 0, as we have from Eq. (2).
However, it is clear that it is not possible to know with accuracy what the P minimum energy state
can be: this undetermination goes within the QM. That is the minimum value of n in Eq. (2)
cannot be determined with accuracy, moreover it can change in different circumstances. Thus we
can write the P inertial-mass formula as follows:
m = 7.372 · 10−48 [g · s] · 10n [1/s] = 7.372 · 10−48+n [g].
(10)
102 [1/s],
Thus, in case the P at the inertial state oscillates, about 100 times per second,
we have
that (10) is as follows:
m = 7.372 · 10−48+2 = 7.372 · 10−46 [g].
(11)
−45
If n was 3, we would have m = 7.372 · 10
[g]. However the inertial mass of the P does not change
significantly.
What do we get from these calculations? The P is not so evanescent, ethereal, it carries an
equivalent mass equal to its energy divided c2 [5, 6]. This should not amaze us. The Universal
Gravitation Law states that the gravity force (GF) takes place only between bodies having a mass.
Einstein’s prediction, based on the Theory of General Relativity, provides that also the light is
conditioned by the action of the GF. In fact observations show that the P feels the GF, as confirmed
firstly by Lord Eddington in 1919 during a sun eclipse. Besides we know that two massless Ps,
meeting, materialise in a couple of quark-antiquark, or even in a couple muon-antimuon which,
summed up, have a mass of 400 electrons. If we apply the Principle of Equivalence Mass-Energy
(E = mc2 ) to the P, we learn that the P may have a minimum equivalent rest-mass equal to
10−48 [g], considering the possibility of one oscillation per second of the examined P.
As we know Feynman stated: “energy and mass differ only by a factor c2 ” [1]. Besides: “along
with E = mc2 , each mass can be expressed in terms of an equivalent-energy” [7]. Thus, the opposite
may be true too: each energy can be expressed in terms of a equivalent-mass.
Progress In Electromagnetics Research Symposium Proceedings, Marrakesh, Morocco, Mar. 20–23, 2011 725
REFERENCES
1. Feynman, R. P., The Feynman Lectures on Physics, Vol. 1, 4–10, 1989; Zanichelli ed., Vol. 1,
17–7, Bologna, 2001.
2. Chandrasekhar, B. S., Why Things Are the Way They Are, Cambridge University Press, 1989;
il Saggiatore ed., 304, Milano, 2004.
3. Galison, P., It Must Be Beautiful: Great Equations of Modern Science, Graham Farmelo ed.,
2002; il Saggiatore ed., 119, Milano, 2005.
4. Kane, G., The Particle Garden, Gordon Kane ed., 1995; TEA ed., 172, Milano, 2002.
5. Puccini, A., “About the zero mass of photom,” Progress In Electromagnetics Research, Vol. 55,
117–146, 2005.
6. Puccini, A., “Light weights,” Progress In Electromagnetics Research B, Vol. 8, 161–177, 2008.
7. Nuclear Physics, Vol. 31, National Academics Press, The National Academy of Science, USA,
1986.