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DE BROGLIE TIRED LIGHT MODEL
AND THE REALITY OF THE QUANTUM WAVES
J.R. Croca
Departamento de Física
Faculdade de Ciências da Universidae de Lisboa
Ed C8 Campo Grande – Lisboa - Portugal
Email: croca@fc.ul.pt
Abstract: In the early sixties of the XX century de Broglie was able to explain the
cosmological observable red shift, without ad hoc assumptions. Starting from basic quantum
considerations he developed his tired light model for the photon. This model explains in a
single and beautiful way the cosmological redshift without need of assuming the Big Bang
and consequently a beginning for the universe. Evidence coming from Earth sciences seems
also to confirm these ideas and furthermore concrete proposal of laboratorial scale
experiments which can test the model are here reviewed.
Key words: Fundamental quantum physics, nonlinear quantum physics, de Broglie tired light
model, fourth order interferometry, local wavelet analysis, redshift, de Broglie aging constant.
1 - Introduction
In the early sixties of the last century de Broglie published two important papers [1, 2] on the
properties of the photon. In this paper we recall his pioneering work, which, unfortunately,
only recently reach my hands due to the difficulty in finding the original papers. Some more
recent works [3] done by myself and other researchers [4] on the subject are here reviewed.
Experimental evidence, whether we like it or not, clearly indicates that the strange quantum
entity we name under the name of photon, just like any other quantum particle, is indeed a
very complex being. Nevertheless it is possible to describe most of its basic properties in
terms of a general causal framework valid for any quantum particle. A first approach [4]
consists in describing a quantum particle in general and the photon in particular by means of
local wavelet analysis [5] that contemplates both the local and extended properties of the
quantum entities. A principal advantage of the model springs from its intrinsic mathematical
simplicity and from the fact that it allows a natural interpretation of the experimental evidence
coming from different branches of sciences. In Earth sciences it allows the explanation of the
minute discrepancies between plate tectonics measurements done with GPS and VLBI. In
astronomy it gives a natural and easy explanation for the cosmological redshift without any
need for assuming the Big Bang for the universe. Furthermore it gives news insights in the
measurements done with the super resolution microscopes of the new generation that falsify
the general validity of Heisenberg uncertainty relations. Finally this causal local model for the
quantum particle allows the design of laboratory scale experiments that can unequivocally test
the validity of the model.
2 – A simplified causal local model for the quantum particle
Any quantum particle, like for instance the photon, can be described, in a first approach, by a
full wave
φ
composed of an extended yet finite region, the wave
θ
, described
Preprint: To appear in Foundations of Physics 1
mathematically by a Morlet gaussian wavelet [5] plus a singularity
ξ
immersed in the wave
so that we have
ξ
θ
φ
+
=
. (2.1)
This singularity or corpuscle
ξ
carrying almost all the energy of the particle is responsible for
the habitual quadratic detection process. The extended wave, practically without energy
guides, through a nonlinear process, the singularity preferentially to the points were the wave
has greater intensity giving origin to the interferometric properties of the quantum particles. In
the linear approximation the wave devoid of singularity is solution to the usual linear
Schrödinger equation, while at the nonlinear approach, the function
φ
representing the
quantum particle, is solution to the nonlinear master equation
t
iV
mm ∂
∂
=+
∇
+∇−
φ
φφ
φφ
φφ
φ
h
hh
2
1
2
1
*)(
*)(
22
22
2
2, (2.2)
which has a solution [3] in one spatial dimension
)(
2/)2(2/)2(
0
22'
2
1
2
0
2'
0
2
1Etpx
EtpxEtpx i
eeea
E−−
−−−−−−
⎥
⎦
⎤
⎢
⎣
⎡+= hhh
σεσε
σπ
φ
, (2.3)
which can be seen as the sum of the two solutions: The first representing, in a preliminary
approach, the singularity
)(
2/)2(
0
2
0
2'
0
2
1Etpx
Etpx i
eea
E−−
−−−
=hh
σε
σπ
ξ
(2.4)
and the second the extended wave
)(
2/)2(
0
22'
2
1Etpx
Etpx i
ee
E−−
−−−
=hh
σε
σπ
θ
. (2.5)
Due to the enormous difference in energy the constant a is very large
, (2.6) 1>>>a
however, this constant may turn to zero in the case when, by splitting or absorption, the full
wave
φ
becomes an empty wave
θ
, that is, a wave devoid of singularity. The width of the
gaussians, 0
σ
and
σ
, are related with the size of the wavelets and verify the relation
0
σ
σ
>>> , (2.7)
meaning that the size of the singularity approaches a Dirac delta function.
The plot of the real part of the function (2.3), representing the particle, is shown in Fig.2.1 In
this representation we see that both the wave and the singularity share the same phase as
stated by de Broglie principle of concordance of phase.
Preprint: To appear in Foundations of Physics 2
Fig.2.1 – Plot of the real part of the quantum particle
2.1 – Behavior of the theta wave
From single particle interferometry it is known that in certain situations it is possible to have
waves without singularity. In this case the real guiding wave, just like any other ordinary
electromagnetic wave, can be reflected diffracted and so. In such conditions when the wave
theta, devoid of singularity, interacts with a 50% beam splitter, half of it is reflected, the other
half being transmitted. If another 50% beam splitter is placed in the transmission path, half of
the incident wave that is, a fourth of the initial wave is transmitted. By placing more equal
beam splitters along the transmission path the intensity of the wave is progressively reduced
till no more wave remains as is indicated in Fig.2.2.
=
0
0
(½)
1
0
(½)
2
0
(½)
3
0
(½)
n
0
(½)
1
0
(½)
2
0
(½)
3
0
(½)
n
Fig.2.2 – The wave devoid of singularity looses amplitude when crossing the beamsplitters
Analytically this situation is represented by
(2.7)
0
)2/1(
θθ
n
=
Preprint: To appear in Foundations of Physics 3
or, for a different generic attenuating coefficient t
(2.8)
10,
0≤≤= ttn
θθ
that is,
, (9)
n
e
µ
θθ
−
=0
with tln
−
=
µ
. (2.10)
In the case of a continuous homogeneous medium the discrete variable need to be changed by
a continuous one
, (2.11)
x
e
µ
θθ
−
=0
with
µ
standing for the average transmission factor.
2.2 – Behavior of the full wave
Suppose now that we place a 50% beamsplitter in front of an incoming full wave
φ
, and that
the singularity is transmitted. Next we place, in front of it, another equal beamsplitter, and,
also in this case, consider that the singularity is transmitted. This process being continued as
long as one wishes. What results shall be expected from this setup?
We know that the singularity
ξ
, being by its very nature indivisible, is either reflected or
transmitted. The guiding wave in each beamsplitter gets its amplitude reduced by half. If this
process of reducing amplitude keeps going on then, after a sufficient number of beamsplitters,
the amplitude of the guiding wave will practically be zero. In such case we would be left with
the singularity without the guiding wave. As a consequence de Broglie basic principle for the
quantum physics, stating that any corpuscle possesses its own associated wave, would be
broken. In order to avoid the breakdown of the whole conceptual structure of the quantum
physics another more complex interacting process must be assumed. It is reasonable to
assume that after having attained a minimum level, compatible with the said basic principle,
the reduction process for the guiding wave stops. After that point on the amplitude of the
guiding wave remains, for all practical purposes, constant. This state of minimum energy for
the guiding wave corresponds to its fundamental state. If the amplitude of the guiding wave,
from this point on, keeps constant, on average, it needs to take energy from the singularity.
So, in each beamsplitter the singularity looses a very minute amount of energy to feed the
guiding wave. This process is shown, schematically, in Fig.2.3.
0
(½)
1
0
0
(½)
2
0
(½)
k
0
(½)
k
⌧
0
⌧
0
⌧
0
⌧
0
⌧
0
b
1
´
⌧
0
b
2
´
Fig.2.3 – Interacting process for the full wave
Preprint: To appear in Foundations of Physics 4
Analytically the whole process can be divided into two parts: The first with the singularity
maintaining the energy constant, while the energy of the guiding wave decreases till attaining
the fundamental energy level. The second when the singularity starts feeding energy to the
guiding wave.
For the guiding wave it reads
(2.10)
⎪
⎩
⎪
⎨
⎧
=>=
≤=
constant,,
,
0
0
kknt
knt
k
n
n
n
θθ
θθ
and for the singularity
(2.11)
⎪
⎩
⎪
⎨
⎧
>=
≤=
knb
kn
n,'
,
0
0
ξξ
ξξ
with b’ standing for the attenuation factor for the singularity.
In order to relate the parameter b’ with t it is necessary to recall the conservation of energy. In
each transition the energy lost by the singularity equals the amount gained by the guiding
wave. We know that
(2.12)
⎩
⎨
⎧
=
=
+
+
kk
kk b
t
ξξ
θθ
'
1
1
with
0
ξ
ξ
=
k. (2.13)
The energy lost by the guiding wave in each transition can be computed from
(2.14)
22
1
2|)1(||||| kkk t
θθθθ
−=−=∆ +
which must be compensated by the energy lost by the singularity
(2.14’)
22
1
2|)'1(||||| kkk b
ξξξξ
−=−=∆ +
that is, we must have
ξ
θ
∆
=
∆
(2.15)
or, explicitly
kk tb
θ
ξ
)1()'1(
−
=
−
, (2.16)
which allows us to express the attenuation factor in terms of the damping constant of the theta
wave
)1(1' tb k
k−−=
ξ
θ
(2.17’)
that is, recalling that k
ξ
ξ
=
0
)1(1'
0
tb k−−=
ξ
θ
. (2.17)
In such case, and naming the fundamental level of the guiding wave by kF
θ
θ
=, the
attenuation for the singularity can be written
Preprint: To appear in Foundations of Physics 5
knt n
F
n≥
⎥
⎦
⎤
⎢
⎣
⎡−−= ,)1(1 0
0
ξ
θ
ξξ
, (2.18”)
or,
kne nt
n
F
≥= ⎥
⎦
⎤
⎢
⎣
⎡−− ,
)1(1ln
00
ξ
θ
ξξ
(2.18’)
that is,
, (2.18) kne nb
n≥= −,
0
ξξ
where the damping constant is given by
))1(1ln('ln 0
ξ
θ
F
tbb −−−=−= . (2.19)
2.3 – Experimental Test
It is possible to imagine a relatively easy experimental setup [4] to test the validity of the
interacting process. For this it is necessary to take in consideration that the interacting process
assumes that the loss of energy from the singularity to the guiding wave is, really, very small.
This is a consequence of the fact that there is, as we shall see, an enormous difference
between the energy of the singularity and the energy of the accompanying wave.
The energy of this wave is so small that is unable to trigger the common quadratic detectors.
In such case it is reasonable to suppose that under usual laboratory scale experiments the
energy of the singularity remains, for all practical purposes, constant, even in the case when
some minute part of it went to the guiding wave. The experimental setup is sketched in
Fig.2.4,
s
D
B
1
B
2
01
1
Fig.2.4 – Experimental setup for testing the photon interacting process
where we can see a modified Mach-Zehnder interferometer with 50% beamsplitters B1 and
B2. Along both arms of the interferometer there are placed an equal number of beamsplitters
with same transmission factor t.
Let us now calculate the expected visibility, of the interference pattern, seen at the detector D.
a) Orthodox approach
Since the number of beamsplitters, in each arm of the interferometer, is equal, this leads to an
overall absorbing effect, given by the A factor, at both the reflected and transmitted beams
01
ψ
and 02
ψ
. So we can write for the output beams
Preprint: To appear in Foundations of Physics 6
(2.20)
⎩
⎨
⎧
=
=
||||
||||
022
011
ψψ
ψψ
A
A
and consequently, the expected intensity distribution for the arriving photons at the detector
shall be given by
(2.21)
2
21 ||
ψψ
+=
o
I
which developed leads to
. (2.22)
2
*
1
*
21
2
2
2
1||||
ψψψψψψ
+++=
o
I
Taking into consideration that for 50% beamsplitters B1 and B2 we have an equal amplitude
|||| 21
ψ
ψ
=
(2.23)
therefore, for this particular case, (2.22) becomes
)
, (2.24)
cos1(||2 2
10
δψ
+=I
where
δ
is the phase difference between the two output beams 1
ψ
and 2
ψ
.
Recalling (2.20) and that
0
2
1
2
02
2
01 |||| I==
ψψ
(2.25)
we finally have
. (2.26)
)cos1(
0
2
δ
+= IAIo
The visibility for this interference pattern is given by Born and Wolf formula
mM
mM II II
V+
−
= (2.27)
which by substitution gives
1
=
o
V. (2.28)
This result means that the visibility expected, by the orthodox approach, is always one no
matter the number of absorbing beamsplitters placed in each arm of the interferometer. This is
a consequence of the fact that the two coherent overlapping beams have the same amplitude.
b) Causal approach
Assuming for the photon the above interaction process the expected visibility can be
decomposed into two parts: The first corresponds to the case where the amplitude of the theta
wave and that of the full wave undergo the same attenuation. The second is related with the
threshold point. The point just after which the guiding wave attains the fundamental energy
level kF
θ
θ
=. From this point on the singularity starts feeding energy to the accompanying
wave, so that, for all practical purposes, the amplitude of the theta wave remains constant.
For the first case we have
(2.29)
2
21 ||
θθ
+=
I
c
I
with
(2.30)
kne n≤== −,|||||| 021
µ
θθθ
Preprint: To appear in Foundations of Physics 7
giving by substitution into (2.29) a visibility equal to one
. (2.30)
1=
I
c
V
After the threshold level k we have
(2.31) const,
02
01 =>
⎪
⎩
⎪
⎨
⎧
=
=
−
−kkn
e
e
k
n
µ
µ
θθ
θθ
which means that the causal intensity
, (2.32)
2
21 ||
θθ
+=
II
c
I
for this part second part, gives by substitution
(2.33)
2
00 ||
δµµ θθ
iknII
ceeeI −− +=
that, after some calculation, leads to
[
]
kneeI knII
c>−+= −− ,cosk)-(nsech1)(|| 222
0
δµθ
µµ
. (2.34)
The visibility of this interference pattern is
. (2.35)
knknV II
c>−= ),(sech
µ
The plot of the visibilities predicted by the two different approaches is shown in Fig.2.5
kn
1
V
III
Fig.2.5 – Plot for the expected visibility: Dotted line orthodox approach. Solid line causal prediction
The visibility predicted by the orthodox approach is one in the two regions. The causal model
in the first region predicts also a visibility one. Only in the second region the predictions are
different. Contrary to the orthodox prediction of constant visibility one, the causal approach
expects a decrease in the visibility attaining, eventually, a zero value.
3 – Causal interpretation of the cosmological redshift
A causal explanation of the cosmological redshift without the Doppler effect was given in
1962 by de Broglie [1]. In his seminal paper de Broglie presents an alternative causal
justification for the observable cosmological redshift in terms of his tired light model for the
photon. Even if he did not elaborate explicitly the aging process for the photon, nevertheless
his formula still stands.
Preprint: To appear in Foundations of Physics 8
The interacting process responsible for the aging of the photon results directly from the
previous considerations. The discrete formula (2.18), derived for the case when the particle
crosses the successive absorbing beamsplitters, can be generalized to include the continuous
homogenous absorbing medium assuming the form
. (3.1)
l
lb
e−
=0
)(
ξξ
In this expression b stands for the mean amplitude attenuation factor of the medium and is
the travelled distance. l
Since
2
||
ξ
∝E
the expression for the energy decreasing, as the photon crosses the cosmic space, is therefore
given by
, (3.2)
lB
eEE −
=0
where B=2b is de Broglie cosmological constant standing for average aging coefficient.
Recalling that
ν
hE = we can also write
, (3.3)
lB
e−
=0
νν
developing the exponential, and staying at the linear approximation, which is quite reasonable
since de Broglie constant is very small, we have
)1(
0lB
−
≅
ν
ν
,
or,
lB−≅
−
0
0
ν
ν
ν
,
which can also be written
0
0
,
ννν
ν
ν
−=∆−≅
∆lB. (3.4)
Recalling with de Broglie [1] that
0
00
,
λλλ
ν
ν
λ
λ
−=∆
∆
−=
∆ (3.5)
we have
lB≅
∆
0
λ
λ
(3.6)
or, since in astronomy the relative wavelength difference is named by Z,
lBZ ≅
∆
=
0
λ
λ
(3.7)
and that Hubble law has the form
l
c
H
Z=, (3.8)
Preprint: To appear in Foundations of Physics 9
with H being Hubble constant and c the velocity of light. In such conditions we have
c
H
B=, (3.9)
which shows that de Broglie average aging constant is given by Hubble constant divided by
the constant c. Recalling that literature presents for the Hubble constant the value
, we get for de Broglie aging constant the figure
118
106.1 −−
×≈ sH
. (3.10)
126
10 −−
=mB
Now in order to estimate the distance of a cosmic light-emitting source, assuming, in a first
approach, that the observable redshift is only due to the aging of the photon, we have only to
use the expression
BZ /
≈
l, (3.11)
or
. (3.11’)
Z
26
10≈l
As we have seen, starting from the causal model of de Broglie for the quantum particle, and
from the inherent complex interaction process, it was possible to arrive directly, without any
ad hoc hypothesis, to a natural explanation for the cosmological redshift, without need to
postulate a hypothetical beginning for the universe.
4 – Ratio between the energy of the guiding wave and the singularity
A first rough estimate of the ratio between the amplitude of the guiding wave, in the
fundamental state, and the amplitude of the singularity can now be made.
Since by (2.19) ))1(1ln( 0tb F−−−=
ξ
θ
and bB 2
=
we have
))1(1ln(2 0tB F−−−=
ξ
θ
,
which means that
2/
0
1)1( B
Fet −
−=−
ξ
θ
.
Remembering that , expanding the exponential, and staying at the linear
approximation we have
126
10 −−
≅mB
27
0
105)1( −
×≈− t
F
ξ
θ
.
Assuming that each transition corresponds, in this case, to a relative large section of space it is
reasonable to make the transition coefficient t relatively small. Therefore, under this
assumption, we get a rough estimate
27
0
10−
≈
ξ
θ
F
or, in terms of energy
Preprint: To appear in Foundations of Physics 10
54
10
0
−
≈
ξ
θ
E
EF,
expression which, as expected, shows that the energy of the guiding wave is indeed much
smaller than the energy of the corpuscle.
5 – Evidence from Earth Sciences corroborating the causal model
Plate tectonics scientists are presently faced with a minute discrepancy between
measurements made with geodetic satellites and those coming from very large baseline
interferometry, VLBI. This small difference remains even after all corrections are made. This
problem is very pertinent because even if the difference between measurements is very small
in a year, over the geological times it can be of real significance. Evidence derived from other
sources lead geologists [6] to believe that geodetic measurements are more precise than those
from VLBI.
Usual VLBI measurements are made assuming that photons, coming from very faraway
cosmic sources, keep unchanged its energy no matter the space they have travelled through
space in order to reach Earth. Since no corrections for the aging of the photon were made it is
natural to expect that when these corrections are introduced the right result shall be obtained.
Furthermore, from this difference, is also possible to estimate the approximate value for de
Broglie aging constant.
Even if it seems amazing that probative evidence for de Broglie causal model for the photon
could be inferred from the macroscopic Earth Sciences, it is not the first time and probably
should not be the last in history of science that such a thing happens.
5.1. Fourth order interferometry
The basis of VLBI lies in fourth order interferometry [7, 8] that, as the name indicates,
correlates four fields. The sketch of the setup is indicated in Fig.5.1
C
1
D
1
C
2
D
2
C
12
y
d
A
B
r
A1
r
B1
r
B2
r
A2
k
A1
k
A2
k
B1
k
B2
Fig.5.1 – Sketch of the essence of fourth order interferometry
where A and B represent two photonic emitting sources, D1 and D2 are detectors and C1, C2
the counters with C12 the coincidence counter. In this particular case of VLBI the photonic
sources are in general cosmic objects, quasars, and the detectors D1 and D2 are, usually, radio
telescopes. The coincidence counter C12 is, normally, named correlator.
From the sketch we see that the fields at the detectors D1 and D2 are
2211 21 ,BABA EEEEEE
+
=
+
= (5.1)
Preprint: To appear in Foundations of Physics 11
therefore, at a certain instant arbitrary of time, and emphasizing the spatial representation
because of its intuitive nature, these fields, for the monochromatic approximation, can be
written
⎪
⎩
⎪
⎨
⎧
=
=
⎪
⎩
⎪
⎨
⎧
=
=
+
+
+
+
).(
0
).(
0
).(
0
).(
0
22
2
22
2
11
1
11
1
BBB
AAA
BBB
AAA
rki
BB
rki
AA
rki
BB
rki
AA
eEE
eEE
eEE
eEE
ϕ
ϕ
ϕ
ϕ
r
r
r
r
r
r
r
r
(5.2)
and by substitution into (5.1) the total fields at the detectors are
⎪
⎩
⎪
⎨
⎧
+=
+=
++
++
).(
0
).(
02
).(
0
).(
01
2222
1111
BBBAAA
BBBAAA
rki
B
rki
A
rki
B
rki
A
eEeEE
eEeEE
ϕϕ
ϕϕ
r
r
r
r
r
r
r
r
(5.3)
to which correspond the intensities seen by the detectors
⎪
⎪
⎩
⎪
⎪
⎨
⎧
⎥
⎦
⎤
⎢
⎣
⎡+++==
⎥
⎦
⎤
⎢
⎣
⎡+++==
+−−+−
+−−+−
)..()..(
*
222
)..()..(
*
111
22222222
11111111
.
.
ϕϕ
ϕϕ
BBAABBAA
BBAABBAA
rkrkirkrki
BABA
rkrkirkrki
BABA
eeIIIIEEI
eeIIIIEEI
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
(5.4)
where we have made
BA
ϕ
ϕ
ϕ
−
=
. (5.5)
From these expressions it is possible to obtain the correlation intensity function measured at
C12
〉
〈
=
2112 III (5.6)
which explicitly reads
〉
⎥
⎦
⎤
⎢
⎣
⎡++
⎥
⎦
⎤
⎢
⎣
⎡++
⎥
⎦
⎤
⎢
⎣
⎡+++
⎥
⎦
⎤
⎢
⎣
⎡++++〈=
+−−−+−−
+−−+−+−−+
+−−+−
+−−+−
)....()....(
)2....()2....(
)..()..(
)..()..(
2
12
2211221122112211
2211221122112211
22222222
11111111
)(
)()(
BBBBAAAABBBBAAAA
BBBBAAAABBBBAAAA
BBAABBAA
BBAABBAA
rkrkrkrkirkrkrkrki
BA
rkrkrkrkirkrkrkrki
BA
rkrkirkrki
BABA
rkrkirkrki
BABABA
eeII
eeII
eeIIII
eeIIIIIII
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
ϕϕ
ϕϕ
ϕϕ
(5.7)
Now assuming, as usual, that the light arriving at the two detectors is not correlated in phase
all terms containing the phase difference in a long run average to zero. So, we simply got
[
]
)..()..(cos2)( 22112211
2
12 BBBBAAAABABA rkrkrkrkIIIII
r
r
r
r
r
r
r
r
−−−++= . (5.8)
In order to further simplify the expression it is convenient to consider the case where
III BA
≈
≈
(5.9)
Preprint: To appear in Foundations of Physics 12
then, the coincidence intensity, that is, the cross correlation function assumes the habitual
form
)cos1(4 2
1
12
δ
+
=
II (5.10)
with the modulating phase
δ
given by
)..()..( 22112211 BBBBAAAA rkrkrkrk
r
r
r
r
r
r
r
r
−−−=
δ
(5.11)
5.2. Orthodox Model
In this approach it is assumed that the photon as it travels astronomical distances through the
“empty” space, the subquantum medium, or the zero point field as is usually called, keeps
always its energy unchanged. Therefore since the velocity of the light is constant we must
write
BBBAAA kkkkkk
r
r
r
r
r
r
==== 2121 ,. (5.12)
By substitution into (5.11) we obtain for the phase predicted by the orthodox model
)()( 2121 BBBAAAo rrkrrk
r
r
r
r
r
r
−−−=
δ
. (5.13)
From Fig.5.1 we see that
⎪
⎩
⎪
⎨
⎧
==−
==−
2
2
21
21
eyyrr
eyyrr
BB
AA rrrr
r
r
r
r
(5.14)
which means
ykk BAo
r
r
r
)( −=
δ
. (5.15)
From the angle
θ
between the two plane waves coming from sources A and B, and for the
orientation indicated in Fig. 5.2
y
x
θ
k
B
k
A
Fig.5.2 – Relative orientation of the two incoming plane waves
we see that
[]
ykeyeekykk BA
θθθ
sinsin)cos1()( 221 −=−−=−
r
r
r
r
r
r, (5.16)
which gives
yk
o
θ
δ
sin
−
=
(5.17)
or, recalling that the angle
θ
between the two incoming plane waves is indeed very small
yk
o
θ
δ
−
=
. (5.18)
Preprint: To appear in Foundations of Physics 13
In this conditions the expected correlation function intensity I12, assuming no aging status for
the photon so that it remains unchanged no matter the distance it travels through the space, is
given by
[
]
)cos(14 2
1
12 ykIIo
θ
+= . (5.19)
From the difference between the full maximum 0
=
δ
and the minimum
π
δ
= it is possible
to get the approximate angular diameter
θ
of the cosmic object, which gives the customary
formula
y
o2
λ
θ
=. (5.20)
Some times this expression is presented in the scientific literature [9] with a different
multiplying constant resulting from considering the sources as circular apertures. In any case
the expression remains essentially the same.
By assuming that the angular diameter of the cosmic object is known then it is possible to
determinate the distance between the two detectors which typically happen to be
radiotelescopes. In such situation the distance between the two detectors, assuming, of course,
that the angular diameter is known, is given by
θ
λ
2
=
o
y. (5.21)
5.3. De Broglie causal approach
Now assuming de Broglie local causal model for the photon important modifications need to
be made in the calculations. From above we know that the change in energy of the photon as
it travels through the subquantum medium, usually called in the in the orthodox literature by
zero point field, can be described, as we have seen previously (3.2), by
, (5.22)
lB
eEE −
=0
or in terms of frequency
lB
e−
=0
ν
ν
, (5.23)
recalling that the velocity c of the light is supposed constant,
. (5.24)
lB
ekk −
=0
In such circumstances, and in order to take into account the change in wavelength due to the
aging of the photon it is necessary to write
⎪
⎩
⎪
⎨
⎧
=
=
⎪
⎩
⎪
⎨
⎧
=
=
−
−
−
−
B
rB
B
B
rB
B
A
rB
A
A
rB
A
kekk
kekk
kekk
kekk
B
B
A
A
ˆ
ˆ
ˆ
ˆ
2
2
1
1
2
2
1
1
0
0
0
0r
r
r
r
(5.25)
with
B
B
B
A
A
Ak
k
k
k
k
k
r
r
== ˆ
,
ˆ. (5.26)
Preprint: To appear in Foundations of Physics 14
The corrected phase, assuming the local causal aging model, is therefore
)
ˆˆ
()
ˆˆ
(2
2
1
1
2
2
1
10000 BB
rB
BB
rB
AA
rB
AA
rB
crkekrkekrkekrkek BBAA r
r
r
r⋅−⋅−⋅−⋅= −−−−
δ
, (5.27)
or
[
]
)(
ˆ
)(
ˆ21
212
2
12
1
1)()(
0BB
rrB
B
rB
A
rrB
AA
rB
crrekererkek BBBAAA
r
r
r
r−−−= −−−−−−
δ
. (5.27’)
In order to obtain a more manageable expression it is convenient to make the realistic
approximation, which corresponds to assume a symmetric geometry
l
≈
≈
21 BA rr , (5.28)
with being the average distance traveled by the photon, from the cosmic object to the
detectors D
l
1 and D2, and naming
2112 ;BBBAAA rrrr
−
=
−
=
ε
ε
, (5.29)
the phase becomes
[
]
)(
ˆ
)(
ˆ2121
0BB
B
BA
B
AA
B
crrekrerkek BA
r
r
r
r
l−−−= −−
−
εε
δ
(4.30)
or recalling that (5.31)
lB
ekk −
=0
we have
)()( 2121 BB
B
BA
B
AAc rrekrerk BA
r
r
r
r
r
r
−−−= −−
εε
δ
. (5.32)
Taking in consideration Fig.5.3
y
d
A
B
r
A1
r
B2
2
A
B
re
A
ε−
1
B
B
re
B
ε−
A
ξ
B
ξ
y
A
y
B
Fig.5.3 – Graphic representation of the variables
it is possible to write
BBAAc ykyk
r
r
r
r
⋅−⋅=
δ
. (5.33)
Naming by
ξ
the difference between the vectors and the corrected ones, we have
⎪
⎩
⎪
⎨
⎧
−=−=
−=−=
−−
−−
)1(
)1(
111
222
BB
AA
B
BB
B
BB
B
AA
B
AA
errer
errer
εε
εε
ξ
ξ
rrr
r
r
r
r
r
(5.34)
From Fig.5.3 we see that
BBAA yyyy
ξξ
r
r
r
r
rr −=+= ;, (4.35)
so the corrected phase assumes the form
Preprint: To appear in Foundations of Physics 15
)()( BBAAc ykyk
ξξδ
r
r
r
r
r
r
−−+= (5.36)
that is, by rearranging the terms
BBAABAc kkykk
ξξδ
r
r
r
r
r
r
r
⋅+⋅+−= )( , (5.37)
or, recalling the previous calculations (5.15) and (5.18) we got
BBAAc kkyk
ξ
ξ
θ
δ
+
+
−
= (5.38)
and since it is reasonable to make
BBAA kk
ξξ
r
r
//
ˆ
;//
ˆ. (5.39)
Because
kkk BA
=
=
(5.40)
we have
)( BAc kyk
ξ
ξ
θ
δ
+
+
−
= (5.41)
or by (5.34)
. (5.42)
))1()1(( 21
BA B
B
B
Ac ererkyk
εε
θδ
−− −+−+−=
In order to further simplify this expression it is convenient to recall the following
approximations
ε
ε
ε
≈
≈
≈
≈BBA rr A
and ,
21 l, (5.43)
then (5.42) can be written
(5.44)
)1(2
ε
θδ
B
cekyk −
−+−= l
or
⎥
⎦
⎤
⎢
⎣
⎡−−−= −)1(2
ε
θ
θδ
B
ceyk l. (5.45)
In order to estimate the value of
ε
in terms of known quantities it is worth to look at next
Fig.5.4
y
d
A
B
r
1
r
2
υυ
ε
Fig.5.4 – Sketch with the representation of the quantities for estimating
ε
which allows us to write
y
ε
θ
=sin (5.46)
and making the usual small angle approximation, we got
y
θ
ε
≈
. (5.47)
Under the above approximations the corrected phase assumes the form
Preprint: To appear in Foundations of Physics 16
⎥
⎦
⎤
⎢
⎣
⎡−−−= −)1(2 yB
ceyk
θ
θ
θδ
l, (5.48)
since the argument of the exponential is much less than one this expression can be further
simplified giving
)21( lByk
c
−
−
=
θ
δ
. (5.49)
Finally, for correlated intensity function, taking in account the aging of the photon, we got
under the simplificative approximations made
[
))21(cos(14 2
1
12 lBykII c−−+=
θ
]
. (5.50’)
or
[
))12(cos(14 2
1
12 −+= lBykII c
θ
]
. (5.50)
From the previous considerations we know that in order to make the concrete measurement
we need to change the length y from zero, corresponding to a null phase, maximal visibility,
to a of minimal visibility which corresponds to a phase value of
π
. That is
π
θ
=
−
|12| lByk , (5.51)
which gives
|12| 1
2−
=lB
yc
θ
λ
, (5.52)
or, recalling the value predicted by the orthodox approach (4.21)
θ
λ
2
=
o
y,
we may write |12| 1
−
=lB
yy oc , (5.53)
or
co yBy |12|
−
=
l. (5.53’)
These expressions implies that the length we got with the usual VLBI method, yo, is different
from the causal value yc, obtained assuming the aging model for the photon. Naturally if de
Broglie aging constant approaches zero, 0
≅
B, the two predictions are precisely the
same, .
co yy ≅
It is worth to draw the attention that in these VLBI measurements, both without correction
and with the aging correction for the light it is supposed that the true angular diameter of the
cosmic object is known. This assumption is, as we are well aware, not entirely correct because
the value of the angular diameter is also inferred from the theory. In the common approach it
is given by
y
o2
λ
θ
=,
which assumes that the distance between the two light sensors y is known. Assuming the
aging model for the photon we have instead
Preprint: To appear in Foundations of Physics 17
|12| 1
2−
=lBy
c
λ
θ
, (5.54)
that depends also on de Broglie aging constant B and on the distance l.
Now to estimate the value of de Broglie aging constant for the photon and compare it with the
previous estimation it is convenient to know the approximate value for this small discrepancy
between the two measurement processes. In order to do that it is necessary to get assistance
from the Earth Sciences [10]. In such circumstances and quoting A. Ribeiro and L. Matias [6]:
“The VLBI geodetic method gives results that are distinctly different from near-Earth
geodetic methods (GPS, SLR, DORIS). Direct comparison between the different methods have
not been investigated in a systematic way. Nevertheless there are studies that compare results
of VLBI with plate kinematic models NUVEL-1 and NUVEL-1A, as well as results of near-
Earth geodetic methods with these kinematic models.
The plate kinematic model NUVEL-1 for relative plate motion was computed by DeMets et al.
(1990) using a large set of tectonophysical measurements: spreading velocities measured
from magnetic anomalies younger than 3 MA, young fracture zone azimuths and earthquake
slip vectors. A recent revision of the magnetic time scale implied a revision of the velocity
values and the relative model NUVEL-1A was defined
(DeMets et al., 1994). In the NUVEL-1A model the velocity magnitudes were multiplied by a
constant factor, 0.9562, giving 4.4% slower velocities than NUVEL-1. According to the same
authors, this correction reduced the discrepancy between the tectonophysical model and the
kinematic models derived by geodetic methods to only 2%, being the NUVEL-1A the faster.
However, considering only the results provided by the VLBI method, Heki (1996) showed that
this technique gave plate velocities 3.4% faster than NUVEL-1, that is, 7.8 % faster than
NUVEL-1A. This means that, using NUVEL-1A as a reference, the VLBI method for
estimating plate kinematics gives velocities that are 10 % faster (
±
a few %) than the other
geodetic methods.
Considering the length of a base line across the Pacific we can evaluate an order of
magnitude of discrepancy between distances measured by VLBI and distances measured by
near-Earth methods. For a base line of 5000 km and a relative plate velocity of 150 mm/year,
7% of plate velocity discrepancy means 1 cm of discrepancy over 1 year. This implies, a
relative distance difference of 2x10-9 over the base line, with VLBI giving higher distances.”
From this information we gather that experimental evidence tells us that , with y
standing for the right distance. This implies
yyy co =
≥
1|12| ≥−= lB
y
yo, (5.55)
or
, (5.56)
1≥lB
which means that
112 ≥−= lB
y
yo (5.57)
giving for the aging constant
Preprint: To appear in Foundations of Physics 18
)1(
2
1y
y
Bo
+= l. (5.58)
This expression can be written in other form. Naming by
∆
the difference between the two
predicted distances, we have
∆
+
=
−
=∆ yyyy oo ;, (5.59)
which by substitution in (4.58) gives
σ
ll 2
11 +=B, (5.60)
with
σ
standing for the relative distance difference, y/
∆
=
σ
.
Now since quasar distances from Earth can be from about 1025 m to 1026 m and that
we got by substitution in (60)
9
102 −
×≈
σ
,
mB /10 26−
≈
which perfectly agrees with the first estimated value, , for the mean aging
constant assuming that the observable astronomical redshift is only due to the aging of the
photon.
mB /10 26−
≈
6 – Conclusion
It was shown that not only the theoretical model for the photon is sound but even more it is
capable, starting from first principles and without any ad hoc assumptions, to explain the
cosmological redshift without need of a beginning for the universe as is assumed by the
nowadays in fashion theory of the Big Bang of religious charisma. On the other hand
convergence with results coming from unexpected and unsuspected source, as is Geology,
indicates the correctness of the model to describe, in a first approach, the structure of the
quantum particles. Furthermore it is worth to call the attention to the important fact that the
causal model for the photon can be tested in laboratory scale experiments with controlled
parameters.
Acknowledgements:
I dedicate this work to my dear friend Franco Selleri.
Part of this work was the result of a very interesting idea with what the geologist A. Ribeiro
challenged me. His idea was that probably the discrepancy between the two sets of geodetic
measurements must, somehow, be explained by basic physical processes, namely by the aging
model for the photon of de Broglie. For his idea, constant support, and helpful discussions I
want to express him my many thanks. I want also to thank L. Matias for helpful discussions
concerning the concrete geodynamic data. Finally I want to thank L. Mendes Victor for the
given support, allowing me to stay at a place where the principal part of the present work was
done. Part of this work was supported by FCT.
Preprint: To appear in Foundations of Physics 19
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J.R. Croca, Apeiron, 4(1997)41
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9 - M. Born and E. Wolf, Principles of Optics, Pergamon Press, (New York, 1983)
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Preprint: To appear in Foundations of Physics 20
