The Photon Puzzle

Modern physics contains two different concepts of light. In the theories of physics light can appear as an electromagnetic wave or as a flux of particles. Those twin concepts reconcile the theories of light that come from electromagnetism and from thermodynamics with each other. We now want to ask whether those two concepts possess the property of mutual consistency. An imaginary experiment will answer that question.

In the laboratory of your imagination set up the following
described apparatus. In an inertial frame that we designate Frame-1 establish a
monochromatic filter, a thin plate that allows only one frequency of light,
represented by í_{f},
to pass through, and place it in front of a photocell that, when struck by
light, causes an electric current to flow through a light bulb. In an inertial
frame that we designate Frame-2, moving through Frame-1 at the speed V,
establish a transmitter that emits light at a frequency of
í_{0} as
measured in Frame-2. To conduct the experiment we have two observers: Archie
sits at rest in Frame-1 and Betty sits at rest in Frame-2. To simplify the
analysis we assume, quite arbitrarily, that the transmitter moves toward the
optical filter.

It just happens that the radiation coming from the transmitter and subjected to a Doppler shift, as seen from Frame-1, passes through the filter and, via the photocell, makes the light bulb glow. In Frame-1, then, the light from the transmitter displays a Doppler upshift such that

(Eq’n 1)

Archie understands that result through the standard analysis of the relativistic Doppler shift. He knows that the frequency which Betty calculates of the radiation coming from the transmitter stands equal to the reciprocal of the time elapsed between the emission of any two successive wavecrests,

(Eq’n 2)

Archie also knows that the relative motion between their frames makes the time ticked off Betty’s clocks dilate relative to the time ticked off his clocks: a given interval that Betty measures corresponds to more time elapsed on his clocks, so that he would measure the time interval as

(Eq’n 3)

But the rate at which Archie receives the waves gets modified by another
effect: in the interval between the emission of two wavecrests the transmitter
moves a distance VΔt_{0}'
from the point at which it emitted the first wavecrest while the first wavecrest
moves a distance cΔt_{0}',
so the distance between the wavecrests comes out as

(Eq’n 4)

Traveling at the speed of light, those wavecrests will pass Archie an interval

(Eq’n 5)

apart. Thus the wavecrests will pass Archie, by his calculation, at a frequency of

(Eq’n 6)

as Equation 1 requires.

Betty sees the situation differently. At any point in her
frame two successive wavecrests pass by her with the interval
Ät_{0} between
them. She knows that after one wavecrest hits the optical filter, the filter
moves a short distance toward her transmitter before the second wavecrest hits
it; thus, the motions of the second wavecrest and of the filter subdivide the
distance cΔt_{0}
in the proportion V:c. The second wavecrest hits the filter an interval
Δt_{1}
after the first one does, so Betty has the distances as cΔt_{1}+VΔt_{1}=cΔt_{0}
and that fact gives her the time interval between the two successive wavecrests
hitting the filter as

(Eq’n 7)

Betty also knows that the time that Archie would measure would span fewer nanoseconds, due to time dilation, so she calculates the time elapsed on Archie’s clock between the arrivals of the successive wavecrests as

(Eq’n 8)

And then she calculates the frequency at which Archie measures the wave hitting the filter by taking the reciprocal of that equation and getting Equation 6, as she should.

Archie and Betty face the real challenge here when they notice the photocell in the apparatus and understand that light has another aspect that they must acknowledge in their analysis. Instead of conceiving the light emanating from the transmitter as a wave of mutually supporting electric and magnetic fields, they conceive it as a stream of photons, particle-like quanta of energy that will interact with electrically charged particles.

Betty notes that each photon in the stream carries an amount of energy given by Planck’s formula,

(Eq’n 9)

Likewise, Archie knows that he must measure the same photon carrying an amount of energy

(Eq’n 10)

in order for it to pass through the filter. Because the photons emanating from the transmitter do, indeed, pass through the filter, both Archie and Betty know that the difference between their inertial frames (the relative speed V) makes the photons emitted in Betty’s frame possess in Archie’s frame the energy that they need to pass through the filter. Now they want to figure out how that difference makes that effect happen.

Archie has a relatively straightforward way of understanding the effect. He assumes that a photon conforms to a minimum indeterminacy wave packet, which means that

(Eq’n 11)

in accordance with Heisenberg’s principle. Thus he writes

(Eq’n 12)

Applying time dilation to interrelate the two intervals in that equation necessitates that he write

(Eq’n 13)

the lesser energy reflecting the slowing of Betty’s clocks (and, therefore, the transmitter) as seen from Archie’s frame. In that interval of time, when the photon emerges from the transmitter, the Maxwell-Bartoli effect (light exerts pressure) enables the transmitter to do work upon the photon. Of course the photon cannot accelerate in response to the force, so the work goes entirely into augmenting the Planck energy of the photon. And by how much? The force equals the rate at which the photon’s energy emerges from the transmitter (divided by the speed of light), so the work done by the force equals the integral of that force over the distance the transmitter moves in the interval of the photon’s creation, so Archie has

(Eq’n 14)

and, in consequence,

(Eq’n 15)

That calculation misses the mark set by Equation 6 by a
factor of (1-v^{2}/c^{2}), so Archie knows that he has done
something wrong in his analysis. He had made a bad assumption and now he must
find it and correct it. So what factor has he not taken into account?

As he examines his list of relativistic effects, Archie
notices that in his frame the transmitter ponders more mass than it does in
Frame-2. In accordance with Einstein’s mass-energy equivalence theorem (E=mc^{2}),
that increase includes the energy stored in the transmitter’s batteries,
whatever form it may take. So assume that Betty makes the transmitter emit a
single photon and that the battery had just enough energy, 2E_{0}, to
generate two identical photons. Archie thus calculates the energy of creating
the photon as

(Eq’n 16)

in accordance with the mass-energy theorem. If that calculation did not give Archie a true picture of Reality, then Betty would send a second photon identical to the first, thereby emptying the transmitter’s battery, but Archie would detect a photon carrying more or less energy than the first did and that fact would violate the Principle of Relativity. So Archie accepts Equation 16 rather than Equation 13 as describing the creation of the photon. When he augments that calculation through the Maxwell-Bartoli effect, he gets

(Eq’n 17)

which conforms to Equation 6 by way of a multiplication by Planck’s constant (in accordance with Equations 9 and 10).

But Equation 16 seems to invalidate Equation 12. If
Δt_{1}
spans more nanoseconds than does Δt_{0},
due to time dilation, and if E_{1}' contains more electron-volts than
does E_{0}, then their respective combinations seem to produce the
possibility of invalidating Heisenberg’s indeterminacy principle. Fortunately
for the laws of Nature, Archie made another error: he confused the indeterminacy
in the time of the photon’s creation with the period of the photon’s vibration.
If, however, Archie calculates the Δt’s
by calculating the reciprocal of Equation 6 and multiplying the result by
Equation 17, he gets Equation 12. That fact tells him that he must take into
account both the creation and the augmentation of the photon in calculating the
indeterminacy in the time: __then__ it corresponds to the photon’s period of
vibration.

Finally, Betty analyzes the experiment through the photon
interpretation. In her frame a photon carrying energy E_{0}=hν_{0}
emerges from the transmitter, crosses space, and disappears into the filter. As
an aid to her imagination Betty conceives the filter as an array of harmonic
oscillators whose natural frequency at rest equals
ν_{f}.
If the photon has a frequency that resonates with those oscillators, it will
pass through the filter; if the photon has a different frequency, the filter
will absorb it or reflect it. Because the harmonic oscillators can serve as the
basic mechanism of clocks, they must oscillate more slowly for Betty than they
do for Archie, conforming to both the time dilation effect and the increase of
mass separately, so Betty calculates in her frame

(Eq’n 18)

In order to go through the filter, then, the photon must carry the energy

(Eq’n 19)

But, of course, it doesn’t get emitted with that much energy.

As the photon strikes the filter it takes a nonzero, albeit minuscule, elapse of time to get incorporated into the device. In that time the filter does work upon the photon by way of the Maxwell-Bartoli force and that work comes out as

(Eq’n 20)

Adding that term to the emission energy E_{0} gives Betty

(Eq’n 21)

which corresponds to Equations 1 and 6, as it should.

Thus, beginning with the wave interpretation, Archie and Betty show its consistency with the particle interpretation in one important aspect, the relativistic Doppler shift. We need that consistency, certainly for the sake of the quantum theory, but also because some problems come to a better solution if we can switch from one interpretation to the other. So now we add this little piece to the larger puzzle of radiation thermodynamics.

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