How to conserve energy if the number of photons changes?

Let us analyze further the photon number change effect which we discovered in the previous blog post.

Let us have an obstacle made of a polarizable material. Let the obstacle be much smaller than the wavelength of a coherent source, e.g., a radio transmitter or a laser. The obstacle makes a very sharp area of deformation in the electromagnetic wave. An observer who moves through this deformation will in the temporal Fourier decomposition of the wave see high frequencies.

Does the effect really exist? It almost certainly does. A coherent laser beam, or a coherent beam from a radio transmitter, is a macroscopic object. It would be very surprising if the measuring device would not see high frequencies in Fourier decompositions of complex waveforms.

Thus, we must conclude that even if a laser only emits photons of energy E, a measuring device which moves at a relativistic constant velocity v may sometimes detect photons which have much higher energy E'.

Classically, energy conservation is not a problem. But how to get conservation in quantum mechanics?

Could it be that in quantum mechanics, several photons can sometimes merge into a photon with higher energy?

Maybe we should abolish the notion that the laser beam consists of photons? The beam is created by atoms decaying into a lower energy state. That is, the beam is created as kind of "photons". But maybe the beam itself does not "consist of photons"? Photons are absorbed from the beam. The beam is destroyed in units of a "photon".

In our earlier blog posts we have suggested that one should assume the minimum amount of information in the course of a quantum mechanical process. The prime example of this is that a photon does not have a definite path. It is a step further if we claim that a laser beam does not have definite "photons" either.

A "photon" would only exist when we perform a measurement. We may observe that a certain atom has decayed into a lower energy state, or has been excited to a higher energy state.

We can call this framework "minimalist quantum mechanics". Quantum mechanics moves closer to classical mechanics if we do not assume the existence of quanta in intermediate states of a physical process. Only measurements will observe definite quanta.

Conservation of energy

How is energy conserved if the number of quanta is not defined?

Maybe the quanta which are eventually absorbed from the wave will always have energy less or equal to the energy that went to create the wave?

How would nature do the bookkeeping to ensure conservation of energy?

An accelerating observer: Unruh and Hawking radiation

The derivation of hypothetical Unruh or Hawking radiation uses a Fourier decomposition of a wave as seen by an accelerating observer. Unruh and Hawking believed that negative frequencies in the decomposition represent photons popping out of nothing.

In this blog we hold the opinion that Unruh and Hawking made an error: a negative frequency is simply a classical wave which has counter-clockwise circular polarization, if we define a positive frequency wave to have clockwise polarization.

The problem of a changing number photons is prominent for an accelerating observer. What kind of quanta will he observe in his detectors if he moves within a coherent electromagnetic wave?

Classically the answer is clear: his detectors will absorb photons of various frequencies. Again we face the problem: how to conserve energy?

Does some extra energy come from the kinetic energy of the measuring device?

Suppose that we have static negative electric charges ordered in a line at equal distances. Let an antenna move along the line at a constant velocity v. The antenna will observe a fluctuating electric field and can extract energy from it.

The energy must come from the kinetic energy of the antenna. We may imagine that the antenna collides with a virtual photon sent by a charge, loses momentum, and kinetic energy is freed.

However, if the antenna moves in a coherent electromagnetic wave, then some of the energy comes from the wave, and it might be that none of the energy comes from the kinetic energy of the antenna.

The number of photons may change between inertial frames: a new "Unruh effect"

Let us work in semiclassical physics. Let us have a monochromatic laser beam which is reflected or is refracted by a small object (is scattered by the object). Let us assume that the object is smaller than the wavelength.

        laser beam

        ~~~~~~~~~~~~~~~~~~~

        ~~~~~~~~~~~~  ● object

        ~~~~~~~~~~~~~~~~~~~

                                          <o> measuring device

What kind of photons an observer may measure in the scattered light?

If the system and the measuring device are fixed in the laboratory frame, then the observer will see all oscillation of fields having the same frequency f as the laser. He will see photons whose energy matches the photons in the laser beam.

But let us then make the measuring device to move at a constant speed v relative to the laboratory frame.

The waveform close to the object has a complex form. The Fourier decomposition of the signal received by the measuring device will have many frequencies, some of them very high.

The observer may see a photon which has much higher energy than the photons emitted by the laser.

How do we interpret this? If the laser sends, say, one photon of energy E in a second, how can the observer see a photon with a much higher energy E'? Is energy conserved?

Seeing a high-energy photon has an extremely low probability, though.

We have discussed the analogous problem in the context of an accelerating laser, or an accelerating observer. There is no obvious way to match individual photons in emission to photons in absorption. We have called this problem the "real Unruh effect".

In purely classical physics there is no problem. Energy in a wave of frequency f can be transformed to energy in a wave of much higher frequency f'. It is quantization which poses the problem here.

The length scale problem is ubiquitous

We have been studying bremsstrahlung in the past weeks. The "length scale" problem is that an electron passing very close to a proton should classically emit photons of very high energy. What wipes away these photons? Our new observation in this blog post shows that a similar problem exists in very mundane scattering of laser light.