## On the photon frequency

We know photons exist. We also know how to manipulate photons collectively. Our technology has now advanced to the level where we can even manipulate the individual photons. But, if you ask me “what is a photon?” I cannot answer that question. No physicist can provide a satisfactory answer at this point. The answer depends on the theory we use.

The situation is very similar with electron. We know electrons exist. We have known this for more than a century. We manipulate electrons, we design and manufacture electronic devices. The modern life would be unthinkable without the manipulation of electrons. But, we really don’t know what an electron is. Electron has 3 basic properties: charge, spin and mass. We can measure these properties but we don’t understand them either. Again, the answer to the question “what is an electron?” depends on the theory we use.

I delayed the release of this post for more than a week. As I worked on this article I realized how little I know on the subject of photons and how controversial the concept of photon is among the physicists and philosophers of science. Yes, there has been some progress since the days of Einstein but photon is still controversial.

Observational facts about the photon

• Photons go through each other. Photons do not interact with each other.
• The speed of the photon emitting particle has no effect on the speed of the emitted photon. For example, the photon emitted from a proton moving almost at the speed of light in the beam pipe of the Large Hadron Collider (LHC) will be slightly faster than the emitting proton, not twice as fast, as you would naively expect.
• The speed of photons in empty space is constant in all frames of reference.
• Let’s imagine that you have the ability to shrink yourself and travel on top of that proton which is moving at a speed very close to the speed of light and observe the release of a photon from that proton. The emitted photon will move away from you at the speed of light. If your friend is observing the same phenomenon from the control room he will think that the photon is moving at the same speed as you. The classical/naive thinking says that the observer traveling on top of that proton should see a stationary photon. But, that’s not what happens. You (traveling on top of the proton traveling almost at the speed of light) will observe that the released photon is moving away from you at the speed of light.
• There is no such thing as stationary photon. Photons always move at the speed of light in empty space relative to the frame of reference where the measurement is made.
• All observers in all frames of reference (a moving particle, control room, Earth or a distant planet) will always measure the exact same speed for the emitted photons as measured in that frame of reference. This is very counter-intuitive indeed.
• Using this experimentally established fact Einstein developed the theory of Special Relativity. The most important result of his theory is the law of the universal speed limit which claims that no physical object or information can move faster than light in empty space. So far, no counter-example has been found.
• Photons interact with the gravitational force. The heavy astronomical objects bend the paths of photons. Curiously, photons are not affected by the other forces of nature. Electromagnetic, weak nuclear and strong nuclear forces have no effect on photons. When we create a high energy density in a very small volume by smashing protons against each other in modern particle colliders such as the LHC various particles and photons will emerge from that vortex of energy. In that sense, you can argue that photons are interacting with the forces other than gravitation but a photon carries no charge of any kind so this argument is not convincing. You can also argue that with gamma ray photons (very energetic, very particle-like photons) we can hit other elementary particles and there is an interaction there. I invite other physicists to clarify this aspect of photons.
• To repeat, a photon carries no charge (no electrical charge, no color charge, no other charges). This is why it is not affected by the electromagnetic, weak nuclear and strong nuclear forces.
• Photons have zero rest mass. Particle physicists do not mention the word “rest” anymore. They simply say “mass.” Photon is a massless particle.
• Massless particles always move at the speed of light in empty space. Remember this.
• A photon is a spin=1 particle. See the section below for more on this.
• A photon cannot be linearly polarized. See the next section for more details on this.
• A photon can coast in empty space without any dissipation until it is absorbed by an atom.
• A single photon cannot split into 2 photons in empty space. If you define “split” as the appearance of a second or third photon then there are possibilities in certain crystals.

Spin and Polarization

Whenever a photon is absorbed by an object, a small amount of angular momentum (same amount independent of the energy of the photon) is imparted to the object. This is a tiny spin. The imparted spin can be either +1 or -1 $\hbar$ unit of angular momentum. A photon that imparts  $-\hbar$ unit of spin is known as left-handed photon. Similarly, a photon that imparts $+\hbar$ unit of spin is known as right-handed photon.

A classical electromagnetic wave consists of many photons. When we say that the classical wave is right circularly polarized we understand that this is a stream of photons having more right-handed photons than left handed photons. The overall angular momentum imparted to the absorbing object will be the net angular momentum of photons contained in that stream.

When we say that the classical electromagnetic wave is linearly polarized we are referring to the fact that there are almost equal number of right-handed and left-handed photons in that particular stream.

Well…it is not that simple! The classical electromagnetic waves are generated by our electronic and electrical devices everyday. Think of radio broadcasts or TV broadcasts in the air or in the cable. Can we really say that those waves consist of photons? The answer is yes but the mathematical demonstration is not easy. Roy J. Glauber‘s work shed light on this topic. Glauber received the 2005 Nobel Prize in physics.

A single photon cannot be linearly polarized.

A single photon is either right-handed or left-handed.

Right-handed: the electric field component of the photon is rotating in the direction of your fingers when the thumb of your right hand is pointing in the direction of propagation.

Left-handed: the electric field component of the photon is rotating in the direction of your fingers when the thumb of your left hand is pointing in the direction of propagation.

Remember that in a circularly polarized classical electromagnetic wave the electric field component E and the magnetic field component B are 90° out of phase with respect to each other. Since the left-handedness or right-handedness of a photon refers to its circular polarization the same property is found in a photon as well. Namely, the E and the B components of a photon are 90° out of phase. This means that when E peaks B is at a minimum; when B peaks E is at a minimum.

An important point

In empty space, the speed of propagation of light waves is independent of the wave frequency. This implies that, as far as the electromagnetic waves are concerned, there is no substructure in empty space. Another consequence is that the light waves have no longitudinal component. Since photon is an electromagnetic wave in essence the same characteristics are observed with the photons as well. The speed of a photon in empty space is the the same as the speed of any electromagnetic wave-packet in empty space. The speed of a photon in empty space is independent of its spectrum (I will discuss the confusion regarding multiple frequencies versus single frequency below). Also, a photon does not have any longitudinal component. These characteristics are common to classical electromagnetic waves and photons.

How are the photons created?

Photons are released from a charged particle when the charged particle goes through either a classical decelerating motion or a quantum mechanical transition.

Digression: quantum mechanical transitions are instantaneous therefore conceptually problematic from the point of view of Special Relativity. Quantum mechanical transitions do not obey the universal speed limit. In other words, the quantum mechanical transition (wavefunction collapse) happens faster than light. This is not possible according to the Special Relativity theory, but obviously, it is happening routinely in this universe. You might want to take a look at “Tension between Quantum Mechanics and Relativity.” I will write more about this subject in the future.

1. Spontaneous radiation: electrons orbiting the nucleus of an atom make spontaneous transitions to a lower energy level. The energy difference is released as a single photon. A single photon is released.  1 photon. This is very important.

Similarly, when a photon is absorbed by the atom one of the electrons will jump to a higher energy state.

2. Annihilation: when electron and positron collide 2 photons are released. In general when a charged fermion collides with its anti-fermion the total energy – the energy contained in the rest masses of these fermions and their kinetic energies – is converted into 2 photons.

3. Bremstrahlung: when a charged particle is forced to follow a curved path in an external magnetic field, the curved path forces the charged particle to release some of its energy as photons (many photons)

4. Cherenkov radiation:  when a charged particle passes through a dielectric medium at a speed greater than the speed of light in that medium the charged particle releases photons (many photons) (remember the speed of light is different in dielectric media, the universal speed limit is the speed of light in empty space)

What is a photon?

Classical physics (Special Relativity is part of classical physics): photon is an electromagnetic wave-packet similar to an electromagnetic pulse. Wave-packets necessarily contain many frequencies (otherwise a wave-packet cannot be formed).

Quantum Mechanics: photon is not a wave-packet because a photon emitted by spontaneous emission is characterized by a single frequency. According to physics a photon cannot be localized. See more on this below.

Quantum Field Theory (QFT)

a) Photon is a ripple in the universal electromagnetic quantum field

b) Photon mediates the electromagnetic force: this implies that a photon has to be emitted and absorbed. The concept of a photon coasting in the universe forever is not acceptable by QFT.

c) According to QFT forces mediated by particles of zero rest-mass have infinite range. Photon has zero rest-mass, that’s why electromagnetic force has infinite range.

d) No comment on the frequency but the assumption of multiple frequencies is implicit in the concept of a field quantum.

e) The so-called “virtual photon” discussed in the context of QFT is not a photon. The “virtual photon” is a misnomer.

Non-spatial and non-local characteristics of photons

Multiple bosons (force carriers) (spin=1 particles)  such as photons can occupy the same space. I mentioned earlier that photons do not interact with each other or any other force except gravitation. This is the reason why many of them can occupy the same space. Contrast this to the situation with fermions (constituents of matter) (spin=1/2 particles).  No two spin=1/2 particles, such as electrons, can occupy the same quantum state

Massless spin=1 particles (gauge bosons) have a non-spatial characteristic at the fundamental level. This has implications for non-locality and the constancy of the speed of light in vacuum.

Can we localize a photon?

According to Quantum Mechanics (a mathematical formalism) photon cannot be (spatially) localized. The reason is a formal mathematical declaration saying that in Quantum Mechanics photon does not have a position wavefunction. But, don’t believe every thing you hear. Always approach formal/mathematical explanations with caution. There are many assumptions and ignored facts behind mathematical formalisms. First of all, there are many everyday examples of (spatially) localized photons:

• a photon can be spatially localized in an optical cavity (laser cavity, for example)
• open/close a small circular aperture of a light source, whatever photons released from that aperture will be finite in duration and spatially localized to one direction. The uncertainty on the position of the photon will increase in time (light spreading)…but..the photon is localized within these uncertainties.
• the photon in an optical fiber is spatially localized as well.

So, it is not true to say that a photon cannot be spatially localized. When a photon is spatially or temporally localized it is no longer a single-frequency electromagnetic vibration. Multiple-frequency is problematic in Quantum Mechanics. The quantum mechanical photon is supposed to have a single frequency.

QFT solves this problem in a similar fashion as it solves the other problems related to wavefunction collapse in Quantum Mechanics. QFT says that photons are ripples in the universal electromagnetic quantum field and therefore they are local by definition. The multiple-frequency is implicit in the definition of a field quantum. In other words, QFT compromises on the single-frequency assumption.

Crux of the problem

The crux of the problem can be traced to ontology. What is real and what is incidental? There are 3 positions.

1. Particle is real. We describe the probabilistic behavior of the particle by its quantum mechanical wave function. The Copenhagen interpretation of Quantum Mechanics is in this category. The Feynman approach to Quantum Mechanics is in this category as well.
2. Quantum field is real, particles are incidental. We describe particle-like phenomena by saying that quantum field is capable of creating or destroying particles at any place any time.
3. Equal ontological status for particle and wave. This approach can be described as particle-wave duality. This concept has been popularized but it has had very little impact on actual physics research.

So, you can take one of these 3 philosophical positions. You can say that photon is a particle then you have to have a quantum mechanical wavefunction for the photon. This is problematic. Photon wavefunction is not well defined. People are still trying to clarify this.

Or, you can say that the universal electromagnetic quantum field is real and photons are just ripples on this field and therefore incidental. This approach has been more successful.

Or, you can say that photon has both particle and wave properties. This sounds very dialectic and therefore appealing but in real physics research not very useful.

Photon Wavefunction

Where is that quantum?

No place for particles in relativistic quantum theories

Photon localization barrier can be overcome

The “field collapse” point of view