What is the mass of a photon?

This question falls into two parts:

Does the photon have mass?  After all, it has energy and energy is equivalent to mass.

Photons are traditionally said to be massless.  This is a figure of speech that physicists use to describe something about how a photon's particle-like properties are described by the language of special relativity.

The logic can be constructed in many ways, and the following is one such.  Take an isolated system (called a "particle") and accelerate it to some velocity v (a vector).  Newton defined the "momentum" p of this particle (also a vector), such that p behaves in a simple way when the particle is accelerated, or when it's involved in a collision.  For this simple behaviour to hold, it turns out that p must be proportional to v.  The proportionality constant is called the particle's "mass" m, so that p = mv.

In special relativity, it turns out that we are still able to define a particle's momentum p such that it behaves in well-defined ways that are an extension of the newtonian case.  Although p and v still point in the same direction, it turns out that they are no longer proportional; the best we can do is relate them via the particle's "relativistic mass" m_rel.  Thus

           p = m_relv .

When the particle is at rest, its relativistic mass has a minimum value called the "rest mass" m_rest.  The rest mass is always the same for the same type of particle.  For example, all protons, electrons, and neutrons have the same rest mass; it's something that can be looked up in a table.  As the particle is accelerated to ever higher speeds, its relativistic mass increases without limit.

It also turns out that in special relativity, we are able to define the concept of "energy" E, such that E has simple and well-defined properties just like those it has in newtonian mechanics.  When a particle has been accelerated so that it has some momentum p (the length of the vector p) and relativistic mass m_rel, then its energy E turns out to be given by

           E = m_relc² ,   and also    E² = p²c² + m²_restc⁴ .           (1)

There are two interesting cases of this last equation:

1. If the particle is at rest, then p = 0, and E = m_restc².
2. If we set the rest mass equal to zero (regardless of whether or not that's a reasonable thing to do), then E = pc.

In classical electromagnetic theory, light turns out to have energy E and momentum p, and these happen to be related by E = pc.  Quantum mechanics introduces the idea that light can be viewed as a collection of "particles": photons.  Even though these photons cannot be brought to rest, and so the idea of rest mass doesn't really apply to them, we can certainly bring these "particles" of light into the fold of equation (1) by just considering them to have no rest mass.  That way, equation (1) gives the correct expression for light, E = pc, and no harm has been done.  Equation (1) is now able to be applied to particles of matter and "particles" of light.  It can now be used as a fully general equation, and that makes it very useful.

Is there any experimental evidence that the photon has zero rest mass?

Alternative theories of the photon include a term that behaves like a mass, and this gives rise to the very advanced idea of a "massive photon".  If the rest mass of the photon were non-zero, the theory of quantum electrodynamics would be "in trouble" primarily through loss of gauge invariance, which would make it non-renormalisable; also, charge conservation would no longer be absolutely guaranteed, as it is if photons have zero rest mass.  But regardless of what any theory might predict, it is still necessary to check this prediction by doing an experiment.

It is almost certainly impossible to do any experiment that would establish the photon rest mass to be exactly zero.  The best we can hope to do is place limits on it.  A non-zero rest mass would introduce a small damping factor in the inverse square Coulomb law of electrostatic forces.  That means the electrostatic force would be weaker over very large distances.

Likewise, the behavior of static magnetic fields would be modified.  An upper limit to the photon mass can be inferred through satellite measurements of planetary magnetic fields.  The Charge Composition Explorer spacecraft was used to derive an upper limit of 6 × 10^-16 eV with high certainty.  This was slightly improved in 1998 by Roderic Lakes in a laboratory experiment that looked for anomalous forces on a Cavendish balance.  The new limit is 7 × 10^-17 eV.  Studies of galactic magnetic fields suggest a much better limit of less than 3 × 10^-27 eV, but there is some doubt about the validity of this method.

Number 625 #2, February 19, 2003 by Phil Schewe, James Riordon, and Ben Stein

A New Limit on Photon Mass

A new limit on photon mass, less than 10^-51 grams or 7 x 10^-19 electron volts, has been established by an experiment in which light is aimed at a sensitive torsion balance; if light had mass, the rotating balance would suffer an additional tiny torque. This represents a 20-fold improvement over previous limits on photon mass.

Photon mass is expected to be zero by most physicists, but this is an assumption which must be checked experimentally. A nonzero mass would make trouble for special relativity, Maxwell's equations, and for Coulomb's inverse-square law for electrical attraction.

The work was carried out by Jun Luo and his colleagues at Huazhong University of Science and Technology in Wuhan, China (junluo@mail.hust.edu.cn, 86-27-8755-6653). They have also carried out a measurement of the universal gravitational constant G (Luo et al., Physical Review D, 15 February 1999) and are currently measuring the force of gravity at the sub-millimeter range (a departure from Newton's inverse-square law might suggest the existence of extra spatial dimensions) and are studying the Casimir force, a quantum effect in which nearby parallel plates are drawn together. (Luo et al., Physical Review Letters, 28 February 2003)

Photon Mass Gets a Boost