Jesse

So you could use that to put an upper limit on the mass of a photon, right? The compton wavelength is h/mc, or (2.21*10^-42 / m) meters, and if the cosmic background radiation has traveled 13.7 billion light years to reach us, or about 1.295*10^26 meters, wouldn't that mean the mass of a photon cannot be larger than 2.21*10^-42/1.295*10^26 = 1.7*10^-68 kg? In contrast, according to this article the upper limit on the mass of a neutrino is 0.2 eV = 3.2*10^-20 J = 3.6*10^-37 kg. Is there any chance we could ever detect the mass of a photon if it was that tiny? If time and space end up being quantized in quantum gravity, would mass be quantized as well, giving a minimum possible mass for any particle?

edit: according to this page, the current upper limit on photon mass is 4*10^-51 kg, so obviously my calculation is not correct...

lpetrich

Observable traveling electromagnetic waves, like the Cosmic Microwave Background, are no counterargument, because photon mass would not keep them from propagating.

And their wavelengths are MUCH shorter than the length scales of some large magnetic fields, like those of Jupiter and the Milky Way, so they offer a relatively large upper limit to the photon's mass.

So we are reduced to arguing over what length scale the Milky Way's magnetic field is produced over.

Jesse

lpetrich:
Observable traveling electromagnetic waves, like the Cosmic Microwave Background, are no counterargument, because photon mass would not keep them from propagating.

Wouldn't it limit the distance they could propogate, though? I had assumed that's what it meant for the electromagnetic force to have a finite range. If we lived in a steady-state universe with an infinite past, wouldn't a finite range for the electromagnetic force mean there would be some distance beyond which no light would reach us?

I think I see what I did wrong in my calculation-- the "m" in the Compton wavelength formula l = h/mc is supposed to be the relativistic mass, not the rest mass, right? I assume that must be true since photons have finite Compton wavelength.

lpetrich:
And their wavelengths are MUCH shorter than the length scales of some large magnetic fields, like those of Jupiter and the Milky Way, so they offer a relatively large upper limit to the photon's mass.

So when we say a given force has a limited range, that range still depends on the wavelength of the force-carrying particles rather than just being a set distance? For instance, the weak force is said to be carried by W and Z particles--would W and Z particles with different wavelengths (due to different relativistic masses) have different ranges? Would the absolute upper limit to the range be when the particles were at rest, since that's when the relativistic mass is minimized? (and I probably need to go review some physics here, but what's the physical meaning of the 'wavelength' of a particle at rest? For the De Broglie wavelength h/mv, 'wavelength' is the actual distance travelled between oscillations, but this will not be true of the Compton wavelength h/mc if the velocity is less than c...)

lpetrich

I'll have to use some mathematics here. The equation of motion of a spin-0 particle with field P is

d^2(P)/dt^2 - d^2(P)/dx1^2 - d^2(P)/dx2^2 - d^2(P)/dx3^2 + m^2*P = S

where hbar = c = 1, t,x1,x2,x3, are space-time, m is the rest mass, and S is the source term. Though no spin-0 particles have ever been observed, nonzero-spin particles behave in a very similar fashion.

To find out how a wave behaves, plug in a generic wave:

P = exp(i*k*x1 - i*w*t)

where k = 2*pi/(wavelength) is the momentum and w = 2*pi*(frequency) is the energy of each quantum.

The Compton wavelength one derives by setting k = m; it is (2*pi)/m in hbar = c = 1 units, or h/(m*c) in more familiar units.

This results in the equation w^2 = k^2 + m^2 or E^2 = p^2 + m^2 the familiar relativistic momentum-energy relation. Thus, mass will not stop a free particle from propagating, which is what is observed for every massive particle known. And massive photons will behave in the exact same way.

Now consider the behavior of a static field with a source. This is straightforward to calculate:

P(x) = Integral(G(x-x')*S(x') dx')

where G(x) is - exp(-m*r)/r the Yukawa potential, whose derivative is

exp(-m*r)*(1/r^2 + m/r)

-- a screened inverse-square force. The exponential-falloff distance is 1/m. This is hbar/(m*c) in ordinary units, or 1/(2*pi) * the Compton wavelength as usually defined.

This is what justifies looking for photon mass limits by studying long-range magnetic fields.

lpetrich

As to the weak interactions, the W and the Z do not become free particles unless one imparts their rest masses of energy to the system -- 80 GeV for the W and 90 GeV for the Z.

At relatively low energies, however, the W and the Z act much like static fields, complete with an exponential falloff in interaction strength with increasing distance, with the falloff distance being hbar/(m(W)*c) and hbar/(m(Z)*c) respectively. In fact, at low energies, the W and Z are usually treated as contact interactions, since that falloff distance is very small compared to typical distances.

Jesse

OK, thanks--so basically it's the ranges of static fields that are limited by force-carrying particles with nonzero rest mass, but waves can still propogate arbitrarily far. I think I'd have to learn some quantum field theory to understand the details, but this clears most of what I was confused about.

Answerer

I recalled someone here once told me that weak and strong force actually had an infinite range but was too weak in comparsion to electromagnetic force and caused it to be undetectable. Although I disagree with him, I can't really say that he is wrong, what do you guys think?

lpetrich

The strong and weak interactions have exponential falloffs, which means that they have nominally infinite ranges -- but effectively finite ranges, about 10^-14 m for the strong force and 10^-17 m for the weak force.