# Proca Equations and nonzero mass of photon

#### helppls

Hi. I'm a 3rd year undergraduate and have to do a 15 page assignment about Proca equations and how those lead to an effective mass of photon that isn't zero. I don't even know where to begin. Can anyone recommend me something about this issue?

#### topsquark

Forum Staff
Hi. I'm a 3rd year undergraduate and have to do a 15 page assignment about Proca equations and how those lead to an effective mass of photon that isn't zero. I don't even know where to begin. Can anyone recommend me something about this issue?
I Googled this and only found an argument for an imaginary mass. Is this what you are talking about? Or are you solving the Proca equation with a photon mass to a first approximation?

-Dan

#### Farsight

Can you provide more information? I'm not sure what you have to do.

Meanwhile, I can give you a few bits and pieces that might prove useful. Consider a photon propagating at c. We say it's massless. As it happens, it has a non-zero "active gravitational mass" and a non-zero "inertial mass", but when we say mass without qualification, we usually mean rest mass. And the photon has no rest mass because it's never at rest.

However there is a trick, where you catch the photon in a gedanken mirror-box. The photon isn't actually at rest, but it's bouncing back and forth so it's effectively at rest. And it increases the mass of the system. The inertia of the box-system is increased. The box is harder to move because the photon is in there. Then when you open the box it's a radiating body that loses mass, just like Einstein's E=mc² paper. (As an aside, look at the last line: "If the theory corresponds to the facts, radiation conveys inertia between the emitting and absorbing bodies". That's why the photon has a non-zero inertial mass. Which is actually a measure of energy.)

Now here's the important thing: If the photon is moving at c, none of its energy-momentum is exhibited as mass. If the photon is effectively moving at a speed of zero, all of its energy-momentum is exhibited as mass. And there's a sliding scale in between. If you slow the photon down a little, some of its energy-momentum is exhibited as "effective mass". (Google it). So if your photon enters a glass block, it slows down a little and the mass of the glass block increases a little. So the photon is behaving a little like a massive boson. See this though. And note that you have to do the work for yourself. If you get too much help, that's not good. But ask around anyway. Ah, I see Kyle Kanos was a bit curt!

#### Farsight

Hmmn. When I google on helppls Proca I can see there's a thread on physicsforums, but when I click on it it says the requested thread could not be found.

#### helppls

Thanks very much for answering. I'm sorry I couldn't be more specific when i posted this, but had only had a very short and dense conversation with my professor and could only remember that I had to show that the potential propagation equations could be rewritten with a friction term - the proca equations- and that somehow related with the fact that the photon could be thought as having a nonzero rest mass.

After some research I understand now that considering a massive photon has consequences like the coulomb potential becoming a yukawa potential and the potential propagation equations becoming the proca equations. Still, the assignment is derive the proca equations by considering a massive photon and use them to get an estimate of the rest mass of the photon. I found a book where the authors do that considering the Meissener effect, but i'm not sure whether that's the only way to do it. And I'm still searching for a derivation of the proca equations

The physics forum thread was deleted because it was considered "Misplaced Homework" sorry about that :/

#### topsquark

Forum Staff
The physics forum thread was deleted because it was considered "Misplaced Homework" sorry about that :/
Yeah they're *sses about that. That's why I don't go there anymore.

Thanks very much for answering. I'm sorry I couldn't be more specific when i posted this, but had only had a very short and dense conversation with my professor and could only remember that I had to show that the potential propagation equations could be rewritten with a friction term - the proca equations- and that somehow related with the fact that the photon could be thought as having a nonzero rest mass.

After some research I understand now that considering a massive photon has consequences like the coulomb potential becoming a yukawa potential and the potential propagation equations becoming the proca equations. Still, the assignment is derive the proca equations by considering a massive photon and use them to get an estimate of the rest mass of the photon. I found a book where the authors do that considering the Meissener effect, but i'm not sure whether that's the only way to do it. And I'm still searching for a derivation of the proca equations
I have two possible responses for you, one of which you probably have. The easy way to do this is to derive the Proca equation from the Lagrangian. Here's a source with the steps worked out. (And yes, I took great pains to find it on PF. 8P )

The other way is much more difficult but more satisfying from a metaphysical stand point. The Proca equation can be derived from the (1, 0) + (0, 1) representation of the Lorentz group. If that is what you are looking for then I'll write it out for you. As soon as I find the *#&$^ text. I just found out that I've misplaced my QFT texts by Weinberg. I'll likely find them in a day or so, so just hang in there for a bit. -Dan #### Farsight helppls said: the assignment is derive the proca equations by considering a massive photon and use them to get an estimate of the rest mass of the photon. Ouch, sounds like mission impossible. IMHO mass is a combination of 1) how much energy is there and 2) how much less than c it's moving, and when you forget this, you get sucked into "non real solutions" aka fantasyland. I presume you've seen section 4 of this. "Implications of a massive photon: variation of c; longitudinal electromagnetic radiation and gravitational deflection; possibility of charged black holes; the existence of magnetic monopoles [21]; modification of the standard model [22], etc. An upper limit for the photon rest mass [23] is mγ 1 × 1049 g6 × 1017 eV. The concept of a nonzero rest mass graviton may be defined [24] in two ways: phenomenologically, by introducing of a mass term in the linear Lagrangian density, as in Proca electrodynamics, and self-consistently, by solving Einstein’s equations in the conformally flat case. The rest mass of the graviton was given in terms of the three fundamental constants: gravitational, Planck, and light velocity..." There are no magnetic monopoles or tachyons. A massless photon moving through space at c is not surrounded by a cloud of massive gravitons. And what's that about a variation in c? It varies anyway. See this. The "coordinate" speed of light varies with gravitational potential, and the massless photon stays massless. Maybe I ought to shut up now before you get too discouraged. But I'm interested to see how this works out for you. #### topsquark Forum Staff The other way is much more difficult but more satisfying from a metaphysical stand point. The Proca equation can be derived from the (1, 0) + (0, 1) representation of the Lorentz group. If that is what you are looking for then I'll write it out for you. As soon as I find the *#&$^ text. I just found out that I've misplaced my QFT texts by Weinberg. I'll likely find them in a day or so, so just hang in there for a bit.
I finally found my Weinberg. I can run you through the Lorentz group derivation now if you need it.

-Dan

#### Pmb

PHF Hall of Fame
I Googled this and only found an argument for an imaginary mass. Is this what you are talking about? Or are you solving the Proca equation with a photon mass to a first approximation?

-Dan
Find the Proga Lagrangian. In it there is a term for the proper mass of the photon. Using the Proca-Lagrangian you can arrive at a set of EM equations which are different than the ones you're already familiar with.

#### topsquark

Forum Staff
Find the Proga Lagrangian. In it there is a term for the proper mass of the photon. Using the Proca-Lagrangian you can arrive at a set of EM equations which are different than the ones you're already familiar with.
What does "proper mass of the photon" mean? If we like we can solve the Proca equation with a massive particle, but it won't act like a photon because it isn't a photon. One of the immediate consequences is that the massive "photon" can't travel at c.

There are only three particles that spring to mind when discussing the Proca equation: The W's and Z of the weak nuclear force.

Beyond that I really can't say.

-Dan