C represents the speed limit of the universe rather than the speed of light

[Sam Gralla]

Proca theory is presented confusingly in almost every reference, and your quote,

"Phase invariance (U(1) invariance) is lost in Proca theory, but the Lorentz gauge is automatically held, and this is indispensable to charge conservation, i.e. the Lorentz condition becomes a condition of consistency for the Proca field."

is a prime example. The logic here is totally backwards and in fact I would say simply incorrect. In standard Maxwell theory charge conservation is guaranteed--you cannot find solutions to Maxwell's equations with non-conserved sources. (This can be viewed as a consequence of the gauge symmetry, if you like.) In proca theory charge conservation is not guaranteed by the equations--you can perfectly well find solutions with non-conserved source. However, if you demand that charge be conserved, then you find that the "Lorentz gauge condition" holds. (I put it in quotes because there is no notion of gauge in proca theory. The condition is just an extra equation that comes out if you demand charge conservation.) You can see how the above quote has the logic backwards. Also I think it is crazy to call the Lorentz gauge condition a "consistency condition" for the field. The "Lorentz gauge condition" enforces charge conservation, but there is nothing inconsistent about not conserving charge! As far as I know there are no consistency problems with the proca equations, with or without the "Lorentz gauge condition".

 

[Demystifier]

Proca theory is presented confusingly in almost every reference, and your quote,

"Phase invariance (U(1) invariance) is lost in Proca theory, but the Lorentz gauge is automatically held, and this is indispensable to charge conservation, i.e. the Lorentz condition becomes a condition of consistency for the Proca field."

is a prime example. The logic here is totally backwards and in fact I would say simply incorrect. In standard Maxwell theory charge conservation is guaranteed--you cannot find solutions to Maxwell's equations with non-conserved sources. (This can be viewed as a consequence of the gauge symmetry, if you like.) In proca theory charge conservation is not guaranteed by the equations--you can perfectly well find solutions with non-conserved source. However, if you demand that charge be conserved, then you find that the "Lorentz gauge condition" holds. (I put it in quotes because there is no notion of gauge in proca theory. The condition is just an extra equation that comes out if you demand charge conservation.) You can see how the above quote has the logic backwards. Also I think it is crazy to call the Lorentz gauge condition a "consistency condition" for the field. The "Lorentz gauge condition" enforces charge conservation, but there is nothing inconsistent about not conserving charge! As far as I know there are no consistency problems with the proca equations, with or without the "Lorentz gauge condition".

After seeing your excellent points on another topic, I came to see your other comments on the Forum (and this one is the only at the moment). Again, your comments are excellent. You are new here, but I hope you will stay here for a longer time a contribute a lot.

 

[homology]

Proca theory is presented confusingly in almost every reference, and your quote,
.

Thanks for clearing that up. I've only just recently even heard of the Proca mass and have started reading some papers on it. That being said I'm a total noob.

Welcome to PF forums :)

 

[Sam Gralla]

Thanks for the welcoming comments, demystifier and homology. I stumbled across these forums a few days ago. They seem well moderated and very active, and it's a great idea to have such a place where people can discuss physics. So, I thought I'd join in.

 

[Neandethal00]

I think I probably am being a bit confusing. So let's make it concrete. If we find the photon has a non-zero mass, then the speed of light would not be the same as the maximum speed limit. The number 'c' in E = mc^2, would be the maximum speed limit, not the speed of light because as far as I know, you can derive special relativity independently of electrodynamics.

First of all, one unwritten universal law is 'everything in this universe has a limit', so does the speed of anything. But the question is 'how do we know maximum speed limit is c?' It is just 'one theory' that tells us c is the highest speed of things. A theory.

If we use E=mc², and assume c is the highest speed of any object, doesn't it also mean 'the rest energy of mass m is twice its maximum kinetic energy?'

 

[_PJ_]

First of all, one unwritten universal law is 'everything in this universe has a limit', so does the speed of anything. But the question is 'how do we know maximum speed limit is c?' It is just 'one theory' that tells us c is the highest speed of things. A theory.

If we use E=mc², and assume c is the highest speed of any object, doesn't it also mean 'the rest energy of mass m is twice its maximum kinetic energy?'

You're assuming rest mass is equivalent to obserrved mass.

 

[DrGreg]

If we use E=mc², and assume c is the highest speed of any object, doesn't it also mean 'the rest energy of mass m is twice its maximum kinetic energy?'

No. You are assuming the formula for kinetic energy is ½mv². But it isn't. That's an approximation that's valid only at low speeds. The accurate formula is

\left( \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} - 1\right)mc^2​

There is no upper limit on kinetic energy.

(I'm using the convention that the vast majority of physicists follow nowadays, that "mass" m means "rest mass".)