Re: Article: A Century of Einstein

From: Gregory L. Hansen (glhansen_at_steel.ucs.indiana.edu)
Date: 09/02/04 

Date: Thu, 2 Sep 2004 01:26:35 +0000 (UTC)

In article <f6tZc.16115$D7.11693@news-server.bigpond.net.au>,
Bill Hobba <bhobba@rubbish.net.au> wrote:
>
>"Gregory L. Hansen" <glhansen@steel.ucs.indiana.edu> wrote in message
>news:ch4me7$e27$1@hood.uits.indiana.edu...
>> In article <ledZc.15367$D7.12100@news-server.bigpond.net.au>,
>> Bill Hobba <bhobba@rubbish.net.au> wrote:
>> >
>> >"Gregory L. Hansen" <glhansen@steel.ucs.indiana.edu> wrote in message
>> >news:ch387q$v1i$4@hood.uits.indiana.edu...
>>
>> >>
>> >> Um... do we not have charge invariance if we have a non-central F=F(v),
>v
>> >> the velocity of the test particle? I suppose we'd have to have q->Q(q)
>> >> where Q is some tensor, and q the coupling constant.
>> >
>> >I think charge invariance is more or less assumed in making sense of the
>> >Lorentz force law.
>>
>> There's lots of ways of looking at things, and it's useful to be able to
>> do that.
>>
>> I don't see how the Lorentz force law, by itself, requires charge
>> conservation. If charge isn't conserved, you could suppose destroying
>> charge just creates a weakening E and probably a B, which are just
>> plugged into the force law.
>
>Sorry by not making myself claer. What I meant is we can reverse the
>arumnet that derives the Lotentz force law and get charge invarience.
>
>>
>> But, as Bilge pointed out, it's the U(1) conserved current. He likes his
>> gauge symmetries.
>>
>
>So do I.

They're nifty, but for many purposes they're not very illuminating. A
Lagrangian that's invariant under a U(1) transformation is just math to
me. I don't know what a U(1) is when I see it; I know how to do a
reflection or a boost or a mirror image, but I don't know how to do a
U(1) except to close my eyes and work the math. But the argument of a
magnetic-like force arising from the principle of relativity has a more
immediate physical picture, and it's also more general in that it applies
to Galilean-invariant systems, to SU(2)_L and SU(3) symmetries, and
anything else that could be written down. For purposes of illuminating
those types of universal relationships in physics, one particular gauge or
another just isn't as useful.

>
>>
>> An argument I'd seen is that if you get/spend some amount of energy to
>> create a charge, and so much to destroy it, you can get any arbitrarily
>> greater amount by moving it through an arbitrarily high potential
>> difference, then make a cyclical machine and violate the first law of
>> thermodynamics.
>
>Wait a minute - if you suddenly 'turn' on a charge then you have given it a
>certain potential energy by its relation to other nearby charges thus such
>would require energy in the first place.

What information would be available to the new charge, such that the
energy matters in its creation? Only an electric field and/or magnetic
field. Whether that's caused by a weak charge nearby, or a strong charge
far away, the new particle cannot know.

>
>> So charge conservation is related to energy conservation,
>> given some other assumptions of charges in fields. That's sort of related
>> to the gauge symmetry in that we can't require that the energy to create
>> or destroy a charge be qV since the V=0 point is arbitrary.
>
>Not as far as I can see. Every conservation law is related to symmetry in a
>systems lagrangian - charge conservation is related to gauge symmetry -
>energy conservation to time symmetry - they are inherently two different
>things.
>
>>
>> Charge conservation is also directly implied in Maxwell's equations. We
>> have
>>
>> div E = 4 pi rho
>>
>> curl B - 1/c dE/dt = 4 pi/c J
>>
>> Take the time derivative of the top equation, the divergence of the
>> bottom, substitute away the E term, and you get the equation of continuity
>> relating charge and current.
>
>Yep. But what property of Maxwell's equations makes it so? Noethers
>theorem tells us every conservation law is related to a symmetry of the
>lagrangian. It turns out gauge symmetry is what is responsible for charge

I think that's backwards. Every symmetry of the Lagrangian has a related
conservation law, but not every conservation law needs a symmetry of the
Lagrangian.

If we give light a mass, then Proca's version of Maxwell's equations will
no longer be gauge symmetric, and Coulomb's law -> V=exp(-mr)/r. Is
charge still conserved?

I tried to work that out today, starting from Jackson's equations of
motion in four-vector form. And, if I can recall it without reworking it,
I got something like

  d(rho)/dt + div J = c/(4*pi) m^2 (dV/dt + 1/c div A)

The right hand side isn't explicitly zero, but the term in parentheses is
zero if we use the Lorentz gauge, which Jackson was. But I'm not sure we
can just set that to zero now. Greiner says the massive spin-1 field will
automatically satisfy the Lorentz condition. I suppose that means it will
satisfy the Lorentz condition and no other gauge? Anyway, setting that to
zero, charge is conserved, but I'd have to think some more before I can
decide whether or what kind of Noether current it comes from. 

-- 
"Don't try to teach a pig how to sing.  You'll waste your time and annoy 
the pig."

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