Massless free field equation -> Maxwell's eqn.

[lark]

Massless free field equation --> Maxwell's eqn.

The massless free field equation is supposed to turn into the empty space Maxwell's equations for spin 1 (like a photon).
But, in the book I'm using, Roger Penrose's "The Road to Reality", there seems to be a typo, because it's not quite working out. Almost but not quite.
Can someone tell me what the problem is?
See http://camoo.freeshell.org/33.24quest.pdf"
for details.
(sorry if you find it inconvenient to click on the link, but I'm not going to rewrite everything into the forum's Latex).
Laura

 

[reilly]

Could you explain the notation, please.
Regards,
Reilly Atkinson

 

[muppet]

I can't help directly... but if this is one of his problems have you tried Penrose's solutions online?

 

[lark]

I can't help directly... but if this is one of his problems have you tried Penrose's solutions online?

I'm the only one who has posted solutions to the problems in the 2nd half of the book!
Laura

 

[lark]

Could you explain the notation, please.
Regards,
Reilly Atkinson

It won't help if I explain it, I could be misinterpreting something and that's part of the question. I'm sure my calculations are OK.
I was hoping somebody would know how the massless free field equation translates into Maxwell's equations, concretely.

 

[Avodyne]

I think it's a little more complicated than what you're doing. First of all, there are two psi fields, one with two unprimed indices and one with two primed indices; each field must be treated separately.

See sections 34 and 35 of Srednicki's field theory book for an introduction to this notation (but with conventions that probably don't match Penrose's). The Srednicki book is available free online in draft form at his web page.

 

[lark]

I think it's a little more complicated than what you're doing. First of all, there are two psi fields, one with two unprimed indices and one with two primed indices; each field must be treated separately.

I tried the one with two unprimed indices. It's supposed to work out to Maxwell's equations. It does almost, but not quite. That's the problem. Can anyone tell me what the typo is, that if fixed, would make it come out to be Maxwell's equations?
The worked out version is in http://camoo.freeshell.org/33.24.pdf and for example, eqn 7 minus eqn 5 in there, is $$\partial E_x/\partial x + \partial E_y/\partial y +\partial B_z/\partial z=0$$. It wants to be $$\nabla\cdot E=0,$$ but it isn't quite. I'm sure I'm not making an algebra mistake, the problem is mis-stated or something.
Laura

 

[George Jones]

But, in the book I'm using, Roger Penrose's "The Road to Reality", there seems to be a typo, because it's not quite working out.

After looking at page 323 of Spinors and Space-Time V1 by Penrose and Rindler, it looks like it should be $\psi_{01} = -C_3$, not $\psi_{01} = -iC_3$. This seems to give stuff like divergences and components of curls, but I haven't worked through the details.

 

[lark]

After looking at page 323 of Spinors and Space-Time V1 by Penrose and Rindler, it looks like it should be $\psi_{01} = -C_3$, not $\psi_{01} = -iC_3$. This seems to give stuff like divergences and components of curls, but I haven't worked through the details.

I made that change. Now I get $$\nabla\cdot E=0$$ and $$\nabla\cdot B=0$$. But the curl equations have the sign exactly reversed! So that if you reverse the sign of $$t$$, it works out right.
I understand now, I think. I got the sign of $$t$$ in $$\nabla^a$$ wrong, it's actually $$-\partial /\partial t+\partial /\partial x+\partial /\partial y+\partial /\partial z$$ not $$+\partial /\partial t+\partial /\partial x+\partial /\partial y+\partial /\partial z$$.
$$Laura$$

 

[George Jones]

I got the sign of $$t$$ in $$\nabla^a$$ wrong,

Maybe the sign of t was okay, and the signs of the spatial derivatives were wrong. I think Penrose uses the + - - - convention for the metric, so raising the standard partials in $$\nabla_a$$ to $$\nabla^a$$ would put minus signs in front of the spatial derivatives.

 

[lark]

Maybe the sign of t was okay, and the signs of the spatial derivatives were wrong. I think Penrose uses the + - - - convention for the metric, so raising the standard partials in $$\nabla_a$$ to $$\nabla^a$$ would put minus signs in front of the spatial derivatives.

Well, whatever. It doesn't matter in this case
I put the whole calculation in http://camoo.freeshell.org/33.24.pdf"
Thanks,
$$Laura$$