Classical Field Theory without Force

[Phrak]

I'm sorry, this line of reasoning is totally wrong, and reaches a completely wrong conclusion. I'm not sure where you got this from, but it is clear you do not understand what 'd' is and how to apply it. Sounds like you should do some more basic reading.

You keep saying this sort of thing, and I appreciate it greatly. Can you be more specific in what you find erronious?

What do you recommend for more basic reading?

Point 3. Could you please answer me about how you arrived at d(A^G) = dA^G - A^dG?

 

[WannabeNewton]

You keep saying this sort of thing, and I appreciate it greatly. Can you be more specific in what you find erronious?

What do you recommend for more basic reading?

Point 3. Could you please answer me about how you arrived at d(A^G) = dA^G - A^dG?

Hey Phrak if you still need to know the general product rule for differential forms is: given an n - form $$\Psi$$ and an m - form $$\Phi$$
$$d(\Psi \wedge \Phi) = d\Psi \wedge \Phi + (-1)^{n}\Psi \wedge d\Phi$$
If you want to prove it then just work with the general equation for the wedge product of two differential forms and the general equation for the exterior derivative of a differential form.

 

[Phrak]

Hey Phrak if you still need to know the general product rule for differential forms is: given an n - form $$\Psi$$ and an m - form $$\Phi$$
$$d(\Psi \wedge \Phi) = d\Psi \wedge \Phi + (-1)^{n}\Psi \wedge d\Phi$$
If you want to prove it then just work with the general equation for the wedge product of two differential forms and the general equation for the exterior derivative of a differential form.

Thanks for adding your thoughts, Wannabe N. Nice handle.

Your equation is

$$d(\Psi \wedge \Phi) = d\Psi \wedge \Phi + (-1)^{n}\Psi \wedge d\Phi \ .$$​

If the roles of $\Psi$ and $\Phi$ are reversed we have

$$d(\Phi \wedge \Psi) = d\Phi \wedge \Psi + (-1)^{m}\Phi \wedge d\Psi \ .$$​

This is an equally true statement; all this has done is to arbitrarily relabel free variables.

Commuting the wege product in each term and negating gives

$$d(\Psi \wedge \Phi) = \Psi \wedge d\Phi + (-1)^{m}d\Psi \wedge \Phi \ .$$​

Realigning terms to make comparison with your equation easy,

$$d(\Psi \wedge \Phi) = (-1)^{m}d\Psi \wedge \Phi \ + \Psi \wedge d\Phi.$$​

Your equation seems only to work when n and m are both even.

 

[element4]

Commuting the wege product in each term and negating gives
$$d(\Psi \wedge \Phi) = \Psi \wedge d\Phi + (-1)^{m}d\Psi \wedge \Phi \ .$$

You cannot do that! And the formula works for any positive integers n and m.
One mistake is, let $$\Psi$$ be a m form and $$\Phi$$ a n form, then

$$\Psi\wedge \Phi = (-1)^{mn}\Phi\wedge\Psi$$.​

If you google "differential geometry lecture notes", or "differential forms lecture notes", then you will find many very good and free notes. A very nice and pedagogic book on these topics is "Gauge Fields, Knots and Gravity" by John Baez and Javier P Muniain.

 

[Phrak]

You cannot do that!

Thank's for the correction.