Reversing Signs in Algebra: Explained for a Beginner

[Ratzinger]

Could someone tell me why I'm allowed to change the sign before the second term in (3)? It's certainly baby stuff for you, but could you tell me anyway?

pdf attached

thanks

 

[quantumdude]

As this is from quantum field theory, I'm going to shuffle this over to Advanced Physics to see if it gets more love there.

I don't know the answer off the top of my head, but notice that the second limit of integration changed from p_0=E_p to p_0=-E_p.

 

[Meir Achuz]

It looks like the variable change p_\mu-->-p_\mu has been made.
d3p/E_p won't change sign, but the other signs would.

 

[Physics Monkey]

Hi Ratzinger,

The integration variable in the second part of the integral has been changed from [ itex] \vec{p}[/itex] to [ itex] -\vec{p} [/itex]. The Jacobian of the transformation is unity, and the factor [ itex] E_p = \sqrt{|\vec{p}|^2 + m^2 } [/itex] doesn't change sign. This accounts for the stuff out front. Now what about the stuff in the exponential? Before the change of variable, the argument of the exponential was [ itex] i p_\mu x^\mu = i( - \vec{p}\cdot \vec{r} + p^0 t) = i( - \vec{p}\cdot \vec{r} + E_p t)[/itex] since the original integral specified that [ itex] p^0 = E_p [/itex]. When you change the sign of [ itex] \vec{p} [/itex], [ itex] E_p [/itex] remains fixed, so the exponential now looks like [ itex] i( \vec{p}\cdot \vec{r} + E_p t ) = - i (- \vec{p}\cdot\vec{r} - E_p t )[/itex]. You would like to interpret this as a four vector dot product of the form [ itex] - i (-\vec{p}\cdot \vec{r} + p^0 t ) = - i p_\mu x^\mu [/itex] which clearly means you must now identify [ itex] p^0 = - E_p [/itex].

Edit: Argh, why doesn't the latex work? See the attachment.

 

[quantumdude]

Edit: Argh, why doesn't the latex work?

Well, I was going to say that it isn't working because you put extra spaces in your tex brackets. So I just edited your post to remove the spaces (and to change them to "itex" brackets, so the LaTeX lines up with your text). But then I saw the error messages and concluded that you must have put those spaces in on purpose.

I'll stop mucking around here and go approve your attachment...

 

[quantumdude]

FYI, I've just learned that the LaTeX problem has been discovered and is being worked on. Sit tight.

 

[Physics Monkey]

Well, I was going to say that it isn't working because you put extra spaces in your tex brackets. So I just edited your post to remove the spaces (and to change them to "itex" brackets, so the LaTeX lines up with your text). But then I saw the error messages and concluded that you must have put those spaces in on purpose.

I'll stop mucking around here and go approve your attachment...

Haha! I knew somebody would say something.

P.S. Thanks for the info about the itex tag.