Reversing Signs in Algebra: Explained for a Beginner

In summary, The conversation is discussing a change in the integration variable for a quantum field theory problem. The variable was changed from [ itex] \vec{p}[/itex] to [ itex] -\vec{p} [/itex], with the Jacobian of the transformation being unity and the factor [ itex] E_p = \sqrt{|\vec{p}|^2 + m^2 } [/itex] remaining unchanged. The exponential argument also changed, requiring a change in the identification of [ itex] p^0 [/itex] to be - [ itex] E_p [/itex].
  • #1
Ratzinger
291
0
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
 

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  • #2
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.
 
  • #3
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.
 
  • #4
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.
 

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  • #5
Physics Monkey said:
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. :tongue:

I'll stop mucking around here and go approve your attachment...
 
  • #6
FYI, I've just learned that the LaTeX problem has been discovered and is being worked on. Sit tight.
 
  • #7
Tom Mattson said:
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. :tongue:

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

Haha! I knew somebody would say something. :rofl:

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

1. What is reversing signs in algebra?

Reversing signs in algebra is the process of changing the positive and negative signs of numbers or variables in an equation or expression.

2. Why is it important to reverse signs in algebra?

Reversing signs is important because it allows us to simplify equations and solve them more easily. It also helps us to correctly combine like terms and manipulate expressions.

3. How do you reverse signs in algebra?

To reverse the sign of a number or variable, simply change the positive sign to a negative or vice versa. For example, changing +5 to -5 or -x to +x.

4. When should you reverse signs in algebra?

You should reverse signs in algebra when you are combining like terms, simplifying an equation, or solving for a variable. It can also be used when graphing equations or solving word problems.

5. What are some common mistakes when reversing signs in algebra?

Some common mistakes include forgetting to change the sign, reversing the sign of only one term in an equation, and mixing up the order of operations. It is important to double check your work and follow the correct order of operations when reversing signs in algebra.

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