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What is the correct way to expand (p_{3}-p_{4})^{2} where p_{3} and p_{4} are 4-vectors, with metric g_{mu nu}=diag[1,-1,-1,-1], p = [w_{p}, p], where p is 3-vector, and w_{p}= (p^{2}+m^{2})^{(1/2)}
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p_{3}.p_{4} = = p_{3}^{0} p_{4}^{0} - p_{3}.p_{4}And do you know how to expand that? In Euclidean 3-space ##\vec a.\vec b=a_xb_x+a_yb_y+a_zb_z##. Do you know the equivalent Minkowski 4-space expression?
The question isRight. So when you replace the vector with a difference of two vectors what do you get?
Instead of guessing, write it out explicitly: ##(p_3 - p_4)^2 = (p_3 - p_4) \cdot (p_3 - p_4)##, and then just expand out the product and do the algebra..what do I get?
How would you write out (p_{3} - p_{4})?Instead of guessing, write it out explicitly: ##(p_3 - p_4)^2 = (p_3 - p_4) \cdot (p_3 - p_4)##, and then just expand out the product and do the algebra.
You don't need to do that. The 4-vector product distributes over addition and subtraction just like the ordinary scalar product does.How would you write out (p_{3} - p_{4})?
How would you write out (p3 - p4)?
You do realize that that equation answers the question you just asked?see for e.g. Equation 46.29
First, vector subtraction against an orthonormal basis is always just what you would expect.How would you write out (p_{3} - p_{4})?
There's a bit of subtlety here, and not guessing, see for e.g. Equation 46.29 here
You wrote down an expression for the dot product of two vectors in terms of the components. So if the vectors, instead of ##p_3## and ##p_4##, are both ##p##, and ##p=p_3-p_4## what do you get?The question is
(p_{3} - p_{4})^2 - whether it is (w_{p3}-w_{p4})^{2} + (p_{3} - p_{4})^{2}
or is it
(w_{p3}+w_{p4})^{2} - (p_{3} + p_{4})^{2}
Hmmmmmm.....what do I get?
Actually, you may not realize, there might be a surprise on the way!You do realize that that equation answers the question you just asked?
Perhaps it would be helpful if you either completed the algebra we've been discussing or said what you think is wrong with 4.29 in the pdf you linked.Actually, you may not realize, there might be a surprise on the way!
The link provided may be using a different convention. Until now, I have provided all the algebra and the question. It was suggested that "...realize that the link you provided...has the solution...?" The original question has two alternatives, and it is useful to see what the forum members come up with. This is not a homework that is to be submitted to the forum in all its completeness, for it to be graded.Perhaps it would be helpful if you either completed the algebra we've been discussing or said what you think is wrong with 4.29 in the pdf you linked.
You mean your expressions in #7? Neither appears to be correct, as PeterDonis said in #8. How did you get them? I haven't tried to replicate 4.29 from your link, but it looks plausible at a quick glance.The link provided may be using a different convention. Until now, I have provided all the algebra and the question. It was suggested that "...realize that the link you provided...has the solution...?" The original question has two alternatives
The issue is that for everyone who has responded on this thread so far, your problem is trivial, yet you seem to have misunderstandings. We feel it is much more useful to help you resolve these than simply give you the answer.The link provided may be using a different convention. Until now, I have provided all the algebra and the question. It was suggested that "...realize that the link you provided...has the solution...?" The original question has two alternatives, and it is useful to see what the forum members come up with. This is not a homework that is to be submitted to the forum in all its completeness, for it to be graded.
I did replicate that equation, just doing the algebra in my head.You mean your expressions in #7? Neither appears to be correct, as PeterDonis said in #8. How did you get them? I haven't tried to replicate 4.29 from your link, but it looks plausible at a quick glance.
Thank you. I was confident up to sign errors on the cross terms - it's been a long day...I did replicate that equation, just doing the algebra in my head.
Have you done the algebra I asked for in post #8? If so, post it. If not, do it and post it.Until now, I have provided all the algebra
Do you mean the question you asked in post #7? Neither of those alternatives is correct.The original question has two alternatives
No, but that doesn't mean we're just going to tell you an answer that you could derive for yourself by doing basic algebra. PF is not an oracle that you can feed guesses to and get yes/no answers back.This is not a homework that is to be submitted to the forum in all its completeness, for it to be graded
Sorry, sign errors in which equation? By the way, the way it is derived in the link is not necessarily consistent.Thank you. I was confident up to sign errors on the cross terms - it's been a long day...
Do the algebra yourself and find out what the correct answer is. As I said, PF is not an oracle.What is the correct answer (of the alternatives posted)?