Solving Lorentz Matrix Product Problem - Help Needed

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Discussion Overview

The discussion revolves around a problem involving the product of Lorentz matrices and the application of the Minkowski metric in a specific mathematical expression. Participants are attempting to resolve discrepancies in their calculations and clarify the implications of certain tensor contractions.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents a matrix product involving the Minkowski metric and expresses confusion over their calculated result, which differs from the expected outcome.
  • Another participant suggests that providing intermediate steps would clarify where the error lies.
  • There is a discussion about the value of the contraction of the Minkowski metric tensors, with one participant assuming it to be +1.
  • A later reply indicates that the contraction does not yield a definitive value and points out a potential typo in the original expression.
  • Participants debate the significance of the contraction of the metric tensors and its impact on the calculations, with one asserting that neglecting certain terms leads to a specific coefficient in front of a term.
  • Another participant proposes that the trace of the delta tensor is crucial for resolving the confusion, suggesting it may be equal to 4.
  • One participant expresses relief upon confirming that the trace of the delta tensor is indeed 4, which resolves their issue with the calculations.
  • A final confirmation is provided that the trace is equal to the total number of dimensions, but this remains a point of clarification rather than consensus.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the initial calculations, and there are multiple competing views regarding the treatment of the Minkowski metric and the implications of the tensor contractions. The discussion remains unresolved in terms of the exact steps leading to the expected result.

Contextual Notes

Participants note potential typos and unclear steps in the calculations, particularly regarding the contraction of the Minkowski metric and the trace of the delta tensor, which are critical to the problem but not fully resolved in the discussion.

Dixanadu
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Hey guys,

So consider the following product of matrices:
(p_{1}^{\mu}\cdot p_{1}^{\prime\nu} -(p_{1}\cdot p_{1}')\eta^{\mu\nu}+p_{1}^{\nu}p_{1}^{\prime\mu})(p_{2\mu}p_{2\nu}'-(p_{2}\cdot p_{2}')\eta_{\mu\nu}+p_{2\nu}p_{2\mu}')

where eta is the Minkowski metric.

I keep getting

2(p_{1}\cdot p_{2})(p_{1}'\cdot p_{2}')+2(p_{1}\cdot p_{2}')(p_{1}'\cdot p_{2}) - 3(p_{1}\cdot p_{1}')(p_{2}\cdot p_{2}')

But apparently its wrong; I'm meant to get just
2(p_{1}\cdot p_{2})(p_{1}'\cdot p_{2}')+2(p_{1}\cdot p_{2}')(p_{1}'\cdot p_{2})

Cant figure it out for the life of me -- someone please help!
 
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It would be easier to see where you go wrong if you include your middle steps. The answer you are supposed to get seems correct.
 
Okay I'll write it out explicitly for you, please bear with me a moment.
 
gnZhjde.png


Here it is...

Btw is \eta^{\mu\nu}\eta_{\mu\nu}=1 or -1 lol XD I've assumed its +1
 
It is neither ...
$$
\eta_{\mu\nu} \eta^{\mu\nu} = \delta^\mu_\mu = \ldots
$$

Edit: Your first expression in your attempt also does not match what you wrote in the OP. Only what you wrote in the attempt makes sense together with the presumtive answer so I am going to edit your OP to reflect this.
 
OMG so those terms vanish? :O
 
Dixanadu said:
OMG so those terms vanish? :O

That would depend on what terms you are referring to. What did you get for the trace of the delta?
 
That's fine there is a typo...there is a cdot somewhere it shouldn't be in the first term.

But mu isn't = nu and there is only one term where the two etas are being contracted. If that term goes to 0 I get

NrwG6pf.png
 
Sorry, but it is not clear what you are doing with your etas. What did you get for ##\eta^{\mu\nu}\eta_{\mu\nu}## in the end? This is of crucial importance for the problem so you need to write these steps out. Neglecting the term where the etas contract you should get -4 in front of the term where the p1s are contracted with each other (and te p2s with each other).
 
  • #10
I think it's easier if I just tell you a few terms. So in each line there is an example of a product of terms:

ovtVlmh.png
 
  • #11
Yes, the question mark is the crucial point here. What is the trace of the delta tensor?
 
  • #12
is it 4..?
 
  • #13
Yes doctor that solves my problem. The terms now cancel if I have a factor of 4 in front of one of them due to the trace of the delta tensor in spacetime. You saved the day once more doctor, you should consider becoming a superhero :D thank you!
 
  • #14
Just for closure: Yes, it is 4. In general it is equal to the total number of dimensions.
 

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