Divergence of the Stress-Energy Tensor

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The discussion focuses on the divergence of the Stress-Energy Tensor in Quantum Field Theory, specifically the transition from line (31) to line (32) in a calculation related to Noether's Theorem. Participants identify a missing factor of 1/2 in line (31) that affects the first two terms within the square brackets. It is confirmed that the second term can be rearranged to match the first, allowing for cancellation with another term in the equation. This manipulation involves renaming summed indices, a technique noted as useful for understanding the derivation. Overall, the conversation emphasizes clarity in mathematical steps and the importance of recognizing such tricks in theoretical physics.
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Im studying Quantum Field Theory as part of my undergraduate course, and am currently looking at Noether's Theorem which has led me to the following calculation of the divergence of the Stress-Energy Tensor. I'm having difficulty in seeing how we get from line (31) to line (32). Is the 2nd term in the square brackets zero? if so, why?

9f06390eaf950b1f7f3fe32ae06e4182.png


Thanks
 
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It appears they forgot 1/2 in line 31 in the square brackets for the first 2 terms.
 
dextercioby said:
It appears they forgot 1/2 in line 31 in the square brackets for the first 2 terms.

Thanks, i thought this could be the case. Then inside the square brackets I am assuming we can rearrange the second term so that the first and second term are the same. so we get 1/2*[...] + 1/2*[...] = [...]. and this [...] cancels with the second term of the whole line, resulting in line (32)?
 
That's right. Just renaming indices summed over, a trick you should know.
 
dextercioby said:
That's right. Just renaming indices summed over, a trick you should know.

indeed, a trick I am quickly beginning to learn! Thanks for your help! :)
 

Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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