I Trouble with Product of Green's Functions

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The discussion focuses on a specific calculation involving the imaginary part of the product of Green's functions, particularly the expression Im(G(k,t)G(k,-t)). The user is struggling to understand why their calculations consistently yield zero for the imaginary part, as the exponentials appear to cancel out. Clarification is sought on the transition from the first line of the expression to the second line in the context of Green's functions. The Heaviside step function's role in the calculation is also implied as a point of confusion. Assistance is requested to resolve this issue and gain a clearer understanding of the underlying mathematics.
thatboi
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Hi all,
Consider the following Green's function:
1698539684976.png

where ##\Theta(t)## is the Heaviside step function and ##\tilde{\Theta}(t)## is defined as
1698539743261.png

I want to understand the following calculation:
1698539807121.png

More specifically, the ##\text{Im}(G(\textbf{k},t)G(\textbf{k},-t))## from the first line to the second line. Everytime I write out the expression, the exponentials seem to just cancel and I get 0 for the imaginary part. Any assistance would be greatly appreciated.
 

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