Undergrad Trouble with Product of Green's Functions

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SUMMARY

The discussion centers on the calculation of the imaginary part of the product of Green's functions, specifically the expression ##\text{Im}(G(\textbf{k},t)G(\textbf{k},-t))##. Participants highlight the role of the Heaviside step function ##\Theta(t)## and its modified version ##\tilde{\Theta}(t)## in this context. The main issue raised is the cancellation of exponentials leading to a result of zero for the imaginary part, prompting requests for clarification on the calculation process.

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  • Understanding of Green's functions in quantum mechanics
  • Familiarity with complex analysis and imaginary components
  • Knowledge of the Heaviside step function and its properties
  • Basic skills in mathematical manipulation of exponential functions
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  • Explore complex analysis techniques for evaluating imaginary parts of functions
  • Investigate common pitfalls in calculations involving exponential functions
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Physicists, mathematicians, and students studying quantum mechanics or field theory who are looking to deepen their understanding of Green's functions and their calculations.

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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