Help with a complex integration in QFT

[thisisphysics]

Homework Statement

Hey! So I'm reading a physics book, and they're evaluating this integral: https://imgur.com/DnpYvs6. I'm confused about their statement on the next page. They say that $$e^{i |p| |x|}$$ is exponentially decreasing for large imaginary values of $$|p|$$. Now, I'm confused on how |p| can be negative; isn't it real-valued since we're taking the modulus? Someone mentioned that the modulus can be defined as the square root of the expectation value of p, which would then be complex, but this is something I've never come across, I'm used to the following definition of the modulus: $$|a+ib| = \sqrt{a^2 + b^2}$$. Is this not true here? Any help appreciated!

Relevant Equations

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[George Jones]

Problem Statement: Hey! So I'm reading a physics book

You should always identify the source (in this case, Lancaster and Blundell), as it is possible to unknowingly omit relevant content (in this case, I don't think that you did).

they're evaluating this integral: . I'm confused about their statement on the next page. They say that $$e^{i |p| |x|}$$ is exponentially decreasing for large imaginary values of $$|p|$$. Now, I'm confused on how |p| can be negative

Note that in the first two lines of (8.18), $\left| \bf{p} \right|$ is a non-negative dummy variable of integration, and thus can be relabeled to anything, e.g., Fred or George or thisisphysics. More conventionally, relabel the dummy variable integration $\left| \bf{p} \right|$ to $u$. What happens?

 

[thisisphysics]

You should always identify the source (in this case, Lancaster and Blundell), as it is possible to unknowingly omit relevant content (in this case, I don't think that you did).
Note that in the first two lines of (8.18), $\left| \bf{p} \right|$ is a non-negative dummy variable of integration, and thus can be relabeled to anything, e.g., Fred or George or thisisphysics. More conventionally, relabel the dummy variable integration $\left| \bf{p} \right|$ to $u$. What happens?

Thank you so much! I mulled over it, and I understand now. I appreciate it.

 

[hilbert2]

Even when a physical quantity appearing in a function has to be real-valued (and in this case, non-negative too), it is sometimes possible to think of an analytical continuation of the function to imaginary values of that variable. It is just a mathematical trick that makes the calculation of some integrals easier. A similar computational trick in QM is to allow complex values of the time variable.