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I am studing QFT using the text book of Srednicki's. And I am stuck on one of calculations of the integrals in his book.

Consider a harmonic oscillator with hamiltonian: enter image description here

We can write the following integral for the transition from ground state to ground state: enter image description here

In his book, he said: Passing to the lagrangian formulation then gives: enter image description here

What does he mean by passing to the lagrangian formulation? And how did he calculate the $Dp$ part?

It's on page 46 of the book: http://chaosbook.org/FieldTheory/extras/SrednickiQFT03.pdf

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Going from the Hamiltonian to the Lagrangian formulation usually means integrating out the momentum variables. In this case the momentum integrations in the phase space path integral is just Gaussian integrations.

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For some reason he left out the $hp$ term in the second equation you quoted. If you put it back as it is in the earlier sections, and then do the $p$ inetegrals you get the $(m/2)(\dot q+h)^2$ term directly.

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