How Do You Calculate <A|0> in a Free Field Theory?

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SUMMARY

The discussion focuses on calculating the vacuum expectation value in the context of free field theory, specifically referencing Srednicki problem 8.8. The solution involves expressing using path integral formulation, where the free field is represented as a collection of harmonic oscillators. The ground state wave functional for each momentum mode is identified as a Gaussian function, leading to the conversion of a product of exponentials into an exponential of a sum through integration.

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Homework Statement


Srednicki problem (8.8)

Under a free-field theory, calculate <A|0> , where |A> is the real sclar field's eigenket

Homework Equations




The Attempt at a Solution



I am trying to write <A|0> into path integral formulation, but it is hard.
 
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As a function of momentum (rather than position), a free field is just a collection of harmonic oscillators. So for a given momentum mode, the ground state wave functional is just an appropriate gaussian. Then take a product over momentum modes, and convert the product of exponentials into the exponential of a sum (=integral).
 

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