Integration in path integral formalism?

  • #1

Main Question or Discussion Point

Hi,

Does anyone know how this integral is calculated

[tex]\int[dx] x_i x_j \exp \{ - (\frac{1}{2} \sum_{rs} A_{rs}x_r x_s+\sum_r L_r x_r ) \}[/tex]

Thanks
 

Answers and Replies

  • #2
fzero
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First note that

[tex]
\int[dx] x_i x_j \exp \{ - (\frac{1}{2} \sum_{rs} A_{rs}x_r x_s+\sum_r L_r x_r ) \} = \frac{\partial}{\partial L_i} \frac{\partial}{\partial L_j} \int[dx]\exp \{ - (\frac{1}{2} \sum_{rs} A_{rs}x_r x_s+\sum_r L_r x_r ) \} . [/tex]

Now the integral on the RHS above can be performed by completing the square in the exponent.
 

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