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Generalized complex Gaussian integrals with linear terms

  1. Aug 13, 2012 #1
    Hi there!

    It's fairly simple to show the following identity of a multivariate Gaussian integral with linear terms, if [itex]H[/itex] is a Hermitean matrix and [itex]{}^*[/itex] denotes complex conjugation:

    [tex]\int \frac{dx_1^* \, dx_1 ... dx_n^* \, dx_n}{(2\pi i)^n} e^{-\sum_{ij} x_i^* H_{ij} x_j + \sum_i (J^*_i x_i + J_i x_i^*)} = (\det H)^{-1} e^{\sum_{ij} J_i^* (H^{-1})_{ij} J_j} [/tex]
    (One just introduces the new variable [itex]y_i = x_i - \sum_j (H^{-1})_{ij} J_j[/itex], uses that [itex]H[/itex] is Hermitean and solves the remaining purely Gaussian integral.)

    In the book "Quantum Many-Particle Systems" by Negele and Orland (p. 34), they say that this identity also holds if [itex]H[/itex] just has a positive Hermitean part but they don't show how to prove it...can someone help me? I've searched on Google and PF for a quite a while but found nothing.

    Thanks a lot!
  2. jcsd
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