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Gradiant and exponent of trace of matrices

  1. Dec 28, 2011 #1
    hi everybody,

    I'm currently work on a simple problem where i want to show that
    [tex]\int d[H] ~ \exp\{-1/(2v^2)tr(H^2)\} \exp\{i tr(HK)\} = \exp\{-1/(2v^2)tr(\bigtriangledown_K^2)\} \int d[H] ~\exp\{i tr(HK)\} [/tex]

    where [itex] d[H] [/itex] is the lebesgue measure on the space of NxN matrices. This is easy and was no problem when one use simples rules form fourier analysis. Now I would like to replace K, since it is hermitian, by a product of a Nx2k matrix [itex] A [/itex] and its hermitian conjugated [itex] A^{\dagger} [/itex]. May anyone of you has an idea how to express the integral above with the aid of differential operators depending on the components of [itex] A [/itex] and [itex] A^{\dagger} [/itex]. Or maybe could someone of you tell me, what is the problem in finding such an expression from mathematical point of view.

    best Timb00
  2. jcsd
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