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Thermal average

  1. Nov 4, 2011 #1
    1. The problem statement, all variables and given/known data
    2. Relevant equations


    Hi!

    I read a book where a free real scalar field with Hamiltonian

    [tex] H = \int \dfrac{\mathrm{d}^{3}p}{(2 \pi)^{3}} \, E_{\vec{p}} a_{\vec{p}}^{\dagger} a_{\vec{p}} [/tex]

    is beeing discussed.

    Note that:

    [tex] E_{\vec{p}} = \sqrt{\vert \vec{p} \vert^{2} + m^{2}} [/tex]

    and

    [tex] \left[ a_{\vec{p}}, a_{\vec{q}}^{\dagger} \right] = (2 \pi)^{3} \delta^{3}(\vec{p} - \vec{q}) [/tex].

    Now there is the statement that the thermal average of the operator
    [tex] a_{\vec{p}}^{\dagger} a_{\vec{q}} [/tex] is given by:

    [tex] \left \langle a_{\vec{p}}^{\dagger} a_{\vec{q}} \right \rangle = \dfrac{1}{Tr \left[ \mathrm{e}^{-\beta H} \right]} Tr\left[\mathrm{e}^{-\beta H} a_{\vec{p}}^{\dagger} a_{\vec{q}} \right] = \dfrac{1}{\mathrm{e}^{\beta E_{\vec{p}}} -1} (2 \pi)^{3} \delta^{3}(\vec{p} - \vec{q}) [/tex]

    I don't know how to obtain this average value.

    3. The attempt at a solution

    When I first start to compute [tex] Tr \left[ \mathrm{e}^{-\beta H} \right] [/tex] I end up with an ill-defined expression:

    [tex] Tr \left[ \mathrm{e}^{-\beta H} \right] = \int \dfrac{\mathrm{d}^{3} k}{(2 \pi)^{3}} \left \langle \vec{k} \right| \mathrm{e}^{-\beta H} \left| \vec{k} \right \rangle = \int \dfrac{\mathrm{d}^{3} k}{(2 \pi)^{3}} (2 \pi)^{3} \delta^{3}(0) [/tex]

    where I used

    [tex] H \left| \vec{k} \right \rangle = \int \dfrac{\mathrm{d}^{3} k}{(2 \pi)^{3}} E_{\vec{p}} a_{p}^{\dagger} a_{\vec{p}} a_{\vec{k}}^{\dagger} \vert 0 \rangle = \int \dfrac{\mathrm{d}^{3} k}{(2 \pi)^{3}}

    E_{\vec{p}} \delta^{3}(\vec{p} - \vec{k}) (2 \pi)^{3} \vert \vec{p} \rangle = E_{\vec{k}} \vert \vec{k} \rangle [/tex]

    I think the problem is that I used momentum states as eigenstates of the Hamiltonian, but they are not normalizable.

    Does anyone have an idea how to compute this thermal average?

    (If I have the simple Hamiltonian [tex] H = \omega a^{\dagger} a [/tex] and consider a complete set of eigenstates |n> it is very easy to compute this average. But in the case described above I don't know how to do that).

    I hope someone can help me.

    Note:

    The same problem is also described here (page 5, eq. (3.10))
    http://www.actaphys.uj.edu.pl/vol38/pdf/v38p3661.pdf [Broken]
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
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