# Homework Help: Thermal average

1. Nov 4, 2011

### parton

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

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

is beeing discussed.

Note that:

$$E_{\vec{p}} = \sqrt{\vert \vec{p} \vert^{2} + m^{2}}$$

and

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

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

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

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

3. The attempt at a solution

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

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

where I used

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

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 $$H = \omega a^{\dagger} a$$ 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