Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 337 / 1 / d

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- Solution d

Solution

For fixed momentum put

E_{k} = v^{2} k^{2} + m (T)^{2} .

In the contour formula,

F (z) = 1/ (E_{k}^{2} - z^{2})

has poles at

z = \pm E_{k}

. Deforming the contour onto those poles and using the oddness of

coth (z / (2 T))

gives the standard Bosonic Matsubara sum

T n \in Z \sum \frac{1}{ω _{n}^{2} + E _{k}^{2}} = \frac{1}{2 E _{k}} coth (\frac{E _{k}}{2 T}) .

Equivalently,

\frac{1}{2 E _{k}} coth (\frac{E _{k}}{2 T}) = \frac{1 + 2 n _{B} ( E _{k} )}{2 E _{k}},

where

n_{B} (E) = 1/ (e^{E / T} - 1)

is the Bose-Einstein distribution. The first term is the zero-point fluctuation and the second is its thermal occupation.

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