Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Orthogonality of Matsubara Plane Waves

  1. Aug 14, 2009 #1
    Hi there!

    In thermal field theory, the Matsubara frequencies are defined by [itex]\nu_n = \frac{2n\pi}{\beta}[/itex] for bosons and [itex]\omega_n = \frac{(2n+1)\pi}{\beta}[/itex] for fermions. Assuming discrete imaginary time with time indices [itex]k=0,\hdots,N[/itex], it is easy to obtain the following orthogonality relation for bosons, just by using the standard formula for the geometric series ([itex]\beta[/itex] is the inverse temperature),

    [tex]\frac{\beta}{N} \sum_{k=0}^{N-1} \mathrm{e}^{\mathrm{i} \frac{\beta}{N} (\nu_n+\nu_m)k} = \begin{cases} \frac{\beta}{N} \sum_{k=0}^{N-1} 1 = \beta & \mathrm{for}\ n=-m \\ \frac{1-\mathrm{e}^{\mathrm{i} \beta (\nu_n+\nu_m)}}{1-\mathrm{e}^{\mathrm{i} \frac{\beta}{N} (\nu_n+\nu_m)}} = 0 & \mathrm{for}\ n\neq -m \end{cases} = \beta\delta_{n,-m}[/tex]

    The second line holds because [itex]\beta (\nu_n+\nu_m)[/itex] is an integer multiple of [itex]2\pi[/itex] and thus the numerator vanishes. But in the case of fermions, I obtain

    [tex]\frac{\beta}{N} \sum_{k=0}^{N-1} \mathrm{e}^{\mathrm{i} \frac{\beta}{N} (\omega_n+\omega_m)k} = \begin{cases} \frac{\beta}{N} \sum_{k=0}^{N-1} 1 = \beta & \mathrm{for}\ n=-(m+1) \\ \frac{1-\mathrm{e}^{\mathrm{i} \beta (\omega_n+\omega_m)}}{1-\mathrm{e}^{\mathrm{i}\frac{\beta}{N} (\omega_n+\omega_m)}} = 0 & \mathrm{for}\ n \neq -(m+1)} \end{cases} = \beta\delta_{n,-(m+1)}[/tex]

    Is this true? The Kronecker delta with [itex]n,-(m+1)[/itex] looks rather strange!

    Thanks for your help!
  2. jcsd
  3. Aug 14, 2009 #2
    It's not that strange, but it's a consequence of writing your exponential with a sum of frequencies instead of a difference. Consider the values around n = 0, [tex]\omega_{-1} = -T\pi[/tex], [tex]\omega_0 = T\pi[/tex], so for your orthogonality to hold, you need n = 0, m = -1, or n = -1, m = 0. If you choose to write the orthogonality with a difference, you should get n = m in both cases.
  4. Aug 24, 2009 #3
    Thanks very much! I ask because I've encountered a sum of the above type (with a plus sign) while computing a two-point Green's function and I was wondering if the energy/frequency was conserved...
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook