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

Infinite Product in partition function

  1. Mar 25, 2014 #1
    Hi I am having trouble following the derivation of the fermionic oscillator partition function in zeta function regularisation. Specifically the following step:

    [itex] Z( \beta ) = e^{ \beta \omega /2 } lim_{ N \rightarrow \infty } \prod_{ k = -N/4 }^ {N/4 } \left[ i( 1-\epsilon \omega ) \frac{ \pi (2k-1) }{ \beta } + \omega \right] [/itex]

    [itex] = e^{ \beta \omega /2 }e^{ - \beta \omega /2} \prod_{k = 1}^{ \infty } \left[ ( \frac{ 2 \pi (k-1/2)}{\beta} )^{2} + \omega^{2} \right] [/itex]

    where [itex] \epsilon = \frac{\beta}{N} [/itex]


    I understand that we multiply the negative and positive parts of the product and take the limit of N however k going to -k gives rise to cross terms that I don't know how to deal with.

    Thanks for your help
     
    Last edited: Mar 25, 2014
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Loading...