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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?
Draft saved Draft deleted



Similar Discussions: Infinite Product in partition function
  1. Pair Production (Replies: 11)

  2. Pair production (Replies: 12)

  3. Muon Production (Replies: 6)

Loading...