Universal chiral partition function

introduction

• grand partition function for n species of right moving (chiral) particles with fugacities z
• N개의 보존 입자가 있고, 에너지의 단위를 \(\hbar\omega=1\)으로 하여, 에너지레벨이  \(E_0,E_1,E_2,\cdots\) 인 시스템을 생각하자.

N개의 입자가 있는 보존 시스템의 분배함수를  \(Z_B(N)\) 이라 두자.

큰 분배함수(grand partition function)는 \(Z_G=\sum_{n=0}^{\infty}Z_B(N)z^N\) 으로 쓸수 있다.

\(n_0,n_1,n_2,\cdots\) 은 각각 에너지가 \(E_0,E_1,E_2,\cdots\)인 입자의 수라고 하자.

 \(Z_B(N)=\sum_{\sum n_r=N}\exp(-\beta\sum_{r}n_r E_r)\) 이므로, 

\(Z_G=\sum_{N=0}^{\infty}Z_B(N)z^N=\sum_{N=0}^{\infty} \sum_{\sum n_r=N}\exp(-\beta\sum_{r}n_r E_r)z^N\)

\(=\prod_{r=0}^{\infty}\sum_{n_r=0}^{\infty} (ze^{-\beta E_r})^{n_r}=\prod_{r=0}\frac{1}{1-ze^{-\beta E_r}}\)

 

 

physical meaning

\(f_{A,B,C}(\tau)=\sum_{n\in \mathbb{Z}_{\geq 0}^r}\frac {q^{\frac{1}{2}n^{t}An+B^{t}\cdot n+C}} {(q)_{n_1}\cdots(q)_{n_r}}\)

A: energy shift due to interaction

B : energy shift due to (global) statistics

C : ground state Casimir energy

 

 

 

special cases

• rank 1 case examples
• Berkovich1998 and Wu's paper

 

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

related items

• fractional and exclusion statistics
• Fermionic summation formula

 

 

encyclopedia

• http://en.wikipedia.org/wiki/
• http://www.scholarpedia.org/

 

 

articles

• Exclusion statistics in conformal field theory and the UCPF for WZW models
  • Peter Bouwknegt, Leung Chim, David Ridout, 1999
• Comment on the paper ``The universal chiral partition function for exclusion statistics
  • K. Schoutens (University of Amsterdam)
• Berkovich, A., 와/과B. M McCoy. 1998. “The universal chiral partition function for exclusion statistics”. hep-th/9808013 (8월 4). http://arxiv.org/abs/hep-th/9808013

• Statistical distribution for generalized ideal gas of fractional-statistics particles
  • Y.S. Wu,, Phys. Rev. Letts. 73 (1994) 922