On Statistical Mechanics of Non-Abelian Chern-Simons Particles

Lee replies to the comment on ”Statistical Mechanics of Non-Abelian ChernSimons Particles” by C. R. Hage PACS number(s): Typeset using REVTEX 1 Lee Replies: I recently evaluated [1] the second virial coefficient for the two-dimensional gas of non-Abelian Chern-Simons particles (NACS) [2] which obey the non-Abelian braid statistics and showed that the coefficient is not periodic in the induced spin. In a comment on the letter Hagen [3] claimed that the virial coefficient obtained in Ref. [1] is incorrect and there exists a periodicity in the flux parameter as in the Abelian theory [4]. Here I show that the additional factor, which is claimed to be omitted in Eq.(28) of Ref. [1] is improper and the periodicity discussed in Ref. [3] is of no significance. In the comment [3] it was claimed that factors of (−1) are omitted in the expression of the two-particle partition function Eq.(26) of Ref. [1] or in that of the second virial coefficient. (Here l is the isospin quantum number of the particle.) However, these factors are unnecessary unless we impose a relationship, between the (intrinsic) statistics of the particles and their isospins, which would be similar to the spin-statistics relation in (3+1) dimensions. Obviously such a relation does not exist. The NACS particles are described in Ref. [1] as point like spinless sources which interact with each other through the non-Abelian Chern-Simons gauge fields. Thus, when we describe the NACS particles in the regular gauges such as the Coulomb gauge or the holomorphic gauge, the wave function for the many NACS particle system is symmetric under an exchange of any pair of the particles. It was also claimed in Ref. [3] that the configuration space wave functions for all states are taken to be symmetric in Ref. [1] and the resultant expression for the second virial coefficient reduces to the bosonic one as κ → ∞. Contrary to the claim, the configuration space wave function is symmetric only when its isospin counterpart is symmetric; it is antisymmetric when its counterpart is antisymmetric. It becomes manifest, as we rewrite the two-particle partition function Eq.(21b) of Ref. [1] as follows Z ′ 2 = ∫