1. Introduction
2. The construction of transformation between two representations
where a, b, ${a}^{\dagger }$, ${b}^{\dagger }$ are annihilation and creation operators of Gentile system and $a={b}^{* }$. $a={b}^{* }$ means that the space of the states we considered consists of two parts: the normal state space and its complex conjugate space. If we denote the space of ${a}^{\dagger },a$ as V and ${b}^{\dagger },b$ as ${V}^{* }$, the space of the entire system can be written as $V\otimes {V}^{* }$. It has been proved that only ${a}^{\dagger },a$ is not enough to represent the whole system just like the Holstein-Primakoff representation which is a modified bosonic realization [7, 8]. An additional constraint must be introduced. If we want to express the system with no constraint, the complex conjugate space is needed [8]. The commutation relation returns to boson and fermion when the maximum occupation number $n\to \infty $ and n = 1. When $n\to \infty $, the phase factor is $\exp ({\rm{i}}2\pi /({\text{}}n+1))=1$. When n = 1, we have $\exp ({\rm{i}}2\pi /({\text{}}{n}+1))=-1$. In these two limits, the space of the states returns to V. The particle number operator is constructed as
This equation tells us how many particles are there in one state. The eigenvalue of the particle number operator is the practical number of particles in one state. The second quantization of quantum field can be expressed in the language of the creation and annihilation operators. For bosons, the exchange of two particles gives nothing (+1), and the commutation relation reads $\left[a,{a}^{\dagger }\right]={{aa}}^{\dagger }-{a}^{\dagger }a=1$. For fermions, the exchange of two particles gives −1 which means $\left\{a,{a}^{\dagger }\right\}={{aa}}^{\dagger }-(-1){a}^{\dagger }a=1$. Comparing these two commutation relations with the basic commutation relation of Gentile statistics, we get the phase of exchanging Gentile particles ${{\rm{e}}}^{{\rm{i}}2\pi /\left({\text{}}{n}+1\right)}$. So the commutation relation equation (
ν is the practical particle number in one state and $\nu \in N$. So we have the change of the winding number $\delta k=g/2$. The relation of two phase factors between two representations is
or
In Bose case, $n\to \infty ,\Rightarrow \alpha =0,k=\left(\nu g\right)/2$. Equations (
3. Basic relations of the second quantization operators
The states in the occupation number representation are ${\left|\nu \right\rangle }_{n}$. In the winding number representation, the states can be expressed as
And the variations of the practical occupation number and the winding number are $\delta \nu =1$, $\delta k=g/2$ which means $\nu =0,1,2\cdots n$ and $k=0,g/2,g\cdots ({ng})/2$. To create and annihilate a particle, we have
where
${a}^{\dagger }$ creates a particle each time and b annihilates one. And
where
4. The coherent states
where the eigenvalue χ is a Grassmann number. Grassmann numbers do not commute with the states. They satisfy
here we take
as an example and we also have
Under these assumptions, we can construct the coherent state as
where M is the normalization constant, the coefficients read
and both the variations $\delta k=\delta p=g/2$. In this case, the state in adjoint space is
where $\bar{\chi }$ is also the Grassmann number in adjoint space. Under the normalization condition ${\left\langle \chi | \chi \right\rangle }_{\alpha ,g}=1$, we have
and also the variation $\delta l=g/2$. Moreover, the relations of the Grassmann number and creation and annihilation operators can be obtained :
As for ${b}^{\dagger },b$, there will be the same relations.
5. Berry phase
and ${J}_{z}=N-n/2$. As known to all, SO(3) group is locally isomorphic to SU(2) group. Generally speaking, the unitary matrix can be written as a three dimensional rotation around certain axis. Anyons exist in two dimensions, so the symmetry is SO(2). Jz is one of the generators of the spin angular momentum algebra in both two dimensions (only one generator) and three dimensions. $U\left(\theta \right)={{\rm{e}}}^{-{\rm{i}}\theta {\text{}}{{\rm{J}}}_{{\text{}}z}}$ is the rotation θ around the z-axis, it is the element of the group SO(2). It represents nontrivial braiding operation in two dimensions. In three dimensions, the spin angular momentum algebra (SO(3)) is $\left[{J}_{i},{J}_{j}\right]={\rm{i}}{\epsilon }_{{ijk}}{\text{}}{J}_{{\text{}}k}$, $i,j,k=x,y,z$ . They do not commute to each other. In two dimensions, we assume that the particles live on x-y plane. There exists only one angular momentum Jz (z-axis perpendicular to the x-y plane). Differing from the spin in three dimensions, the algebra is trivial since there is only one generator Jz in two dimensions. It commutes with itself obviously. So there is no restriction of Jz in two dimensions and it can take any value. This fact indicates that the statistics of two dimensional particles may be fractional statistics.
According to equation (
The positive and negative value of ${\eta }_{\nu }$ correspond to two closed trajectories in two opposite directions.
Then we get a restrict of the winding number
The positive and negative value of k means braiding clockwise and counterclockwise. For instance, when n = 3 and g = 2, we have $\nu =0,1,2,3$ and $\alpha =1/6$. So the restrict of k is $k=6\nu -9=-9,-3,3,9$. The difference is that equation (