The present thesis is concerned with the local quantum physics of relativistic particles and fields in three space-time dimensions, whose statistics is to be described by a representation of the braid group -- so-called plektons or, if the representation is Abelian, anyons. In particular the issue of the existence of free anyonic fields is addressed. In our context, these are operators affiliated with the `local´ field algebras and creating only single particle vectors from the vacuum. (Localizability here refers to regions extending to infinity in some spacelike direction.) Under a mild regularity condition for these fields, we can derive commutation relations which are not compatible with braid group statistics. Further, model-independent results concerning the PCT operator and the connection of spin and statistics are obtained. Assuming the observable algebra to satisfy the Bisognano-Wichmann theorem, a PCT theorem for plektons is derived. For anyons it is shown that the Bisognano-Wichmann property of the observable algebra lifts, in a `twisted´ version, to the field algebra. In particular, the PCT-operator coincides, up to a `twist operator´, with the modular conjugation of the field algebra associated to a standard wedge region. From this fact, the spin-statistics theorem for anyons is derived. The structure of the Hilbert space of scattering states of plektons and anyons is further clarified. It is known to be fixed by the relevant representations of the braid group and by the rules which determine the charges contained in the scattering states (the `fusion´ rules). We present Møller operators W+ and W- from the spaces of outgoing and incoming scattering states into a reference Hilbert space, which translate the physical ray representation of the Poincaré group on the spaces of scattering states into a canonical ray representation on the reference Hilbert space. A new result concerns the `incoming free PCT operator´, i.e. the product of the PCT operator with the S-matrix S = ( W+)* W- : It is exhibited how this model-independent operator acts in the reference Hilbert space. In the case of anyons, the fusion rules are trivial and the relevant representations of the braid group are classified by a finite set of complex numbers, namely the statistics parameters of the particles under consideration. Hence in the anyonic case the above results yield explicit formulae for the ray representation of the Poincaré group and for the incoming free PCT-operator. In view of the Bisognano-Wichmann property, we have thereby constructed the product of the Tomita operator associated to any given wedge region with the S-matrix, i.e. the `incoming free´ Tomita-operator of the anyonic field algebra.The present thesis is concerned with the local quantum physics of relativistic particles and fields in three space-time dimensions, whose statistics is to be described by a representation of the braid group -- so-called plektons or, if the representation is Abelian, anyons. In particular the issue of the existence of free anyonic fields is addressed. In our context, these are operators affiliated with the `local´ field algebras and creating only single particle vectors from the vacuum. (Localizability here refers to regions extending to infinity in some spacelike direction.) Under a mild regularity condition for these fields, we can derive commutation relations which are not compatible with braid group statistics. Further, model-independent results concerning the PCT operator and the connection of spin and statistics are obtained. Assuming the observable algebra to satisfy the Bisognano-Wichmann theorem, a PCT theorem for plektons is derived. For anyons it is shown that the Bisognano-Wichmann property of the observable algebra lifts, in a `twisted´ version, to the field algebra. In particular, the PCT-operator coincides, up to a `twist operator´, with the modular conjugation of the field algebra associated to a standard wedge region. From this fact, the spin-statistics theorem for anyons is derived. The structure of the Hilbert space of scattering states of plektons and anyons is further clarified. It is known to be fixed by the relevant representations of the braid group and by the rules which determine the charges contained in the scattering states (the `fusion´ rules). We present Møller operators W+ and W- from the spaces of outgoing and incoming scattering states into a reference Hilbert space, which translate the physical ray representation of the Poincaré group on the spaces of scattering states into a canonical ray representation on the reference Hilbert space. A new result concerns the `incoming free PCT operator´, i.e. the product of the PCT operator with the S-matrix S = ( W+)* W- : It is exhibited how this model-independent operator acts in the reference Hilbert space. In the case of anyons, the fusion rules are trivial and the relevant representations of the braid group are classified by a finite set of complex numbers, namely the statistics parameters of the particles under consideration. Hence in the anyonic case the above results yield explicit formulae for the ray representation of the Poincaré group and for the incoming free PCT-operator. In view of the Bisognano-Wichmann property, we have thereby constructed the product of the Tomita operator associated to any given wedge region with the S-matrix, i.e. the `incoming free´ Tomita-operator of the anyonic field algebra.

Jan 1, 19990 citations

Anyons as Elementary Field Quanta

WORLD SCIENTIFIC eBooksFeb 1, 20050 citations

Fock representations of multicomponent (particularly non-Abelian anyon) commutation relations

Alexei Daletskii, A. A. Kalyuzhnyi, Eugene Lytvynov +1 more

Let $H$ be a separable Hilbert space and $T$ be a self-adjoint bounded linear operator on $H^{\otimes 2}$ with norm $\le1$ , satisfying the Yang--Baxter equation. Bo\.zejko and Speicher (1994) proved that the operator $T$ determines a $T$ -deformed Fock space $\mathcal F(H)=\bigoplus_{n=0}^\infty\mathcal F_n(H)$ . We start with reviewing and extending the known results about the structure of the $n$ -particle spaces $\mathcal F_n(H)$ and the commutation relations satisfied by the corresponding creation and annihilation operators acting on $\mathcal F(H)$ . We then choose $H=L^2(X\to V)$ , the $L^2$ -space of $V$ -valued functions on $X$ . Here $X:=\mathbb R^d$ and $V:=\mathbb C^m$ with $m\ge2$ . Furthermore, we assume that the operator $T$ acting on $H^{\otimes 2}=L^2(X^2\to V^{\otimes 2})$ is given by $(Tf^{(2)})(x,y)=C_{x,y}f^{(2)}(y,x)$ . Here, for a.a.\ $(x,y)\in X^2$ , $C_{x,y}$ is a linear operator on $V^{\otimes 2}$ with norm $\le1$ that satisfies $C_{x,y}^*=C_{y,x}$ and the spectral quantum Yang--Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function $C_{xy}$ in the case $d=2$ determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its $T$ -deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.Let $H$ be a separable Hilbert space and $T$ be a self-adjoint bounded linear operator on $H^{\otimes 2}$ with norm $\le1$, satisfying the Yang--Baxter equation. Bo\.zejko and Speicher (1994) proved that the operator $T$ determines a $T$-deformed Fock space $\mathcal F(H)=\bigoplus_{n=0}^\infty\mathcal F_n(H)$. We start with reviewing and extending the known results about the structure of the $n$-particle spaces $\mathcal F_n(H)$ and the commutation relations satisfied by the corresponding creation and annihilation operators acting on $\mathcal F(H)$. We then choose $H=L^2(X\to V)$, the $L^2$-space of $V$-valued functions on $X$. Here $X:=\mathbb R^d$ and $V:=\mathbb C^m$ with $m\ge2$. Furthermore, we assume that the operator $T$ acting on $H^{\otimes 2}=L^2(X^2\to V^{\otimes 2})$ is given by $(Tf^{(2)})(x,y)=C_{x,y}f^{(2)}(y,x)$. Here, for a.a.\ $(x,y)\in X^2$, $C_{x,y}$ is a linear operator on $V^{\otimes 2}$ with norm $\le1$ that satisfies $C_{x,y}^*=C_{y,x}$ and the spectral quantum Yang--Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function $C_{xy}$ in the case $d=2$ determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its $T$-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.

Nov 7, 201910 citations

Anyons as Elementary Field Quanta

WORLD SCIENTIFIC eBooksMar 1, 19980 citations

Some Properties of Anyons

H. Mori

Springer proceedings in physicsJan 1, 19920 citations

Let

H

be a separable Hilbert space and

T

be a self-adjoint bounded linear operator on

H^{\otimes 2}

with norm

\le1

, satisfying the Yang--Baxter equation. Bo\.zejko and Speicher (1994) proved that the operator

T

determines a

T

-deformed Fock space

\mathcal F(H)=\bigoplus_{n=0}^\infty\mathcal F_n(H)

. We start with reviewing and extending the known results about the structure of the

n

-particle spaces

\mathcal F_n(H)

and the commutation relations satisfied by the corresponding creation and annihilation operators acting on

\mathcal F(H)

. We then choose

H=L^2(X\to V)

, the

L^2

-space of

V

-valued functions on

X

. Here

X:=\mathbb R^d

and

V:=\mathbb C^m

with

m\ge2

. Furthermore, we assume that the operator

T

acting on

H^{\otimes 2}=L^2(X^2\to V^{\otimes 2})

is given by

(Tf^{(2)})(x,y)=C_{x,y}f^{(2)}(y,x)

. Here, for a.a.\

(x,y)\in X^2

C_{x,y}

is a linear operator on

V^{\otimes 2}

with norm

\le1

that satisfies

C_{x,y}^*=C_{y,x}

and the spectral quantum Yang--Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function

C_{xy}

in the case

d=2

determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its

T

-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.