Using the theory of fibre bundles, we provide several equivalent intrinsic descriptions for the Hilbert spaces of $n$ ``free'' nonrelativistic and relativistic plektons in two space dimensions. These spaces carry a ray representation of the Galilei group and a unitary representation of the Poincaré group respectively. In the relativistic case we also discuss the situation where the braid group is replaced by the ribbon braid group.
Using the theory of fibre bundles, we provide several equivalent intrinsic descriptions for the Hilbert spaces of n ``free'' nonrelativistic and relativistic plektons in two space dimensions. These spaces carry a ray representation of the Galilei group and a unitary representation of the Poincaré group respectively. In the relativistic case we also discuss the situation where the braid group is replaced by the ribbon braid group.
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 Galilean symmetry of a free particle in one-dimensional space is examined under the scope of non-Abelian quantum kinematics. Within the Hilbert space that carries the regular ray representation of the Galilei group the Schrödinger operator appears as one of the three fundamental invariants of the extended kinematic algebra. By means of a superselection rule the physical Hilbert subspaces of the system are identified, in which a complementary ray representation of the Galilean transformation produces the time-dependent Schrödinger equation, and the Feynman space-time propagator. The quantization approach used in this paper is purely group theoretic and relativistic.
This paper is devoted to the study of the Galilei group and its representations. The Galilei group presents a certain number of essential differences with respect to the Poincaré group. As Bargmann showed, its physical representations, here explicitly constructed, are not true representations but only up-to-a-factor ones. Consequently, in nonrelativistic quantum mechanics, the mass has a very special role, and in fact, gives rise to a superselection rule which prevents the existence of unstable particles. The internal energy of a nonrelativistic system is known to be an arbitrary parameter; this is shown to come also from Galilean invariance, because of a nontrivial concept of equivalence between physical representations. On the contrary, the behavior of an elementary system with respect to rotations, is very similar to the relativistic case. We show here, in particular, how the number of polarization states reduces to two for the zero-mass case (though in fact there are no physical zero-mass systems in nonrelativistic mechanics). Finally, we study the two-particle system, where the orbital angular momenta quite naturally introduce themselves through the decomposition of the tensor product of two physical representations.
We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or section along paths in this bundle. The evolution of a pure state is determined through the bundle (analog of the) Schrödinger equation. Now the dynamical variables and density operator are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this work. This is the fifth, closing, part of our investigation. We briefly discuss the observer's role in the theory and different realizations of the space–time model used as a base space in the bundle approach to quantum mechanics. The exact conditions for the equivalence of Hilbert bundle and Hilbert space formulations of the theory are pointed out. A table of comparison between both descriptions of nonrelativistic quantum mechanics is presented. We discuss some principal moments of the Hilbert bundle description and show that as a scheme it is more general than the Hilbert space one. Different directions for further research are pointed out.
We realize the physical N-anyon Hilbert spaces, introduced previously via unitary representations of the group of diffeomorphisms of the plane, as N-fold braided-symmetric tensor products of the 1-particle Hilbert space. This perspective provides a convenient Fock space construction for nonrelativistic anyon quantum fields along the more usual lines of boson and fermion fields, but in a braided category, and clarifies how discrete (lattice) anyon fields relate to anyon fields in the continuum. We also see how essential physical information is encoded. In particular, we show how the algebraic structure of the anyonic Fock space leads to a natural anyonic exclusion principle related to intermediate occupation number statistics, and obtain the partition function for an idealized gas of fixed anyonic vortices.
After a schematic examination of the ’’physical’’ representations of the Galilei group, a quantum mechanical description is drawn from a ’’laboratory’’ one in the case of a free elementary system with spin, assuming that the kinematical group is the Galilei group. In fact, the ’’laboratory states’’ of the system naturally fit in a Galilei−Hilbert bundle. Then, according to the general theory of the unitary representations of groups in the framework of Hilbert bundles which is outlined in this paper, a unitary representation of the Galilei group is constructed which is shown to contain all the physical Galilei representations. The quantum mechanical description which has been obtained in this way gives rise to an algebra of observables by means of a general procedure which connects a unitary representation of a Lie group with a Hilbert representation of the corresponding Lie algebra. The inner energy is shown to be a superobservable for the system under consideration. Moreover, the kinematical properties of such a system are independent of the values of the inner energy.