A.J. Macfarlane

On q-analogues of the quantum harmonic oscillator and the quantum group SU(2)q

The quantum group SU(2)q is discussed by a method analogous to that used by Schwinger to develop the quantum theory of angular momentum. Such theory of the q-analogue of the quantum harmonic oscillator, as is required for this purpose, is developed.

Invariant tensors for simple groups

The forms of the invariant primitive tensors for the simple Lie algebras Al, Bl, Cl, and Dl are investigated. A new family of symmetric invariant tensors is introduced using the non-trivial cocycles for the Lie algebra cohomology. For the Al algebra it is explicitly shown that the generic forms of these tensors become zero except for the l primitive ones and that they give rise to the l primitive Casimir operators. Some recurrence and duality relations are given for the Lie algebra cocycles. Tables for the 3- and 5-cocycles for su(3) are su(4) are also provided. Finally, new relations involving the d and f su(n) tensors are given.

Generalised harmonic oscillator systems and their Fock space description

Abstract We describe a new generalisation of the quantum harmonic oscillator, which involves a deformation parameter q and a modification parameter ν, and its Fock space Fq,ν. We discuss spectrum generating algebras and hidden supersymmetry properties of Fq,ν and of various limiting cases, such as the Calogero model.

Geometrical foundations of fractional supersymmetry

A deformed q-calculus is developed on the basis of an algebraic structure involving graded brackets. A number operator and left and right shift operators are constructed for this algebra, and the whole structure is related to the algebra of a q-deformed boson. The limit of this algebra when q is an nth root of unity is also studied in detail. By means of a chain rule expansion, the left and right derivatives are identified with the charge Q and covariant derivative D encountered in ordinary/fractional supersymmetry, and this leads to new results for these operators. A generalized Berezin integral and fractional superspace measure arise as a natural part of our formalism. When q is a root of unity the algebra is found to have a nontrivial Hopf structure, extending that associated with the anyonic line. One-dimensional ordinary/fractional superspace is identified with the braided line when q is a root of unity, so that one-dimensional ordinary/fractional supersymmetry can be viewed as invariance under translation along this line. In our construction of fractional supersymmetry the q-deformed bosons play a role exactly analogous to that of the fermions in the familiar supersymmetric case.

Algebraic structure of parabose Fock space. I. The Green’s ansatz revisited

A description of the Fock space of a parabose oscillator of order p based on commutation relations bilinear in creation and annihilation operators rather than trilinear ones is developed. A new statement of the well‐known Green’s ansatz is introduced to help understand the significance of the bilinear commutation relations. It is also applied to show how all parabose oscillators of even order (respectively, odd order) can be obtained as irreducible constituents of the Fock space associated with a Green’s ansatz of two (three) terms. Analogs of the form in which the parabose Green’s ansatz is presented are provided for one parafermi oscillator and for systems of many parabose and parafermi oscillators of the same order. Methods used in the paper allow various new results for q‐deformed parabose oscillators to be derived.

The quantum mechanics of the supersymmetric nonlinear σ-model

The classical and quantum mechanical formalisms of the models are developed. The quantisation is done in such a way that the quantum theory can be represented explicitly in as simple a form as possible, and the problem of ordering of operators is resolved so as to maintain the supersymmetry algebra of the classical theory.

On characteristic equations, trace identities and Casimir operators of simple Lie algebras

Two approaches are developed to exploit, for simple complex or compact real Lie algebras g, the information that stems from the characteristic equations of representation matrices and Casimir operators. These approaches are selected so as to be viable not only for “small” Lie algebras, but also to be suitable for treatment by computer algebra. A very large body of new results emerges in the forms of (a) identities of a tensorial nature, involving structure constants etc. of g, (b) trace identities for powers of matrices of the adjoint and defining representations of g, (c) expressions of nonprimitive Casimir operators of g in terms of primitive ones. The methods are sufficiently tractable to allow not only explicit proof by hand of the nonprimitive nature of the quartic Casimir of g2, f4, e6, but also, e.g., of that of the tenth order Casimir of f4.

On Integrable Models Related to the osp(1, 2) Gaudin Algebra

The osp(1,2) Gaudin algebra is defined and integrable models described by it are considered. The models include the osp(1,2) Gaudin magnet and the Dicke model related to it. Detailed discussion of the simplest cases of these models is presented. The effect of the presence of fermions on the separation of variables is indicated.

Quantum theory of compact and non-compact σ models

Abstract We discuss quantization of SO( N + 1) σ models and CP N models, and of certain non-compact counterparts, SO( N , 1) and QP N respectively, of these, both in canonical operator formalism and the covariant path integral formulation, showing the equivalence of the two approaches. We discuss also a class of supersymmetric σ models formulated in d ⩽ 3 dimensions and apply the results to the SO( N + 1) and SO( N , 1) cases. This allows us to calculate the Witten index in each case. For SO(2 l + 1,1) we thereby find supersymmetry breaking. However, for SO(2 l , 1), we find supersymmetry is unbroken. Moreover, there is no unique ground state, invariant under SO(2 l , 1), rather an infinite multiple of zero energy states, carrying a unitary irreducible representation of the non-compact SO(2 l , 1) group. We discuss also field theoretic aspects of the models in d ⩾ 2 dimensions, stressing differences of the non-compact to the compact cases. These include infrared instead of ultraviolet asymptotic freedom, lack of an energy gap, failure (in the QP N case) of the auxiliary vector field to become dynamical. A further conclusion that is argued concerns the absence of a consistent particle interpretation for the QP N model in exactly two dimensions. For d > 2 the non-compact symmetry of QP N is broken down to the compact subgroup.

Quantum group structure in a fermionic extension of the quantum harmonic oscillator

We discuss the role osp(12) plays for a simple generalisation SE of the harmonic oscillator, showing how a quantum group structure, with universal R-matrix, arises out of it in the treatment of composite “many SE” systems.