q-deformed phase space and its lattice structure

A q-deformed two-dimensional phase space is studied as a model for a noncommutative phase space. A lattice structure arises that can be interpreted as a spontaneous breaking of a continuous symmetry. The eigenfunctions of a Hamiltonian that lives on such a lattice are derived as wavefunctions in ordinaryx-space.     

q-deformed Lorentz-algebra in Minkowski phase space

In the present paper we show that the Lorentz algebra as defined in [5] is isomorphic to an algebra closely related to a q-deformed algebra. On this algebra we define a Hopf algebra structure and show its action on q-spinor modules. This algebra is related to the q-deformed Minkowski space algebra by a non invertible factorisation. Received: 12 June 1998 / Published online: 5 October 1998     

The geometry of a q-deformed phase space

The geometry of the q-deformed line is studied. A real differential calculus is introduced and the associated algebra of forms represented on a Hilbert space. It is found that there is a natural metric with an associated linear connection which is of zero curvature. The metric, which is formally defined in terms of differential forms, is in this simple case identifiable as an observable. Received: 26 November 1998 / Published online: 27 April 1999     

Free particle in q-deformed configuration space

The SO _q (N)-invariant Schrödinger equation for the free particle is formulated in polar coordinates as a partial differential equation in noncommutative geometry. For each value of the total angular momentum, a Hilbert space of radial functions is constructed as the space of normalizable functions respective to the q-integral. The spectrum of the Hamiltonian is found to be discrete.     

q-deformed Minkowski space based on a q-Lorentz algebra

The Hilbert space representations of a non-commutative -deformed Minkowski space, its momenta and its Lorentz boosts are constructed. The spectrum of the diagonalizable space elements shows a lattice-like structure with accumulation points on the light-cone. Received: 23 January 1997 / Published online: 10 March 1998     

Magnetic phase structure on the Penrose lattice

The Ising model on a two-dimensional Penrose tiling is studied by means of the Migdal-Kadanoff scheme. This approximate renormalization method closely follows the inflation rules of the tiling, which are easily described in terms of Robinson triangles, and lead to the consideration of four types of nearest neighbor couplings. The ferromagnetic phase transition is similar to the usual one encountered on periodic lattices. When the couplings have both signs, the presence of frustration without randomness yields a fairly intricate phase diagram, essentially made up of two regions with a very complicated border. Region I consists of quasiferromagnetic models, which exhibit long-range order below some finite critical temperature. The models of region II are paramagnetic at nonzero (low) temperature, but may become ordered (reen-trant phases) in a higher temperature range.     

The phase structure of mixed lattice gauge theories

The mixed compact-non-compact U(1) model is shown to be equivalent to a compact U(1) Higgs model. It is argued that the mixed SO(3)-SU(2) model is dual to an SO(3) gauge theory coupled to a scalar field in the fundamental representation. The degrees of freedom are Z(2) monopoles and charges or, in a dual picture, monopoles and loops. This picture is supported by a Monte Carlo calculation. The implications for the SU(2) transition region are discussed.     

The q-deformed vector and q-deformed outer product

In this paper the q-deformed vector is introduced and the q-deformed outer product is investigated.     

Twisted mass quarks and the phase structure of lattice QCD

The phase structure of zero temperature twisted mass lattice QCD is investigated. We find strong metastabilities in the plaquette observable in correspondence of which the untwisted quark mass assumes positive or negative values. We provide interpretations of this phenomenon in terms of chiral symmetry breaking and the effective potential model of Sharpe and Singleton.Received: 24 August 2004, Revised: 29 October 2004, Published online: 25 January 2005     

Discrete phase space II. Relativistic lattice wave equations and divergence-free green's functions

Therelativistic lattice Klein-Gordon equation, Dirac equation, electromagnetic equations, and gauge field equations are presented as partialdifference equations. Various lattice Green's functions are constructed (except for non-abelian gauge fields). It is proved that many of the lattice Green's functions are non-singular or divergence-free. Moreover, it is conjectured that all lattice Green's functions are non-singular.     

Theory of q-deformed forms. I. q-deformed alternating tensor and q-deformed wedge product

In this paper we deal with the q-deformed alternating tensor and prove the associativity of the q-deformed wedge product. Moreover, we construct the theory of q-deformed homology in order to prove the q-deformed Stokes theorem. Lastly we prove the q-deformed Poincaré lemma.     

Theory of q-deformed forms. II. q-deformed differential forms and q-deformed Hamilton equation

In this paper we introduce the q-deformed differential forms and quantum-algebra-valued q-deformed forms. We use these to obtain the q-inner derivative and investigate its properties. As a physical application we discuss the q-deformed Hamilton equation.     

Product structure of heat phase space and branching Brownian motion

A generical formalism for the discussion of Brownian processes with non-constant particle number is developed, based on the observation that the phase space of heat possesses a product structure that can be encoded in a commutative unit ring. A single Brownian particle is discussed in a Hilbert module theory, with the underlying ring structure seen to be intimately linked to the non-differentiability of Brownian paths. Multi-particle systems with interactions are explicitly constructed using a Fock space approach. The resulting ring-valued quantum field theory is applied to binary branching Brownian motion, whose Dyson-Schwinger equations can be exactly solved. The presented formalism permits the application of the full machinery of quantum field theory to Brownian processes.     

The chiral WZNW phase space and its Poisson-Lie groupoid

The precise relationship between the arbitrary monodromy dependent 2-form appearing in the chiral WZNW symplectic form and the ‘exchange r-matrix' that governs the corresponding Poisson brackets is established. Generalizing earlier results related to diagonal monodromy, the exchange r-matrices are shown to satisfy a new dynamical generalization of the classical modified Yang-Baxter equation, which is found to admit an interpretation in terms of (new) Poisson-Lie groupoids. Dynamical exchange r-matrices for which right multiplication yields a classical or a Poisson-Lie symmetry on the chiral WZNW phase space are presented explicitly.