Quaternionic Lorentz Group and Dirac Equation

We formulate Lorentz group representations in which ordinary complex numbers are replaced by linear functions of real quaternions and introduce dotted and undotted quaternionic one-dimensional spinors. To extend to parity the space-time transformations, we combine these one-dimensional spinors into bi-dimensional column vectors. From the transformation properties of the two-component spinors, we derive a quaternionic chiral representation for the space-time algebra. Finally, we obtain a quaternionic bi-dimensional version of the Dirac equation.     

Quantum Minkowski Space and Generators of Quantum Lorentz Group

From the basic 4 × 4 R matrix associated with the quantum Lorentz group SL_q(2, C) and its various fusion matrices, the covariant differential calculus on the quantum Minkowski space and the R commutation relation for the covariant generators of quantum Lorents group are presented.     

Differential Calculus on Quantum Lorentz Group

In this paper, we discuss the bicovariant differential calculus on quantum Lorentz group, and provide corresponding de Rham complex and Maurer-Cartan formulae.     

The Boltzmann Equation for a One-Dimensional Quantum Lorentz Gas

We study the macroscopic behavior of a quantum particle under the action of randomly distributed scatterers on the real line. Each scatterer generates a δ-potential. We prove that, in the low density limit, the Wigner function of the system converges to a probability distribution satisfying a classical linear Boltzmann equation, with a scattering cross section computed according to the Quantum Mechanical rules. Received: 2 April 1998 / Accepted: 12 February 1999     

Quantum Random Walks and Thermalisation

It is shown how to construct quantum random walks with particles in an arbitrary faithful normal state. A convergence theorem is obtained for such walks, which demonstrates a thermalisation effect: the limit cocycle obeys a quantum stochastic differential equation without gauge terms. Examples are presented which generalise that of Attal and Joye (J Funct Anal 247:253–288, 2007).     

Open Quantum Random Walks

A new model of quantum random walks is introduced, on lattices as well as on finite graphs. These quantum random walks take into account the behavior of open quantum systems. They are the exact quantum analogues of classical Markov chains. We explore the “quantum trajectory” point of view on these quantum random walks, that is, we show that measuring the position of the particle after each time-step gives rise to a classical Markov chain, on the lattice times the state space of the particle. This quantum trajectory is a simulation of the master equation of the quantum random walk. The physical pertinence of such quantum random walks and the way they can be concretely realized is discussed. Differences and connections with the already well-known quantum random walks, such as the Hadamard random walk, are established.     

Random Time-Dependent Quantum Walks

We consider the discrete time unitary dynamics given by a quantum walk on the lattice \mathbb Z^d{\mathbb {Z}^d} performed by a quantum particle with internal degree of freedom, called coin state, according to the following iterated rule: a unitary update of the coin state takes place, followed by a shift on the lattice, conditioned on the coin state of the particle. We study the large time behavior of the quantum mechanical probability distribution of the position observable in \mathbb Z^d{\mathbb {Z}^d} when the sequence of unitary updates is given by an i.i.d. sequence of random matrices. When averaged over the randomness, this distribution is shown to display a drift proportional to the time and its centered counterpart is shown to display a diffusive behavior with a diffusion matrix we compute. A moderate deviation principle is also proven to hold for the averaged distribution and the limit of the suitably rescaled corresponding characteristic function is shown to satisfy a diffusion equation. A generalization to unitary updates distributed according to a Markov process is also provided.     

Quantum Walks on Hypergraphs

In this work we introduce the concept of a quantum walk on a hypergraph. We show that the staggered quantum walk model is a special case of a quantum walk on a hypergraph.

     

Weyl Geometries, Fisher Information and Quantum Entropy in Quantum Mechanics

It is known that quantum mechanics can be interpreted as a non-Euclidean deformation of the space-time geometries by means Weyl geometries. We propose here a dynamical explanation of such approach by deriving Bohm potential from minimum condition of Fisher information connected to the entropy of a quantum system.     

On a new condition distinguishing Weyl and Lorentz space-times

The paper is concerned with the derivability of a Lorentz instead of only a Weyl manifold as space-time structure from postulates about free fall and light propagation. For this purpose it identifies a property distinguishing both kinds of space-times. The property is one of a particular metric of the conformal class of the Weyl manifold. viz. that in suitably chosen locally geodesic coordinates theg _i4 components,i=1, 2, 3 vanish along the time axis. Although seemingly somewhat hidden, one is led to this property in looking for a metric which can play a distinguished role. We demonstrate that for a Lorentzian manifold such a condition is always given; thus it is a necessary one. It is sufficient since for a Weyl space it has the consequence that the metric connection of the selectedg is projectively equivalent to the Weyl connection. Thus, if a Weyl space-time complies with it, it is a reducible one. The results of this paper lay the ground for deriving in a second step this condition from a simple, empirically testable postulate about free-fall worldlines and “radar” measurements by light signals.     

Hash Function Based on Quantum Walks

Higher security and lower collision rate have always been people’s pursuits in the construction of hash functions. We consider a quantum walk where a walker is driven by two coins alternately. At each step, a message bit decides whether to swap two coins. In this way, a keyed hash function is constructed. Theoretically infinite possibilities of the initial parameters as the key ensure the security of the proposed hash function against the unforgery and collision resistance. Finally, we establish a generic quantum walk-based hash function model and give a guide in constructing hash functions in quantum walk architecture. It also provides a clue for the construction of other quantum walk-based cryptography protocols.

     

Quantum Entanglement Under Lorentz Transformation

We study the properties of quantum entanglement in moving frames, with a non-maximally entangled bipartite state: $|\phi\rangle=\sqrt{1-\varepsilon}|{\uparrow\uparrow}\rangle +\sqrt{\varepsilon}|{\downarrow\downarrow}\rangle$ , (0<ε<1). We calculate the concurrence of this state under Lorentz transformation and show that if the momenta part of the spin-1/2 pair is appropriately correlated, the concurrence is invariant ( $\mathcal {C}(\rho)=2\sqrt{\varepsilon-\varepsilon^{2}}$ ); despite the entanglement of this state is not maximal, there is no transfer of entanglement between spin and momentum.     

Quantum deformation of Lorentz group

A one parameter quantum deformationS L(2,) ofSL(2,) is introduced and investigated. An analog of the Iwasawa decomposition is proved. The compact part of this decomposition coincides withS U(2), whereas the solvable part is identified as a Pontryagin dual ofS U(2). It shows thatS L(2,) is the result of the dual version of Drinfeld's double group construction applied toS U(2). The same construction applied to any compact quantum groupG _c is discussed in detail. In particular the explicit formulae for the Haar measures on the Pontryagin dualG _d ofG _c and on the double groupG are given. We show that there exists remarkable 1-1 correspondence between representations ofG and bicovariant bimodules (tensor bundles) overG _c . The theory of smooth representations ofS L(2,) is the same as that ofSL(2,) (Clebsh-Gordon coefficients are however modified). The corresponding tame bicovariant bimodules onS U(2) are classified. An application to 4D ₊ differential calculus is presented. The nonsmooth case is also discussed.     

Weyl gauging and spontaneously broken Lorentz symmetry: Some alternative models

Some new effective actions are suggested for theories in which the affine connection is not completely specified by the metric. The new actions lead to models in which the metric, torsion, and Weyl vector fields all propagate. The dimensionally reduced versions do not contain third derivatives of the gauge potentials in the field equation. Some simple models which exhibit simultaneous breaking of Weyl andD-dimensional Lorentz symmetry are investigated. It is shown that it is possible for this effect to occur in any model in which the field action contains the Einstein-Hilbert term. This is due to the fact that the contortion occurs in this object as part of an indefinite quadratic form.