Results 1  10
of
25
From quantum cellular automata to quantum lattice gases
 Journal of Statistical Physics
, 1996
"... A natural architecture for nanoscale quantum computation is that of a quantum cellular automaton. Motivated by this observation, in this paper we begin an investigation of exactly unitary cellular automata. After proving that there can be no nontrivial, homogeneous, local, unitary, scalar cellular a ..."
Abstract

Cited by 109 (21 self)
 Add to MetaCart
A natural architecture for nanoscale quantum computation is that of a quantum cellular automaton. Motivated by this observation, in this paper we begin an investigation of exactly unitary cellular automata. After proving that there can be no nontrivial, homogeneous, local, unitary, scalar cellular automaton in one dimension, we weaken the homogeneity condition and show that there are nontrivial, exactly unitary, partitioning cellular automata. We find a one parameter family of evolution rules which are best interpreted as those for a one particle quantum automaton. This model is naturally reformulated as a two component cellular automaton which we demonstrate to limit to the Dirac equation. We describe two generalizations of this automaton, the second of which, to multiple interacting particles, is the correct definition of a quantum lattice gas. KEY WORDS: quantum cellular automaton; quantum lattice gas; quantum computation. to appear in J. Stat. Phys.
Quantum geometry with intrinsic local causality
, 1997
"... The space of states and operators for a large class of background independent theories of quantum spacetime dynamics is defined. The SU(2) spin networks of quantum general relativity are replaced by labelled compact twodimensional surfaces. The space of states of the theory is the direct sum of the ..."
Abstract

Cited by 40 (17 self)
 Add to MetaCart
The space of states and operators for a large class of background independent theories of quantum spacetime dynamics is defined. The SU(2) spin networks of quantum general relativity are replaced by labelled compact twodimensional surfaces. The space of states of the theory is the direct sum of the spaces of invariant tensors of a quantum group Gq over all compact (finite genus) oriented 2surfaces. The dynamics is background independent and locally causal. The dynamics constructs histories with discrete features of spacetime geometry such as causal structure and multifingered time. For SU(2) the theory satisfies the Bekenstein bound and the holographic hypothesis is recast in this formalism.
Causal evolution of spin networks
 Nucl. Phys
, 1997
"... A new approach to quantum gravity is described which joins the loop representation formulation of the canonical theory to the causal set formulation of the path integral. The theory assigns quantum amplitudes to special classes of causal sets, which consist of spin networks representing quantum stat ..."
Abstract

Cited by 31 (6 self)
 Add to MetaCart
A new approach to quantum gravity is described which joins the loop representation formulation of the canonical theory to the causal set formulation of the path integral. The theory assigns quantum amplitudes to special classes of causal sets, which consist of spin networks representing quantum states of the gravitational field joined together by labeled null edges. The theory exists in 3+1, 2+1 and 1+1 dimensional versions, and may also be interepreted as a theory of labeled timelike surfaces. The dynamics is specified by a choice of functions of the labelings of d+1 dimensional simplices,which represent elementary future light cones of events in these discrete spacetimes. The quantum dynamics thus respects the discrete causal structure of the causal sets. In the 1 + 1 dimensional case the theory is closely related to directed percolation models. In this case, at least, the theory may have critical behavior associated with percolation, leading to the existence of a classical limit.
Simplicial Euclidean and Lorentzian Quantum Gravity
"... One can try to define the theory of quantum gravity as the sum over geometries. In two dimensions the sum over Euclidean geometries can be performed constructively by the method of dynamical triangulations. One can define a propertime propagator. This propagator can be used to calculate generalized ..."
Abstract

Cited by 19 (5 self)
 Add to MetaCart
One can try to define the theory of quantum gravity as the sum over geometries. In two dimensions the sum over Euclidean geometries can be performed constructively by the method of dynamical triangulations. One can define a propertime propagator. This propagator can be used to calculate generalized HartleHawking amplitudes and it can be used to understand the the fractal structure of quantum geometry. In higher dimensions the philosophy of defining the quantum theory, starting from a sum over Euclidean geometries, regularized by a reparametrization invariant cut off which is taken to zero, seems not to lead to an interesting continuum theory. The reason for this is the dominance of singular Euclidean geometries. Lorentzian geometries with a global causal structure are less singular. Using the framework of dynamical triangulations it is possible to give a constructive definition of the sum over such geometries, In two dimensions the theory can be solved analytically. It differs from twodimensional Euclidean quantum gravity, and the relation between the two theories can be understood. In three dimensions the theory avoids the pathologies of threedimensional Euclidean quantum gravity. General properties of the fourdimensional discretized theory have been established, but a detailed study of the continuum limit in the spirit of the renormalization group and asymptotic safety is till awaiting.
Nonperturbative dynamics for abstract (p,q) string networks, Phys.Rev. D58
, 1998
"... We describe abstract (p,q) string networks which are the string networks of Sen without the information about their embedding in a background spacetime. The nonperturbative dynamical formulation invented for spin networks, in terms of causal evolution of dual triangulations, is applied on them. The ..."
Abstract

Cited by 11 (6 self)
 Add to MetaCart
We describe abstract (p,q) string networks which are the string networks of Sen without the information about their embedding in a background spacetime. The nonperturbative dynamical formulation invented for spin networks, in terms of causal evolution of dual triangulations, is applied on them. The formal transition amplitudes are sums over discrete causal histories that evolve (p,q) string networks. The dynamics depend on two free SL(2,Z) invariant functions which describe the amplitudes for the local evolution moves.
2000 Statistical Lorentzian geometry and the closeness of Lorentzian manifolds
 6944–6958, and grqc/0002053
"... I introduce a family of closeness functions between causal Lorentzian geometries of finite volume and arbitrary underlying topology. When points are randomly scattered in a Lorentzian manifold, with uniform density according to the volume element, some information on the topology and metric is encod ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
I introduce a family of closeness functions between causal Lorentzian geometries of finite volume and arbitrary underlying topology. When points are randomly scattered in a Lorentzian manifold, with uniform density according to the volume element, some information on the topology and metric is encoded in the partial order that the causal structure induces among those points; one can then define closeness between Lorentzian geometries by comparing the sets of probabilities they give for obtaining the same posets. If the density of points is finite, one gets a pseudodistance, which only compares the manifolds down to a finite volume scale, as illustrated here by a fully worked out example of two 2dimensional manifolds of different topology; if the density is allowed to become infinite, a true distance can be defined on the space of all Lorentzian geometries. The introductory and concluding sections include some remarks on the motivation for this definition and its applications to quantum gravity. PACS numbers 04.20.Gz, 02.40.k Running head: Statistical Lorentzian geometry 1 I.
Unitarity in one dimensional nonlinear quantum cellular automata
"... Unitarity of the global evolution is an extremely stringent condition on finite state models in discrete spacetime. Quantum cellular automata, in particular, are tightly constrained. In previous work we proved a simple Nogo Theorem which precludes nontrivial homogeneous evolution for linear quantum ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
Unitarity of the global evolution is an extremely stringent condition on finite state models in discrete spacetime. Quantum cellular automata, in particular, are tightly constrained. In previous work we proved a simple Nogo Theorem which precludes nontrivial homogeneous evolution for linear quantum cellular automata. Here we carefully define general quantum cellular automata in order to investigate the possibility that there be nontrivial homogeneous unitary evolution when the local rule is nonlinear. Since the unitary global transition amplitudes are constructed from the product of local transition amplitudes, infinite lattices require different treatment than periodic ones. We prove Unitarity Theorems for both cases, expressing the equivalence in 1+1 dimensions of global unitarity and certain sets of constraints on the local rule, and then show that these constraints can be solved to give a variety of multiparameter families of nonlinear quantum cellular automata. The Unitarity Theorems, together with a Surjectivity Theorem for the infinite case, also imply that unitarity is decidable for one dimensional cellular automata.
Strings as perturbations of evolving spinnetworks
"... A connection between nonperturbative formulations of quantum gravity and perturbative string theory is exhibited, based on a formulation of the nonperturbative dynamics due to Markopoulou. In this formulation the dynamics of spin network states and their generalizations is described in terms of hi ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
A connection between nonperturbative formulations of quantum gravity and perturbative string theory is exhibited, based on a formulation of the nonperturbative dynamics due to Markopoulou. In this formulation the dynamics of spin network states and their generalizations is described in terms of histories which have discrete analogues of the causal structure and many fingered time of Lorentzian spacetimes. Perturbations of these histories turn out to be described in terms of spin systems defined on 2dimensional timelike surfaces embedded in the discrete spacetime. When the history has a classical limit which is Minkowski spacetime, the action of the perturbation theory is given to leading order by the spacetime area of the surface, as in bosonic string theory. This map between a nonperturbative formulation of quantum gravity and a 1+1 dimensional theory generalizes to a large class of theories in which the group SU(2) is extended to any quantum group or supergroup. It is argued that a necessary condition for the nonperturbative theory to have a good classical limit is that the resulting 1+1 dimensional theory defines a consistent and stable perturbative string theory.
The future of spin networks
, 1997
"... Since Roger Penrose rst introduced the notion of a spin network as a simple model of discrete quantum geometry, they have reappeared in quantum gauge theories, quantum gravity, topological quantum eld theory and conformal eld theory. The roles that spin networks play in these contexts are brie y des ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
Since Roger Penrose rst introduced the notion of a spin network as a simple model of discrete quantum geometry, they have reappeared in quantum gauge theories, quantum gravity, topological quantum eld theory and conformal eld theory. The roles that spin networks play in these contexts are brie y described, with an emphasis on the question of the relationships among them. It is also argued that spin networks and their generalizations provide a language which may lead to a uni cation of the di erent approaches to quantum gravity and quantum geometry. This leads to a set of conjectures about the form of a future theory that may be simultaneously an extension of the nonperturbative quantization of general relativity and a nonperturbative formulation of string theory.