Probability Representation of Quantum Observables and Quantum States

We introduce the probability distributions describing quantum observables in conventional quantum mechanics and clarify their relations to the tomographic probability distributions describing quantum states. We derive the evolution equation for quantum observables (Heisenberg equation) in the probability representation and give examples of the spin-1/2 (qubit) states and the spin observables. We present quantum channels for qubits in the probability representation.     

Internal geometry of hadron resonances

Motivated by previous work on high-energy quantum mechanics, a simple model is devised to study the internal geometry of hadron resonances. In this model we assume new basic canonical commutation relations between the (internal) coordinate and momentum operators of the hadronic quantum system. By systematically imposing Lie algebra commutation relations between these and other observables, we discuss the free and bound particle problems, identifying in each case the corresponding internal symmetries. For the bound particle problem, which models quark confinement, this symmetry turns out to be characterized by Dirac's two-oscillator representation of theO(3, 2) de Sitter group.     

Generalizations of Heisenberg uncertainty relation

A survey on the generalizations of Heisenberg uncertainty relation and a general scheme for their entangled extensions to several states and observables is presented. The scheme is illustrated on the examples of one and two states and canonical quantum observables, and spin and quasi-spin components. Several new uncertainty relations are displayed. Received 10 October 2001 / Received in final form 6 March 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: dtrif@inrne.bas.bg     

Quantum localization observables and accelerated frames

We define quantum observables associated with Einstein localization in space-time. These observables are built on Poincaré and dilatation generators. Their commutators are given by spin observables defined from the same symmetry generators. Their shifts under transformations to uniformly accelerated frames are evaluated through algebraic computations in conformal algebra. Spin number is found to vary under such transformations with a variation involving further observables introduced as irreducible quadrupole momenta. Quadrupole observables may be dealt with as non commutative polarizations which allow one to define step operators increasing or decreasing the spin number by unity. Received: 11 December 1997 / Revised: 1st July 1998 / Accepted: 1st September 1998     

Combinatorial Space from Loop Quantum Gravity

The canonical quantization of diffeomorphism invariant theories of connections in terms of loop variables is revisited. Such theories include general relativity described in terms of Ashtekar-Barbero variables and extension to Yang-Mills fields (with or without fermions) coupled to gravity. It is argued that the operators induced by classical diffeomorphism invariant or covariant functions are respectively invariant or covariant under a suitable completion of the diffeomorphism group. The canonical quantization in terms of loop variables described here, yields a representation of the algebra of observables in a separable Hilbert space. Furthermore, the resulting quantum theory is equivalent to a model for diffeomorphism invariant gauge theories which replaces space with a manifestly combinatorial object.     

General approach to quantum entanglement

We examine quantum entanglement as a physical phenomenon independent of specific problems of quantum information technologies. Within the dynamic symmetry approach, we briefly discuss the role of quantum fluctuations in formation of entangled states, including single-particle entanglement, relativity of entanglement with respect to the choice of basic observables, and stabilization of robust entanglement.     

Quantum electrodynamics on a space-time lattice

A Minkowski-lattice version of quantum electrodynamics (or rather its simplified version, with matter described by a scalar field) is constructed. Quantum fields are consequently described in a gauge-independent way, i.e. the algebra of quantum observables of the theory is generated by gauge-invariant operators assigned to zero-, one-, and two-dimensional elements of the lattice. The operators satisfy canonical commutation relations. The uniqueness of representation of this algebra is proved. Field dynamics is formulated in terms of difference equations imposed on the field operators. It is obtained from a discrete version of the path-integral. The theory is local and causal.     

On the general covariance and strong equivalence principles in quantum general relativity

The various physical aspects of the general relativistic principles of covariance and strong equivalence are discussed, and their mathematical formulations are analyzed. All these aspects are shown to be present in classical general relativity, although no contemporary formulation of canonical or covariant quantum gravity has succeeded to incorporate them all. This has, in part, motivated the recent introduction of a geometro-stochastic framework for quantum general relativity, in which the classical frame bundles that underlie the formulation of parallel transport in classical general relativity are replaced by quantum frame bundles. It is shown that quantum frames can take over the role played by complete sets of observables in conventional quantum theory, so that they can mediate the natural transference of the general covariance and the strong equivalence principles from the classical to the quantum general relativistic regime. This results in a geometrostochastic mode of quantum propagation in general relativistic quantum bundles, which is mathematically implemented by path integration methods based on parallel transport along horizontal lifts of geodesics for the vacuum expectation values of a quantum gravitational field in a quantum spacetime supermanifold. The covariance features of this field are embedded in a quantum gravitational supergroup, which incorporates Poincaré as well as diffeomorphism invariance, and resolves the issue of time in quantum gravity.     

Symmetry and Evolution in Quantum Gravity

We propose an operator constraint equation for the wavefunction of the Universe that admits genuine evolution. While the corresponding classical theory is equivalent to the canonical decomposition of General Relativity, the quantum theory contains an evolution equation distinct from standard Wheeler–DeWitt cosmology. Furthermore, the local symmetry principle—and corresponding observables—of the theory have a direct interpretation in terms of a conventional gauge theory, where the gauge symmetry group is that of spatial conformal diffeomorphisms (that preserve the spatial volume of the Universe). The global evolution is in terms of an arbitrary parameter that serves only as an unobservable label for successive states of the Universe. Our proposal follows unambiguously from a suggestion of York whereby the independently specifiable initial data in the action principle of General Relativity is given by a conformal geometry and the spatial average of the York time on the spacelike hypersurfaces that bound the variation. Remarkably, such a variational principle uniquely selects the form of the constraints of the theory so that we can establish a precise notion of both symmetry and evolution in quantum gravity.     

The quantum geometry of spin and statistics

Both, spin and statistics of a quantum system can be seen to arise from underlying (quantum) group symmetries. We show that the spin–statistics theorem is equivalent to a unification of these symmetries. Besides covering the Bose–Fermi case we classify the corresponding possibilities for anyonic spin and statistics. We incorporate the underlying extended concept of symmetry into quantum field theory in a generalised path integral formulation capable of handling general braid statistics. For bosons and fermions the different path integrals and Feynman rules naturally emerge without introducing Grassmann variables. We also consider the anyonic example of quons and obtain the path integral counterpart to the usual canonical approach.     

The physical meaning of canonical quantization in terms of reference systems

The canonical approach to general relativity in terms of reference systems is discussed to show that Einstein's principles of equivalence and general relativity imply the physical insignificance of quantized general relativity. In particular it is demonstrated that even the (anholonomic) flat-space canonical formalism leads to physically uninterpretable results. This lack of quantum content of general relativity is reflected by Rosenfeld's uncertainty relations and can especially be removed by modifying general relativity in the spirit of classical Einstein-Cartan theory with teleparallelism.     

Symmetry groups and non-abelian cohomology

We consider the implementation of symmetry groups of automorphisms of an algebra of observables in a reducible representation whose multipliers in general are non-commuting operators in the commutant of the representation. The multipliers obey a non-abelian cocycle relation which generalizes the 2-cohomology of the group. Examples are given from the theory of spin algebras and continuous tensor products. For typeI representations we show that the multiplier can be chosen to lie in the centre, giving an isomorphism with abelian theory.Dedicated to Res Jost and Arthur Wightman     

Construction of field algebras with quantum symmetry from local observables

It has been discussed earlier that (weak quasi-) quantum groups allow for a conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid-) statistics and locality was established. This work addresses the reconstruction of quantum symmetries and algebras of field operators. For every algebraA of observables satisfying certain standard assumptions, an appropriate quantum symmetry is found. Field operators are obtained which act on a positive definite Hilbert space of states and transform covariantly under the quantum symmetry. As a substitute for Bose/Fermi (anti-) commutation relations, these fields are demonstrated to obey a local braid relation.     

Midisuperspace Models of Canonical Quantum Gravity

A midisuperspace model is a field theoryobtained by symmetry reduction of a parent gravitationaltheory. Such models have proven useful for exploring theclassical and quantum dynamics of the gravitational field. I present three recent classes ofresults pertinent to canonical quantization of vacuumgeneral relativity in the context of midisuperspacemodels. (1) I give necessary and sufficient conditions such that a given symmetry reduction can beperformed at the level of the Lagrangian or Hamiltonian.(2) I discuss the Hamiltonian formulation of modelsbased upon cylindrical and toroidal symmetry. In particular, I explain how these models can beidentified with parametrized field theories of wavemaps; thus a natural strategy for canonical quantizationis available. (3) The quantization of a parametrized field theory, such as the midisuperspace modelsconsidered in (2), requires construction of a quantumfield theory on a fixed (flat) spacetime that allows fortime evolution along arbitrary foliations of spacetime. I discuss some recent results on thepossibility of finding such a quantum fieldtheory.     

Making classical and quantum canonical general relativity computable through a power series expansion in the inverse cosmological constant

We consider general relativity with a cosmological constant as a perturbative expansion around a completely solvable diffeomorphism invariant field theory. This theory is the lambda --> infinity limit of general relativity. This allows an explicit perturbative computational setup in which the quantum states of the theory and the classical observables can be explicitly computed. An unexpected relationship arises at a quantum level between the discrete spectrum of the volume operator and the allowed values of the cosmological constant.     

相对论粒子的自旋算符

发展了关于相对论态自旋算符的系统理论.考虑了具有非零静质量的粒子情况.对带自旋的相对论粒子,通常的自旋算符需换为相对论的自旋算符.在Poincar啨群不可约表示的框架里,构造了适用于粒子任意正则态的自旋算符,称为运动自旋.本文的讨论限于量子力学.随后将在量子场论中对此作进一步深入研究.     

Consistent canonical quantization of general relativity in the space of vassiliev invariants

We present a quantization of the Hamiltonian and diffeomorphism constraint of canonical quantum gravity in the spin network representation. The novelty consists in considering a space of wave functions based on the Vassiliev invariants. The constraints are finite, well defined, and reproduce at the level of quantum commutators the Poisson algebra of constraints of the classical theory. A similar construction can be carried out in 2+1 dimensions leading to the correct quantum theory.