Some applications of dilatation invariance to structural questions in the theory of local observables

In a dilatation-invariant theory it is shown that there is a unique locally normal dilatation-invariant state. Furthermore a gauge transformation of a local algebra cannot be implemented by a unitary operator from the local algebra. If the local field algebras are factors then so are the local observable algebras. The superselection structure of the theory can be determined locally.     

Momentum and Hamiltonian operators in generalized coordinates

The self-adjointness of momentum operators in generalized coordinates, questioned by Domingos and Caldeira is shown. The momentum operators of a particle and the kinetic part of its Hamiltonian operator constructed from them are characterized as self-adjoint operators and geometrical objects in coordinate-free form. Local coordinates of ann-dimensional Riemannian manifold are taken as the generalized coordinates of the particle. As an example the curvilinear coordinates of Euclidean space are treated. The coefficients of connection and curvature are given on the manifold for which the assumed momentum operators exist. It is found that if our momentum operators form a complete set of mutually commuting observables, the manifold is locally Euclidean, i.e., there exists a local coordinate system such that we obtain the usual Schrödinger correspondence rule.     

A note on essential duality

By considering some simple models, it is shown that the essential duality condition for local nets of von Neumann algebras associated with Wightman fields need not be fulfilled if Lorentz covariance is dropped. These models illustrate a point made by Borchers in the proof of his two-dimensional CPT theorem for local nets: The Lorentz covariant net constructed from the wedge algebras of a given two-dimensional net may not be unique. It is also shown that in higher dimensions, the Lorentz boosts constructed by means of the modular groups of wedge algebras may act nonlocally in the directions parallel to the edge of the wedge.     

On local functions of fields

Properties of local functions of fields are discussed. A condition, called the Borchers condition, is introduced which is weaker than duality but allows the construction of a maximal local extension of a system of local algebras. This extension will satisfy duality. The local structure of the generalized free field is studied, and it is shown that duality does not hold for the local algebras associated with certain generalized free fields, whereas the Borchers condition is satisfied for all generalized free fields. The appendix contains an elementary proof of duality for the free field.     

Hyper-parahermitian manifolds with torsion

Necessary and sufficient conditions for the existence of a hyper-parahermitian connection with totally skew-symmetric torsion (HPKT-structure) are presented. It is shown that any HPKT-structure is locally generated by a real (potential) function. An invariant first order differential operator is defined on any almost hyper-paracomplex manifold showing that it is two-step nilpotent exactly when the almost hyper-paracomplex structure is integrable. A local HPKT-potential is expressed in terms of this operator. Examples of (locally) invariant HPKT-structures with closed as well as non-closed torsion 3-form on a class of (locally) homogeneous hyper-paracomplex manifolds (some of them compact) are constructed.     

Punctured Haag Duality in Locally Covariant Quantum Field Theories

We investigate a new property of nets of local algebras over 4-dimensional globally hyperbolic spacetimes, called punctured Haag duality. This property consists in the usual Haag duality for the restriction of the net to the causal complement of a point p of the spacetime. Punctured Haag duality implies Haag duality and local definiteness. Our main result is that, if we deal with a locally covariant quantum field theory in the sense of Brunetti, Fredenhagen and Verch, then also the converse holds. The free Klein-Gordon field provides an example in which this property is verified.     

Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics

Given the local observables in the vacuum sector fulfilling a few basic principles of local quantum theory, we show that the superselection structure, intrinsically determined a priori, can always be described by a unique compact global gauge group acting on a field algebra generated by field operators which commute or anticommute at spacelike separations. The field algebra and the gauge group are constructed simultaneously from the local observables. There will be sectors obeying parastatistics, an intrinsic notion derived from the observables, if and only if the gauge group is non-Abelian. Topological charges would manifest themselves in field operators associated with spacelike cones but not localizable in bounded regions of Minkowski space. No assumption on the particle spectrum or even on the covariance of the theory is made. However the notion of superselection sector is tailored to theories without massless particles. When translation or Poincaré covariance of the vacuum sector is assumed, our construction leads to a covariant field algebra describing all covariant sectors.Research supported by Ministero della Pubblica Istruzione and CNR-GNAFA     

A gravitating SO(3, 1) gauge field

In this article, we postulate SO(3, 1) as a local symmetry of any relativistic theory. This is equivalent to assuming the existence of a gauge field associated with this noncompact group. This SO(3, 1) gauge field is the spinorial affinity which usually appears when we deal with weighting spinors, which, as is well known, cannot be coupled to the metric tensor field. Furthermore, according to the integral approach to gauge fields proposed by Yang, it is also recognized that in order to obtain models of gravity we have to introduce ordinary affinities as the gauge field associated with GL(4) (the local symmetry determined by the parallel transport). Thus if we assume both L(4) and SO(3, 1) as local independent symmetries we are led to analyze the dynamical gauge system constituted by the Einstein field interacting with the SO(3, 1) Weyl-Yang gauge field. We think this system is a possible model of strong gravity. Once we give the first-order action for this Einstein-Weyl-Yang system we study whether the SO(3, 1) gauge field could have a tetrad associated with it. It is also shown that both fields propagate along a unique characteristic cone. Algebraic and differential constraints are solved when the system evolves along a null coordinate. The unconstrained expression for the action of the system is found working in the Bondi gauge. That allows us to exhibit an explicit expression of the dynamical generator of the system. Its signature turns out to be nondefinite, due to the nondefinite contribution of the Weyl-Yang field, which has the typical spinorial behavior. A conjecture is made that such an unpleasant feature could be overcome in the quantized version of this model.     

Positivity of Wightman functionals and the existence of local nets

The paper is concerned with the existence of a local net of von Neumann algebras associated with a given Wightman field. For fields satisfying a generalizedH-bound the existence of such a net is shown to be equivalent to a certain positivity property of the Wightman distributions.     

On Hot Bangs and the Arrow of Time in Relativistic Quantum Field Theory

 A recently proposed method for the characterization and analysis of local equilibrium states in relativistic quantum field theory is applied to a simple model. Within this model states are identified which are locally (but not globally) in thermal equilibrium and it is shown that their local thermal properties evolve according to macroscopic equations. The largest space–time regions in which local equilibrium states can exist are timelike cones. Thus, although the model does not describe dissipative effects, such states fix in a natural manner a time direction. Moreover, generically they determine a distinguished space–time point where a singularity in the temperature (a hot bang) must have occurred if local equilibrium prevailed thereafter. The results illustrate how the breaking of the time reflection symmetry at macroscopic scales manifests itself in a microscopic setting. Received: 17 January 2003 / Accepted: 5 March 2003 Published online: 17 April 2003 Communicated by H. Araki and K. Fredenhagen     

Gauge fields, point interactions and few-body problems in one dimension

Point interactions for the second derivative operator in one dimension are studied. Every operator from this family is described by the boundary conditions which include a 2x2 real matrix with the unit determinant and a phase. The role of the phase parameter leading to unitarily equivalent operators is discussed in the present paper. In particular, it is shown that the phase parameter is not redundant (contrary to previous studies) if nonstationary problems are concerned. It is proven that the phase parameter can be interpreted as the amplitude of a singular gauge field. Considering the few-body problem we extend the range of parameters for which the exact solution can be found using the Bethe Ansatz.     

On intrinsic randomness of dynamical systems

We discuss the problem of nonunitary equivalence, via positivity-preserving similarity transformations, between the unitary groups associated with deterministic dynamical evolution and semigroups associated with stochastic processes. Dynamical systems admitting such nonunitary equivalence with stochastic Markov processes are said to beintrinsically random. In a previous work, it was found that the so-called Bernoulli systems (discrete time) are intrinsically random in this sense. This result is extended here by showing that a more general class of dynamical systems—the so-calledK systems andK flows—are intrinsically random. The connection of intrinsic randomness with local instability of motion is briefly discussed. We also show that Markov processes associated through nonunitary equivalence tononisomorphic K flows are necessarily non-isomorphic.Dr. Goldstein's research was supported in part by NSF Grant No. PHY78-03816.     

Relativistic temperature

A measurement theory for the temperature of relativistic systems is developed. The resulting operational approach is shown to be quasi-local and therefore may be applicable in general Riemannian manifolds even when there are temperature gradients which induce heat flows. The surprising feature of our analysis is that it leads to a bifurcation of the temperature concept into two distinctly different measurable quantities: one a frame invariant scalar field which a local co-moving observer would tend to identify with the local temperature and employ in the definition of entropy, the other a frame dependent, but nevertheless locally determinable quantity which governs the flow of heat and the ability to extract work. The two quantities differ by the bookkeeping methodology employed to calibrate the thermometer. A simple relationship between the two temperatures can be established if a preferred Killing vector field is available in the Riemannian manifold.     

Uncertainty Relations for Coherence

Quantum mechanical uncertainty relations are fundamental consequences of the incompatible nature of noncommuting observables. In terms of the coherence measure based on the Wigner-Yanase skew information, we establish several uncertainty relations for coherence with respect to von Neumann measurements, mutually unbiased bases(MUBs), and general symmetric informationally complete positive operator valued measurements(SIC-POVMs),respectively. Since coherence is intimately connected with quantum uncertainties, the obtained uncertainty relations are of intrinsically quantum nature, in contrast to the conventional uncertainty relations expressed in terms of variance,which are of hybrid nature(mixing both classical and quantum uncertainties). From a dual viewpoint, we also derive some uncertainty relations for coherence of quantum states with respect to a fixed measurement. In particular, it is shown that if the density operators representing the quantum states do not commute, then there is no measurement(reference basis) such that the coherence of these states can be simultaneously small.     

Invariants for a Class of Nongeneric Three-Qubit States

We investigate the equivalence of quantum states under local unitary transformations. A complete set of invariants under local unitary transformations are presented for a class of non-generic three-qubit mixed states. It is shown that two such states in this class are locally equivalent if and only if all these invariants have equal values for them.     

Anomalous angular momenta in a quantised theory of monopoles

In this paper a gauge theory with a classical solution corresponding to a magnetic monopole is quantised. By careful handling of the zero frequency modes it is shown that the monopole is capable of absorbing both momentum and charge. The angular momentum operator is considered and it is shown that if the original theory contains an isodoublet scalar field, the quantum excitations may be half odd integer eigenvalue eigenstates of this operator.     

Nonlinear connections for curved twistor spaces

A holomorphic connection on (1, 0)-vector fields which is intrinsically defined on any curved twistor space is described. Although it is a local operator, it is given in terms of the nonlocal geometry of the twistor space corresponding to the local geometry of the spacetime. The connection is represented in local coordinates by a system ofnonlinear first-order partial differential operators. It has torsion but no curvature. A parallelism is given explicitly, and an example is computed.     

Equivalence between non-localizable and local fields

We discuss the nature of non-localisable fields constructed as certain limits of sequences of local fields. For sequences for which the corresponding Wightman functions converge we construct a PCT operator; if the sequences converge strongly in a given Hilbert space then a scattering theory can be constructed for the non-localisable limit field. Such fields are shown to have the sameS-operator as any local field which has the defining sequence of local fields in its Borchers class, and has the same in field. We give non-trivial examples of this equivalence between local and non-localisable fields.     

Uniform topologies and strong operator topologies on polynomial algebras and on the algebra of CCR

In the paper uniform topologies and strong operator topologies on the free polynomial algebra in n Hermitian indeterminants, on the polynomial algebra in n commuting Hermitian indeterminants and on the *-algebra generated by the CCR (finite number of degrees of freedom) are investigated. It is proved that the strongest locally convex topology on these algebras is a uniform topology and a strong operator topology. For the polynomial algebra in one variable it is shown that on each algebraical realization as an Op*-algebra by an unbounded operator, the strongest locally convex topology coincides with the uniform topology. If in addition the realization is closed, then also the strong operator topology is equal to the strongest locally convex topology.     

Space-like symmetrization operator and local commutativity axiom

The correct definition of the space-like symmetrization operator is given. Using this operator, an extension of the Fock space is constructed. In this space an operator function, which satisfies the local commutativity axiom in a more complete sense than the free field, is defined. It is shown that this operator function satisfies all the other axioms of the field theory, except the spectral one.