Algebra of Observables and Charge Superselection Sectors for QED on the Lattice

Quantum Electrodynamics on a finite lattice is investigated within the hamiltonian approach. First, the structure of the algebra of lattice observables is analyzed and it is shown that the charge superselection rule holds. Next, for every eigenvalue of the total charge operator a canonical irreducible representation is constructed and it is proved that all irreducible representations corresponding to a fixed value of the total charge are unique up to unitary equivalence. The physical Hilbert space is by definition the direct sum of these superselection sectors. Finally, lattice quantum dynamics in the Heisenberg picture is formulated and the relation of our approach to gauge fixing procedures is discussed. Received: 21 October 1996 / Accepted: 10 February 1997     

Quantum Charges and Spacetime Topology: The Emergence of New Superselection Sectors

A new form of superselection sectors of topological origin is developed. By that it is meant a new investigation that includes several extensions of the traditional framework of Doplicher, Haag and Roberts in local quantum theories. At first we generalize the notion of representations of nets of C*–algebras, then we provide a brand new view on selection criteria by adopting one with a strong topological flavour. We prove that it is coherent with the older point of view, hence a clue to a genuine extension. In this light, we extend Roberts’ cohomological analysis to the case where 1–cocycles bear non-trivial unitary representations of the fundamental group of the spacetime, equivalently of its Cauchy surface in the case of global hyperbolicity. A crucial tool is a notion of group von Neumann algebras generated by the 1–cocycles evaluated on loops over fixed regions. One proves that these group von Neumann algebras are localized at the bounded region where loops start and end and to be factorial of finite type I. All that amounts to a new invariant, in a topological sense, which can be defined as the dimension of the factor. We prove that any 1–cocycle can be factorized into a part that contains only the charge content and another where only the topological information is stored. This second part much resembles what in literature is known as geometric phases. Indeed, by the very geometrical origin of the 1–cocycles that we discuss in the paper, they are essential tools in the theory of net bundles, and the topological part is related to their holonomy content. At the end we prove the existence of net representations. Dedicated to Klaus Fredenhagen on the occasion of his sixtieth birthday     

Conformal Covariance and Positivity of Energy in Charged Sectors

It has been recently noted that diffeomorphism covariance of a Chiral Conformal QFT in the vacuum sector automatically ensures Möbius covariance in all charged sectors. In this article it is shown that diffeomorphism covariance and positivity of the energy in the vacuum sector even ensure the positivity of energy in the charged sectors.The main observation of this paper is that the positivity of energy — at least in case of a Chiral Conformal QFT — is a local concept: it is related to the fact that the energy density, when smeared with some local nonnegative test functions, remains bounded from below (with the bound depending on the test function).The presented proof relies in an essential way on recently developed methods concerning the smearing of the stress-energy tensor with nonsmooth functions.     

General Covariance in General Relativity?

The mathematical approach to General Relativity insists that all coordinate systems are equal. However physicists and astrophysicists in fact almost always use preferred coordinate systems not merely to simplify the calculations but also to help define quantities of physical interest. This suggests we should reconsider and perhaps refine the dogma of General Covariance.     

Superselection variables and generalized multipliers

We discuss and solve the problem of phase factor families that arises in the representation theory of the symmetries of an elementary physical system, the latter being expressed in terms of its propositional lattice.     

Superselection rules in quantum theory: Part I. A new proposal for state restriction violation

It is argued that preparation of a quantum state characterized by density operator not commuting with a superselection operatorQ does not by itself constitute an instance of superselection rule violation. It would, however, be an instance of state restriction violation. It is held that superselection rule violation is only possible with simultaneous observable and state restriction violations. It is shown that it is a priori conceivable to subdivide an ensemble whose satisfies[, Q] = 0 into subensembles whose density operators violate the state restrictions. The dynamics of the subdivision process is not considered.     

The Principle of Covariance and the Hamiltonian Formulation of General Relativity

The implications of the general covariance principle for the establishment of a Hamiltonian variational formulation of classical General Relativity are addressed. The analysis is performed in the framework of the Einstein-Hilbert variational theory. Preliminarily, customary Lagrangian variational principles are reviewed, pointing out the existence of a novel variational formulation in which the class of variations remains unconstrained. As a second step, the conditions of validity of the non-manifestly covariant ADM variational theory are questioned. The main result concerns the proof of its intrinsic non-Hamiltonian character and the failure of this approach in providing a symplectic structure of space-time. In contrast, it is demonstrated that a solution reconciling the physical requirements of covariance and manifest covariance of variational theory with the existence of a classical Hamiltonian structure for the gravitational field can be reached in the framework of synchronous variational principles. Both path-integral and volume-integral realizations of the Hamilton variational principle are explicitly determined and the corresponding physical interpretations are pointed out.     

Extensions of Conformal Nets¶and Superselection Structures

Starting with a conformal Quantum Field Theory on the real line, we show that the dual net is still conformal with respect to a new representation of the M?bius group. We infer from this that every conformal net is normal and conormal, namely the local von Neumann algebra associated with an interval coincides with its double relative commutant inside the local von Neumann algebra associated with any larger interval. The net and the dual net give together rise to an infinite dimensional symmetry group, of which we study a class of positive energy irreducible representations. We mention how superselection sectors extend to the dual net and we illustrate by examples how, in general, this process generates solitonic sectors. We describe the free theories associated with the lowest weight n representations of , showing that they violate 3-regularity for $n > 2. When n≥ 2, we obtain examples of non M?bius-covariant sectors of a 3-regular (non 4-regular) net. Received: 19 March 1997 / Accepted: 1 July 1997     

Superselection rules in quantum theory: Part II. Subensemble selection

A dynamical analysis of standard procedures for subensemble selection is used to show that the state restriction violation proposal in Part I of the paper cannot be realized by employing familiar correlation schemes. However, it is shown that measurement of an observable not commuting with the superselection operator is possible, a violation of the observable restrictions. This is interpreted as supporting the position that each of these restrictions is sufficient but not necessary for the superselection rule. The results do constitute a proposal for superselection rule violation in theories requiring both restrictions, e.g., the axiomatic treatment by Bogolubov, Logunov, and Todorov. It is also concluded that superselection rules place restrictions on procedures for selective state preparations using correlations. More generally, it is conjectured that a mathematically conceivable decomposition of a given density operator does not necessarily represent a possibility for partitioning of the corresponding ensemble into subensembles by any physically realizable means.     

Nonlinear neural networks. I. General theory

A neural network is called nonlinear if the introduction of new data into the synaptic efficacies has to be performed through anonlinear operation. The original Hopfield model is linear, whereas, for instance, clipped synapses constitute a nonlinear model. Here a general theory is presented to obtain the statistical mechanics of a neural network with finitely many patterns and arbitrary (symmetric) nonlinearity. The problem is reduced to minimizing a free energy functional over all solutions of a fixed-point equation with synaptic kernelQ. The case of clipped synapses with bimodal and Gaussian probability distribution is analyzed in detail. To this end, a simple theory is developed that gives the spectrum ofQ and thereby all the solutions that bifurcate from the high-temperature phase.     

Strength distributions and statistical spectroscopy. I. General theory

The strength distribution for an arbitrary excitation is given in terms of a double expansion, and its sum rules by single expansions, in polynomials defined by the initial and final energy spectra. In model spaces which are not too large, a rapid convergence, to within fluctuations, is assured by the action of a central limit theorem, as is shown in particular by considering the response of the system under infinitesimal deformations of the Hamiltonian. When larger spaces are decomposed into subspaces defined by a partitioning of the single-particle space a similar convergence results. At the same time, close contact is made with, and important corrections are found to, intuitive procedures which are often used for approximating strength distributions. The general features of the distribution are often easily understood in termsof a simple geometry made effective in the model space by the central limit theorem, and further features by exploiting the connection of this geometry to the unitary group of transformations in the single-particle space. Extensions are given for multipole strengths and sum rules, appropriate when the angular momenta (and isospins) are specified for the states involved in the transitions. Measures for the RMS fluctuations in the sum-rule quantities, and correlations between them, are given by combining the low-order-polynomial (statistically smoothed) strengths with an assumed Porter-Thomas distribution for the (high-order) strength fluctuations.     

Superselection sectors with braid group statistics and exchange algebras

The theory of superselection sectors is generalized to situations in which normal statistics has to be replaced by braid group statistics. The essential role of the positive Markov trace of algebraic quantum field theory for this analysis is explained, and the relation to exchange algebras is established.     

Radiative energy transfer I. General equations

A set of equations is derived which makes possible to study the radiative energy transfer process whereby the photons emitted by the energy donor are absorbed by the energy acceptor and so increase the efficiency of the overall energy transfer. It is shown that the coefficients describing the radiative transfer which appear in the expressions for the intensities of the energy donor and the energy acceptor are not the same, due to the fact that part of the fluorescence absorbed by the acceptor comes from radiation which is not detected as donor emission when there is no acceptor present. The general equations derived are applied to two particular cases commonly considered: measurements in reflection, where the fluorescence emission is observed from the same face of the absorption and measurements in transmission where the fluorescence emission is observed from the opposite face of the cell.     

Symmetry breaking boundaries I. General theory

We study conformally invariant boundary conditions that break part of the bulk symmetries. A general theory is developed for those boundary conditions for which the preserved subalgebra is the fixed algebra under an abelian orbifold group. We explicitly construct the boundary states and reflection coefficients as well as the annulus amplitudes. Integrality of the annulus coefficients is proven in full generality.     

Scattering from non-overlapping potentials. I. General formulation

The problem of scattering from an assembly of non-overlapping spherical potentials is solved in partial-wave basis for each of the constituent potentials. The resulting scattering operator is a quotient of two infinite matrices and depends on “on-shell” partial wave amplitudes of the individual potentials. It suggests in general a truncation scheme which essentially considers only those partial waves effective for each collision at the given energy. The multiple-scattering series is recovered and limiting cases of low energy and high energy are considered. Applications to high-energy scattering of elementary particles on nuclei are briefly discussed.