Space-time ergodic properties of systems of infinitely many independent particles

We investigate the ergodic properties of the equilibrium states of systems of infinitely many particles with respect to the group generated by space translations and time evolution. The particles are assumed to move independently in a periodic external field. We show that insofar as good thermodynamic behavior is concerned these properties provide much sharper discrimination than the ergodic properties of the time evolution alone. In particular, we show that though the infinite ideal gas is mixing in the space-time framework, it has vanishing space-time entropy and fails to be a space-timeK-system. On the other hand, if the particles interact with fixed convex scatterers (the Lorentz gas) the system forms a space-timeK-system. Also, the space-time entropy of a system of the type we consider is shown to equal its time entropy per unit volume.Research supported in part by the National Science Foundation Grant No. GP-16147 A No. 1.     

Interacting relativistic boson fields in the De Sitter universe with two space-time dimensions

The positive temperature Gibbs state of a scalar boson field with a relativistic local self-interaction in two space-time dimensional Minkowski universe as constructed in [1] is not relativistic invariant. We prove in this paper that the corresponding state in the De Sitter universe is actually relativistic invariant if the temperature is given byT=1/2πR whereR is the constant radius of curvature of the De Sitter universe. Moreover the construction gives that the Schwinger functions or imaginary time Wightman functions are the moments of a generalized Markoff process on the sphere of radiusR.     

Electromagnetic fluctuations in the dielectric slab resonator

Electromagnetic fluctuations between two parallel (distanceL) infinite extended and perfectly conducting plates are studied. The space between the plates may be homogeneously filled with an isotropic dielectric and magnetic medium. For small values ofLT (T=temperature) the influence of the imaginary part of the dielectric function on the spectral correlations of the radiation field is discussed.The local spectral energy density is calculated. It is shown that the energy density isr-independent for media without dispersion. In this case ther-dependent part of the trace of the electric correlation tensor cancels with the analogous part of the magnetic correlation tensor.For the empty slab the complete spatial and temporal correlation tensors are derived. Two representations suitable for small and large values ofLT are discussed. These representations are expansions arround the 2-dimensional radiation field, i.e. no modes with wave vectors perpendicular to the plates are excited and the half space limit, respectively. Curves of the relative deviation of the electric correlation functions from the infinite isotropic space limit and curves of the degree of anisotropy of the radiation field as functions of the distance to the metallic boundary are displayed.     

An algebraic characterization of vacuum states in Minkowski space

An algebraic characterization of vacuum states on nets ofC ^*-algebras over Minkowski space is given and space-time translations are reconstructed with the help of the modular structures associated with such states. The result suggests that a condition of geometrical modular action might hold in quantum field theories on a wider class of spacetime manifolds.     

A Kinetic Model of Quantum Jumps

A new class of models describing the dissipative dynamics of an open quantum system S by means of random time evolutions of pure states in its Hilbert space is considered. The random evolutions are linear and defined by Poisson processes. At the random Poissonian times, the wavefunction experiences discontinuous changes (quantum jumps). These changes are implemented by some non-unitary linear operators satisfying a locality condition. If the Hilbert space of S is infinite dimensional, the models involve an infinite number of independent Poisson processes and the total frequency of jumps may be infinite. We show that the random evolutions in are then given by some almost-surely defined unbounded random evolution operators obtained by a limit procedure. The average evolution of the observables of S is given by a quantum dynamical semigroup, its generator having the Lindblad form.⁽¹⁾ The relevance of the models in the field of electronic transport in Anderson insulators is emphasised.     

Deformations of Quantum Field Theories and Integrable Models

Deformations of quantum field theories which preserve Poincaré covariance and localization in wedges are a novel tool in the analysis and construction of model theories. Here a general scenario for such deformations is discussed, and an infinite class of explicit examples is constructed on the Borchers-Uhlmann algebra underlying Wightman quantum field theory. These deformations exist independently of the space-time dimension, and contain the recently studied warped convolution deformation as a special case. In the special case of two-dimensional Minkowski space, they can be used to deform free field theories to integrable models with non-trivial S-matrix.     

Spin substructure of space-time

Space-time is provided with an underlying SL(2,C) spin space with complex noncommutative spinor coordinatesC ^A which satisfy . It is shown that the orbital angular momentum operator has a realization in this space as a derivative operator which can take on half-integral spin values, and the graded Lie algebra which describes the structure of the spin space is discussed. The spin-space translations mix fermi and bose fields and produce space-time translations which are not nil-potent. A hermitean metric with a line element which is invariant under such localx-dependent translations is introduced.     

The :φ2: Field in theP(φ)2 model

Euclidean Field Theory techniques are used to study the Schwinger functions and characteristic function of the :φ²: field in evenP(φ)₂ models. The infinite volume limit is obtained for Half-Dirichlet boundary conditions by means of correlation inequalities. Analytic continuation yields Lorentz invariant Wightman functions. It is shown that, in the infinite volume limit, <:φ(x)²:>≧0 for both the Half and the Full-Dirichlet (λφ⁴)₂ model. This result also holds for a finite volume with periodic boundary conditions.     

General Dynamical Equations for Free Particles and?Their Galilean Invariance

Dynamical equations describing evolution of state functions in space-time of a given metric are important components of physical theories of particles. A method based on a group of the metric is used to obtain an infinite set of general dynamical equations for a scalar and analytical function representing free and spinless particles. It is shown that this set of equations is the same for any group of the metric that consists of an invariant Abelian subgroup of translations in time and space. For Galilean space-time, such group is the extended Galilei group. Using this group, it is proved that the infinite set of equations has only one subset of Galilean invariant dynamical equations, and that the equations of this subset are Schr?dinger-like equations.     

On the stochastic quantization of field theory

We give a rigorous construction of a stochastic continuumP()₂ model in finite Euclidean space-time volume. It is obtained by a weak solution of a non-linear stochastic differential equation in a space of distributions. The resulting Markov process has continuous sample paths, and is ergodic with the finite volume EuclideanP()₂ measure as its unique invariant measure. The procedure may be called stochastic field quantization.Laboratoire Associé 280 au CNRSSupported in part by GNSM and INFN     

Dispersion in momentum space and the existence ofU(t,t')

The operatorU(t,t') giving transition probabilities between finite times or connecting free and interacting fields does not exist (apart from the ultraviolet divergence problem) because of the 3-translation invariance of current quantum field theory. To remedy this, the idealization that one has an infinite timeT = to prepare initial, or measure final,n-particle momentum eigenstates is discarded here. It is shown that random space-time (which itself eliminates ultraviolet divergences from field theory) implies and fixes uniquely a random momentum space if free particle momentaK are determined by time-of-flight measurements withT < . In particular, the dispersion ofK m/T, where is the space-time dispersion andm is the particle mass. Stochastic momentum space is incorporated into field theory in a preliminary way; because 3-translation form-invariance is slightly violated, the unitaryU-operator expressed as the usualT-exponential exists and the limitU S ast ,t' – is welldefined withoutad hoc tricks like the adiabatic cut-off. A frame-dependence is necessarily introduced into fields andU-operator, and the transformation properties expressing Lorentz covariance are of the same more general type encountered in previous work on quantum field theory over stochastic spacetime.     

Constants of motion in local field theory (Coleman's theorem revisited)

We analyze the space integralsQ=d ³ x(x) of finitely localized densities . It turns out that the time translated operatorsQ(t) are polynomials int ifQ annihilates the vacuum. In particular,Q(t) =Q in models with short-range forces and complete particle interpretation. These results are valid in the Haag-Araki framework of field theory as well as in the Wightman formalism. Lorentz covariance is not needed in the proofs.     

Rationality of Conformally Invariant Local Correlation Functions on Compactified Minkowski Space

Rationality of the Wightman functions is proven to follow from energy positivity, locality and a natural condition of global conformal invariance (GCI) in any number D of space-time dimensions. The GCI condition allows to treat correlation functions as generalized sections of a vector bundle over the compactification of Minkowski space M and yields a strong form of locality valid for all non-isotropic intervals if assumed true for space-like separations. Received: 20 October 2000 / Accepted: 5 December 2000     

Perturbation theory of Wightman functions

A perturbative expansion of the Wightman functions, and more generally of vacuum expectation values of products of time-ordered and anti-time-ordered products, is derived for ₄ ⁴ field theory. The result is expressed as a sum over generalized Feynman graphs. The derivation is based exclusively on the equation of motion and the Wightman axioms. Neither canonical commutation relations nor asymptotic conditions are needed at any point. In the zero-mass case the individual graphs are infrared divergent, but the sum over all graphs of a given order is convergent.     

Physics of the Schwinger model

The Coulomb gas of massless fermions (Schwinger model) is solved in a one-dimensional space of finite lengthL using the boson representation of fermions. Special attention is paid to boundary effects and global degrees of freedom. It is shown that the mean current is not conserved, but oscillates. The theory is constructed in all charge sectors. The Wightman functions are calculated and the limitL is discussed.Work performed within the research program of the Sonderforschungsbereich 125, Aachen-Jülich-Köln     

Quantization of space-time and the corresponding quantum mechanics

An axiomatic framework for describing general space-time models is presented. Space-time models to which irreducible propositional systems belong as causal logics are quantum (q) theoretically interpretable and their event spaces are Hilbert spaces. Such aq space-time is proposed via a canonical quantization. As a basic assumption, the time t and the radial coordinate r of aq particle satisfy the canonical commutation relation [t,r]=±i . The two cases will be considered simultaneously. In that case the event space is the Hilbert space L²(³). Unitary symmetries consist of Poincaré-like symmetries (translations, rotations, and inversion) and of gauge-like symmetries. Space inversion implies time inversion. Thisq space-time reveals a confinement phenomenon: Theq particle is confined in an size region of Minkowski space at any time. One particle mechanics overq space-time provides mass eigenvalue equations for elementary particles. Prugoveki's stochasticq mechanics andq space-time offer a natural way for introducing and interpreting consistently such aq space-time andq particles existing in it. The mass eigenstates ofq particles generate Prugoveki's extended elementary particles. When 0, these particles shrink to point particles and is recovered as the classical (c) limit ofq space-time. Conceptual considerations favor the case [t,r]=+i , and applications in hadron physics give the fit 2/5 fermi/GeV.This paper is a revised version of the author's work, Quantization of Space-time and the Corresponding Quantum Mechanics (Part I), report KFKI-1981-48.     

Special conformal symmetries in general relativity

A geometrical discussion of special conformal vector fields in space-time is given. In particular, it is shown that if such a vector field is admitted, it is unique up to a constant scaling and the addition of a homothetic or a Killing vector field. In the case when the gradient of the conformal scalar associated with is non-null it is shown that other homothetic and affine symmetries are necessarily admitted by the space-time, that an intrinsic family of 2-dimensional flat submanifolds is determined in the space-time, that is, in general, hypersurface orthogonal and that the space-time, if non-flat, is necessarily (geodesically) incomplete. Other geometrical features of such space-times are also considered.     

A general class of quantum fields without cut-offs in two space-time dimensions

We consider a selfinteracting boson field in two space-time dimensions, with interaction densities of the form:V((x)): where (x) is a scalar boson field, andV() is a real positive function of exponential type. We define the space cut-off interaction by and prove thatH _r =H ₀+V _r , whereH ₀ is the free energy, is essentially self adjoint. This permits us to take away the space cut-off and we obtain a quantum field free of cut-offs.At leave from Mathematical Institute, Oslo University.This research partially sponsored by the Air Force Office of Scientific Research under Contract AF 49(638)1545.