Saturation of confining forces in a relativistic quark model for the baryons

A relativistic quark model for the baryons with saturating three-particle forces is investigated. The properties of relativistic three-fermion amplitudes are analyzed with respect to Lorentz transformations and permutations. Six different classes of possible structures in spin space are found. They serve as an appropriate basis for the classification and calculation of spin-dependent interactions. The quantum numbers and amplitudes for the orbital part are determined for Euclidean relative vectors with help of the irreducible representations of the groups SU(4) SO(4) SO(3). These kinematical results together with the Green's function techniques of relativistic quantum field theory are applied to a Bethe-Salpeter model for the binding of three heavy quarks inside a baryon. We give as an example a confining saturating interaction which yields baryon quantum numbers similar to those of the non-relativistic harmonic oscillator model. However, the spin structure of the amplitudes obtained in this way differs from the boosted non-relativistic ones. This feature is important, since the phenomenological discussion of photoproduction and strong decays of the baryon resonances shows that at least sizable corrections to the non-relativistic amplitudes are necessary.     

Models of Local Relativistic Quantum Fields with Indefinite Metric (in All Dimensions)

A condition on a set of truncated Wightman functions is formulated and shown to permit the construction of the Hilbert space structure included in the Morchio--Strocchi modified Wightman axioms. The truncated Wightman functions which are obtained by analytic continuation of the (truncated) Schwinger functions of Euclidean scalar random fields and covariant vector (quaternionic) random fields constructed via convoluted generalized white noise, are then shown to satisfy this condition. As a consequence such random fields provide relativistic models for indefinite metric quantum field theory, in dimension 4 (vector case), respectively in all dimensions (scalar case). Received: 25 April 1996 / Accepted: 29 July 1996     

Calculating the Relativistic Wave Functions of 11S0 and 13S1 Positronium

Based cn the relativistic Bethe-Salpeter (BS) equation, the positronium wavefunctions in Euclidean momentum space are obtained. Meanwhile the energy levels of positronium ground states 11S0 and 13S1 are fitted to be 6.7934 eV and 6.7929 eV respectively, which qualitatively agree with the previous theoretical values. It is shown that the BS theory is valid and reliable to treat positronium.     

Two-fermion bound states within the Bethe-Salpeter approach

To solve the spinor-spinor Bethe-Salpeter equation in Euclidean space we propose a novel method related to the use of hyperspherical harmonics. We suggest an appropriate extension to form a new basis of spin-angular harmonics that is suitable for a representation of the vertex functions. We present a numerical algorithm to solve the Bethe-Salpeter equation and investigate in detail the properties of the solution for the scalar, pseudoscalar and vector meson exchange kernels including the stability of bound states. We also compare our results to the nonrelativistic ones and to the results given by light-front dynamics.     

Calculating the Relativistic Wave Functions of 11S0 and 13S1 Positronium

Based on the relativistic Bethe-Salpeter (BS) equation, the positronium wavefunctions in Euclidean momentum space are obtained. Meanwhile the energy levels of positronium ground states 1¹S₀ and 1³S₁ are fitted to be 6.7934 eV and 6.7929 eV respectively, which qualitatively agree with the previous theoretical values. It is shown that the BS theory is valid and reliable to treat positronium.     

New framework for the Feynman path integral

The well-known Fourier integral solution of the free diffusion equation in an arbitrary Euclidean space is reduced to Feynmannian integrals using the method partly contained in the formulation of the Fresnelian integral. By replacing the standard Hilbert space underlying the present mathematical formulation of the Feynman path integral by a new Hilbert space, the space of classical paths on the tangent bundle to the Euclidean space (and more general to an arbitrary Riemannian manifold) equipped with a natural inner product, we show that our Feynmannian integral is in better agreement with the qualitative features of the original Feynman path integral than the previous formulations of the integral.     

A variational principle for bound states in quantum field theory

We develop a Rayleigh-Ritz variational method for estimating relativistic, multi-particle bound state energies in any (weak-coupling) quantum field theory. A comparison is made with bound state energies derived from the Bethe-Salpeter equation in the Wick-Cutkosky model. Possible applications to QCD are discussed.     

Scattering Theory for Quantum Fields¶with Indefinite Metric

In this work, we discuss the scattering theory of local, relativistic quantum fields with indefinite metric. Since the results of Haag–Ruelle theory do not carry over to the case of indefinite metric [4], we propose an axiomatic framework for the construction of in- and out-states, such that the LSZ asymptotic condition can be derived from the assumptions. The central mathematical object for this construction is the collection of mixed vacuum expectation values of local, in- and out-fields, called the “form factor functional”, which is required to fulfill a Hilbert space structure condition. Given a scattering matrix with polynomial transfer functions, we then construct interpolating, local, relativistic quantum fields with indefinite metric, which fit into the given scattering framework. Received: 13 September 1999/ Accepted: 1 August 2000     

Parity conservation and the Euclidean field formulation of relativistic quantum field theories

A method to construct Euclidean covariant fields corresponding to a relativistic quantum field theory with arbitrary spins is presented. The constructed fields act on a state space with an indefinite inner product, they commute (or anticommute) totally and (except for hermitian Fermion fields) adjoint relativistic fields correspond to adjoint Euclidean fields. The cases where this method can be applied include all Gårding-Wightman theories invariant under space inversion.     

Realization of the three-dimensional quantum Euclidean space by differential operators

The three-dimensional quantum Euclidean space is an example of a non-commutative space that is obtained from Euclidean space by q-deformation. Simultaneously, angular momentum is deformed to , it acts on the q-Euclidean space that becomes a -module algebra this way. In this paper it is shown, that this algebra can be realized by differential operators acting on functions on . On a factorspace of a scalar product can be defined that leads to a Hilbert space, such that the action of the differential operators is defined on a dense set in this Hilbert space and algebraically self-adjoint becomes self-adjoint for the linear operator in the Hilbert space. The self-adjoint coordinates have discrete eigenvalues, the spectrum can be considered as a q-lattice. Received: 27 June 2000 / Published online: 9 August 2000     

Stochastic physical origin of the quantum operator algebra and phase space interpretation of the Hilbert space formalism: The relativistic spin zero case

Based on a recent association of quantum observable algebra with stochastic processes in the frame of the causal stochastic interpretation of quantum mechanics, a relativistic Hilbert space is defined for the Klein-Gordon case. It is demonstrated that unitary transformations in Hilbert space reflect canonical transformations in the associated phase space, manifesting thus an underlying symplectic structure.     

Parametrization of realistic Bethe-Salpeter amplitude for the deuteron

The parametrization of the realistic Bethe-Salpeter amplitude for the deuteron is given. Eight components of the amplitude in the Euclidean space are presented as an analytical fit to the numerical solution of the Bethe-Salpeter equation in the ladder approximation. The applicability of the parametrization to the observables of the deuteron is briefly discussed.     

THE ELECTROMAGNETIC FORM FACTOR OF THE O- MESON IN THE STRATON MODEL

In the straton model,the wave functions of the 0^- meson can be obtained numerical-ly from the Wick rotated Bethe-Salpeter equation.The problem of comparing experi-ment with the mesonic electromagnetic form factor calculated by analytical continua-tion of the wave functions from the Euclidean space back to the Minkowski space isan unsolved problem.On the basis of analyzing the analytic property of the formfactor,we proved that by choosing a special reference system in which the photon isspace-like,one may calculate the physical space-like electromagnetic form factor dire-ctly from the Euclidean B-S wave functions of the meson in the Euclidean space.As an example,we calculated the electromagnetic form factor of the pseudoscalarmeson by using the wave functions corresponding various choices of parameters.Pre-liminary results show that the theoretical calculation may be in accordance with experi-ment by appropriately choosing the parameters.     

Relativistic dynamics on a null plane

In view of possible applications to the quark model and to hadron spectroscopy, we investigate relativistic Hamiltonian quantum theories of finitely many degrees of freedom. We make use of the fact that if null planes are used as initial surfaces, the structure of the theory closely resembles nonrelativistic quantum mechanics: the inner variables that describe the structure of the system uncouple from the motion of the system as a whole. The dynamical content of such a theory resides in the operators M, $\text{j}$ of mass and spin that act in the space carrying the inner degrees of freedom. Relativistic invariance is equivalent to the requirement that M and $\text{j}$ generate a unitary representation of U(2). In contrast to this requirement, the condition that the wavefunctions of the system transform covariantly strongly restricts the dynamics. It is proven that for systems containing two constituents, covariance is equivalent to an algebraic relation that involves M and $\text{j}$ — the angular condition. A class of solutions of the angular condition is provided by a particular type of local manifestly covariant wave equations. One nontrivial solution of this class, a relativistic oscillator is given in detail. Confinement models of this type represent an interesting alternative to the solutions of the angular condition that result from the perturbation expansion of a local field theory through the three-dimensional quasipotential versions of the Bethe-Salpeter equation.     

On a Liouville-master equation formulation of open quantum systems

The dynamics of open quantum systems is formulated in terms of a probability distribution on the underlying Hilbert space. Defining the time-evolution of this probability distribution by means of a Liouvillemaster equation the time-dependent wave function of the system becomes a stochastic Markov process in the sense of classical probability theory. It is shown that the equation of motion for the two-point correlation function of the random wave function yields the quantum master equation for the statistical operator. Stochastic simulations of the Liouville-master equation are performed for a simple example from quantum optics and are shown to be in perfect agreement with the analytical solution of the corresponding equation for the statistical operator.     

Relativistic Gamow Vectors

Whereas in Dirac quantum mechanics and relativistic quantum field theory one uses Schwartz space distributions, the extensions of the Hilbert space that we propose uses Hardy spaces. The in- and out-Lippmann-Schwinger kets of scattering theory are functionals in two rigged Hilbert space extensions of the same Hilbert space. This hypothesis also allows to introduce generalized vectors corresponding to unstable states, the Gamow kets. Here the relativistic formulation of the theory of unstable states is presented. It is shown that the relativistic Gamow vectors of the unstable states, defined by a resonance pole of the S-matrix, are classified according to the irreducible representations of the semigroup of the Poincaré transformations (into the forward light cone). As an application the problem of the mass definition of the intermediate vector boson Z is discussed and it is argued that only one mass definition leads to the exponential decay law, and that is not the standard definition of the on-the-mass-shell renormalization scheme.     

Symmetry groups, density-matrix equations and covariant Wigner functions

A representation theory for Lie groups is developed taking the Hilbert space, say , of the w^*-algebra standard representation as the representation space. In this context the states describing physical systems are amplitude wave functions but closely connected with the notion of the density matrix. Then, based on symmetry properties, a general physical interpretation for the dual variables of thermal theories, in particular the thermofield dynamics (TFD) formalism, is introduced. The kinematic symmetries, Galilei and Poincaré, are studied and (density) amplitude matrix equations are derived for both of these cases. In the same context of group theory, the notion of phase space in quantum theory is analysed. Thus, in the non-relativistic situation, the concept of density amplitude is introduced, and as an example, a spin-half system is algebraically studied; Wigner function representations for the amplitude density matrices are derived and the connection of TFD and the usual Wigner-function methods are analysed. For the Poincaré symmetries the relativistic density matrix equations are studied for the scalar and spinorial fields. The relativistic phase space is built following the lines of the non-relativistic case. So, for the scalar field, the kinetic theory is introduced via the Klein–Gordon density-matrix equation, and a derivation of the Jüttiner distribution is presented as an example, thus making it possible to compare with the standard approaches. The analysis of the phase space for the Dirac field is carried out in connection with the dual spinor structure induced by the Dirac-field density-matrix equation, with the physical content relying on the symmetry groups. Gauge invariance is considered and, as a basic result, it is shown that the Heinz density operator (which has been used to develope a gauge covariant kinetic theory) is a particular solution for the (Klein–Gordon and Dirac) density-matrix equation.     

Euclidean Fermi fields with a hermitean Feynman-Kac-Nelson formula. I

We construct free, Euclidean, spin one-half, quantum fields with the following properties: (i) CAR; (ii) Symanzik positivity; (iii) Osterwalder-Schrader positivity; (iv) no doubling of particle or spin states. They admit the recovery of the relativistic Dirac field by the Osterwalder-Schrader technique. We then formally parametrize interacting theories by a natural class of Hermitean, Euclidean actions, and obtain a simple, Hermitean, Feynman-Kac-Nelson formula. The interacting theory formally obeys all the properties (i)–(iv), and admits the reconstruction of a physical Hilbert space, including a Hermitean, contraction semigroup for the Wick rotated time evolution. We propose a system of axioms for the interacting theory.     

The FKG inequality for the Yukawa2 quantum field theory

We establish the FKG correlation inequality for the Euclidean scalar Yukawa₂ quantum field model and, when the Fermi mass is zero, for pseudoscalar Yukawa₂. To do so we approximate the quantum field model by a lattice spin system and show that the FKG inequality for this system follows from a positivity condition on the fundamental solution of the Euclidean Dirac equation with external field. We prove this positivity condition by applying the Vekua-Bers theory of generalized analytic functions.Research partially supported by the National Research Council of Canada.Alfred P. Sloan Foundation Fellow.