Quantum Field Models Research Articles

Nontrivial scattering amplitudes for some local relativistic quantum field models with indefinite metric

Abstract We study models of self-interacting massless spin 1 local relativistic quantum fields with indefinite metric in space-time dimension four. We prove that these models for large times converge to free fields and we derive explicit formulae for their (nontrivial, gauge invariant) scattering amplitudes. These scattering amplitudes have properties expected for S-matrix theory.

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).

On the thermal effective expansion of a scalar theory

We consider a scalar massless quantum field model, at finite temperature T, both renormalizable and asymptotically free. Focusing on the thermal equilibration process of some Higgs particle, several new aspects are brought out, regarding the hard thermal loop resummation program.

A Class of Representations of the ∗-Algebra of the Canonical Commutation Relations over a Hilbert Space and Instability of Embedded Eigenvalues in Quantum Field Models

In models of a quantum harmonic oscillator coupled to a quantum field with a quadratic interaction, embedded eigenvalues of the unperturbed system may be unstable under the perturbation given by the interaction of the oscillator with the quantum field. A general mathematical structure underlying this phenomenon is clarified in terms of a class of Fock space representations of the ∗-algebra of the canonical commutation relations over a Hilbert space. It is also shown that each of the representations is given as a composition of a proper Bogolyubov (canonical) transformation and a partial isometry on the Fock space of the representation.

Relationship between the classical transport coefficients and hydrodynamic kernels for quantum-field models

A study is made of the classical limit of nonlocal hydrodynamics, describing quantum systems of many particles. The quantum-field model is not specified, it being assumed only that there are local laws governing the conservation of energy — momentum and the number of particles. Changing over to the classical limit corresponds in nonlocal hydrodynamics to the limit of slow processes and long waves. It is shown that a hydrodynamics with viscosity and self-diffusion but without heat conduction is obtained in this case because the velocity of the medium is determined in terms of the energy flux, which is natural for quantum-field systems. Heat conduction can be introduced if velocity is instead determined in terms of particle flux.

Nonlocal hydrodynamics in the quantum field modelϕ 4

The procedure of passing from quantum statistical mechanics to the hydrodynamics previously developed by the author is now applied to the quantum field model ϕ4. For a certain class of external forces, the equations of many-body systems in quantum theory appear to be equivalent to the equations of nonlocal hydrodynamics. The hydrodynamic nonlocalities arising in constituent relations are expressed through the Green's functions for currents. Some properties of the nonlocal kernels, in particular, the conditions related to dissipation and T-invariance of the model ϕ4 (an analogue of Onsager's relations), are deduced from the general symmetry properties. In hydrodynamics, nonlocality allows causality and dissipativity to be consistently combined. The connection between the classical transport coefficients and the hydrodynamic kernels is established. An algorithm for calculating constituent relations by perturbation theory, using the technique of temperature Green's functions, is described.

ISOLATION AND EXPULSION OF DIVERGENCES IN QUANTUM FIELD THEORY

Divergences that arise in the quantization of scalar quantum field models by means of a lattice-space functional integration may be attributed to a single integration variable, and this fact is demonstrated by showing that if the integrand for that single integration variable is appropriately changed, then a perturbation expansion becomes order-by-order finite and divergence free. The paper concludes with a brief review of a current proposal of how an auxiliary, nonclassical potential added to the lattice action of a relativistic scalar field quantization may automatically render an analogous change of the integrand, and thus may lead, as well, to nontrivial and divergence-free results.

On the light cone singularity of the thermal effective expansion

In the case of a scalar massless quantum field model, at finite temperature T, we analyse the singular structure of the effective perturbation theory on the light cone.

Properties of hydrodynamic nuclei for quantum field models

The transition to the hydrodynamic limit for a many-particle quantum system withN local conservation laws is made in a specified class of external effects. It is shown that the hydrodynamic equations are nonlocal in time and space and the hydrodynamic model is equivalent to the initial quantum statistical model. The nuclei appearing in the material relations are expressed in terms of the Green functions for the currents. It follows from the properties of the Green functions that the hydrodynamic model satisfies the dissipation conditions. When the quantum field model isT-invariant, the nuclei are related by reciprocal relations (analogous to the Onsager relations).

Quantum-field model of the deuteron

A quantum-field model of the deuteron is investigated for the case of a scalar field with point interaction. In the strong-coupling limit, exact solutions of the nonlinear field equations for the deuteron are found. The results are compared with the experimental values for the energy, radius, and mass of the deuteron. A scheme for eliminating the ultraviolet divergences in strong-coupling theory is proposed.

Lattice Wess-Zumino-Witten model and quantum groups

Quantum groups play the role of symmetries of integrable theories in two dimensions. They may be detected on the classical level as Poisson-Lie symmetries of the corresponding phase spaces. We discuss specifically the Wess-Zumino-Witten conformally invariant quantum field model combining two chiral parts which describe the left- and right-moving degrees of freedom. On one hand, the quantum group plays the role of the symmetry of the chiral components of the theory. On the other hand, the model admits a lattice regularization (in Minkowski space) in which the current algebra symmetry of the theory also becomes quantum, providing the simplest example of a quantum group symmetry coupling space-time and internal degrees of freedom. We develop a free field approach to the representation theory of the lattice sl (2)-based current algebra and show how to use it to rigorously construct an exact solution of the quantum SL (2) WZW model on lattice.

Exact solution for an integrable quantum field model

An integrable one-dimensional quantum field model is introduced. The exact eigenstates of the Hamiltonian are constructed. Furthermore, the existence of bound states is shown. It is concluded that this model is completely integrable.

The bound states of the [formula omitted] relativistic quantum field models with weak coupling obtained by the variational perturbation method

The two-particle bound states of the simplest quantum field theory models with effective interaction, the weakly coupled P(ϕ) 2 models, are obtained by the variational perturbation method. This method leads to an eigenvalue equation for the bound state mass, which can be seen as a relativistic generalization of the Schrödinger equation, proposed by the P(ϕ) 2 models at first perturbation orders. Moreover it gives the zero-time eigenvectors of the bound states at first perturbation orders.

Quantum-field model of the injected atomic beam in the micromaser.

A general theory of the micromaser is described. The model is based on treating the input atomic beam as a two-component quantum field so that the two-level atoms in the beam are ``quanta'' of this field. This approach makes it possible to formulate a general quantum Langevin description of the dynamics with the input atomic field as a source of quantum noise. The passage to a master-equation description is then effected by use of an adjoint-operator method, and by introducing a general class of statistical states for the atomic beam known as generalized shot noise. The result is a (non-Markovian) master equation for the field inside the cavity which is valid for a broad range of statistical properties of the input atomic beam. The approximate steady-state solutions to this master equation for the photon statistics of the cavity field for sub-Poissonian (antibunched) atomic beams found by other researchers are regained. The theory is then extended to treat super-Poissonian (bunched) atomic beams. An exact result is found in the limit of a strongly bunched beam in which the cavity-field state is shown to converge to a mixture of a thermal-field state and the state produced by a random beam of twice the intensity. A physical explanation of this result is also presented.

Hidden quantum group structure in a relativistic quantum integrable model

A relativistic quantum field model, recently solved exactly [Y.K. Zhou, Phys. Lett. B 276 (1992) 135], is shown to have an underlying quantum group structure. The baxterisation of the FRT algebra enables us to construct through q-oscillators an exact lattice regularised version of the model. Moreover this model is found to be hybridisation of a new quantum integrable bosonic model.

Von Hove model of quantum field theory-from the point of view of infinite dimensional stationary Markov process theory

There are two ways to quantize the classical von Hove model. One way is to quantize the classical von Hove field equation, and the other is to quantize the classical energy functional. In this paper, their relation to each other is pointed out. As an application the second-order approximate quantization of (φ4)2-kink field is studied.

Infinite-parametric extension of the conformal algebra in D > 2 space-time dimensions

On the basis of the analytic continuations of semisimple Lie algebras discovered recently by us we construct manifestly quasi-conformal infinite-dimensional algebras AC(so(4, 1)) and PAC(so(3, 2)) extending the conformal algebras in three-dimensional euclidean and Minkowski space-time like the Virasoro algebra extends so(2, 1). Their higher spin generalizations are also constructed. A counterpart of the central extension for D > 2 and possible applications in exactly solvable conformal quantum field models in D > 2 are discussed.

The triviality of the Abelian Thirring quantum field model

By using a Grassmannian polymer representation for the fermionic functional determinant the authors argue the triviality of the vectorial four-fermion interaction for spacetime with dimensionality greater than two.

Stochastic field theory, holomorphic quantum mechanics, and supersymmetry

Holomorphic quantum mechanics are studied from the point of view of stochastic quantization in Minkowski space which involves the introduction of two stochastic fields, one in the external space and the other in the internal space. The equilibrium condition is given by Z2 symmetry between the external and internal fields. In the nonequilibrium case, N=2 Wess–Zumino quantum fields are arrived at giving rise to supersymmetry. This helps to define the supercharge operator Q when the Hamiltonian is given by H=Q2 and an index theorem is derived for an interacting case when the superpotential is given by V(φ)=λφn, φ being complex with n>2. It is found that the vacuum is degenerate and is in conformity with the result obtained by Jaffe, Lesniewski, and Lewenstein [Ann. Phys. 178, 313 (1987)] in the two-dimensional N=2 Wess–Zumino quantum field model.

Possible role of oxygen valence in superconducting ceramics

Superconducting ceramics are discussed in terms of the joint occurrence of oxygen ions with different valencies. The relevant behaviour can be described by a simple quantum-field model.