On the existence of equilibrium states in local quantum field theory

It is shown that any local quantum field theory admits thermodynamical equilibrium states (KMS-states) for all positive temperatures provided it satisfies a nuclearity condition, proposed by Wichmann and one of the authors, which restricts the admissible number of local degrees of freedom.Dedicated to E. H. Wichmann on the occasion of his 60th birthday     

Adiabatic theorem in axiomatic quantum field theory

It is shown that Møller matricesS _± and scattering matrixS in axiomatic field theory can be expressed through their adiabatic analogs. In particular, it is proved under certain conditions that \(S_ - = \mathop {s\lim }\limits_{\alpha \to 0} S_\alpha (0,\infty )W_\alpha \) whereW _α is a trivial phase factor [i.e. a unitary operator of the form exp i / α ∝r(k)a ⁺ (k)a(k)dk]. Corresponding results in Hamiltonian approach are discussed.     

On some aspects of the definition of scattering states in quantum field theory

The problem of extending quantum-mechanical formal scattering theory to a more general class of models that also includes quantum field theories is discussed, with the aim of clarifying certain aspects of the definition of scattering states. As the strong limit is not suitable for the definition of scattering states in quantum field theory, some other limiting procedure is needed. Two possibilities are considered, the abelian limit and adiabatic switching. Formulas for the scattering states based on both methods are discussed, and it is found that generally there are significant differences between the two approaches. As an illustration of the applications and the features of these formulas, S-matrix elements and energy corrections in two quantum field theoretical models are calculated using (generalized) old-fashioned perturbation theory. The two methods are found to give equivalent results.     

Perturbative quantum field theory at positive temperatures: An axiomatic approach

It is shown that the perturbative expansions of the correlation functions of a relativistic quantum field theory at finite temperature are uniquely determined by the equations of motion and standard axiomatic requirements, including the KMS condition. An explicit expression as a sum over generalized Feynman graphs is derived. The canonical formalism is not used, and the derivation proceeds from the beginning in the thermodynamic limit. No doubling of fields is invoked. An unsolved problem concerning existence of these perturbative expressions is pointed out.     

Zero-time one-particle states in quantum field theory

In the framework of the weakly-coupledP()₂-models we construct perturbation approximations of vectors of a dense set of the state space, especially vectors of the one-particle state subspace, by polynomials of zero-time fields acting on the vacuum state, with rigorous control of the remainders.     

Atomicity and maximality in axiomatic quantum theory

The “weak-maximality” condition is proved to be equivalent to atomicity of the lattice of “propositions” (“decision effects”) in quantum axiomatics, satisfying certain simple conditions. In particular, it is shown that these conditions are fulfilled in Ludwig's axiomatic formulation of quantum mechanics. It is further proved that atomicity of the lattice of propositions follows from the condition of “strong maximality”. The maximality conditions have a clear physical interpretation. They are also fulfilled in the Hilbert space formulation of quantum mechanics. Since the atomicity property is used in theories based on Type I factors, the connection between atomicity and maximality seems of general interest. Useful theorems are proved.     

Bound states and scattering from bound states in the light-front field theory dynamics

Starting from the Weinberg rules we derive a covariant form of the relativistic Schrödinger equation and formulate the bound state problem in the light-front field theory dynamics. We present an explicit rule for embodying the two-body subsystem in the three-body space and demonstrate that the cluster decomposition property is explicitly preserved in the light front field theory dynamics. As an application of these results we write amplitudes forπd→nN ^*, πd→πpn, andπd→πd, in the impulse approximation, in terms of the internal bound state wave functions and two-body reducedt-matrix elements.     

Functional approach to scattering in quantum field theory

A functional approach to scattering theory in quantum field theory is developed by deriving an explicit functional expression fortransition amplitudes. In applications, the formalism avoids dealing with noncommutativity problems of field operators, avoids solving the field equations, avoids dealing with the often quite complicated continual (path) integrals, and avoids combinatoric problems associated with Feynman rules and the old-fashioned Wick's theorem. Finally, it avoids explicitly taking mass shell limits as in the LSZ formalism. The basic idea of the formalism is to use the quantum action principle followed by a systematic analysis of the concept of stimulated emissions as applied to particles of any spin, and is a generalization of an earlier method applied by the author to the much simpler situation of quantum mechanics.     

UV/IR duality in noncommutative quantum field theory

We review the construction of renormalizable noncommutative Euclidean ϕ⁴-theories based on the UV/IR duality covariant modification of the standard field theory, and how the formalism can be extended to scalar field theories defined on noncommutative Minkowski space.     

Local preparability of states and the split property in quantum field theory

Extending similar results of Buchholz, Doplicher, and Longo, it is shown that the existence of local operations preparing a given local state implies the split property for the local net of observable algebras, i.e., the existence of type I factors interpolating between the observable algebras of regions strictly contained in each other. It is shown that local preparations, if they exist, may be taken to be nonselective.     

The Haag-Ruelle formulation of scattering in hyperfunction quantum field theory

The Källén-Lehmann representation for two-point Wightman Fourier hyperfunctions and the cluster property for truncated vacuum expectation values are established in the framework of hyperfunction quantum field theory. With some additional assumptions these properties allow one to verify the Haag-Ruelle asymptotic conditions.     

A note on the inverse scattering problem in quantum field theory

We study the set of local fields describing the dynamics of a scalar, massless particle. It turns out that these fields are relatively local to the free, massless, scalar fieldA if the massless particle does not interact. This leads to a simple algebraic characterisation of interacting fields in the above framework.     

Physical principles in quantum field theory and in covariant harmonic oscillator formalism

It is shown that both covariant harmonic oscillator formalism and quantum field theory are based on common physical principles which include Poincaré covariance, Heisenberg's space-momentum uncertainty relation, and Dirac's “C-number” time-energy uncertainty relation. It is shown in particular that the oscillator wave functions are derivable from the physical principles which are used in the derivation of the Klein-Nishina formula.     

Definition and existence of multichannel scattering states

The field theoretical formulation of quantum mechanics is used to consider the nonrelativistic multichannel scattering theory. With the help of appropriately constructed time dependent creation operators, Hilbert vectors are formed whose limits in time can be defined as multichannel scattering states in the usual sense. The existence of these states is proved under certain assumptions for the potential, by showing the convergence of the above mentioned operators. The commutation relations for the limits of these operators are given.     

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.     

Isolated one particle states in boson quantum field theory models

We give sufficient conditions to ensure the existence of isolated one-particle states in the joint energy-momentum spectrum of an Osterwalder-Schrader scalar boson quantum field theory. We do not require the existence of sharp-time fields or weak coupling.     

A variational approach to bound states in quantum field theory

We consider here in a toy model an approach to bound state problem in a nonperturbative manner using equal time algebra for the interacting field operators. The potential is replaced by offshell bosonic quanta inside the bound state of nonrelativistic particles. The bosonic dressing is determined through energy minimisation, and mass renormalisation is carried out in a nonperturbative manner. Since the interaction is through a scalar field, it does not include spin effects. The model however nicely incorporates an intuitive picture of hadronic bound states in which the gluon fields dress the quarks providing the binding between them and also simulate the gluonic content of hadrons in deep inelastic collisions.     

A curiosity concerning the role of coherent states in quantum field theory

In the usual Fock quantisation of fields in Minkowski space-time, one has the result that the expectation value of the quantum Hamiltonian in any coherent state equals the energy of the classical field at which the state is peaked. It is shown that this property can be used tocharacterise the usual Fock representation. It is also pointed out that the entire analysis goes through for a substantially more general class of systems including, in particular, Bose fields in arbitrary stationary space-times.