Uniqueness of the physical vacuum and the Wightman functions in the infinite volume limit for some non polynomial interactions

We consider quantum field theoretical models inn dimensional space-time given by interaction densities which are bounded functions of an ultraviolet cut-off boson field. Using methods of euclidean Markov field theory and of classical statistical mechanics, we construct the infinite volume imaginary and real time Wightman functions as limits of the corresponding quantities for the space cut-off models. In the physical Hilbert space, the space-time translations are represented by strongly continuous unitary groups and the generator of time translationsH is positive and has a unique, simple lowest eigenvalue zero, with eigenvector , which is the unique state invariant under space-time translations. The imaginary time Wightman functions and the infinite volume vacuum energy density are given as analytic functions of the coupling constant. The Wightman functions have cluster properties also with respect to space translations.     

Scale and Lorentz Transformations at the Light-Front

The light-front (LF) quantization is applied for the model of massive scalar field with self-interaction. We check some of the LF postulates by considering the Wightman function for this model. The scale symmetry imposed only on the LF quantization hypersurface and the Lorentz symmetry assumed for all points in Minkowski’s space-time lead to a strong constraint for the Wightman functions, which is satisfied only by a free and massless scalar field. This result agrees with the recent Weinberg’s result for a scale-symmetric theory. This means that one cannot expect the unitary equivalence of the Fock space for scalar fields with different masses at the LF hypersurface.     

Necessary and sufficient conditions for integral representations of Wightman functionals at Schwinger points

Given a set of Wightman functions one would like to associate to it a field on Euclidean space admitting a simultaneous diagonalization. We investigate when this can be done in such a way that the Schwinger functions are the expectation values of this commutative field with a bounded metric operator commuting with the field. This requires as a tool the characterization of those linear functionals on the symmetric tensor algebra over a space of test functions which can be represented by complex measures on the corresponding space of distributions.     

Rigorous construction of planar diagram field theories in four dimensional Euclidean space

Asymptotically free quantum field theories with planar Feynman diagrams [such as SU(∞) gauge theory] are considered in 4 dimensional Euclidean space. It is shown that if all particles involved have non-vanishing masses and if the coupling constant(s) γ (org ²) are small enough (λ≦λ^crit), then an absolutely convergent procedure exists to obtain Green functions that uniquely solve the Dyson-Schwinger equations.     

Would-Be Local Fields

General quantum field theory is formulated for the case when the Wightman distributions can grow in momentum space as the exponential of a covariant polynomial. Appropriate spaces of test functions are introduced, and it is shown that the vacuumexpectation values can be written in terms of various associated tempered distributions, which enjoy some of the properties of ordinary Wightman distributions; in particular, they can be represented as boundary values of functions holomorphic in the usual extended tubes. Notions of locality for the tempered distributions can be introduced, which are sufficient to imply the PCT theorem and theorems on the connection between spin and statistic for the non-tempered fields. It is shown how a Haag-Ruelle theory of asymptotic states and fields may be set up. A possible line of generalisation is illustrated by the special example of fields of the type χ (□) A (x), where A is a tempered field, and χ an entire analytic function of finite exponential order.     

Prime field decompositions and infinitely divisible states on Borchers' tensor algebra

We generalize some notions of probability theory and theory of group representations to field theory and to states on the Borchers algebraS. It is shown that every field (relativistic and Euclidean, ...) can be decomposed into a countable number of prime fields and an infinitely divisible field. In terms of states this means that every state onS is a product of an infinitely divisible state and a countable number of prime states, and in this formulation it applies equally well to correlation functions of statistical mechanics and to moments of linear stochastic processes overS orD. Necessary and sufficient conditions for infinitely divisible states are given. It is shown that the fields of the ø ₂ ⁴ -theory are either prime or contain prime factors. Our results reduce the classification problem of Wightman and Euclidean fields to that of prime fields and infinitely divisible fields. It is pointed out that prime fields are relevant for a nontrivial scattering theory.     

Remarks on some new models of interacting quantum fields with indefinite metric

We study quantum field models in indefinite metric. We introduce the modified Wightman axioms of Morchio and Strocchi as a general framework of indefinite metric quantum field theory (QFT) and present concrete interacting relativistic models obtained by analytical continuation from some stochastic processes with Euclidean invariance. As a first step towards scattering theory in indefinite metric QFT, we give a proof of the spectral condition on the translation group for the relativistic models.     

Scaling Transform and Stretched States in Quantum Mechanics

We consider the Husimi Q(q, p)-functions which are quantum quasiprobability distributions on the phase space. It is known that, under a scaling transform (q; p) → (?q; ?p), the Husimi function of any physical state is converted into a function which is also the Husimi function of some physical state. More precisely, it has been proved that, if Q(q, p) is the Husimi function, the function ?² Q(?q; ?p) is also the Husimi function. We call a state with the Husimi function ?² Q(?q; ?p) the stretched state and investigate the properties of the stretched Fock states. These states can be obtained as a result of applying the scaling transform to the Fock states of the harmonic oscillator. The harmonic-oscillator Fock states are pure states, but the stretched Fock states are mixed states. We find the density matrices of stretched Fock states in an explicit form. Their structure can be described with the help of negative binomial distributions. We present the graphs of distributions of negative binomial coefficients for different stretched Fock states and show the von Neumann entropy of the simplest stretched Fock state.     

Wavefront Sets in Algebraic Quantum Field Theory

The investigation of wavefront sets of n-point distributions in quantum field theory has recently acquired some attention stimulated by results obtained with the help of concepts from microlocal analysis in quantum field theory in curved spacetime. In the present paper, the notion of wavefront set of a distribution is generalized so as to be applicable to states and linear functionals on nets of operator algebras carrying a covariant action of the translation group in arbitrary dimension. In the case where one is given a quantum field theory in the operator algebraic framework, this generalized notion of wavefront set, called “asymptotic correlation spectrum”, is further investigated and several of its properties for physical states are derived. We also investigate the connection between the asymptotic correlation spectrum of a physical state and the wavefront sets of the corresponding Wightman distributions if there is a Wightman field affiliated to the local operator algebras. Finally we present a new result (generalizing known facts) which shows that certain spacetime points must be contained in the singular supports of the 2n-point distributions of a non-trivial Wightman field. Received: 27 July 1998 / Accepted: 3 March 1999     

BPHZ renormalization of λφ4 field theory in curved spacetime

A proof is given to all orders in perturbation theory of the renormalizability of λφ⁴ field theory in curved spacetime. The proof is based on the BPHZ definition of a renormalized Feynman integrand and uses dimensional regularization to ensure that products of Feynman propagators are well-defined distributions. The explicit structure of the pole terms in the Feynman integrand is obtained using a local momentum space representation of the Feynman propagator and is shown to be of a form which can be cancelled by counterterms in the scalar field Lagrangian. The proof given is, technically, only valid for metrics which have been analytically continued to Euclidean (++++) signature.     

Axiomatic Conformal Theory in Dimensions >2 and AdS/CT Correspondence

We formulate axioms of conformal theory (CT) in dimensions >2 modifying Segal’s axioms for two-dimensional CFT. (In the definition of higher-dimensional CFT, one includes also a condition of existence of energy-momentum tensor.) We use these axioms to derive the AdS/CT correspondence for local theories on AdS. We introduce a notion of weakly local quantum field theory and construct a bijective correspondence between conformal theories on the sphere S^d and weakly local quantum field theories on \({H^{d+1}}\) that are invariant with respect to isometries. (Here \({H^{d+1}}\) denotes hyperbolic space = Euclidean AdS space.) We give an expression of AdS correlation functions in terms of CT correlation functions. The conformal theory has conserved energy-momentum tensor iff the AdS theory has graviton in its spectrum.     

Renormalized Normal Product and equations of motion of Φ4-coupling

The notion of a Renormalized Normal Product (RNP) in Euclidean space of 1 ≤ r ≤ 4 dimensions is introduced for a Φ⁴-model in a nonperturbative approach. The essential ingredients used for this purpose are the composite operators defined in perturbation theory and the renormalized G-convolution product constructed in the axiomatic field theory framework in Euclidean momentum space. Convergent equations of motion for the connected Green's functions are established where the interaction term is represented by the RNP. The corresponding renormalization constants are defined as boundary values of the RNP by imposing “physical” renormalization conditions. In the special case of 2-dimensions it is proved that these equations conserve analyticity and algebraic properties (in complex Minkowski space of 2-momenta) coming from the first principles of general local field theory, together with properties of asymptotic behaviour at infinity (in Euclidean space of 2-momenta).     

On certain non-relativistic quantized fields

We consider a Euclidean invariant interaction Hamiltonian which is a polynomial in smeared Fermion field operators (the smearing function providing an ultraviolet cut-off). By considering Guenin's perturbation series for the time-development of the theory, we show that time-displacements define a one-parameter group of automorphisms of the field algebra att=0, which acts continuously in the time-parameter. Results are obtained for any dimension of space and for both relativistic and nonrelativistic forms for the free Hamiltonian. In special cases the total Hamiltonian is a positive self-adjoint operator in Fock space, thus defining a concrete non-relativistic quantized field with non-trivial particle production.Supported in part by the United States Air Force under contract AFOSR 500-66.     

Fock representations and BRST cohomology inSL(2) current algebra

We investigate the structure of the Fock modules overA ₁ ⁽¹⁾ introduced by Wakimoto. We show that irreducible highest weight modules arise as degree zero cohomology groups in a BRST-like complex of Fock modules. Chiral primary fields are constructed as BRST invariant operators acting on Fock modules. As a result, we obtain a free field representation of correlation functions of theSU(2) WZW model on the plane and on the torus. We also consider representations of fractional level arising in Polyakov's 2D quantum gravity. Finally, we give a geometrical, Borel-Weil-like interpretation of the Wakimoto construction.     

Perturbative QED in terms of gauge invariant fields

A formulation of QED using only gauge invariant fields acting on a physical state space is discussed. The fields are the electromagnetic tensor F_μν and a non-local electron field ψ_f depending on a quadruple {f^μ} of auxiliary functions. The f-ambiguity is physically meaningful: the f^μ contain information on the asymptotic configuration of the electromagnetic field accompanying charged particles. Equations of motion are introduced and solved perturbatively, in the sense that expressions for the Wightman functions of the theory are derived. No information on the commutation relations between the basic fields is needed.     

Dimensional regularization of massless theories in spherical space-time

The use of space-time curvature as an infra-red cut-off has been suggested for massless theories. In this paper we investigate the renormalization of massless theories in a spherical space-time (Euclidean version of de Sitter space) using dimensional regularization. Naive expectations are confirmed, namely that the coupling constant and wave-function renormalizations are independent of the curvature. Furthermore the curvature does not induce divergent mass terms or vacuum field values as would be possible on purely dimensional grounds. Although we have investigated only scalar field theories, φ⁴ theory in four dimensions and φ³ theory in six, these results are encouraging for an application of the method to gauge theories.Formally massless theories are conformally invariant so the formulation of the theory in a spherical space ought to be equivalent to its formulation in flat space. In fact the renormalization procedure breaks conformal invariance and removes this equivalence. We show that to achieve the flat space limit it is necessary to invoke the aid of the renormalization group. Thus the zero curvature limit can be achieved for infra-red stable theories (φ₄⁴) but not for infra-red unstable theories (φ₆³ as might be expected.     

Boundary Structure and Module Decomposition of the Bosonic Z2 Orbifold Models with R=1/2k

The Z₂ bosonic orbifold models with compactification radius R²=1/2k are examined in the presence of boundaries. Demanding the extended algebra characters to have definite conformal dimension and to consist of an integer sum of Virasoro characters, one arrives at the right splitting of the partition function. This is used to derive a free field representation of a complete, consistent set of boundary states, compatible with the modular transformations of the characters. Finally the modules of the extended symmetry algebra that correspond to the finitely many characters are identified inside the direct sum of Fock modules that constitute the space of states of the theory.     

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.     

Covariant quantization of “massive” spin-\frac{3}{2} fields in the de sitter space

We present a covariant quantization of the free “massive” spin- $\frac{3}{2}$ fields in four-dimensional de Sitter space-time based on analyticity in the complexified pseudo-Riemannian manifold. The field equation is obtained as an eigenvalue equation of the Casimir operator of the de Sitter group. The solutions are calculated in terms of coordinate-independent de Sitter plane-waves in tube domains and the null curvature limit is discussed. We give the group theoretical content of the field equation. The Wightman two-point function $S^{i \bar{j}}_{\alpha\alpha'}(x,x')$ is calculated. We introduce the spinor-vector field operator Ψ _α (f) and the Hilbert space structure. A coordinate-independent formula for the field operator Ψ _α (x) is also presented.     

Equivalence between non-localizable and local fields

We discuss the nature of non-localisable fields constructed as certain limits of sequences of local fields. For sequences for which the corresponding Wightman functions converge we construct a PCT operator; if the sequences converge strongly in a given Hilbert space then a scattering theory can be constructed for the non-localisable limit field. Such fields are shown to have the sameS-operator as any local field which has the defining sequence of local fields in its Borchers class, and has the same in field. We give non-trivial examples of this equivalence between local and non-localisable fields.