Constructive Quantum Field Theory Research Articles

RENORMALIZATION GROUP METHODS IN CONSTRUCTIVE FIELD THEORIES

Mathematical construction of quantum field theory is reviewed with emphasis on the conceptual structure of the construction and on the role of rigorous renormalization group analysis without technical details. After explaining a rigorous formulation of a renormalization group method in a weak coupling region, a new approach in a strong coupling region is proposed in the context of the hierarchical approximation. An idea of proving triviality in d≥4 dimensions utilizing this new proposal concludes this review.

On the finiteness of the elementary-particle mass spectrum

A nonstandard approach to the construction of quantum field theory inspired by Markov’s idea of the finiteness of the elementary-particle mass spectrum is discussed.

Exponential Integrability and Application to Stochastic Quantization

We study the exponential integrability problem for I n (f n ), i.e., E exp{I n (f n )} < ∞, where I n (f n ) is a multiple Ito-Wiener integral on some Gaussian space arising from the constructive quantum field theory and from stochastic quantization. We also study a class of singular infinite-dimensional stochastic differential equations whose drift coefficients are measurable and unbounded. Using a condition due to Kazamaki, we prove the existence of a weak solution assuming some integrability conditions on the drift coefficient. Then we apply the exponential integrability theorem and the existence theorem to study infinite-dimensional stochastic differential equations of stochastic quantization.

On the Existence and Uniqueness of Ground States of a Generalized Spin-Boson Model

A generalization of the standard spin-boson model is considered. The HamiltonianH(α) of the model with a coupling parameterα∈Racts in the tensor product H⊕Fbof a Hilbert space H and the boson (symmetric) Fock space FboverL2(Rν). The existence and uniqueness of ground states ofH(α) are investigated. The degeneracy of the ground states is also discussed. The results obtained arenonperturbative. The methods used are those of constructive quantum field theory and the min-max principle. An exact asymptotic formula for the ground state energy ofH(α) as |α|→∞ is also established.

An axiomatic approach to Quantum Gauge Field Theory

In the present article we display a new constructive quantum field theory approach to quantum gauge field theory, utilizing the recent progress in the integration theory on the moduli space of generalized connections modulo gauge transformations. That is, we propose a new set of Osterwalder Schrader like axioms for the characteristic functional of a measure on the space of generalized connections modulo gauge transformations rather than for the associated Schwinger distributions. We show non-triviality of our axioms by demonstrating that they are satisfied for twodimensional Yang-Mills theory on the plane and the cylinder. As a side result we derive a closed and analytical expression for the vacuum expectation value of an arbitrary product of Wilson-loop functionals from which we derive the quantum theory along the Glimm and Jaffe algorithm which agrees exactly with the one as obtained by canonical methods.

Correlations of quantum fields on Robertson - Walker spacetimes

It is a well known fact that quantum fields on Minkowski spacetime are correlated for each pair of spacetime regions. In Robertson - Walker spacetimes there are spacelike separated regions with disjoint past horizons but the absence of correlations in that case was never proved. We derive in this paper formulae for correlations of quantum fields on Robertson - Walker spacetimes. Such correlations could have reasonably influenced the formation of structure in the early universe. We use methods of algebraic and constructive quantum-field theory.

Introduction to Algebraic and Constructive Quantum Field Theory

The authors present a rigorous treatment of the first principles of the algebraic and analytic core of quantum field theory. Their aim is to correlate modern mathematical theory with the explanation of the observed process of particle production and of particle-wave duality that heuristic quantum field theory provides. Many topics are treated here in book form for the first time, from the origins of complex structures to the quantization of tachyons and domains of dependence for quantized wave equations. This work begins with a comprehensive analysis, in a universal format, of the structure and characterization of free fields, which is illustrated by applications to specific fields. Nonlinear local functions of both free fields (or Wick products) and interacting fields are established mathematically in a way that is consistent with the basic physical constraints and practice. Among other topics discussed are functional integration, Fourier transforms in Hilbert space, and implementability of canonical transformations. The authors address readers interested in fundamental mathematical physics and who have at least the training of an entering graduate student. A series of lexicons connects the mathematical development with the underlying physical motivation or interpretation. The examples and problems illustrate the theory and relate it to the scientific literature.Originally published in 1992.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Solving two-dimensional phi4 field (complex scalar) theory by discretized light-front quantization.

The discretized light-front quantization method is applied to ${\ensuremath{\varphi}}^{4}$ field (complex scalar) theory in 1 + 1 dimensions. The interaction Hamiltonian is constructed in its normal-ordered form, and calculations are performed with and without finite-mass renormalization in the charge-0 sector of the field. It is found that, like real scalar theory, finite-mass renormalization prevents the phase transition by restricting the theory to the weak-coupling region. A comparison of the results with and without mass renormalization demonstrates the same estimate of the critical coupling for which the mass gap vanishes. The invariant mass of various states is calculated as a function of bare coupling. In the weak-coupling region where extrapolation to the continuum limit is easily found, there is evidence for scattering, but there is no two-particle bound state in agreement with the well-known result established for constructive quantum field theory. Also, no multiparticle bound states are found. The essential outcome is that the results valid for real-scalar theories are found to be valid for complex scalar theory also.

On a class of stochastic reaction-diffusion equations in two space dimensions

The authors prove, for any finite time, the existence of a weak dynamics for a class of reaction-diffusion systems in two space dimensions perturbed by a white noise. This class includes models which may exhibit complex behaviour in space and time. Their approach, based on the methods of constructive quantum field theory, extends previous results established for gradient-type systems.

Commutator Theory on Hilbert Space

Commutator theory has its origins in constructive quantum field theory. It was initially developed by Glirnm and Jaffe [7] as a method to establish self-adjointness of quantum fields and model Hamiltonians. But it has subsequently proved useful for a variety of other problems in field theory [17] [15] [8] [3], quantum mechanics [5], and Lie group theory [6]. Despite all these detailed applications no attempt appears to have been made to systematically develop the theory although reviews have been given in [22] and [9]. The primary aim of the present paper is to partially correct this situation. The secondary aim is to apply the theory to the analysis of first and second order partial differential operators associated with a Lie group.

Solving two-dimensional phi4 theory by discretized light-front quantization.

The recently proposed discretized light-front quantization (DLFQ) method is applied to ${\ensuremath{\varphi}}^{4}$ field theory in 1+1 dimensions. We start with the normal-ordered Hamiltonian and perform calculations with and without finite-mass renormalization in order to elucidate its role. We find that finite-mass renormalization prevents the phase transition by restricting the theory to the weak-coupling region. Comparison with results obtained without mass renormalization demonstrates that both treatments can yield the same estimate of the critical coupling for which the mass gap vanishes. This DLFQ estimate of the critical coupling may be compared with other estimates. The invariant mass of various states is calculated as a function of bare coupling. In the weak-coupling region where we can easily extrapolate to the continuum limit we find evidence for scattering but there is no two-particle bound state in agreement with the well-known result established for constructive quantum field theory. In addition, we find no multiparticle bound states.

Integral representations of the Schwinger functions for wick polynomials in the free field

Integral representations for the Schwinger functions of products of Wick polynomials in the free field are obtained, i.e., Euclidean realizations of Wightman quantum fields specified by Wick polynomials in the free field are obtained. These Euclidean realizations are used to propose a new mathematically rigorous approach to the construction of quantum field theory with polynomial interaction in a finite volume of d-dimensional space-time even without ultraviolet cutoff. In particular, the generating functional of the Schwinger functions is constructed for purely imaginary values of the coupling constant. The theory constructed in this manner takes into account explicitly the presence of ultraviolet divergences and if expanded with respect to the coupling constant gives a renormalized perturbation-theory series.

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Pseudoscalar interaction of coupled quantum-mechanical oscillators with independent Fermi systems

Using the techniques of constructive quantum field theory we analyze the dynamics of a cubic lattice of quantum mechanical oscillators with nearest-neighbor coupling that interact with a corresponding lattice of finite-volume truncations of independent relativistic Fermi fields. Since the model is nonrelativistic, we rely on a nonrelativistic version of the Osterwalder–Schrader (OS) reconstruction theorem. Also, the absence of Nelson’s symmetry in the (un)Euclidean picture is not serious because the transfer matrix in a given space direction is simple enough to make the verification of spatial OS positivity easy. After establishing for our model many of the basic results that hold for the more standard models, we give a proof of the Fortuin–Kasteleyn–Ginibre (FKG) inequality that is essentially independent of the dimension of the Fermi systems.

Algebras of local observables on a manifold

We propose a generalization of the Haag-Kastler axioms for local observables to Lorentzian manifolds. The framework is intended to resolve ambiguities in the construction of quantum field theories on manifolds. As an example we study linear scalar fields for globally hyperbolic manifolds.

A class of gaussian fields of Markov type

We establish necessary and sufficient conditions for a certain class of multicomponent Gaussian generalized stochastic fields to have the Markov property proposed by Wong. In an earlier paper by the present author, Wong's notion of Markov property was compared with another notion of Markov property due to Nelson, and it was shown that, in the case of Euclidean covariant Gaussian generalized stochastic fields, the former notion is implied by the latter. Besides the characterization mentioned above, results in the present paper further confirm this assertion. With the recent intervention of Markov generalized stochastic fields in Constructive Quantum Field Theory, it is clear that a study of the former will be beneficial to an understanding of the latter; hence the results of this paper are of physical significance.

Ultradistributions and quantum fields: Fourier-Laplace transforms and boundary values of analytic functions

The distribution theory is the mathematical framework of the axiomatic quantum field theory of A.S. Wightman. The axioms are satisfied in the case of the free field and in some non-trivial models studied in the constructive quantum field theory introduced by J. Glimm and A. Jaffe (two and three dimensions). No non-trivial example in four dimensions is known. The vacuum expectation values in the Wightman theory can have at most polynomial increase in momentum space. A. Jaffe has extended the axioms in order to allow non-polynomial increase in momentum space. In this paper we discuss the ultradistribution framework which is the most general framework for Jaffe fields (strictly localizable fields). The ultradistributions have been introduced by A. Beurling, G. Björck and independently by C. Roumieu. Ultradistribution theory is a natural generalization of the distribution theory. We study the Fourier-Laplace transform of ultradistributions, extenting results of A. Jaffe [8,9] in several directions. A Bochner-Schwartz theorem for ultradistributions is also shown to be valid. We expect ultradistribution theory to play a role in constructive quantum field theory.

On the construction of quantized gauge fields. I. General results

In this paper gauge theories are analyzed from the point of view of constructive quantum field theory. Diamagnetic inequalities for general lattice gauge theories are proven. They say that in a local field theory the ground-state energy density rises when the fields (scalars, Dirac fields, etc.) are minimally coupled to an external gauge field. A mass generation mechanism is described. Abelian Higgs models (scalar QED) on the lattice are investigated in more detail: Strong diamagnetic inequalities; infrared bounds; correlation inequalities yielding monotonicity in the space-time cutoff, in the mass of the photon and in the electric charge; gauge invariance and Osterwalder-Schrader positivity,etc., are shown to hold uniformly in the lattice spacing. For the two-dimensional Higgs model on the lattice the θ-vacua are constructed, a first-order phase transition at θ = π and the confinement of fractionally charged quarks are described. Details and applications to the two-dimensional Higgs model in the continuum limit appear in subsequent papers.

Uninvertible linear canonical transformations

Reducible quasi-Fock representations of canonical commutation relations (CCR) are studied which result from uninvertible linear canonical transformations of the Fock representation. Necessary and sufficient conditions are proved for: (i) invertibility of the linear canonical transformation, (ii) quasi-Fock character of the obtained representation of CCR, (iii) coincidence of the vacuum subspaces of two representations of this type, (iv) coincidence of W ∗-algebras of operators of these representations. The results obtained are interesting for constructive quantum field theory with asymptotic fields giving rise to the quasi-Fock representations of CCR.

A rigorous approach to Debye screening in dilute classical coulomb systems

The existence and exponential clustering of correlation functions for a dilute classical coulomb system are proven using methods from constructive quantum field theory, the sine gordon transformation and the Glimm, Jaffe, Spencer expansion about mean field theory. This is a vindication of a belief, of long standing amongst physicists, known as Debye screening. This states that, because of special properties of the coulomb potential, the configurations of significant probability are those in which the long range parts ofr −1 are mostly cancelled, leaving an “effective” exponentially decaying potential acting between charge clouds.