Evaluation of functional integrals for an ultralocal static field theory

Exact and perturbative solutions of an ultralocal, static, real φ^m field theory, obtained by functional integral methods are given and discussed. It is shown that this unrenormalizable theory leads to a free one in the local limit, whereas a small non-locality leads to a small effective coupling constant in an effective φ⁴ Lagrangian. It is proved that the usual perturbation seeries is the asymptotic expansion of the exact result.     

Spontaneous symmetry breaking in the present and early universe

Spontaneous symmetry breaking in λφ⁴ theory is formulated in terms of the operator φ², and in a manner which requires no specific expectation value to be assigned to φ. At the one-loop order of perturbation theory, a renormalized effective action for a field ζ, linearly related to φ², is obtained as a gradient expansion. Potential advantages of this formulation in applications to phase transitions in the early universe are discussed. They include the possibilities (i) of obtaining a well-defined semiclassical equation of motion, and (ii) of following the evolution of a field theory from an initial symmetrical high temperature state without the introduction, ad hoc, of regions in which 〈φ〉 ≠ 0.     

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.     

On constrained instantons

A simple method is presented for doing systematic constrained instanton calculations in models such as φ⁴ or Higgs theories where the presence of a mass term prevents the existence of a classical solution. As an application, instanton estimates of the large-order behavior of the perturbation series in massive φ₄⁴ theory are derived. (These estimates agree with those of Frishman and Yankielowicz.)     

Improved renormalization group equations with dimensional regularization and the ε expansions

Due to the absence of dimensional cut-off parameters in the dimensional regularization scheme, vanishing of the renormalized mass of the scalar boson implies vanishing of its renormalized mass; thus the masses of both bosons and fermions in renormalizable field theories can be made finite by multiplicative mass renormalizations. The improved renormalization group equations in D dimensions are derived in such a way that both the large (or the small) momentum limits and the Wilson ? expansions can be uniformly treated for the fermion as well as the boson cases. We discuss the improved equations for φ₆³ theory, φ₄⁴ theory, quantumelectrodynamics, massive vector-gluon model, and non-Abelian guage theories incorporating fermions. For the latter three classes of theories, the gauge dependent problem of the coefficient functions in the improved renormalization group equations is discussed.     

Infinite order radiative corrections and renormalization

By means of the Ward identity and the correlation expansion derived earlier, we calculate virtual radiative corrections to an infinite order. Partial factorization of the correlation effects, as in the simple exchange processes, makes infinite summation possible. Furthermore, because the Ward identity is satisfied order by order and because we are able to carry out the infinite order summation and obtain a closed form, renormalization turns out to be very simple and transparent. Here the calculations are performed for a scalar: ?²: φ model, but are easily generalized to other similar models. We also indicate why this rearrangement of the ordinary perturbation expansion is suitable for strong coupling theories, ordinary local field theories as well as dual models.     

Soliton-like solutions of some nonlinear theories and transformations among them

The observation that the soliton-like solutions of a given second-order nonlinear differential equation define the separatrix of the equivalent autonomous system is used to obtain the one-soliton solutions for theφ ⁴ theories (the usual and the one with the wrong sign of the mass term), theφ ⁶, theφ ⁸, the sine-Gordon theories and the KdV equation. Transformations are given which transform the sine-Gordon equation into an equation belonging to theφ ²ⁿ class of theories. A procedure is evolved for obtaining the two-soliton solutions for the sine-Gordon theory without the use of Backlund transformations; it is suggested that this procedure may be useful for investigating the existence of similar solutions for theories of the polynomial type.     

Dimensional regularisation in the presence of large background fields

The method of dimensional regularisation is introduced and discussed in detail for the case of quantum field theories with large (i.e., not infinitesimal) classical background fields. Examples considered are the φ⁴-theory on four-dimensional compact curved spaces as well as Yang-Mills gauge theories on S⁴ with an arbitrary multi-instanton background field.     

The vacuum structure of light-frontφ 1+1 4 -theory

We discuss the vacuum structure ofφ ⁴-theory in 1+1 dimensions quantised on the light-frontx ⁺=0. To this end, one has to solve a non-linear, operator-valued constraint equation. It expresses that mode of the field operator having longitudinal light-front momentum equal to zero, as a function of all the other modes in the theory. We analyse whether this zero mode can lead to a non-vanishing vacuum expectation value of the fieldφ and thus to spontaneous symmetry breaking. In perturbation theory, we get no symmetry breaking. If we solve the constraint, however, non-perturbatively, within a meanfield type Fock ansatz, the situation changes: while the vacuum state itself remains trivial, we find a non-vanishing vacuum expectation value above a critical coupling. Exactly the same result is obtained within a light-front Tamm-Dancoff approximation, if the renormalisation is done in the correct way.     

Conformal field theories near a boundary in general dimensions

The implications of restricted conformal invariance under conformal transformations preserving a plane boundary are discussed for general dimensions d. Calculations of the universal function of a conformal invariant ξ which appears in the two-point function of scalar operators in conformally invariant theories with a plane boundary are undertaken to first order in the ge = 4 − d expansion for the operator φ² in φ⁴ theory. The form for the associated functions of ξ for the two-point functions for the basic field φ^α and the auxiliary field λ in the N → ∞ limit of the O(N) nonlinear sigma model for any d in the range 2 < d < 4 are also rederived. These results are obtained by integrating the two-point functions over planes parallel to the boundary, defining a restricted two-point function which may be obtained more simply. Assuming conformal invariance this transformation can be inverted to recover the full two-point function. Consistency of the results is checked by considering the limit d → 4 and also by analysis of the operator product expansions for φ^αφ^β and λλ. Using this method the form of the two-point function for the energy-momentum tensor in the conformal O(N) model with a plane boundary is also found. General results for the sum of the contributions of all derivative operators appearing in the operator product expansion, and also in a corresponding boundary operator expansion, to the two-point functions are also derived making essential use of conformal invariance.     

Nonperturbative dynamics of scalar field theories through the Feynman-Schwinger representation

We present a summary of results obtained for scalar field theories usingt he Feynman-Schwinger (FSR) approach. Specifically, scalar QED and X²φ theories are considered. The motivation behind the applications discussed in this paper is to use the FSR method as a rigorous tool for testing the quality of commonly used approximations in field theory. Exact calculations in a quenched theory are presented for one-, two-, and three-body bound states. Results obtained indicate that some of the commonly used approximations, such as Bethe-Salpeter ladder summation for bound states and the rainbow summation for one-body problems, produce significantly different results from those obtained from the FSR approach. We find that more accurate results can be obtained using other, simpler, approximation schemes.     

Anomalies of currents in the quantized sine-gordon equation

We investigate the meaning of the (infinitely many) conserved non-trivial currents of the classical Sine-Gordon equation for conventional quantum perturbation theory of a scalar field φ 0000with selfinteraction (cos(Bφ) ? 1 + (ßφ)²/2). Radiation corrections produce for all currents anomalies with contributions on the mass shell.     

A phase cell cluster expansion for Euclidean field theories

We adapt the cluster expansion first used to treat infrared problems for lattice models (a mass zero cluster expansion) to the usual field theory situation. The field is expanded in terms of special block spin functions and the cluster expansion given in terms of the expansion coefficients (phase cell variables); the cluster expansion expresses correlation functions in terms of contributions from finite coupled subsets of these variables. Most of the present work is carried through in d space time dimensions (for φ₂⁴ the details of the cluster expansion are pursued and convergence is proven). Thus most of the results in the present work will apply to a treatment of φ₃⁴ to which we hope to return in a succeeding paper. Of particular interest in this paper is a substitute for the stability of the vacuum bound appropriate to this cluster expansion (for d = 2 and d = 3), and a new method for performing estimates with tree graphs. The phase cell cluster expansions have the renormalization group incorporated intimately into their structure. We hope they will be useful ultimately in treating four dimensional field theories.     

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.     

Gradient properties of the renormalisation group equations in multicomponent systems

The existence of a metric, which enables the renormalisation group β functions of a multicomponent field theory to be written as a gradient, has very important implications for the asymptotic behavior of the renormalisation group equations. It is shown that a very simple metric exists in a field theory with n-component Bose fields and arbitrary φ⁴ interaction, when the β functions are calculated perturbatively up to and including the 2-loop diagrams. This same metric is shown to work for all irreducible diagrams, but it must and can be modified to accommodate reducible 3-loop contributions. The prospects and outlook of this aspect of the renormalisation group are discussed.     

Classical solutions and the energy-momentum tensor

A hyperelliptic two-meron solution of the massless scalar φ^N theory in $\text{n =}\text{2N}\text{(N ? 2)}$ Euclidean dimensions is given. This solution (which interpolates between the two-meron solution and the instanton solution of this theory) is used to illustrate several theory-independent statements which can be made about the energy-momentum tensor for instanton, meron and elliptic meron solutions of all scale invariant classical field theories.     

Critical exponents predicted by grouping of Feynman diagrams in φ4 model

Different perturbation theory treatments of the Ginzburg‐Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ⁴ model with O(n) symmetry. As a result, equations for calculation of the two‐point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.     

External gravitational interactions in quantum field theory

We consider the renormalization of Green's functions of λφ⁴ quantum field theory in an external gravitational field specified by the metric tensor g^μν(y). Green's functions Γ^(n,3) describing the interaction of j scalar particles to arbitrary order n in the gravitational field are shown to be made finite by the standard renormalizations of the flatspace theory and a renormalization of the coefficient of the improvement term in the action functional. These results in φ⁴ theory can be extended to all renormalizable field theories.