General Quantum Field Theory Research Articles

The averaged null energy condition for general quantum field theories in two dimensions

It is shown that in any local quantum field theory in two-dimensional Minkowski space–time possessing a mass gap and an energy-momentum tensor, the averaged null energy condition is fulfilled for the set of those vector states which correspond to energetically strongly damped, local excitations of the vacuum. This set of physical vector states is translation invariant and dense. The energy-momentum tensor of the theory is assumed to be a Wightman field which is local relative to the observables, generates locally the translations, is divergence-free, and energetically bounded. Thus the averaged null energy condition can be deduced from completely generic, standard assumptions for general quantum field theory in two-dimensional flat space–time.

ADVANCE IN DYNAMICAL SPONTANEOUS SYMMETRY BREAKING

Recently, a condition is derived for a nontrivial solution of the Schwinger–Dyson equation to be accompanied by a Goldstone bound state in a special quantum electrodynamics model. This result is extended and a new form of the Goldstone theorem is obtained in a general quantum field theory framework.

Universality of low-energy scattering in three-dimensional field theory

The universal low-energy behavior, $2mc/\mathrm{ln}|s\ensuremath{-}{4m}^{2}|$ of the scattering function of particles of positive mass m near the threshold ${s=4m}^{2},$ and $\ensuremath{\pi}/\mathrm{ln}|s\ensuremath{-}{4m}^{2}|$ for the corresponding S-wave phase shift, is established for weakly coupled field theory models with a mass $m>0$ in space-time dimension 3; c is a numerical constant independent of the model and couplings. This result is a nonperturbative property based on an exact analysis of the scattering function in terms of a two-particle irreducible (or Bethe-Salpeter) structure function. It also appears as generic by the same analysis in the framework of general relativistic quantum field theory.

The one-loop functional as a Berezinian

We employ notions familiar from supersymmetry for constructing the one-loop functional of general quantum field theories with bosons and fermions (spin ≤ 1/2). To demonstrate the advantages of such an approach for calculating one-loop divergences, we analyse a simple Yukawa theory with two different versions of a super-heat-kernel expansion. These methods also simplify the calculation of one-loop divergences for non-renormalizable meson–baryon Lagrangians occurring in chiral perturbation theory.

Constructing a c-function for SUSY gauge theories

Recently a non-perturbative formula for the RG flow between UV and IR fixed points of the coefficient in the trace of the energy momentum tensor of the Euler density has been obtained for N=1 SUSY gauge theories by relating the trace and R-current anomalies. This result is compared here with an earlier perturbation theory analysis based on a naturally defined metric on the space of couplings for general renormalisable quantum field theories. This approach is specialised to N=1 supersymmetric theories and extended, using consistency arguments, to obtain the Euler coefficient at fixed points to 4-loops. The result agrees completely, to this order, with the exact formula.

CLIFFORD ALGEBRAS IN FINITE QUANTUM FIELD THEORIES I.

Finite quantum field theories may be constructed from the most general renormalizable quantum field theory by forbidding, order by order in the perturbative loop expansion, all ultraviolet-divergent renormalizations of the physical parameters of the theory. The relevant finiteness conditions resulting from this requirement relate all dimensionless couplings in the theory. At first sight, Yukawa couplings which are equivalent to the generators of some Clifford algebra with identity element represent a very promising type of solutions of the condition for one-loop finiteness of the Yukawa couplings. However, under a few reasonable and simplifying assumptions about their particular structure, these Clifford-like Yukawa couplings prove to be in conflict with the requirements of one- and two-loop finiteness of the gauge coupling and of the absence of gauge anomalies, at least for all simple gauge groups up to and including rank 8.

An Analogue of the Kac--Wakimoto Formula¶and Black Hole Conditional Entropy

A local formula for the dimension of a superselection sector in Quantum Field Theory is obtained as vacuum expectation value of the exponential of the proper Hamiltonian. In the particular case of a chiral conformal theory, this provides a local analogue of a global formula obtained by Kac and Wakimoto within the context of representations of certain affine Lie algebras. Our formula is model independent and its version in general Quantum Field Theory applies to black hole thermodynamics. The relative free energy between two thermal equilibrium states associated with a black hole turns out to be proportional to the variation of the conditional entropy in different superselection sectors, where the conditional entropy is defined as the Connes-Stoermer entropy associated with the DHR localized endomorphism representing the sector. The constant of proportionality is half of the Hawking temperature. As a consequence the relative free energy is quantized proportionally to the logarithm of a rational number, in particular it is equal to a linear function the logarithm of an integer once the initial state or the final state is taken fixed.

Jets and quantum field theory

We discuss quantum-field-theoretic interpretation of the family of observables (the so-called C-algebra) introduced by F.V. Tkachov [preprint FERMILAB-PUB-95/191-T] for a systematic description of multijet structure of multiparticle final states at high energies. We argue that from the point of view of general quantum field theory, all information about the multijet structure is contained in the values of a family of multiparticle quantum correlators that can be expressed in terms of the energy-momentum tensor.

TOWARDS A CLASSIFICATION OF FINITE FIELD THEORIES

We consider the finiteness conditions on the Yukawa couplings of a general quantum field theory for gauge groups SU (n)(n>6) and a rather general particle content. It is shown that in the class of theories considered (149 different particle contents), only two models are able to fulfill the finiteness conditions. Only one of these is supersymmetric. For the nonsupersymmetric one the appropriate Yukawa couplings are constructed explicitly.

Intrinsic algebraic characterization of space-time structure

Starting from a partially ordered set ofC*-algebras ℛi representing algebras of observables of physical subsystems, we derive a topological Hausdorff space ℳ as a candidate for some generalized “space-time” with the help of which one can define a net\(O \to \mathcal{A}(O),O \subseteq \mathcal{M}\), of algebras. This opens a way to define a physical theory without an underlying metaphysical manifold, an aspect which may be relevant for the unification of general relativity and quantum field theory.

NO-GO THEOREMS FOR NONSUPERSYMMETRIC FINITE QUANTUM FIELD THEORIES

We consider the finiteness conditions on the Yukawa couplings of a general quantum field theory for groups SU (N). Their gauge invariance leads us to the necessary structure of the couplings, and for some cases the nonexistence of non-trivial solutions is proved. Somewhat miraculously a special role of SU(5) emerges as a possible case of evading these no-go theorems.

Tensor products of quantized tilting modules

LetU k denote the quantized enveloping algebra corresponding to a finite dimensional simple complex Lie algebraL. Assume that the quantum parameter is a root of unity ink of order at least the Coxeter number forL. Also assume that this order is odd and not divisible by 3 if typeG 2 occurs. We demonstrate how one can define a reduced tensor product on the familyF consisting of those finite dimensional simpleU k-modules which are deformations of simpleL and which have non-zero quantum dimension. This together with the work of Reshetikhin-Turaev and Turaev-Wenzl prove that (U k,F) is a modular Hopf algebra and hence produces invariants of 3-manifolds. Also by recent work of Duurhus, Jakobsen and Nest it leads to a general topological quantum field theory. The method of proof explores quantized analogues of tilting modules for algebraic groups.

Continuum and discretum—Unified field theory and elementary constants

Unitary field theories and “SUPER-GUT” theories work with an universal continuum, the structured spacetime of R. Descartes, B. Spinoza, B. Riemann, and A. Einstein, or a (Machian (1–3) ) structured vacuum according the quantum theory of unitary fields (Dirac, (4,5) and Heisenberg (6–8) ). The atomistic aspect of the substantial world is represented by the fundamental constants which are invariant against “all transformations” and which “depend on nothings” (Planck (9–11) ). A satisfactory unitary theory has to involve these constants like the mathematical numbers. Today, Planck's conception of the three elementary constants ħ, c, and G may be the key to general relativistic quantum field theory like unitary theory. However, the elementary constants are a question of measurement-theory, also. According to Popper's theory (12–16) of induction, such unitary theories are “universal explaining theories.” The fundamental constants involve the complementarity between the universal statements in unitary theory and the “basic statements” in the language of classical observables.

Universal conditions of finiteness for general quantum field theories

Starting from the observation that a certain quantity F is common to the lowest order finiteness conditions in a general quantum field theory of gauge fields, scalars and fermions, we are led - making massive use of algebraic computer methods- to a new, extremely restrictive condition and eventually to the conjecture that F = 1 for finite theories. All such theories are classified and contain many more interesting candidates for finite theories (∼ 4000 types) than the known supersymmetric ones (∼ 200).

Hyperfunction quantum field theory: Basic structural results

The choice of the class E′ of generalized functions on space-time in which to formulate general relativistic quantum field theory (QFT) is discussed. A first step is to isolate a set of conditions on E′ that allows a formulation of QFT in otherwise the same way as the original proposal by Wightman [Ark. Fys. 28, 129 (1965)], where E′ is the class of tempered distributions. It is stressed that the formulation of QFT in which E′ equals the class of Fourier hyperfunctions on space-time meets the following requirements: (A) Fourier hyperfunctions generalize tempered distributions thus allowing more singular fields as suggested by concrete models; (B) Fourier hyperfunction quantum fields are localizable both in space-time and in energy-momentum space thus allowing the physically indispensable standard interpretation of Poincaré covariance, local commutativity, and localization of energy-momentum spectrum; and (C) in Fourier hyperfunction quantum field theory almost all the basic structural results of ‘‘standard’’ QFT (existence of a PCT operator, spin-statistics theorems, existence of a scattering operator, etc.) hold. Finally, a short introduction to that part of Fourier hyperfunction theory needed in this context is given.

Axial anomaly and Schwinger terms in two-dimensional general quantum field theory.

We compute the vacuum expectation value (\ensuremath{\Omega},${\mathit{j}}^{\mathrm{\ensuremath{\mu}}}$(x)${\mathit{j}}^{\ensuremath{\nu}}$ (y)\ensuremath{\Omega})\ensuremath{\equiv}${J}^{\ensuremath{\mu}\ensuremath{\nu}}$(x-y) for a relativistic, Hermitian, not necessarily local vector field in two-dimensional space-time when ${\ensuremath{\partial}}_{\ensuremath{\mu}}$${J}^{\ensuremath{\mu}\ensuremath{\nu}}$ =${\ensuremath{\partial}}_{\ensuremath{\mu}}$${J}^{\ensuremath{\nu}\ensuremath{\mu}}$=${\ensuremath{\partial}}_{\ensuremath{\mu}}$${\ensuremath{\epsilon}}_{\ensuremath{\alpha}}^{\ensuremath{\mu}}$${J}^{\ensuremath{\alpha}\ensuremath{\nu}}$ =${\ensuremath{\partial}}_{\ensuremath{\mu}}$${\ensuremath{\epsilon}}_{\ensuremath{\alpha}}^{\ensuremath{\mu}}$${J}^{\ensuremath{\nu}\ensuremath{\alpha}}$=0 holds in S'(${\mathrm{openR}}^{2}$), the space of tempered distributions. As the space-time symmetry group of the model we take the inhomogeneous proper orthochronous Lorentz group ${P}_{+}^{\ensuremath{\uparrow}}$. (\ensuremath{\Omega},${\mathrm{Tj}}^{\ensuremath{\mu}}$(x)${j}^{\ensuremath{\nu}}$(y)\ensuremath{\Omega}) is verified to have anomalous (axial-)vector Ward identities due to nonvanishing equal-time current-current commutators. Additional conditions on ${j}^{\ensuremath{\mu}}$ are specified which imply that ${j}^{\ensuremath{\mu}}$ is a free zero-mass Wightman field with ${\ensuremath{\partial}}_{\ensuremath{\mu}}$${j}^{\ensuremath{\mu}}$=${\ensuremath{\partial}}_{\ensuremath{\mu}}$${\ensuremath{\epsilon}}_{\ensuremath{\nu}}^{\ensuremath{\mu}}$${j}^{\ensuremath{\nu}}$=0.

Finite quantum field theories.

We investigate the relation between finiteness of a four-dimensional quantum field theory and global supersymmetry. To this end we consider the most general quantum field theory and analyze the finiteness conditions resulting from the requirement of the absence of divergent contributions to the renormalizations of the parameters of the theory. In addition to the gauge bosons, both fermions and scalar bosons turn out to be a necessary ingredient in a nontrivial finite gauge theory. In all cases discussed, the supersymmetric theory restricted by two well-known constraints on the dimensionless couplings proves to be the unique solution of the finiteness conditions.

Finiteness of quantum field theories and supersymmetry

The consequences of finiteness are studied for a general renormalizable quantum field theory by analysing the finiteness conditions resulting from the requirement of absence of divergent contributions to the renormalizations of the parameters of an arbitrary gauge theory. In all cases considered, the well-known two-loop finite supersymmetric theories prove to be the unique solution of the finiteness criterion.

Two-loop renormalization group equations in a general quantum field theory: (III). Scalar quartic couplings

The two-loop β-functions for the scalar quartic couplings are computed in a general renormalizable quantum field theory with scalar, spin- 1 2 , and (vector) gauge fields associated with a general gauge group G, using dimensional regularization and modified minimal subtraction ( −MS). A more explicit form is given for the two-loop β-function of the quartic coupling of the Higgs doublet in the minimal QCD electroweak theory based on SU(3) × SU(2) × U(1).

Two-loop renormalization group equations in a general quantum field theory (II). Yukawa couplings

The two-loop β-functions for the Yukawa couplings are computed in a general renormalizable quantum field theory with scalar, spin 1 2 , and (vector) gauge fields associated with a general gauge group G. A more explicit form is given for the two-loop β-functions for the Yukawa couplings of the Higgs doublet in the minimal QCD-electroweak theory based on SU(3) × SU(2) × U(1).