Current Commutation Research Articles

Broken scale invariance in inelastic lepton—Nucleon scattering

Broken scale invariance in inelastic lepton—nucleon scattering is discussed studying current commutators near to light-cone and their equal time limits. Models, based onWilson’s ideas, are proposed here to provide a more general frame than canonical models.

Relations Among the Structure Functions of Deep-Inelastic Neutrino-Nucleon Scattering

Light-cone current commutators are used to derive relations between the structure functions of the vector-vector part of neutrino-nucleon scattering and the structure functions of the vector-axial-vector part.

Inequalities for structure functions of deep inelastic lepton-nucleon scattering giving tests of basic algebraic structures

Inequalities for the structure functions of deep inelastic electro-and-neutrino-production on nucleons are derived (a) from the quark parton model or equivalently from a hypothesis of Fritzsch and Gell-Mann on the leading light cone singularity of current commutators. (b) from the quark model with neutral gluons using current algebra techniques to the same extent as for the Callan-Gross relation. Present data satisfy these inequalities, and give stringent predictions for future neutrino-production experiments. On the other hand, three models with integrally charged partons, the SUB-model, the Drell-Levy-Yan model and the Bjorken-Glashow model are found to be in trouble by comparison with experiment.

Deep inelastic neutrino-nucleon interactions

Deep inelastic neutrino-nucleon interactions are discussed under the assumptions of conserved axial vector current and equal-time commutation relations of currents. It is shown that the structure functionsW1 andW2 for vector and axial vector currents are equal in the average sense with respect to the variablev and are equal in the limitv → ∞. The resulting expressions are then used to predict relations connecting deep inelastic neutrino and electron scattering.

Delta-function derivatives of arbitrary order in equal-time charge current commutator

Delta-function derivatives of arbitrary order in equal-time charge current commutator

Light-Cone Commutators and a Sum Rule for High-Energy Neutrino Scattering

Using canonical expressions for certain current commutators on the light cone, we derive a sum rule which relates the structure function ${W}_{3}$ (arising from vector-axial-vector interference) measured in high-energy neutrino experiments to the structure functions measured in electroproduction.

Ambiguity in soft-pion limit and time-space equal-time commutators of currents

Ambiguity in soft-pion limit and time-space equal-time commutators of currents

Generalized equal-time commutators

An expansion of causal distributions is proposed which shows explicitly the structure of these distributions in the « equal-time » limit. It is shown that such an expansion always exists for matrix elements of a commutator in a local axiomatic field theory. The expansion is suitable for the study of the range of applicability of the manipulations usually performed with the equal-time commutators (like construction of field-charge commutators and differentiation of retarded commutators). Furthermore, we show how one can extend these manipulations to cases where equal-time commutators (in the usual sense) do not exist. The expansion is used to study some of the restrictions which are implied by locality on the structure of current commutators.

Factorisable representations of current groups and the Araki-Woods imbedding theorem

In order to study the current commutation relations of quantum field theory, Araki and Woods [2] and Araki [1] introduced the notion of current groups and factorisable representations of such groups. Araki and Woods [2] and Streater [5] estabhshed that such representations admit a natural imbedding in a symmetric Fock space exp H over a Itflbert space H. If G is a locally compact group, then a suitable space of Borel functions on a Borel space with values in G is made into a group under pointwise multiphcation. This is called a current group of G. Araki [1] established that the factorisable representations of the current group are based on certain coeycle valued measures. In this paper we show the existence of a measure on the Borel space over which the current group is constructed, relative to which a coeycle valued density exists. This yields a certain natural topology for the current group under which the faetorisable representation is continuous. In order to take into account all the coeycles of first order in the construction of fae, torisable representations it turns out that projective representations should also be considered. Finally, the Araki-Woods imbedding is expheitly constructed in terms of the cocycles. At this stage it may be worth remarking that our methods differ very much from that of Araki and Woods. We rely more on measure theory and not at all on lattice theory.

Smoothness near the light-cone and superconvergence

It is shown that the derivation of the standard sum rules requires more than the null plane structure of the commutators. The usual form is obtained only for a definite smooth behaviour of the current commutator near the light cone, which corresponds to the hypothesis of superconvergence for strong and and semistrong amplitudes.

Absence of Noncanonical Behavior in Quantum Electrodynamics

Using the explicit electromagnetic-field-dependence property of the electromagnetic current according to Schwinger and Johnson, a perturbation-theoretic calculation of equal-time canonical current commutators is carried out to show that there are no noncanonical derivative terms bilinear in the electromagnetic field, contrary to what has been claimed in the past. This result is in accord with that of the Bjorken-Johnson-Low definition for equal-time commutators. It is pointed out that the conventional distribution-theoretic approach to handling singular expressions of field multilinears at the same space-time point fails to take into account field-dependence properties of the kind present in the Schwinger-Johnson ansatz.

Theory of pi K scattering near threshold

A simple theory id developed for the process pi K to pi K near threshold. It is similar to one recently proposed for pi pi to pi pi , and rests on an assumption of smoothness of the scattering amplitudes as functions of the energy-momentum invariants. The Adler PCAC consistency condition is imposed, but instead of current commutators, rho-meson dominance on the mass shell is used to fix overall normalization. The two approaches are connected by the KSRF relation, but because this model depends on two parameters (mass, width) more information is obtained. Thus the model contains P and D waves, and gives improved scattering-length predictions. Comparison with results from crossing-symmetric Regge models suggests that these need important additional terms.

Soft pion production in deep inelastic electromagnetic and weak processes

On the basis of the PCAC hypothesis and current algebra, we calculate the cross section of one and two soft pion production in the deep inelastic scattering of electrons and neutrinos on nucleons. The experimental measurement of these processes would make it possible to investigate the one-nucleon matrix elements of the axial and isovector current commutators and contribute to elucidation of the scale invariance mechanism.

Local one particle approximation of the current commutators

The spectral functions of the current commutators matrix elements between vacuum and one particle boson states are calculated in a local one particle approximation. In the equal time limit, the restrictions imposed by the current algebra commutation relations without operator Schwinger terms are systematically discussed. It is found, that in this approximation, and when one of the currents is conserved, the PCAC and vector meson dominance models cannot be considered simultaneously. The comparison of the results with experiment gives preference to the first of the two models.

Time-Space Equal-Time Commutators of Currents and Scaling Properties of Current Divergences

On the basis of rather general considerations, it has been recently suggested that the equal-time commutators between the time component of an axial-vector current and the space component of a current should be modified. Such modifications lead to a natural explanation of the ${K}_{l3}$ form factors and the ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}$ and $\ensuremath{\eta}\ensuremath{\rightarrow}3\ensuremath{\pi}$ decays. Moreover, examples may be constructed in which these modifications do not invalidate the usual good results of current algebra. In this paper we make use of scaling considerations to note that such modifications imply the presence of an $\mathrm{SU}(3)\ensuremath{\bigotimes}\mathrm{SU}(3)$-breaking term in the Hamiltonian density with dimension $d\ensuremath{\ge}2$. Conversely, if $d<2$ such modifications must be absent. We conclude by presenting some related comments.

Current Commutators near the Light Cone

Current Commutators near the Light Cone

Light-Cone Structure of Current Commutators in the Gluon-Quark Model

The formal structure of current commutators near the light cone (currents separated by nearly lightlike distances) is studied for quark models with interactions mediated by SU(3) \ifmmode\times\else\texttimes\fi{} SU(3) singlet vector or scalar mesons. The idea in the end is to abstract the general structure features, and thereafter to discard the specifics of the underlying model. The approach adopted here is formal, in the sense that all manipulations are based on canonical equal-time commutation relations and on the corresponding canonical equations of motion. The light-cone commutator is expressed in terms of certain bilocal operators and has a definite tensor and SU(3) \ifmmode\times\else\texttimes\fi{} SU(3) structure that forms the heart of the abstraction. The analysis is greatly facilitated by a theorem which shows that the gluons can be treated in the external-field approximation for purposes of determining the leading light-cone singularities. It is an important result that the tensor and SU(3) \ifmmode\times\else\texttimes\fi{} SU(3) structure of the light-cone commutator turns out to be the same for the gluon models as for the free-quark model. The practical implications of this structure, already discussed by Gell-Mann, are therefore preserved in the gluon model. We review the applications. Some of the delicate points connected with current conservation are discussed, and it is shown how to write the commutator in a form where current conservation is manifest. We also discuss the structure of the vacuum matrix elements of the commutator. Finally, we investigate the commutation relations among the bilocal operators and show that the algebra closes for bilocals defined on a single lightlike ray. In an Appendix our various results are compared with those obtained by canonical light-cone quantization procedures.

Equal-time current commutator and sum rules in axiomatic field theory

Equal-time current commutator and sum rules in axiomatic field theory

Baryon Currents in the Fermi-Yang Model

All possible baryon currents consisting of trilinear local products of anticommuting nucleon fields are constructed and classified with respect to $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2)$. In contrast to the quark-model baryon currents investigated by Hwa and Nuyts, the Fermi-Yang model (nucleon field) baryon currents may be isospin $\frac{3}{2}$ as well as \textonehalf{}. Commutators of the baryon currents with axial charges are investigated, as well as anticommutators of baryon currents with the Hermitian conjugates of their own divergences. Although isospin mixing does occur for some of the axial-charge---baryon-current commutators, it is found that those terms in the commutators carrying the appropriate isospin are always proportional to the baryon current being commuted. For an isospin-\textonehalf{} baryon current ${B}^{\ensuremath{\mu}}(x)$, it is found that $\ensuremath{\delta}({x}_{0}\ensuremath{-}{y}_{0}){{B}^{\ensuremath{\mu}}(x),{\ensuremath{\partial}}^{\ensuremath{\nu}}{{B}_{\ensuremath{\nu}}}^{\ifmmode\dagger\else\textdagger\fi{}}(y)}=\ensuremath{\delta}(x\ensuremath{-}y)\mathcal{O}(\ensuremath{\mu},x)+(\mathrm{Schwinger}\mathrm{term})$, where $〈0|\mathcal{O}(\ensuremath{\mu},x)|0〉=0$. This property of $\mathcal{O}(\ensuremath{\mu},x)$ is equivalent to a spectral-function sum rule, the single-particle approximation of which relates the positive- and negative-parity spin-\textonehalf{}, isospin-\textonehalf{} nucleon resonances.

Fixed poles and light cone singularities

Deviations from strictly canonical light cone singularities of the e.m. current commutator, such as logarithmic factors or fractional power behavior could, in principle, be detected by looking for J ≈ 0 fixed pole contributions to the imaginary part of the Compton amplitude in the Regge limit, i.e. electro-production structure function W 2 and total photoabsorption cross section.