Light-cone Representation Research Articles

Integrability of a family of quantum field theories related to sigma models

A method is introduced for constructing lattice discretizations of large classes of integrable quantum field theories. The method proceeds in two steps: The quantum algebraic structure underlying the integrability of the model is determined from the algebra of the interaction terms in the light-cone representation. The representation theory of the relevant quantum algebra is then used to construct the basic ingredients of the quantum inverse scattering method, the lattice Lax matrices and R-matrices. This method is illustrated with four examples: The sinh-Gordon model, the affine sl ( 3 ) Toda model, a model called the fermionic sl ( 2 | 1 ) Toda theory, and the N = 2 supersymmetric sine-Gordon model. These models are all related to sigma models in various ways. The N = 2 supersymmetric sine-Gordon model, in particular, describes the Pohlmeyer reduction of string theory on AdS 2 × S 2 , and is dual to a supersymmetric non-linear sigma model with a sausage-shaped target space.

Spontaneously broken quark helicity symmetry

We discuss the origin of chiral-symmetry breaking in the light-cone representation of QCD. In particular, we show how quark helicity symmetry is spontaneously broken in SU ( N) gauge theory with massless quarks if that theory has a condensate of fermion light-cone zero modes. The symmetry breaking appears as induced interactions in an effective light-cone Hamiltonian equation based on a trivial vacuum. The induced interaction is crucial for generating a splitting between pseudoscalar and vector meson masses, which we illustrate with spectrum calculations in some 1 + 1-dimensional reduced models of gauge theory.

2004 REVIEW OF LIGHT CONE FIELD THEORY

We review the basic concepts of the Poincare group in light cone quantum mechanics and light cone field theory. We discuss recent progress in light cone representations of the Bethe–Salpeter equation, with application to the pion form factor, and the quark Schwinger–Dyson equation. Applications are discussed for deeply virtual Compton scattering, particularly for skewed parton distributions of the pion, and rare B to lepton pair decays. Meson light cone wave functions and recent experimental studies of light cone wave functions are also covered. This review was motivated by the LANL Light Cone 2002 International Workshop.

End-point singularities of Feynman graphs on the light cone

We show that some Lorentz components of the Feynman integrals calculated in terms of the light-cone variables may contain end-point singularities which originate from the contribution of the big-circle integral in the complex k−-plane. These singularities appear in various types of diagrams (two-point functions, three-point functions, etc.) and provide the covariance of the Feynman integrals on the light cone. We propose a procedure for calculating Feynman integrals which guarantees that the end-point singularities do not appear in the light-cone representations of the invariant amplitudes.

Light-cone wavefunction representation of generalized parton distributions

Light-cone wavefunction representation of generalized parton distributions

Light-cone representation of the quark Schwinger-Dyson equation

We use a light-cone approach to solve the Schwinger-Dyson equation for the quark propagator. We show how this method can be used to solve the equation beyond the space-like region, to which one is usually restricted with the usual Euclidean-space approach. We work in the Landau gauge, and use an infrared-enhanced model for the gluon propagator and include instanton effects to get both confinement and vacuum condensates. With our models, reasonable fits to known quantities are obtained, resulting in a light-cone quark propagator that can be used for hadronic physics at all momentum transfers.

Lessons from gauge (and nongauge) theories in low (< 12) dimensions

In this contribution I shall review some of the lessons that I think have been learned over the years from studies of model problems using the light-cone representation. The selection of works which I shall discuss is made to illustrate certain specific points and is not meant to be a review of all the valuable work done in this area. There has been a very large amount of work in the area, much of it very valuable, and a complete review would exceed the time and space allocations for the contribution and also the competence of the author.

Light-cone representation of the spin and orbital angular momentum of relativistic composite systems

Light-cone representation of the spin and orbital angular momentum of relativistic composite systems

Light-cone representation of the spin and orbital angular momentum of relativistic composite systems

The matrix elements of local operators such as the electromagnetic current, the energy momentum tensor, angular momentum, and the moments of structure functions have exact representations in terms of light-cone Fock state wavefunctions of bound states such as hadrons. We present explicit light-cone wavefunctions for the two-particle Fock state of the fermion in QED and the Yukawa theory which preserve all the Lorentz properties and give explicit realization of the spin sum rules. The role of orbital angular momentum in understanding the “spin crisis” problem for relativistic systems is clarified. We show how the perturbative structure can be used to simulate much more general bound state systems while preserving all Lorentz properties. We thus obtain a theoretical laboratory to test the consistency of formulae which have been proposed to probe the spin structure of hadrons.

Light-cone representation of the spin and orbital angular momentum of relativistic composite systems

The matrix elements of local operators such as the electromagnetic current, the energy momentum tensor, angular momentum, and the moments of structure functions have exact representations in terms of light-cone Fock state wavefunctions of bound states such as hadrons. We illustrate all of these properties by giving explicit light-cone wavefunctions for the two-particle Fock state of the electron in QED, thus connecting the Schwinger anomalous magnetic moment to the spin and orbital momentum carried by its Fock state constituents. We also compute the QED one-loop radiative corrections for the form factors for the graviton coupling to the electron and photon. Although the underlying model is derived from elementary QED perturbative couplings, it in fact can be used to simulate much more general bound state systems by applying spectral integration over the constituent masses while preserving all of the Lorentz properties, giving explicit realization of the spin sum rules and other local matrix elements. The role of orbital angular momentum in understanding the "spin crisis" problem for relativistic systems is clarified. We also prove that the anomalous gravitomagnetic moment B(0) vanishes for any composite system. This property is shown to follow directly from the Lorentz boost properties of the light-cone Fock representation and holds separately for each Fock state component. We show how the QED perturbative structure can be used to model bound state systems while preserving all Lorentz properties. We thus obtain a theoretical laboratory to test the consistency of formulae which have been proposed to probe the spin structure of hadrons.

Chiral symmetry breaking in the light-cone representation [formula omitted

In this paper I shall discuss the way in which vacuum structure and condensates occur in the light-cone representation. I shall particularly emphasize the mechanism by which the mass squared of a composite such as the pion comes to depend linearly on the bare mass of its Fermion constituents. I shall give details in two dimensions then discuss the case of four dimensions more speculatively.

Light-cone form of field dynamics in anti-de Sitter space-time and AdS/CFT correspondence

The light-cone form of field dynamics in anti-de Sitter space-time is developed. Using field theoretic and group theoretic approaches the light-cone representation for generators of the anti-de Sitter algebra acting as differential operators on bulk fields is found. We also present a light-cone reformulation of the boundary conformal field theory representations. Making use of these explicit representations of the AdS algebra as an isometry algebra in the bulk and the algebra of conformal transformations at the boundary a precise correspondence between the bulk fields and the boundary operators is established.

Mass operator in the light-cone representation

I argue that for the case of fermions with nonzero bare mass there is a term in the matter density operator in the light-cone representation which has been omitted from previous calculations. The new term provides agreement with previous results in the equal-time representation for mass perturbation theory in the massive Schwinger model. For the DLCQ case the physics of the new term can be represented by an effective operator which acts in the DLCQ subspace, but the form of the term might be hard to guess and I do not know how to determine its coefficient from symmetry considerations.

The Indispensability of Ghost Fields in the Light-Cone Gauge Quantization of Gauge Fields

We continue McCartor and Robertson's recent demonstration of the indispensability of ghost fields in the light-cone gauge quantization of gauge fields. It is shown that the ghost fields are indispensable in deriving well-defined antiderivatives and in regularizing the most singular component of the gauge field propagator. To this end it is sufficient to confine ourselves to noninteracting abelian fields. Furthermore, to circumvent dealing with constrained systems, we construct the temporal gauge canonical formulation of the free electromagnetic field in auxiliary coordinates xµ=(x−,x+,x1,x2), where x−=x0cosθ − x3sinθ, x+=x0 sinθ+x3 cos θ and x− plays the role of time. In so doing we can quantize the fields canonically without any constraints, unambiguously introduce “static ghost fields” as residual gauge degrees of freedom and construct the light-cone gauge solution in the light-cone representation by simply taking the light-cone limit . As a by product we find that, with a suitable choice of vacuum, the Mandelstam-Leibbrandt form of the propagator can be derived in the θ=0 case (the temporal gauge formulation in the equal-time representation).

Light-Cone Quantization of the Schwinger Model

We consider constructing a canonical quantum theory of the light-cone gauge ($A_-$=0) Schwinger model in the light-cone representation. Quantization conditions are obtained by requiring that translational generators $P_+$ and $P_-$ give rise to Heisenberg equations which, in a physical subspace, are consistant with the field equations. A consistent operator solution with residual gauge degrees of freedom is obtained by solving initial value problems on the light-cones. The construction allows a parton picture although we have a physical vacuum with nontrivial degeneracies in the theory.

Diffractive production of Drell–Yan pairs and heavy flavors

Partonic interpretation of high-energy reactions is known to depend on a reference frame. Particularly, in the rest frame of the target Drell–Yan process looks like a freeing of the projectile l l ̄ fluctuation, rather than q q ̄ →l l ̄ annihilation. The light-cone representation for Drell–Yan reaction is very similar to that in DIS and exposes a substantial contamination of soft interactions which turns out to be dominant in diffractive production of lepton pairs and in nuclear shadowing. We estimate the fraction of diffractive events in the total Drell–Yan cross section, which scales in M 2, and find a substantial deviation from factorization. An analogous approach is developed for diffractive production of heavy flavors.

Quark orbital angular momentum in a light-cone representation

We perform an analysis of the quark angular momentum in a light-cone representation by taking into account the effect due to the Melosh-Wigner rotation and find that there is a relativistic correction factor connecting the quark orbital angular momentum with the quark model spin distribution: $L_q(x)={<M_L(x)>}\Delta q_{QM}(x)$. The quark orbital angular momentum $L_q(x)$ and the quark helicity distribution $\Delta q(x)$ are connected to the quark model spin distribution $\Delta q_{QM}(x)$ by a relation: $\frac{1}{2}\Delta q(x)+ L_q(x)=\frac{1}{2}\Delta q_{QM}(x)$, which means that one can decompose the quark model spin contribution $\Delta q_{QM}(x)$ by a quark helicity term $\Delta q(x)$ {\it plus} an orbital angular momentum term $L_q(x)$. There is also a new relation connecting the quark orbital angular momentum with the measurable quark helicity distribution and transversity distribution ($\delta q(x)$): $\Delta q(x)+L_q(x)=\delta q(x)$, from which we may have new sum rules connecting the quark orbital angular momentum with the nucleon axial and tensor charges.

Massless fields in plane wave geometry

Conformal isometry algebras of plane wave geometry are studied. Then, based on the requirement of conformal invariance, a definition of masslessness is introduced and gauge invariant equations of motion, subsidiary conditions, and corresponding gauge transformations for all plane wave geometry massless spin fields are constructed. Light cone representation for elements of conformal algebra acting as differential operators on wavefunctions of massless higher spin fields is also evaluated. Interrelation of plane wave geometry massless higher spin fields with ladder representation of u(2,2) algebra is investigated.

Quantum Mechanics of Gauge Fixing

In the framework of the canonical Weyl gauge formulation of QED, the quantum mechanics of gauge fixing is discussed. Redundant quantum mechanical variables are eliminated by means of unitary transformations and Gauss′s law. This results in representations of the Weyl-gauge Hamiltonian which contain only unconstrained variables. As a remnant of the original local gauge invariance global residual symmetries may persist. In order to identify these and to handle infrared problems and related "Gribov ambiguities," it is essential to compactify the configuration space. Coulomb, axial, and light-cone representation of QED are derived. The naive light-cone approach is put into perspective. Finally, the Abelian Higgs model is studied; the unitary gauge representation of this model is derived and implications concerning the symmetry of the Higgs phase are discussed.

Schwinger model in the light-cone representation

I present a solution to the Schwinger model in the light-cone representation which corrects an error in a previous work. I emphasize the details of the mechanism by which the physical vacuum is different than the perturbative vacuum. I suggest that the method of analyzing vacuum structure presented here may be of use in more complicated theories such as QCD.