Light-like Plane Research Articles

Space-like maximal surfaces containing entire null lines in Lorentz-Minkowski 3-space

Consider a surface $S$ immersed in the Lorentz-Minkowski 3-space $\mathbf{R}^{3}_{1}$. A complete light-like line in $\mathbf{R}^{3}_{1}$ is called an \textit{entire null line} on the surface $S$ in $\mathbf{R}^{3}_{1}$ if it lies on $S$ and consists of only null points with respect to the induced metric. In this paper, we show the existence of embedded space-like maximal graphs containing entire null lines. If such a graph is defined on a convex domain in $\mathbf{R}^{2}$, then it must be contained in a light-like plane (cf. Remark~3.3). Our example is critical in the sense that it is defined on a certain non-convex domain.

A note on Lamarle formula in Minkowski $3$-space

The Lamarle formula is known as a simple relation between the Gaussian curvature and the distribution parameter of a non-developable ruled surface. In this paper, we obtain the Lamarle formula of a non-developable ruled surface with pseudo null base curve and null director vector field in Minkowski $3$-space. We also obtain the corresponding striction line and distribution parameter of such surface. We prove that there is no Lamarle formula when the director vector field is spacelike and its derivative is null, because the ruled surface in that case is a lightlike plane. Finally, we give some examples.

Zero mean curvature surfaces in Lorentz--Minkowski 3-space which change type across a light-like line

It is well-known that space-like maximal surfaces and time-like minimal surfaces in Lorentz--Minkowski $3$-space $\boldsymbol{R}^{3}_{1}$ have singularities in general. They are both characterized as zero mean curvature surfaces. We are interested in the case where the singular set consists of a light-like line, since this case has not been analyzed before. As a continuation of a previous work by the authors, we give the first example of a family of such surfaces which change type across a light-like line. As a corollary, we also obtain a family of zero mean curvature hypersurfaces in $\boldsymbol{R}^{n+1}_{1}$ that change type across an ($n-1$)-dimensional light-like plane.

Killing scalar of non-linear σ-model on G/H realizing the classical exchange algebra

The Poisson brackets for non-linear σ-models on G/H are set up on the light-like plane. A quantity which transforms irreducibly by the Killing vectors, called Killing scalar, is constructed in an arbitrary representation of G. It is shown to satisfy the classical exchange algebra.

Classical exchange algebra of the superstring on S5 with AdS-time

A classical exchange algebra of the superstring on S5 with AdS-time is shown on the light-like plane. To this end, we use the geometrical method of which consistency is guaranteed by the classical Yang–Baxter equation. The Dirac method does not work, since there are constraints in which first-class and second-class constraints are mixed and one can hardly disentangle with each other keeping the isometry.

Constrained WZWN models on [formula omitted] and exchange algebra of G-primaries

Consistently constrained WZWN models on G/{S⊗U(1)n} is given by constraining currents of the WZWN models with G. Poisson brackets are set up on the light-like plane. Using them we show the Virasoro algebra for the energy–momentum tensor of constrained WZWN models. We find a G-primary which satisfies a classical exchange algebra in an arbitrary representation of G. The G-primary and the constrained currents are also shown to obey the conformal transformation with respect to the energy–momentum tensor. It is checked that conformal weight of the constrained currents is 0. This is necessary for the consistency for our formulation of constrained WZWN models.

LIGHTLIKE DEVELOPABLES IN MINKOWSKI 3-SPACE

We say that a surface in Minkowski 3-space is a lightlike developable if any pseudonormal vector of the regular part of the surface is lightlike. We show that such a surface is a part of a lightlike plane, the lightcone, the tangent surface of a spacelike curve in a lightlike plane, the tangent surface of a lightlike curve or the glue of such surfaces. The most interesting surfaces in such the class of surfaces is the tangent surface of a lightlike curve. We give a classification of the singularities for the tangent surface of a generic lightlike curve. As a consequence, the H3 type singularity appears in generic.

Wightman functions for interacting spinor fields on a lightlike plane

The objects under discussion are the Wightman functions of the «good» components {ie277-1} of a spinor field ψ(x), restricted to a lightlike plane. We assume that on such a plane free and interacting spinor fields ψ+(χ) obey the same anticommutation relations. We impose general conditions on the Wightman functions for interacting fields and show that these quantities are free of ambiguities for lightlike separation of the arguments.

Canonical formalism on a lightlike hyperplane

The canonical formalism for a neutral scalar field on a lightlike hyperplane is studied in terms of Dirac's theory for constrained Hamiltonian systems. The field and its canonical momentum on the plane are restricted to appropriate functional spaces and are expanded into wave packets. The formalism becomes simpler by the absence of unphysical modes forP−=0. After quantization we discuss how the Wightman function loses its dynamical information when it is restricted to the lightlike plane.

A Lightlike Formulation of the Quark Model

·The quark model for hadrons is form~lated on tht basis of quark fields quantized on a lightlike plane. Hadron resonance states are constructed from the vacuum in terms of creation operators of quarks and antiquarks, where the. number of additional q-q pairs are indefinite. Under the non-exoticity assumption for SU(3) li9SU(3) z irreducible representations, the structure of matrix elements of axial lightlike charge couplings is investigated, and related sum rules of Adler-Weisberger type are obtained. The problem of identifying the quarks on a lightlike plane with charged partons is also discus~ed.

Quantization Quark Fields on a Lightlike Plane

We investigate some problems in quantizing quark fields on a lightlike hyperplane. The generators of the stability group of the quantization plane and the lightlike charges play a central role. The set of these observables is denoted by '=! +· The properties of creation and anniliilation operators of quarks are investigated in detail. It is shown that one-partiCle states' can be determined by the measurement of some observables belonging to '=! +· Also discus~ed is the problem of the independent preparation of two-partiCle states on a lightlike plane. § l. Introduction The original idea of formulating field theories on a hyperplane tangent to the light-cone goes back to Dirac.n In hadron physics the technique of taking the infinite momentum limie> appeared as a precursor of the· formulation in which fields are quantized on such a hyperplane (abbreviated hereafter as the lightlike plane). Soon the kinematical properties of the Poincare generators in this new approach became clear, 3> and the utility of choosing such generators in hadron physics was stressed. The importance of the notions of lightlike charges and an algebra they generate was also fully recognized.•> Feynman rules in this formula­ tion of quantum electrodynamics were obtained, 5> and mathematically rigorous treat­ ments have been investigated. 6> The lightlike quantization is convenient for studying the structure functions in the scaling region of electrol).-proton inclusive experiments.n The parton model proposed by Feynman 8> is easier to investigate in this formulation than in the original infinite momentum frame. It seems very likely that partons are to be identified with ,the quarks quantized on a lightlike plane. 9> The motivation of the present author to this quantization came from the possible utility of lightlike chiral algebra for hadron resonances.10l· One· aim of this work is to present the ligh tlike formulation in a way suitable for applications to the problem of hadron resonances and to prepare the same language for the quark model and the parton model. The formulation of field theory is characterized by the structure of the stability group of a plane on which fields are quantized. By the stability group of a plane we mean the subgroup of the Poincare group which keeps the plane invariant. (It may be modified by a cer.t~in unitary transformation.) In the lightlike for

Angular constraints and the Melosh transformation

The Melosh transformation between current and constituent quarks is discussed when formulated on a light-like plane, as is natural for the description of collinear symmetries like SU(6) W. There is no unique definition of the transformation in this context, but further angular constraints are introduced to ensure that the states coupled by SU(6) W transformations have the correct spin properties. It is then shown that there is no transformation generated by single quark operators which satisfies these requirements, save if only terms to first order in the transverse momentum are kept. Some of the resultant inconsistencies are illustrated by a discussion of baryon magnetic moments. It is concluded that there is no satisfactory explanation of μ p /μ n = − 3 2 from this approach to the Melosh transformation. A non-field theoretic version of the free quark model which solves these angular constraints is exhibited.

On a class representations of the bilocal current algebra on a light-like ray

A class of representation of the algebra if bilocal currents on a plane perpendicular to a light-like ray is constructed. The saturation of these representations with states of definite momenta conjugate to the configuration space variables on the light-like plane reveals a Fock-space nature of the representation space, generated by quark and anti-quark fields obeying Fermi statistics, projected on the characteristics light-like ray. The mass spectrum, however, is not determined by the algebra. Representations on a subspace of the constructed Fock space are discussed for special momentum configurations.

Equal-Time and Null-Plane Commutators of Conserved Currents on the Light Cone

The usual manifestly conserved expansions of current densities on the light cone do not lead to the correct commutation relations at equal times or on a lightlike plane. Therefore a piece must be added which is compatible with current conservation only for a specific light-cone singularity, namely, ${({z}^{2}\ensuremath{-}i\ensuremath{\epsilon}{z}_{0})}^{\ensuremath{-}1}$, which obeys ${\ensuremath{\square}}^{(z)}{({z}^{2}\ensuremath{-}i\ensuremath{\epsilon}{z}_{0})}^{\ensuremath{-}1}=0$. A close relation exists between such a nonmanifestly conserved addition to the operator expansion and the well-known phenomenon of a noncausal part in the invariant function of the matrix element between one-particle states.