Null Plane Research Articles

Constraints and Hamiltonian in light-front quantized field theory

Self-consistent Hamiltonian formulation of scalar theory on the null plane is constructed following Dirac method. The theory contains also {\it constraint equations}. They would give, if solved, to a nonlinear and nonlocal Hamiltonian. The constraints lead us in the continuum to a different description of spontaneous symmetry breaking since, the symmetry generators now annihilate the vacuum. In two examples where the procedure lacks self-consistency, the corresponding theories are known ill-defined from equal-time quantization. This lends support to the method adopted where both the background field and the fluctuation above it are treated as dynamical variables on the null plane. We let the self-consistency of the Dirac procedure determine their properties in the quantized theory. The results following from the continuum and the discretized formulations in the infinite volume limit do agree.

A New Picture for Light Front Dynamics, II

The standard picture of light front dynamics employs the null plane ct + z = 0; whereas the new picture employs a set of null planes ξ · x = 0, where ξ is a lightlike vector. The state vectors and operators in the new picture, also called the ξ-picture, are related to the state vectors and operators in the standard picture by a unitary transformation which depends on the unit vector u = − ξ/|ξ|, where ξ is the three-vector part of ξ. The new picture provides a formalism for dealing with the interaction dependence that occurs in the transverse components of light front angular momentum operators. The present work extends the previous work with the same title to particles with spin. This extension is developed within the context of a two-particle model for the pion-nucleon system. This model is derived from a quantum field theory which describes the interaction between nucleons, pions, and sigma mesons through the virtual processes N ↔ N + π, π ↔ π + σ and N ↔ N + σ. The two-particle model is derived by extending to light front dynamics the Okubo-Glöckle-Müller approach for constructing direct interaction instant form models from quantum field theories. The two-particle light front model for pion-nucleon scattering is transformed to the ξ-picture and a set of coupled integral equations are solved numerically to obtain the s-wave pion-nucleon phase shifts. These numerical results justify an approximation for the two-particle interaction which when used in conjunction with a noninteracting two-particle spin operator leads to a Poincaré invariant potential model for the pion-nucleon system.

Value of Gottfried sum rule from the current anticommutator on the null plane.

By extending the sum rules from the current anticommutator on the null plane to the ${K}^{\ifmmode\pm\else\textpm\fi{}}$ case, we relate the Gottfried sum rule to kaon-nucleon scattering. With the use of this relation, together with Adler-Weisberger relations, we estimate the value of the Gottfried sum rule as 0.26\ifmmode\pm\else\textpm\fi{}0.03 which is consistent with the recent experimental value obtained by the New Muon Collaboration.

Null-plane phenomenology for the pion decay constant and radius.

The pion decay constant and the electromagnetic form factor are computed by using quark diagrams and projecting the bound-state wave function on the null plane. We show that the resulting formulas are the same as those of the Hamiltonian front-form scheme. The connection between the radius (${r}_{\ensuremath{\pi}}$) and the pion decay constant (${f}_{\ensuremath{\pi}}$) is studied using different models of confinement.

Null-plane model of three bosons with zero-range interaction

The relativistic formulation of the three-boson model interacting via a zero-range two-body force in the null plane is given. The Faddeev-like equation for the three-boson bound state is obtained and compared with the non-relativistic formula. The three- boson bound state energy is calculated numerically as a function of the two-body boson energy.

Quantized self-dual Maxwell field on a null surface

The canonical formalism for a self-dual Maxwell field on a null plane is reviewed. After solution of the second class constraints, the transition to the quantum theory is carried out using a representation in which the self-dual Maxwell field is diagonal. The Gauss law constraint allows us to consider the physical state vectors to be holomorphic functionals of one complex function. Application of reality conditions allows us to define an inner product such that the Hermitean adjoint operators are identified with the classical complex conjugate operators. In going over to the Fourier expansion of the operators, we find that the inner product is formally convergent for positive frequency functionals and formally divergent for the negative frequency functionals. Following similar results of Ashtekar, Rovelli, and Smolin, negative frequency states are functional distributions identified with the helicity opposite to that of the positive frequency states.

STRANGE BARYON SPECTRA IN A BETHE-SALPETER DYNAMICAL MODEL

A Bethe-Salpeter equation (BSE) has been derived using a covariant confining interaction motivated from QCD. Recently, this interaction has been applied to a qqq system under the null plane ansatz (NPA), which reduces the resulting equations to three-dimensional form. For equal mass quarks, encouraging results were obtained for baryon masses and quantum numbers. In the present paper, this formalism is generalized to unequal mass kinematics and applied to the spectroscopy of strange baryons (Λ, Σ, Ξ, Ω). All input parameters are fixed from previous calculations. The reduced qqq equation in the NPA is solved algebraically in a six-dimensional harmonic oscillator basis, on lines analogous to the corresponding techniques developed for the equal mass qqq problem. The results on the Λ, Σ spectra are in good agreement with data, where available, up to levels N=5. Our results are compared with those from another relativized potential of baryon masses.

Perturbative renormalization of null-plane QED.

It has been recognized for some time that quantization on a null plane has several unique and remarkable advantages for the elucidation of quantum field theories. To date these unique features have not been exploited to solve strongly coupled, four-dimensional gauge theories. This is the first in a series of papers aimed at systematically formulating renormalizable gauge theories on the null plane. In order to lay down the groundwork for upcoming nonperturbative studies, it is indispensable to gain control over the perturbative treatment first. A discussion of one-loop renormalization of QED in the Hamiltonian formalism is presented. In this approach, one is faced with severe infrared divergences characteristic of the light-cone gauge. We show how to treat these divergences in a coherent fashion, and thus recover the usual results of the renormalization procedure such as Ward identities and coupling-constant renormalizations.

The effect of viscosity on the resistive tearing mode with the presence of shear flow

The effect of small viscosity on the ‘‘constant ψ’’ tearing mode in the presence of shear flow is analyzed using the boundary-layer approach. It is found that the influence of viscosity depends upon the parameter G′(0)/F′(0), where G′(0) and F′(0) denote the flow shear and magnetic field shear at the magnetic null plane, respectively. When ‖G′(0)/F′(0)‖≪1, the tearing mode growth rate is suppressed by the viscosity, but not completely stabilized. When ‖G′(0)/F′(0)‖∼𝒪(1) and the viscosity is comparable to the resistivity, the growth rate vanishes as [1−G′(0)2/F′(0)2]1/3, when G′(0)2→F′(0)2 from below. In the case where 1−[G′(0)2/F′(0)2]<0 matching cannot be achieved and the tearing mode vanishes.

Light-front dynamics of the covariant harmonic oscillator.

Following Leutwyler and Stern, the two-particle covariant harmonic-oscillator model of Feynman, Kislinger, and Ravndal is reformulated within the framework of light-front dynamics. The light-front dynamics of the oscillator is obtained by constraining the manifestly covariant formulation to the null plane, ct+z=0. An inner product for the null-plane wave functions is developed, and it is shown that the light-front mass operator M and spin operator scrJ are Hermitian with respect to this inner product. Interaction dependent, nonlocal relative position and momentum operators Q and P, are introduced, and the mass and spin operators are expressed in terms of them. It is found that the part of ${\mathit{M}}^{2}$ that contains the dynamics is proportional to a nonrelativistic harmonic-oscillator Hamiltonian in P and Q, and that the spin operator scrJ=Q\ifmmode\times\else\texttimes\fi{}P. The eigenstates of Q and P are determined, and are used to construct position and momentum representations for the null-plane oscillator. In these representations the light-front dynamics takes on the appearance of nonrelativistic quantum mechanics. In particular, the nonlocal inner product for the null-plane wave functions goes over to the local nonrelativistic form. The obvious relevance of these results to the quark model of the hadrons is briefly discussed.

Null-plane quantization of fermions.

Massive Dirac fermions are canonically quantized on the null plane using the Dirac-Bergmann algorithm. The procedure is carried out in the framework of quantum electrodynamics as an illustration of a rigorous treatment of interacting fermion fields.

Spatial properties of second-order vestibulo-ocular relay neurons in the alert cat

Second-order vestibular nucleus neurons which were antidromically activated from the region of the oculomotor nucleus (second-order vestibuloocular relay neurons) were studied in alert cats during whole-body rotations in many horizontal and vertical planes. Sinusoidal rotation elicited sinusoidal modulation of firing rates except during rotation in a clearly defined null plane. Response gain (spike/s/deg/s) varied as a cosine function of the orientation of the cat with respect to a horizontal rotation axis, and phases were near that of head velocity, suggesting linear summation of canal inputs. A maximum activation direction (MAD) was calculated for each cell to represent the axis of rotation in three-dimensional space for which the cell responded maximally. Second-order vestibuloocular neurons divided into 3 non-overlapping populations of MADs, indicating primary canal input from either anterior, posterior or horizontal semicircular canal (AC, PC, HC cells). 80/84 neurons received primary canal input from ipsilateral vertical canals. Of these, at least 6 received input from more than one vertical canal, suggested by MAD azimuths which were sufficiently misaligned with their primary canal. In addition, 21/80 received convergent input from a horizontal canal, with about equal number of type I and type II yaw responses. 4/84 neurons were HC cells; all of them received convergent input from at least one vertical canal. Activity of many vertical second-order vestibuloocular neurons was also related to vertical and/or horizontal eye position. All AC and PC cells that had vertical eye position sensitivity had upward and downward on-directions, respectively. A number of PC cells had MADs centered around the MAD of the superior oblique muscle, and 2/3 AC cells recorded in the superior vestibular nucleus had MADs near that of the inferior oblique. Thus, signals with spatial properties appropriate to activate oblique eye muscles are present at the second-order vestibular neuron level. In contrast, none of the second-order vestibuloocular neurons had MADs near those of the superior or inferior rectus muscles. Signals appropriate to activate these eye muscles might be produced by combining signals from ipsilateral and contralateral AC neurons (for superior rectus) or PC neurons (for inferior rectus). Alternatively, less direct pathways such as those involving third or higher order vestibular or interstitial nucleus of Cajal neurons might play a crucial role in the spatial transformations between semicircular canals and vertical rectus eye muscles.

Resistive tearing instability with equilibrium shear flow

The effect of equilibrium velocity shear on the resistive tearing instability has been systematically studied, using the boundary layer approach. Both the ‘‘constant-ψ’’ tearing mode, which has a growth rate that scales as S−3/5, and the ‘‘nonconstant-ψ’’ tearing mode [Δ′(αS)−1/3≫1], which has a growth rate that scales as S−1/3, are analyzed in the presence of flow. Here S is the usual ratio of the resistive diffusion and Alfvén times. It is found that the shear flow has a significant influence on both the external ideal region and the internal resistive region. In the external ideal region, the shear flow can dramatically change the value of the matching quantity Δ′. In the internal resistive region, the tearing mode is sensitive to the flow shear at the magnetic null plane: G′(0). When G′(0) is comparable to the magnetic field shear, F′(0), the scalings of the constant-ψ tearing mode are changed and the Δ′>0 instability criterion is removed, provided G′(0)G″(0)−F′(0)F″(0)≠0. The scalings of the nonconstant-ψ tearing mode remain unchanged. When the flow shear is larger than the magnetic field shear at the magnetic null plane, both tearing modes are stabilized. Finally, the transition to ideal instability is discussed.

Fock space description of simple spinors

Cartan’s simple—often called pure—spinors corresponding to even-dimensional complex vector spaces are defined in terms of the associated maximal totally null planes. Their geometrical properties are derived and described using notions familiar to physicists: Dirac and Weyl spinors, gamma matrices, tensors formed bilinearly from pairs of spinors, and creation and annihilation operators of Fermi states. A new theorem characterizes a simple spinor φ by the properties of the vector tψBγμφ, where ψ is an arbitrary spinor and B is the matrix connecting the gamma matrices with their transposes. The Cartan constraint equations on the components of simple spinors are given a new, geometrically transparent derivation based on the action on simple spinors of a maximal Abelian subgroup of the group Spin.

Hamiltonian analysis of linearised topologically massive gravity

The Dirac constrained Hamiltonian formalism is applied to linearised topologically massive gravity, first in normal coordinates and then in null plane coordinates. This is done by introducing velocity fields which allow the theory to be written in terms of at most quadratic time derivatives. In both normal and null plane coordinates the analysis confirms the usual result that the theory describes a single propagating massive field. In null coordinates an extra second-class constraint eliminates the momentum from the set of independent initial data, leaving only a single Hamiltonian degree of freedom.

The octahedron theorem in Minkowski three-space a metric vector space proof of Miquel's theorem in the Laguerre plane

As is well known [1, p.480], the cycles and spears of the real Laguerre plane can be represented by the points and null planes of three-dimensional Minkowski space. Miquel's theorem in the Laguerre plane can then be expressed as an intrinsically interesting Minkowski space theorem the octahedron theorem. We outline the correspondence between the two theorems, and then give a metric vector space proof of the octahedron theorem, thereby providing an alternate proof of Miquel's theorem. We then discuss the generalization of both theorems to more general spaces.

Hamiltonian analysis of linearised gravity on a null plane

The Hamiltonian dynamics of N-dimensional linearised gravity is analysed using Dirac's constraint formalism and a null hyperplane as an initial data surface. In addition to the usual 2N first class constraints, there are N(N-3)/2 second class constraints which are a general feature of null plane dynamics. The number of canonical degrees of freedom is therefore N(N-3)/2, precisely half the number found using a spacelike initial value surface.

Quasi-regular singularities based on null planes

We examine quasi-regular singularities that take the form of invariant two-dimensional null planes in Minkowski space-time, thus extending earlier studies of “conical” singularities based on timelike and spacelike planes. The result is described in terms of a deficit parameter. We also examine the form of the Riemann curvature tensor at the singularity.

Minimal surfaces and strings from spinors a realization of the cartan programme

It is shown ow the old Enneper-Weierstrass integral parametrization ofminimal surfaces inR3 and the Eisenhart ones inR3,1, when expressed through bilinear spinor polynomia, may be considered as deriving from a particularlocal realization of the possibility envisaged by Cartan: to consider ordinary vectors as generated from isotropic planes in complex spaces, in the frame of the bijective Cartan map connecting pure spinor directions to totally null planes in complex spaces. In the case ofR3 the correspondingglobal realization of the Cartan map extends the Enneper-Weierstrass parametrization to the Gauss-conformal map of the minimal surface to S2, which may be identified with the Riemann celestial sphere. Forreal spinors minimal surfaces are substituted bystrings both inR2,1 andR3,1; inR2,1 strings are globally mapped to a torus (inR4). InR3,1 (and its conformal extensions) a prescription is given to obtain strings as integrals of real, bilinear spinor null vectors, from the Enneper-Weierstrass spinor representation of minimal surfaces, through the use of unitary transformations in spinor space which allows its restriction to the real (Majorana spinor-space). It is shown that the Nambu action, or the area of the world surface described by the space-time string, is minimized by the Lagrangian density expressed as a quadrilinear spinor product formally reminding Fermi and Thirring interaction Lagrangians.

Collisionless tearing instability in a non-Maxwellian neutral sheet: An integrodifferential formulation

An integro-differential formalism is developed to study the collisionless tearing instability in a non-Maxwellian neutral sheet. The exact unperturbed particle orbits are used analytically in the orbit integrals. The treatment is linear, self-consistent, and kinetic for both ions and electrons. The analysis is carried out for low-frequency, purely growing electromagnetic perturbations (‖ω‖≪ωci). Using the Galerkin method, the integro-differential equation is solved to obtain the dispersion relation and the eigenmode structure. A sufficient condition for instability is given on the basis of a quadratic form and the eigenvalues of a self-adjoint integro-differential operator. The formalism is applied to a specific model distribution. For the case where the electrons and ions are both non-Maxwellian, it is found that the instability is dominated by the axis-crossing electrons and that the eigenmode is strongly localized to a region of the order of ( ρezp)1/2 at the null plane, where ρe is a measure of the electron gyroradius in the asymptotic magnetic field and zp is the sheet thickness. It is shown that the growth rate can be enhanced by several orders of magnitude over the isotropic case and that short wavelength perturbations are strongly preferred. The dispersion relation has the general form γ/kve∥=const where k∥B0 and ve∥ is the electron thermal velocity along B0, the equilibrium magnetic field.