Infinite-component Wave Equations Research Articles

Boson Mass Spectrum and Pion Form Factor fromSO(5,2). Generalization toSO(m,2)

The $\mathrm{SO}(5,2)$ representation whose basis functions are the solutions of the Bethe-Salpeter equation for two scalar quarks interacting via the exchange of a scalar zero-mass boson to form a bound state of zero mass is written as an infinite-dimensional representation for bosons. A general second-order, infinite-component wave equation is given with a reasonable mass spectrum. The ground-state form factor corresponding to this representation is calculated by identifying the electromagnetic current with the current of the equation, in which case the pion form factor is given by $G(t)=\frac{[\frac{1\ensuremath{-}b{(cosh\ensuremath{\theta})}^{2}t}{2m}]}{{[\frac{1\ensuremath{-}{(cosh\ensuremath{\theta})}^{2}t}{4{m}^{2}}]}^{\frac{5}{2}}}$. The mass spectrum and the form factor can easily be generalized to the group $\mathrm{SO}(m,2)$ by a simple substitution, and the mass spectra and form factors obtained before in the framework of the group $\mathrm{SO}(4,2)$ appear as special cases.

Magnetic Moments and Charge Radii for States Described by an Infinite Component Wave Equation

Magnetic Moments and Charge Radii for States Described by an Infinite Component Wave Equation

Dynamical Groups and the Bethe-Salpeter Equation

We have considered a class of equations whose solutions are the $n$-dimensional spherical harmonics. We have proven that the set of solutions can be accomodated into a single irreducible representation of the Lie algebra ${B}_{\frac{(n+1)}{2}}$ for $n$ odd, or ${D}_{\frac{(n+2)}{2}}$ for $n$ even, and also into a single irreducible representation of the group $\mathrm{SO}(n, 2)$. Examples of such equations are the Schr\"odinger equation for the H atom as well as its $(n\ensuremath{-}1)$-dimensional analog, the Schr\"odinger equation with $O(4)$-invariant potentials, as well as their $(n\ensuremath{-}1)$-dimensional analogs; and the Schr\"odinger equation of an $n$-dimensional rigid rotator. In the previous $(n\ensuremath{-}1)$-dimensional cases and in the case of the $n$-dimensional rotator, the solutions can be accomodated into a single irreducible representation of the algebra ${B}_{\frac{(n+1)}{2}}$ for $n$ odd, or ${D}_{\frac{(n+2)}{2}}$ form $n$ even, and the group $\mathrm{SO}(n, 2)$ is the dynamical group of the equation. In the cases of the H atom and $O(4)$-symmetric potentials, we have the algebra ${D}_{3}$ and the group SO(4,2). The class also includes the Bethe-Salpeter (BS) equation for two scalar quarks interacting via the exchange of a scalar boson of zero mass to form a bound state of zero mass, in which case we have---after the Wick rotation---the algebra ${B}_{3}$ and the dynamical group SO(5,2). When the mass of the bound state is different from zero, the SO(5,2) representation splits into $\ensuremath{\Sigma}\ensuremath{\bigoplus}\mathrm{SO}(4, 2)$. The BS equation is transformed to an infinite-component wave equation in the case that the mass of the bound state is zero and in the case that the mass is different from zero. In the first case the known eigenvalue spectrum is obtained. In the second case perturbation theory is applied, and the eigenfunctions and eigenvalues to first and second order in perturbation theory, respectively, are given. Finally, in an Appendix, the H atom and the BS equation representations are written in the canonical basis.

Construction of Local Quantum Fields Describing Many Masses and Spins

We discuss the possibility of constructing, out of particle creation and destruction operators, local quantum fields that transform as representations of the homogeneous Lorentz group. Our immediate goal is to write down a consistent local quantum field theory which can simultaneously describe many particles with different masses and spins. In the case that the field is a finite-dimensional irreducible Lorentz tensor, we are able to carry through our program with no restrictions on the masses considered as functions of the spin, provided the usual connection between spin and statistics is satisfied. However, when the field transforms as a unitary irreducible representation of the homogeneous Lorentz group (an infinite-dimensional representation), the requirement of locality, along with the physical assumption that the masses are bounded below, m(j)≥m0>0, leads to the restriction that the masses are independent of the spin. This property is shown to hold when the transformation law of the field is taken to be an irreducible finite-dimensional representation ⊗ a unitary irreducible representation. The physical consequences of this result and possible methods for evading it are discussed. Finally, an Appendix is included, where the related problem of orthogonality properties of timelike solutions to infinite-component wave equations is examined. In particular, we show that when the solutions of such wave equations transform as unitary irreducible representations of the homogeneous Lorentz group, only the Majorana representations support a scalar product, which is orthogonal for different spins.

Spacelike Solutions of Infinite-Component Wave Functions

A necessary and sufficient condition for the existence of spacelike solutions is given for infinite-component wave equations which are linear in the momentum and for which the charge operator is positive definite. The condition is used to establish the existence of spacelike solutions for wave equations of the Abers-Grodsky-Norton type, i.e., wave equations of the form $({\ensuremath{\gamma}}_{\ensuremath{\mu}}{p}_{\ensuremath{\mu}}\ensuremath{-}M)\ensuremath{\psi}=0$, where ${\ensuremath{\gamma}}_{\ensuremath{\mu}}$ are the Dirac $\ensuremath{\gamma}$'s, and $M$ is an $\mathrm{SL}(2,C)$ invariant with nontrivial dependence on ${\ensuremath{\gamma}}_{\ensuremath{\mu}}$. An interpretation of the spacelike solutions is given for one simple model and suggests that the occurrence of the spacelike solutions in general is simply a reflection of the composite nature of the particles being described. It is also shown that if the current operator has no explicit momentum dependence and the particles are assigned to a representation of $\mathrm{SL}(2,C)$, then current conservation is equivalent to a wave equation.

Group Dynamics of Elementary Particles

AbstractIt is proposed that particle interactions can be described by means of simple group operations in non‐compact Lie groups. Prescriptions for the structure of these group operations are formulated which are motivated by the study of simple models, in particular of the dynamics of the hydrogen atom. If they are fulfilled we call the structure “group dynamics”.Neglecting at first internal symmetries, a few simple models are investigated in which group dynamics is possible. The group O(3,1) turns out to describe the coupling of pseudoscalar mesons, O(4,2) that of photons to baryons quite well: The pionic decay rates of baryon resonances up to spin 19/2 and the electromagnetic form factors of the nucleons are predicted in good agreement with experiment.The internal symmetry SU(3) is included in the O(3.1) model of the pionic coupling in the simplest way, by assuming O(3,1) × SU(3) to be the dynamical group. This gives rise to a minimal symmetry breaking of the amplitudes and relates it to the mass differences in SU(3) multiplets: The amplitude consists of a product of a SU(3) Clebsch Gordan coefficient and a universal function of the velocity of the final baryon. The way the particle masses enter the decay rates is uniquely prescribed in this approach. The agreement with experiment is excellent.Finally, the connection of this purely algebraic approach with related ideas, like the use of infinite component wave equations, is discussed.

The Degenerate Vacuum and the Infinite Component Wave Equation

In the field theoretical viewpoint, an infinite component wave equation, which is of first order in p '' is derived for the one-particle wave function. The mass formulas are as follows : m=JJ.o(J +~) for the simplified Heisenberg nwdel, m= ±JJ.o(J +D for the original Heisenberg model, where J denotes the intrinsic spin of one-particle states. For this purpose, we have relaxed the requirement for the vacuum in the following way: Instead of the existence of the invari­ ant vacuum, we assume that the vacuum corresponds to an irreducible unitary representation of Poincare group with P'=O. That is, we allow the existence of the infinitely degenerate vacuum.

Infinite-Component Wave Equations with Hydrogenlike Mass Spectra

Continuing a previous work, various models of relativistic wave equations are considered which have an infinite number of components. By combining a unitary representation of $SO(4, 2)$ and the ordinary finite (Dirac) representation of the Lorentz group, it is possible to construct equations which produce hydrogenlike mass spectra. However, they are also accompanied by redundant, or unphysical, solutions. In the nonrelativistic limit, on the other hand, the equations obtained can be shown to be mathematically equivalent to the Schr\"odinger equation for the hydrogen atom. This suggests that the method of infinite-component wave equations may be a useful tool in exploring the physics of strong interactions. A general discussion is made about the principles and problems that will be relevant in pursuing such a program.

Remarks on the Method of Infinite-Component Wave Equation for Relativistic Model with Internal Structure

Remarks on the Method of Infinite-Component Wave Equation for Relativistic Model with Internal Structure

Infinite Multiplets and the Hydrogen Atom

Section I of this report deals with attempts to construct infinite-component wave equations with mass spectra that are free of unphysical features. An example is given of a second-order relativistic wave equation whose mass spectrum, both the discrete and the continuous part, is precisely the same as that of the nonrelativistic hydrogen atom. Section II deals with the nonrelativistic hydrogen atom, without any approximations. It is shown that the Schr\"odinger equation is equivalent to a nonrelativistic analog of Majorana-Nambu wave equations. The Hilbert space of all the states, including the continuum, is profitably and economically used as a representation space for a unitary, irreducible representation of the group $\mathrm{SO}(4,2)$. The dipole operator is an $\mathrm{SO}(4,2)$ generator. The theory is generalized to arbitrary frames of reference by application of Galilei transformations. The generators of Galilei transformations also belong to the $\mathrm{SO}(4,2)$ algebra. What results is a field theory of the hydrogen atom. The exact electromagnetic interaction, including all multipoles, takes the form of a local interaction between the electromagnetic field and the infinite-component hydrogen field. In Sec. III the significance of all this for hadron physics is discussed.

Internal Movement of Hadrons and Infinite-Component Wave Equation

Internal Movement of Hadrons and Infinite-Component Wave Equation