A Quest for Symmetry

The following sections are included:

• CONTENTS

• Foreword

• REMINISCENCES

It is proposed that the strength of the coupling constants is different for the strangeness non-conserving and strangeness conserving currents in the scheme of Fermi interactions of an ordinary charged current-current type.

First, the consistency with experimental results is analyzed by introducing phenomenologically the direct interaction. Then, the possibility of the derivation of this interaction as the effective interaction of the primary Fermi interactions is discussed.

The calculation of the matrix element of the process γ+d → n+p by dispersion techniques is considered. There are twelve invariant amplitudes ; the covariant form of the transition amplitude is related to the noncovariant (Pauli matrix) form, and we further relate this to partial wave amplitudes, keeping, however, only the dipole amplitudes. The Born terms of the dipole amplitudes are derived, and the dispersion relations for the dipole amplitudes are written down and solved in a low-energy approximation in which the n-p final-state rescattering is taken into account, but no other higher -order effects. In an Appendix these calculations are performed directly in the nonrelativistic limit to illustrate the essential simplicity of the technique. No light is shed on the well-known discrepancy between theory and experiment for the threshold M1 amplitude; the nearest (anomalous) singularities, at least, will have to be included in order for the dispersion calculation to be sufficiently accurate. But we remark that the form of the amplitude implies a correlation between the threshold value of the amplitude and its energy dependence, a correlation that would be interesting to check experimentally.

The exact transition amplitude for the photodisintegration of the deuteron at the zero-energy limit of the incident γ ray is presented as a function of the electric charges and the magnetic moments of the proton, neutron, and deuteron, the effective range for triplet n-p scattering, and the binding energy of the deuteron. The method used in this article is based on the theory of composite particles in quantum theory which has been developed by Nishijima, Zimmermann, and Haag. The low-energy limit presented here is the generalization of the Kroll-Ruderman theorem of pion photoproduction to the problem, including a composite particle.

No abstract received.

The notion of supermultiplets first developed by Wigner for the theory of nuclear structure is applied to the structure of elementary particles. The group structure is assumed to be SUe. The quark model is assumed for the entire discussion, although some of these results can be obtained from other models. It is found that the octet of pseudoscalar mesons along with the octet and the singlet of vector mesons form a supermultiplet. Okubo's speculated mass form for the vector mesons is derived. It is also found that the octet of baryons along with a singlet particle of spin form a supermultiplet. The type of baryonic coupling for the electromagnetic and weak current is derived.

No abstract received.

No abstract received.

A relativistic formulation of the SU(6) symmetry scheme is presented, starting with the basic assumption that the fields corresponding to elementary particles are tensors of M(12) [or Ũ(12) or SU(12)£]. In particular a mixed second-rank tensor and a totally symmetric third -rank tensor are associated with the meson and baryon fields, respectively. It is shown that if these fields are required to satisfy prescribed free-field equations of motion, then one is led to a particle supermultiplet structure which corresponds to the 35⊕1 and 56-dimensional representations of SU(6) for the mesons and baryons. It is also shown that the spin-dependent and SU(3)-spin-dependent mass splittings can be included in the theory and that solutions in terms of physical particle fields can be obtained. Effective trilinear meson-meson and meson-baryon vertex functions, using these solutions and an interaction Lagrangian which is invariant under M(12), are calculated in the lowest order perturbation. We would like to note especially the following results: (a) From the known pion-nucleon coupling constant, the width of the pion-nucleon (3,3) resonance is calculated to be 94 MeV. (b) The ratio of the magnetic form factors for the neutron and proton is for all momentum transfers and μp = (1+2M_P/m_ρ) nuclear magnetons. (c) The charge form factor of the neutron is zero for all momentum transfers.

No abstract received.

The algebraic formulation of strong coupling is applied to the strong coupling of the SU ₂-symmetric model in which all the partial waves of the π mesons are included. The strong-coupling group is a semidirect product the SU₂⊗SU₂ internal symmetry group and an Abelian group which is generated by an infinite number of commuting generators corresponding to the vertices of the π mesons in different orbital angular momentum states. A physically interesting irreducible representation of the group is obtained which consists of a series of irreducible representations of the P-wave strong-coupling group. The Regge recurrences of isobars appear in series. Each degenerate multiplet of isobars is specified by three quantum numbers—spin s, isospin i, and additional quantum number ν—which satisfy the angular momentum triangular relation. The following form of mass formula is obtained : M(s,i,v)=M₀+m₀[x (x−1)s(s+1)+(1−x)i(i+1)+xν(ν+1)].

The extension of Veneziano's form V(s,t) to the N-particle amplitude is given.

Realizations of Sugawara's field theory of currents on functionals are discussed from an algebraic viewpoint. It is shown that the realization can also be obtained from a nonlinear Lagrangian of scalar fields. A solvable model of field theory of currents is also investigated in detail. It is shown that the degenerate vacuum is the ground state of the system in this model.

We propose a perturbative approach in which the Veneziano representation plays the role of a Born term. We interpret Veneziano's formula as describing only the contribution of one-particle intermediate states. We then add to it the contribution of many-particle intermediate states by means of Feynman-like diagrams. The rules for writing the integrals corresponding to any planar diagram are given. Crossing symmetry, duality, and Reggeization are explicitly taken into account. We find the asymptotic behavior of each Feynman-like diagram. We sum them and prove that the whole amplitude has Regge behavior. The new trajectory, however, is no longer linear, and it incorporates correctly the elastic unitarity constraint. We argue that this approach will ultimately provide a framework in which generalized unitarity (in Cutkosky's sense) can be imposed.

A general formulation of duality theory is presented that includes nonplanar Feynman-like diagrams. All diagrams, planar as well as nonplanar, are so classified that the diagrams in a given class are mutually connected by duality. A prescription is given for constructing an integral representation of the scattering amplitude for each class. Some fundamental properties of the duality relations are discussed.

We re-examine Nielsen's interpretation of the Veneziano amplitudes from the point of view of the functional formulation. It is shown that the sum of a large number of “fishnet” Feynman graphs of very high order can be approximated by generalized Veneziano amplitudes.

A formulation of dual-symmetric theory is given in terms of functional integrations. Known formulas of the theory, such as the N-particle Veneziano amplitude and both planar and nonplanar one-loop amplitudes, are explicitly given in terms of the new formulation. The new formulation is also shown to be equivalent to the other formulations such as the harmonic-oscillator formulation.

It is shown that dual-resonance amplitudes can be expressed in terms of rudimental amplitudes defined by functional Integrals which correspond to transition amplitudes of quantum-mechanical systems of strings with imaginary time. The equivalence between the path-integral and operator formulation of quantum mechanics is used to establish the connection between this approach and the usual operator approach. The factorization of rudimental amplitudes is studied to obtain the Feynman-like rules for dual-resonance amplitudes. This allows us to express N-Reggeon vertices in terms of rudimental amplitudes, and to determine the propagator, which is shown to be the usual spurious-free twisted propagator. N-loop orientable diagrams are calculated. In general, the functional Integrals considered can be calculated by solving appropriate Neumann's boundary-value problems of corresponding bounded Riemann surfaces. This provides a generalization of the analog model to the case of external Reggeons which are described by extended momentum distributions on the boundaries.

Generalizations of the dual model are discussed in the context of functional integral formulation of the theory. We notice that all the symmetries of dual amplitudes are led from the symmetries of the Lagrangian used for the measure of functional integration. Irreducible fields under conformal transformations are classified by their dimension and conformal spin. Since the factorizability of dual amplitudes requires a local Lagrangian, we construct the general form of local Lagrangian density such that the action integral is conformal invariant. We found that if one restricts the form of the Lagrangian to bilinear form of the fields, the kinetic term of the Lagrangian is possible to construct only for conformal spin 0 and field. The former yields the usual dual model, while the latter yields the Bardakçi -Halpern model and the Neveu-Schwarz one. Quantization of conformal spin field is discussed in detail. The generating functional of this field is constructed in terms of the functional integral technique and used to construct the general N-particle amplitudes.

Possible new invariances of generalized dual models are discussed in context of the functional integral formulation. The operators relevant to new gauges of those models, such as those obtained by Neveu and Schwarz, are derived as infinitesimal generators of new field transformations which leave action integral invariant.

The recently proposed Lagrangian of a free string is quantized by means of path integrals. It is shown how Veneziano amplitudes are obtained in this picture by coupling the string with external sources.

The strong-coupling theory in static models is formulated in terms of functional integrations. The method is demonstrated for the charged-scalar model. The expression of elastic and inelastic meson-nucleon scattering amplitudes is obtained to leading order in the strongcoupling expansion (1/g expansion), while the isobar energy levels are obtained up to the next to the leading order.

A relativistic strong coupling theory of Higgs' Lagrangian is developed. The basic idea is the same as the old strong coupling theory, i.e. the separation of collective coordinates (string coordinates in our case and rotator coordinates in the old string coupling) from meson fields. We obtain a system of a relativistic string interacting with a massive scalar and vector meson fields. In the strong coupling limit the masses of the meson fields become infinite. We prove, therefore, that if one introduces an appropriate cut-off, the system becomes a quantized relativistic string.

The method of collective coordinates developed in the study of strong -coupling theory is used for the quantization of the kink solution of a two-dimensional nonlinear field theory. The position of the kink is treated as a collective coordinate, which represents the position of a particle. It is separated from the rest of the coordinates, which represent the internal degrees of freedom of an extended particle. Two similar but different methods are presented; the one is nonrelativistic and suited for the weak-coupling limit, while the other is relativistic.

The quantum mechanics of solitary-wave classical solutions of nonlinear wave equations is discussed in detail for the kink solution of two-dimensional φ⁴ field theory. The formalism provides a natural interpretation of an extended particle, the soliton, for the classical kink. The perturbation theory around the extended particle is developed and used to calculate the radiative corrections for the mass of soliton up to one loop. The mass renormalization is discussed in detail to show that the mass counterterm to the nonsoliton sector also does the job for the present case, i.e. one-soliton sector. Although our formalism is not manifestly Lorentzcovariant, the Lorentz covariance is shown explicitly by calculating the soliton energy for a fixed momentum. The paper also contains the perturbation calculation of matrix element of φ fields between one-soliton states.

No abstract received.

The importance of the surface term in the action integral in gauge theories is pointed out. It contains additional dynamical variables other than those contained in the Lagrangian density. The additional variables play the role of collective coordinates for the quantization of 'tHooft -Polyakov monopole solution. In general they are necessary if the generalized total charge of the system is non-zero. They serve to select a boundary condition of the solution of classical equations of motion, about which the quantum mechanical perturbative expansion should be done. We obtained the Schwinger quantization condition for the dyon: Qg = integer.

The WKB method for systems with many degrees of freedom is developed. Using a given imaginary-time (Euclidean) classical solution of the equations of motion, we explicitly construct the WKB wave function in the classically forbidden region of configuration space. Similarly, we construct the wave function for the allowed region using a real-time (Minkowski) solution. For this purpose we use the collective-coordinate method previously developed for solitons in quantum field theory. The present WKB method is an extention of that by Banks, Bender, and Wu to systems with many degrees of freedom and field theories. This paper is intended to present ideas and the general formalism: two applications are briefly discussed: the quantization condition for periodic solutions and vacuum tunneling in field theories.

The A₀ = 0 canonical formalism is shown to be completely consistent even though Gauss's law is not verified as a field equation. This is so because the Hilbert space of states must also involve states coupled with external static charge distributions. Indeed these cannot be handled by adding the standard term because it vanishes identically in the A₀ = 0 gauge for static charges. The corresponding charge densities are instead the eigenvalues of the operator of infinitesimal time-independent gauge transformations which commute with the Hamiltonian. The implications of this viewpoint are discussed in connection with Gribov's phenomenon, the θ vacuum, perturbation theory, and quark confinement. The constant of motion due to gauge invariance in gauge theories plays the same role as the constant of motion due to translational invariance in soliton quantization.

The physics of vacuum tunneling for gauge field theories is studied following methods developed in a series of earlier papers by explicitly constructing the eigenstates of the Hamiltonian which correspond to the ground state in the presence of an arbitrary external static charge distribution. This is achieved in the A₀ = 0 canonical formalism where this charge distribution was shown before to be a cyclic degree of freedom. The corresponding wave functionals represent the θ vacuum and very-heavy-quark states. We work in the Schrödinger picture where the field configurations are functions only of the space coordinates. Hence we have one less space dimension than the usual Euclidean approach, and instantons are paths in configuration space which connect different classical vacuums through classically forbidden regions. There we solve the Schrödinger equation exactly to the first two orders in ℏ using the one-instanton solution. The resulting wave functionals are concentrated around the instanton paths which are the locus of maximum tunneling probability. Around classical vacuums the wave functional is shown to possess, besides the usual perturbative Gaussian, a part which is an exponentially increasing function of the fluctuation. The energy and eigenfunctional are determined by matching this wave functional to the WKB expression in an overlap region where both are valid. Special care must be taken concerning the continuous symmetries of the problem since the instantons and the classical vacuum break different subgroups of the symmetry group. Although the formalism applies to any field theory with instantons, the present discussion of matching strictly holds if there are no massless particles. It has thus the same limitations as the dilute-gas approximation (DGA) in the Euclidean approach, the results of which are comprised in the outcome of the present discussion. For the massless case the DGA result for the energy also comes out if one ignores the fact that the matching problem differs a priori in an essential way from the massive case. In field-theoretical language, one should notice that we match two different perturbative expansions, namely the expansions around the instanton and around the classical vacuum. This is rather new in quantum field theory and may have other applications, for instance in connection with the Gribov phenomenon, as discussed in an earlier paper.

We develop a collective field theory of a U(N) gauge field, which involves gauge-invariant operators only. We treat gauge-invariant path-ordered phase factors as collective fields on string space. The theory is formulated in such a way that in the N → ∞ limit the collective field theory exactly approaches the original U(N) gauge field theory. Even for finite N it is expected that it provides correct excitation energy and degeneracy for low-lying states. The final form of the collective field theory is a field theory of closed strings. Although our formalism is Lorentz noncovariant, it is manifestly gauge invariant. The present paper is devoted mainly to the formalism, although a few remarks for the actual calculations are given in the final section.

We formulate a general method of collective fields in quantum theory, which represents a direct generalization of the Bohm-Pines treatment of plasma oscillations. The present method provides a complete procedure for reformulating a given quantum system in terms of a most general (overcomplete) set of commuting operators. We point out and exemplify how this formalism offers a new powerful method for studying the large-N limit. For illustration we discuss the collective motions of N identical harmonic oscillators. As a much more important application, we show how, based on the present formalism, one solves the planar limit of a non-trivial SU(N) symmetric quantum theory.

We discuss the algebra and the representations of SO(2N) groups used in the construction of grand unified theories in a basis in which its connections with the SU(N) grand unification is most transparent. Specializing to the case of N = 5, we discuss the problem of fermion masses for various Higgs representations. Applying our considerations to SO(12) grand unification, we comment on the nature of weak interactions of the extra generation of fermions present in the 32-dimensional spinor representation of this group.

We use the stochastic quantization method of Parisi and Wu to understand the quenched momenta prescription for large N theories. The main advantage of our procedure is its simplicity. It leads to the prescription in a straightforward manner without the explicit use of the perturbation expansion.

Starting from Witten's large-N power counting we derive an equation identical to the so-called bootstrap condition of strong-coupling theory. The large-N baryons are therefore characterized by representations of the strong-coupling group (SCG). It is pointed out that the bootstrap relation is quite general and valid when the semiclassical expansion about soliton solutions is at work. The collective coordinates of the soliton correspond to the coordinates of induced representations of the SCG. One of the interesting representations of the SCG is the quark representation and this makes a bridge between the Skyrme solitons and the nonrelativistic quark model. We explicitly show that the induced representation is derived from N static quarks with N→∞. We further emphasize the generality and power of the algebraic method. For this purpose we present a modified chiral bag model which exhibits the algebraic relations in large N and approaches the Skyrme-soliton picture in the zero-bag-radius limit.

It is shown that chiral symmetry and chiral anomaly are inherent to the incommensurability of a quasi one-dimensional charge-density-wave system. This chiral anomaly induced by the applied electric field is interpreted as an acceleration mechanism of the sliding charge-density wave and is connected with the Thomas-Fermi screening effect. The explicit breaking of chiral symmetry due to the external potential is proved to have a sinusoidal dependence on the phase order parameter. Possible observable effects are also discussed.

Symmetry-related features of charge-density-wave transport phenomena are studied using a nonmean-field effective Lagrangian approach. It is pointed out that a local chiral symmetry (based on the Kač-Moody algebra) emerges in the low-energy structure of one-dimensional electron-phonon systems. From this symmetry follow directly power-law correlations of both electrons and phonons. The Peierls instability is suppressed owing to one-dimensional fluctuations. Still the charge-density wave arises and the chiral anomaly can account for acceleration of a sliding charge-density wave along with a phonon-drag effect. The problem of pinning of charge-density waves is discussed in relation to explicit breakings of the chiral symmetry.

We present field theoretical descriptions of massless (2 +1) dimensional nonrelativistic fermions in an external magnetic field, in terms of a fermionic and bosonic second quantized language. An infinite dimensional algebra, W_∞, appears as the algebra of unitary transformations which preserve the lowest Landau level condition and the particle number. In the droplet approximation it reduces to the algebra of area-preserving difleomorphisms, which is responsible for the existence of a universal chiral boson lagrangian independent of the electrostatic potential. We argue that the bosonic droplet approximation is the strong magnetic field limit of the fermionic theory. The relation to the c=1 string model is discussed.

We construct a W_∞ gauge field theory of electrons in the lowest Landau level. For this purpose we introduce an external gauge potential A such that its W_∞ gauge transformations cancel against the gauge transformation of the electron field. We then show that the electromagnetic interactions of electrons in the lowest Landau level are obtained through a non-linear realization of A in terms of the U(1) gauge potential A^μ. As applications we derive the effective Lagrangians for circular droplets and for the ν = 1 quantum Hall system.

We first present a general method for extracting collective variables out of non-relativistic fermions by extending the gauge theory of collective coordinates to fermionic systems. We then apply the method to a system of non-interacting flavored fermions confined in a one-dimensional flavor-independent potential. In the limit of a large number of particles we obtain a Lagrangian with the Wess-Zumino-Witten term, which is the well-known Lagrangian describing the non-Abelian bosonization of chiral fermions on a circle. The result is universal and does not depend on the details of the confining potential.

The following sections are included:

• BIBLIOGRAPHY

• VITAE