Broken Symmetry

The following sections are included:

• FOREWORD

• RESEARCH IN ELEMENTARY PARTICLE THEORY

• CONTENTS

The following sections are included:

• Introduction

• Preliminary Considerations

• Onsager's Problem

• Kramers-Wannier's method of approach

• Additional Remarks

• References

The following sections are included:

• Introduction

• Formal introduction of the proper time

• Theory of Fock and its extension

• Problem of vacuum polarization

• Appendix. Solution for an arbitrary constant external field

• References

The following sections are included:

• Introduction and Summary

• The Second Order Potential

• The Fourth Order Potential

• The Relation between the Second and Fourth Order Potential

• Further Discussions

• References

The following sections are included:

• Appendix

• References

Please refer to full text.

Parametric representations for the Green's function of field theory are derived in perturbation theory. These representations are valid for each term in the perturbation series that corresponds to a Feynman diagram, and reflect its analytic property and threshold characteristics. As an example, the three-body (vertex) function is shown to satisfy a dispersion relation when two of the three momenta are fixed, with the correct location of the singularities expected from the thresholds of the competing real processes.

The dispersion relations for form factors are studied in perturbation theory. In particular, the relations are proved for a realistic nucleon in the form usually assumed in literature. It is demonstrated, however, that in more general cases the mass spectrum in the dispersion relations depends on the masses of the contributing particles in a peculiar way, which is different from the simple relation hitherto believed. Such a situation arises, for example, in the case of some hyperon models and bound systems such as the deuteron. A formula concerning the final state interaction to be used in conjunction with the dispersion relations is also derived from the invariance of the theory under space-time reversal.

Ideas and techniques known in quantum electrodynamics have been applied to the Bardeen-Cooper-Schrieffer theory of superconductivity. In an approximation which corresponds to a generalization of the Hartree-Fock fields, one can write down an integral equation defining the self-energy of an electron in an electron gas with phonon and Coulomb interaction. The form of the equation implies the existence of a particular solution which does not follow from perturbation theory, and which leads to the energy gap equation and the quasi-particle picture analogous to Bogoliubov's.

The gauge invariance, to the first order in the external electromagnetic field, can be maintained in the quasi-particle picture by taking into account a certain class of corrections to the charge-current operator due to the phonon and Coulomb interaction. In fact, generalized forms of the Ward identity are obtained between certain vertex parts and the self-energy. The Meissner effect calculation is thus rendered strictly gauge invariant, but essentially keeping the BCS result unaltered for transverse fields.

It is shown also that the integral equation for vertex parts allows homogeneous solutions which describe collective excitations of quasi-particle pairs, and the nature and effects of such collective states are discussed.

Please refer to full text.

The following sections are included:

• References

• DISCUSSION

It is suggested that the nucleon mass arises largely as a self-energy of some primary fermion field through the same mechanism as the appearance of energy gap in the theory of superconductivity. The idea can be put into a mathematical formulation utilizing a generalized Hartree-Fock approximation which regards real nucleons as quasi-particle excitations. We consider a simplified model of nonlinear four-fermion interaction which allows a γ₅-gauge group. An interesting consequence of the symmetry is that there arise automatically pseudoscalar zero-mass bound states of nucleon-antinucleon pair which may be regarded as an idealized pion. In addition, massive bound states of nucleon number zero and two are predicted in a simple approximation.

The theory contains two parameters which can be explicitly related to observed nucleon mass and the pion-nucleon coupling constant. Some paradoxical aspects of the theory in connection with the γ₅ transformation are discussed in detail.

Continuing the program developed in a previous paper, a "superconductive" solution describing the proton-neutron doublet is obtained from a nonlinear spinor field Lagrangian. We find the pions of finite mass as nucleon-antinucleon bound states by introducing a small bare mass into the Lagrangian which otherwise possesses a certain type of the γ₅ invariance. In addition, heavier mesons and two-nucleon bound states are obtained in the same approximation. On the basis of numerical mass relations, it is suggested that the bare nucleon field is similar to the electron-neutrino field, and further speculations are made concerning the complete description of the baryons and leptons.

A formally γ₅-invariant system consisting of a Dirac field and a massless pseudoscalar field allows chirality conservation in the sense that its expectation value is a constant of motion. This leads to the consequence that in any reaction a change in the fermion chirality (~helicity×velocity) is compensated for by the emission of a massless boson at zero energy, which can be expressed by a simple formula relating the radiative amplitude to the elastic amplitude. Assuming the pion-nucleon system to be γ₅-invariant when the pion mass can be neglected, the formula is applied to the processes N+π → N+π and N+2π. A reasonable agreement with experiment is obtained in a case dominated by the 3–3 resonance.

We consider some extensions of the relation which has been obtained by Nambu and Lurié between an elastic process and the accompanying soft pion emission process under the assumption of approximate γ₅ invariance. In particular, a generalized formula is found which enables one to describe the electropion production e+N → N+e+π and the neutrino-pion production ν+N → N+e+π at the threshold of Nπ system in terms of the vector (Hofstadter) and the axial-vector form factors. Explicit forms of the cross sections are given.

We calculate the reduction of the energy gap at zero temperature for a bulk superconductor in the presence of a static external magnetic field, to the second order in the field strength. Simple formulas are obtained for the long (λ ≫ ξ₀) and short (λ ≪ ξ₀) wavelength limits, where ξ₀ is the coherence length.

We use the general gauge-invariant formulation of the Bardeen-Cooper-Schrieffer (BCS)-Bogoliubov theory developed by one of the authors, but the result is also shown to agree with that of the BCS variational procedure applied in the presence of the field. The gauge invariance is maintained by virtue of the collective excitations as in the Meissner effect. The mathematical proof of gauge invariance is carried out in a completely general manner using Ward identities.

We adapt the results obtained in a previous paper on the magnetic field dependence of the energy gap in superconductivity for bulk specimen to thin-film superconductors, using the model of discrete quantization in momentum space. Only the case of parallel and constant external magnetic field along the film surfaces is considered. A series of elementary theorems and some specific calculations lead to the conclusions: (1) A second-order phase transition should be observed at all temperatures for thin film thicknesses. (2) A simple scaling rule exists concerning the field and temperature dependence on the energy gap. (3) The critical field H_c depends on thickness L and reduced temperature t like H_C ~ L^-3/2[ln(1/t)]^1/2 for not too thin films (L ≳ 0.5×10^-5 cm). The behavior changes as the film becomes very thin or as the temperature becomes moderately low. A crude comparison with available experimental data seems to bear out our conclusions qualitatively.

With a view to avoiding some of the kinematical and dynamical difficulties involved in the single-triplet quark model, a model for the low-lying baryons and mesons based on three triplets with integral charges is proposed, somewhat similar to the two-triplet model introduced earlier by one of us (Y. N.). It is shown that in a U(3) scheme of triplets with integral charges, one is naturally led to three triplets located symmetrically about the origin of I₃-Y diagram under the constraint that the Nishijima-Gell-Mann relation remains intact. A double SU(3) symmetry scheme is proposed in which the large mass splittings between different representations are ascribed to one of the SU(3), while the other SU(3) is the usual one for the mass splittings within a representation of the first SU(3).

The following sections are included:

• References

Please refer to full text.

We consider a class of wave equations which couple an infinite number of tensors or spinors of all ranks. Such a system of equations naturally possesses an infinite number of mass levels, and each eigenfunction implicitly contains a built-in form factor. Two simple examples of first order differential equations are examined. One is based on the set of all finite representations

The following sections are included:

• Introduction

• Choice of the dynamical group G and its representation

• Models based on SL(2,c)
  • Form factors
  • Wave equations
  • Minimal electromagnetic interaction

• Models based on SO(4, 2) ~ SU(2,2)

• Models based on SU(3,1)

• Problems of quantized fields

• Scattering processes

• Application to hadron physics

• References

• Discussion

The following sections are included:

• References

• Note added in proof

It is shown that the tree approximation currently utilized in connection with phenomenological chiral Lagrangians corresponds to a semiclassical approximation applied to the S-matrix in quantum field theory. Some invariance properties of the S-matrix are derived in this framework.

The following sections are included:

• References

• DISCUSSION

The following sections are included:

• Introduction

• Spontaneous breakdown of symmetries

• PCAC, Current Algebra, and Soft Pion

• Unified description of hadron symmetries

• Further search for spontaneous breakdown

• FOOTNOTES

The following sections are included:

• Introduction

• Factorized Veneziano model

• Quarks and the dual model

• Statistical approximation

• Electromagnetic interactions and inelastic e - p scattering

Taking the Liouville theorem as a guiding principle, we propose a possible generalization of classical Hamiltonian dynamics to a three-dimensional phase space. The equation of motion involves two Hamiltonians and three canonical variables. The fact that the Euler equations for a rotator can be cast into this form suggests the potential usefulness of the formalism. In this article we study its general properties and the problem of quantization.

The Nielsen-Olesen interpretation of dual strings as Abrikosov flux lines is extended to the case of open-ended strings by adapting Dirac's description of magnetic monopoles to a London-type theory. The mathematical formalism turns out to be similar to that of Kalb and Ramond. Translated to hadron physics, it implies that the quarks will act as carriers of magnetic charge, permanently bound in pairs by the string bonds. However, massive axial-vector gluons can be created by hadrons.

In this talk I will make a few scattered observations on mathematical formalisms which are aimed at describing stringlike objects and their interactions. The attempt was originally motivated by the string model of hadrons, which may well be a phenomenological manifestation of a gauge theory of strong interactions. Here, however, I will put emphasis on broader aspects of the problem, keeping in mind that there may exist as yet unknown phenomena which are to be described by these mathematical formalisms. Of course it is sheer speculation at the moment.

There are three main topics to be discussed. They are: relativistic hydrodynamics of vortices, its broken symmetry schemes and non-Abelian versions, and a unified algebraic characterization of string and gauge field equations.

The Weinberg-Salam theory of electromagnetic and weak interactions admits classical configurations in which a pair of magnetic monopoles is bound by a flux string of the Z⁰ field. They give rise to Regge trajectories of excitations with a mass scale in the TeV range.

It is shown that a Hamilton–Jacobi-type formalism can be set up to deal with the classical dynamics of relativistic strings and other one-dimensional extended systems. A special feature is that the formalism involves two evolution parameters which are treated on an equal footing. The corresponding Hamilton–Jacobi functions turn out to be vector potentials or Clebsch potentials, and in this sense we find a link between the string model and gauge field theory.

The following sections are included:

• The paradigms of particle physics
  • Model building in terms of particles
  • Renormalization theory
  • Symmetries and symmetry breaking
  • The gauge principle

• Problems and issues
  • General questions on quantum mechanics
  • Limits of renormalizable field theory
  • Quantum mechanics of extended objects
  • Supersymmetry

• Speculative outlook
  • Mass and mass spectra
  • Broken symmetries
  • Phenomenology of grand unification
  • Beyond the Planck limit

• References

The following sections are included:

• Introduction

• Superconductivity

• Helium 3

• Nuclear physics

• Particle physics

• Acknowledgements

• References

The following sections are included:

• INTRODUCTION

• MATHEMATICAL PRELIMINARIES

• HEAT EQUATION

• THE WAVE EQUATION

• OTHER PROBLEMS
  • Higher dimensional wave equations
  • Partition function

• ACKNOWLEDGMENTS

• REFERENCES

The following sections are included:

• Modes of quest in particle physics

• Evolution of the Lawrence-Yukawa paradigm

• The rise of the Dirac mode

• Speculative comments

• References

The fermionic and bosonic spectra that follow from a BCS mechanism have a rather simple and universal structure. It is shown, in typical examples, that the effective Hamiltonian that reflects such a structure can be factorized in a way similar to the case of supersymmetric systems. The underlying spectrum generating superalgebras are identified.

It is proposed that thermodynamic analogies may be useful in understanding the properties of quantum field theory. As an example, an attempt is made to translate the charge renormalization in gauge theories to a thermodynamic statement about an ideal gas.

The characteristic features of the BCS mechanism, one mathematical and the other physical, are explored in relativistic field theories. The first is called quasi-supersymmetry among fermions and composite bosons, and various examples can be found at various energy scales. The other has to do with the notion of a hierarchical chain of symmetry breakings and that of a self-sustaining (bootstrap) mechanism. The latter is applied to electroweak unification, and some predictions are made.

The BCS theory of fermionic pairing and condensation is used to understand the interacting boson model. Results from BCS are incorporated into an effective Hamiltonian that after symmetry-breaking and second-order corrections yields an IBM-type Hamiltonian with coefficients determined by well-known nuclear constants. The O(6) and O(5) chains are shown to be largely of spontaneous origin. Supersymmetry aspects of the model are also discussed.

An overview is given concerning the concept of dynamical symmetry breaking and its examples in condensed matter, nuclear, and particle physics, including some speculations about the nature of the Higgs field in the Standard Model of electroweak unification.

The following sections are included:

• Bio data

• List of Publications