In this paper we show how to specify the particle content of a nonrelativistic quantum theory of $N$ identical spinless particles in terms of observables like the particle number density and the flux density of particles. Our approach to this problem is through a study of the irreducible representations of the local, equal-time current algebra. It is shown how these representations define a functional representation of the current algebra, and that the Hamiltonian can be written in terms of the currents in a nonsingular fashion in any irreducible representation.

It is pointed out that the third law of thermodynamics, which has been verified experimentally for systems with electromagnetic interactions, is not part of traditional classical theory, and indeed is violated by hypothetical systems, such as an ideal gas, which exhibit equipartition of energy. In the context of quantum theory, the law may be understood from the description of thermodynamic systems as quantum systems having discrete energy levels. Along the same lines as a recent derivation of the blackbody radiation spectrum from classical theory involving classical electromagnetic zero-point radiation, it is shown that the third law holds in classical theory for all thermodynamic systems which interact (no matter how weakly) with electromagnetic radiation.

It is shown that classical electrodynamics with charged particles of finite mass is invariant with respect to a one-parameter group of "dilatation transformations" if the theory is expressed in covariant Hamiltonian form. The "mass transformations," which seem somewhat puzzling in the usual discussions of this invariance group, appear in a natural way. A corresponding conservation law is derived and discussed.

It is shown that, analogously to the situation in the usual classical theories, the solution of the Hamilton-Jacobi equations of general relativity, $S$, satisfies the relation $S=\int \mathcal{L}{d}^{4}x$, where, however, $\mathcal{L}$ is the Lagrangian density of Dirac, rather than the more usual Einstein Lagrangian. This result is then used to prove that if we write $S=\int \mathcal{S}\left(g_{\mathrm{mn}}\right)d^{3}x$, the integrand $\mathcal{S}$ is a three-dimensional scalar density which is a homogeneous functional of $g_{\mathrm{mn}}$ of degree 1.

An expansion for the scattering amplitude at high energy is given. The expression given by Ross, with his $\gamma \left(\theta \right)=0\mathrm{}$, appears as the first term, and the higher-order terms may therefore be used to estimate the accuracy of the approximation $\gamma \left(\theta \right)=0\mathrm{}$. Schiff's large-angle formula is also obtained. The derivations are simple, and physical motivation is emphasized.

It is shown that by bombarding a Li nucleus by neutrinos coming out of kaon decay, we may observe directional photons if Oakes's theory of $\mathrm{CP}$ nonconservation is correct.

A bootstrap theory of the hadrons satisfying crossing symmetry and asymptotic unitarity which was recently proposed by the author is shown to lead to an interesting and testable prediction for high-energy wide-angle proton Compton scattering.

It is shown that the conventional current-algebra derivation of the $\Delta I=\frac{1}{2}$ rule for nonleptonic $K$ decay is invalid within the model of $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$ summetry-breaking given by Gell-Mann, Oakes, and Renner. A method for retaining a $\Delta I=\frac{1}{2}$ rule and this model without invoking either neutral currents or octet dominance is discussed.

A continuous dispersion sum rule is used to estimate the $N\Lambda K$ coupling constant. Together with the recent zero-range $K$-matrix parametrization of low-energy $\overline{K}N$ scattering, the sum rule yields ${g_{p\Lambda k}}^2\simeq 6.9$ and gives excellent results for ${\alpha }_{K+p}-\frac{1}{2}{\alpha }_{K+n}$ (${\alpha }_{K+p}=\frac{\mathrm{Re}{C}_{K+p}}{\mathrm{Im}{C}_{K+p}}$) at high energy. Until the real part of the scattering amplitude is known more accurately in the intermediate region, the present calculation does not absolutely favor or disfavor the multiple-channel effective-range parametrization.

Using two general principles to define an allowed vertex, the numerical selection rule is derived which is equivalent to the general principles for any vertex containing less than 14 quarks and antiquarks.

The physical interpretation and evaluation of the two-temperature statistical model for particle spectra in proton-proton collisions is considered. It is shown that the longitudinal temperature can be considered as a kinematic property of the interaction process as described by a single-temperature momentum distribution in the rest frame of the fireball that results from the collision of the incident particle. The two-peaked angular distribution (in $log{tan\frac{1}{2}\theta }^{*}$) of the produced particles is a pronounced feature of the two-temperature model, which is a one-fireball model. The pion inelasticity distribution is calculated and shown to agree with experiment.

The eigenamplitudes of the unitary doubled-resonance amplitude for the two-channel case are computed and discussed. In addition to poles, the eigenamplitudes also have square-root branch points in the complex energy plane.

A high-energy small-angle eikonal model suggested previously for a diffraction regime in inelastic $\mathrm{ep}$ scattering is reviewed and applied to weak (neutrino-induced) inelastic production. In neutrino production channels, the observed trend for dominance of $N^{*}$ (plus associated mesons) over $\rho$-type events is qualitatively contained in the model. A reasonable estimate of the linear rise with neutrino energy of $\sigma (\nu p\to {\mu }^{-}+\mathrm{hadrons})$ is discussed, especially from general considerations.

We point out that formulas given by Arnowitt, Nath, Srivastava, and Friedman for masses of scalar mesons in the Veneziano model are incorrect; we derive new formulas for these mesons, and compare with experiment.

Electromagnetic corrections to $\pi \alpha$ scattering for spacelike $q^2$ are calculated to see whether they can be used to determine $r_{\pi }$. The low-momentum $t$ dependence of the off-shell strong amplitude used in this calculation is determined with the help of crossing symmetry, current algebra, and the hypothesis of partially conserved axial-vector current. The relativistic corrections are found to be just as important as those which are also included in nonrelativistic treatments. When only the nonrelativistic terms are used, the large value of $r_{\pi }$ (2-3 F) obtained recently from ${\pi }^{\pm }\alpha$ scattering is reproduced. Including the relativistic terms brings the resulting value of $r_{\pi }$ down to approximately 1 F. This is consistent with the values obtained by using inelastic electron-proton scattering; however, the ${\pi }^{\pm }\alpha$ results are quite sensitive to the details of the $\pi \alpha$ nuclear interaction and to the relativistic effects, which are only roughly approximated.

We study the radiative corrections to $K_{l3\mathrm{}}$ decay, assuming that the strong interactions introduce a natural cutoff for the electromagnetic interactions of the hadrons. We take the strong interactions into account by the electromagnetic form factors of the hadrons, for which we assume the vector-meson-dominance model. The radiative corrections are found to be large for some energy configurations, but the correction to the $K_{e3\mathrm{}}$ decay rate is about -2%. This is larger than the -0.45% value previously estimated by Ginsberg. The correction to the $K_{\mu 3}$ decay rate is about -0.4%, and the correction to the polarization of the muon is about -1%.

Given the parton model of the nucleon, the cross section for photoproduction of high-energy muon pairs of small invariant mass and high transverse momentum is estimated.

A nonlinear chiral $S{U}_3\otimes S{U}_3$ gauge-invariant phenomenological Lagrangian with the symmetry breaking through mass terms has been constructed. The model is then employed to study the dynamical structures of the $K_{l4\mathrm{}}$ decay axial-vector form factors. The predicted decay rates are found to be in good agreement with experiments as well as with other calculations based on current algebra and dispersion theory.

We study the ratio of the cross section for elastic positron-proton scattering to that for elastic electron-proton scattering. In first-order $\alpha$, the leading terms of this ratio for small-angle scattering are given. As could be expected, the first term in this "expansion" is the same as that obtained in Dirac theory, and the first plus the second are unchanged from the structureless-proton result. Only in the third term do we begin to find proton structure, this in the form of a sum rule over photoproduction cross sections. Use is made of an off-mass-shell analog of the Compton low-energy theorem, and as a by-product we show that the two-photon contribution to our ratio is finite in the limit of zero electron mass.

A nonet-symmetrical and PCAC-violating term is introduced into a Lagrangian with $S{U}_3\otimes S{U}_3$ chiral symmetry and PCAC (hypothesis of partially conserved axial-vector current). Breaking of the nonet symmetry of this term is achieved through the renormalization of the fields. This renormalization is necessary because of the requirement that the kinetic-energy terms of the Lagrangian be correctly normalized after the removal of the tadpole terms. Using as input the decay rate for ${\pi }^0\to 2\gamma$, we predict $\Gamma (\eta \to 2\gamma )=0.41\mathrm{}\pm 0.08$ keV and $\Gamma (X^0\to 2\gamma )=6\mathrm{}\pm 1$ keV. The result for $\Gamma (\eta \to 2\gamma )$ agrees with experiment within 2 standard deviations. It is argued that errors of about 20% should be expected in our calculations, mainly because of the neglect of the effects due to the finite width of the various particles.