The dual representation of $K\to 3\pi$ decay amplitudes is studied. It is found that $K\to 3\pi$ decay amplitudes in a generalized Veneziano model are incompatible with current-algebra relations. For example, pion poles and kaon poles are not dual to other poles. An example of realistic $K\to 3\pi$ decay amplitudes which contain pion and kaon poles, which contain both $\left|\Delta \mathbf{I}\right|=\frac{1}{2} \mathrm{and} \frac{3}{2}$ parts, and which are compatible with current-algebra relations is obtained. Our results seem to suggest that we should prefer the charged-current × charged-current nonleptonic weak Hamiltonian to Hamiltonians with pure $\left|\Delta \mathbf{I}\right|=\frac{1}{2}$.

Starting from the assumption that the inelastic states in the unitarity relation can effectively be represented by a set of quasi-two-particle states, a $K$-matrix formalism is set up for high-energy elastic scattering and diffraction-dissociation processes. Using arguments similar to those of Freund, it is shown that the Pomeranchuk contribution to elastic scattering and diffraction dissociation can be generated by multiple exchange of an exchange-degenerate quantum number carrying Regge trajectory $R$, by considering at the same time a formation of a sequence of excited intermediate states of the colliding particles between the individual $R$ exchanges. This unitarization procedure leads to an imaginary as well as a real contribution for vacuum exchange, corresponding basically to sums of double and triple $R$-exchange contributions, respectively. At the same time, the $K$-matrix formalism produces an absorptive correction to the input Born terms. The consequences of the proposed model are worked out, particularly as regards the asymptotic behavior of total cross sections and the interpretation of the crossover phenomenon.

Using mutilated five-point functions, we construct an amplitude for $S^{\mu }+\pi \to \pi \pi$, where $S^{\mu }$ is an isoscalar current. The amplitude satisfies crossing symmetry and the Adler condition, and contains an infinite sequence of poles in the current momentum squared. We find that it reduces to the Veneziano amplitude on the $\omega$ mass shell. For fixed $q^2$ and $t$, as $s\to \infty$ the amplitude behaves as $\Gamma (1-\alpha \mathrm{}(t\left)\right)B(1-\alpha (q^2),{\alpha }_{\rho }\mathrm{}(t)-{\gamma }_1)s^{\alpha \mathrm{}\left(t\right)\mathrm{}}$. We find by normalizing the amplitude that ${\gamma }_1=-1\mathrm{}$, and thus the amplitude has fixed multiplicative poles at ${\alpha }_{\rho }\mathrm{}\left(t\right)=-1,-2,\dots$, when we are not at an $\omega$ daughter pole. We then compare this model with vector dominance.

Superconvergence relations are known to obey algebraic equations involving the spin and the mass. States of the same helicity fall into reducible representations of $\mathrm{SU}\left(2\right)\mathrm{}\otimes \mathrm{SU}\left(2\right)\mathrm{}$. When the helicity is ±½, the two representations are related both by parity operation and by spin raising or lowering operators in the helicity space. The combination of the two yields sufficient constraints to solve for a number of properties of low-lying resonances, with results in good agreement with experiment.

A method for considering the spontaneous breakdown of chiral symmetry without recourse to a particular Lagrangian model is presented. The physical or asymptotic fields are assigned to linear representations of chiral ${\mathrm{SU}}_3\times {\mathrm{SU}}_3$ together with appropriate $c$-number addition to the scalar fields. This $c$-number addition manifests the spontaneous breakdown of the symmetry. Expansion of the interpolating fields in terms of asymptotic fields is introduced, which, together with the spontaneously broken chiral symmetry and the arbitrariness in the choice of interpolating fields, produces sum rules relating leptonic, semileptonic, and strong-coupling constants of $0^{\pm }$ and $1^{\pm }$ mesons. Results similar to those of current algebra are obtained with some notable differences. Among the most important is the appearance of the soft-meson amplitude as a consequence of our mechanism of spontaneous breakdown. We are thereby led, in an exceedingly simple way, to generalized soft-meson sum rules. In particular, new results are given for the semileptonic decays of the $K$ meson, and generalized relations among strong-coupling constants and leptonic decays of $1^{\pm }$ mesons are derived.

An inherent ambiguity of broken $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$ symmetry is discussed. It is shown to arise from a discrete unitary transformation in the $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$ space. For Hamiltonian models for which the symmetry-breaking term transforms like the $(3,\overline{3})+(\overline{3},3)$ representation, we find that, in general, there are two such terms which describe the same physical system. Some consequences of this result are discussed.

The spin structure of three-particle vertices may be determined from the quark model. Using these ${\mathrm{SU}\left(6\right)\mathrm{}}_W$ vertices in the Van Hove model, we derive a Reggeized scattering amplitude. In addition to Regge poles, there are necessarily fixed Regge cuts in both fermion and boson exchange amplitudes. The magnitudes of the pole and cut terms in an entire class of $\mathrm{SU}\left(6\right)\mathrm{}$-related reactions are determined by their magnitudes in a single reaction. As an example, we explain the observed presence or absence of wrong-signature nonsense dips in a class of reactions involving vector-meson exchange.

As a natural extension of the formulation of asymptotic $\mathrm{SU}\left(3\right)\mathrm{}$ symmetry previously presented, a new formulation of asymptotic $\mathrm{SU}\left(2\right)\mathrm{}$ symmetry is proposed. Intuitively speaking, we assume that both the $\mathrm{SU}\left(3\right)\mathrm{}$ and $\mathrm{SU}\left(2\right)\mathrm{}$ symmetries are well realized among particles of extremely high momenta where masses are not important. This point of view is formulated by assuming that, only in the asymptotic limit, the matrix elements of the $\mathrm{SU}\left(2\right)\mathrm{}$ generators $V_{{\pi }^{+}}$ and $V_{{\pi }^{-}}$ and also the $\mathrm{SU}\left(3\right)\mathrm{}$ generator $V_K$ [which, in the symmetry limit, are the isotopic-spin raising and lowering and the $\mathrm{SU}\left(3\right)\mathrm{}$ raising operators, respectively] behave, to a reasonably good approximation, as if the symmetries were not broken. Mass sum rules are obtained by using these asymptotic $\mathrm{SU}\left(2\right)\mathrm{}$ and $\mathrm{SU}\left(3\right)\mathrm{}$ symmetries together with exotic charge commutators which involve the time derivative of $V_{{\pi }^{\pm }}$ and which do not depend explicitly on the specific parameters of symmetry breaking. Assuming that the basic (and not effective) $\mathrm{SU}\left(2\right)\mathrm{}$-breaking interaction transforms like an $\mathrm{SU}\left(3\right)\mathrm{}$ octet, the exotic commutators are $[{\dot{V}}_{{\pi }^{+}},V_{{\pi }^{+}}]=[V_{K^0},{\dot{V}}_{{\pi }^{-}}]=[V_{K^{+}},{\dot{V}}_{{\pi }^{+}}]=0\mathrm{}$, etc. From them we reproduce almost all the $\mathrm{SU}\left(2\right)\mathrm{}$ sum rules previously obtained on the assumption of effective octet dominance. However, contrary to previous results, the ${\pi }^{+}$ and ${\pi }^0$, for example, are no longer degenerate in mass. The mass difference is explained in terms of the ${\eta }^0-{\pi }^0$ and ${\eta }^{\prime 0}-{\pi }^0$ mixings. A study is also made of the exotic commutators involving the axial-vector charges. The commutator $[{\dot{V}}_{{\pi }^{+}},A_{{\pi }^{+}}]=0\mathrm{}$ is the least model-dependent one. From this we obtain an intermultiplet baryon mass formula involving the $a$ decuplet and the $b$ octet: ${\left({\delta }_a\right)}^2\simeq {\left(p_b\right)}^2-{\left(n_b\right)}^2$ ($a$ and $b$ are arbitrary). ($p_b$) and ($n_b$) denote the masses of the proton and neutron of the $b$ octet, respectively. ${\left({\delta }_a\right)}^2$ denotes the equal-squared-mass spacing of the $a$ decuplet, i.e., ${\left({\delta }_a\right)}^2={\left({{\Delta }_a}^{++}\right)}^2-{\left({{\Delta }_a}^{+}\right)}^2={\left({{\Delta }_a}^{+}\right)}^2-{\left({{\Delta }_a}^0\right)}^2={\left({{\Xi }_a}^{*0\mathrm{}}\right)}^2-{\left({{\Xi }_a}^{*-}\right)}^2$. The case of $a={\frac{3}{2}}^{+}$ and $b={\frac{1}{2}}^{+}$ coincides with the good sum rule of $\mathrm{SU}\left(6\right)\mathrm{}$. For bosons, the exotic commutators such as $[{\dot{V}}_{{\pi }^{-}},A_{K^0}]=[{\dot{V}}_{{\pi }^{+}},A_{K^{+}}]=0\mathrm{}$, which are more model dependent than the $[{\dot{V}}_{{\pi }^{+}},A_{{\pi }^{+}}]=0\mathrm{}$, produce the following general intermultiplet mass sum rule: ${\left({K_{\alpha }}^0\right)}^2-{\left({K_{\alpha }}^{+}\right)}^2=\mathrm{const}\simeq 0.004$ ${\mathrm{}\left(\mathrm{GeV}\right)\mathrm{}}^2$ ($\alpha$ is arbitrary). Here $\left({K_{\alpha }}^0\right)$ denotes the mass of the $K^0$ meson belonging to the $\alpha$ octet. For the $0^{-}$ and $1^{-}$ octets, it implies ${\left(K^0\right)}^2-{\left(K^{+}\right)}^2={\left(K^{*0\mathrm{}}\right)}^2-{\left(K^{*+}\right)}^2$, which is not inconsistent with present experiment. We also show that both the commutators, $[{\dot{V}}_{{\pi }^{+}},V_{{\pi }^{+}}]=0\mathrm{}$ and $[{\dot{V}}_{{\pi }^{+}},A_{{\pi }^{+}}]=0\mathrm{}$, give, in general, the same mass sum rule when they are applied to the same $\mathrm{SU}\left(2\right)\mathrm{}$ multiplet. Sum rules for the axial-vector semileptonic hyperon decay couplings in broken $\mathrm{SU}\left(3\right)\mathrm{}$ and $\mathrm{SU}\left(2\right)\mathrm{}$ symmetries are also derived.

We consider interacting quantum electrodynamics with the electric current $j^{\mu }$ unspecified and compute the homogeneous Lorentz group transformation laws of the "vector" potential $A^{\mu }$ in the radiation gauge. We show that $A^{\mu }$ decomposes uniquely into a direct sum of an infinite number of linear, infinite-dimensional, nonunitary, indecomposable representations of the homogeneous Lorentz group. One of these representations has spin-multiplicity 2, belongs to a class of representations recently analyzed by Gel'fand and Ponomarev, and may be the first recognized physical example of this class of representations. Finally, the noninteracting limit is shown to reduce correctly to the free-field case in which the transformation properties of the "vector" potential are already known.

The relationship between the eikonal approximation of Glauber and the impact-parameter approximation of Blankenbecler and Goldberger is clarified. This is done by comparing the approximate forms for the scattering amplitude for nonrelativistic scattering by a potential with the Born expansion, which at high energy assumes the natural form of a series of terms in inverse powers of the center-of-mass momentum $p$. It is found that both approximations reproduce the first Born term and, to order $\frac{1}{p}$, the on-energy-shell part of the second Born term, for all momentum transfers. Furthermore, the Glauber approximation correctly gives the leading contribution (in powers of $\frac{1}{p}$) to the on-energy-shell part of every term in the Born series, whereas the Blankenbecler-Goldberger approximation fails to do this for Born terms beyond the second. However, since the comparison to the Born series shows that both approximations neglect contributions of order $\frac{1}{p^2}$, neither approximation, in the simple forms commonly used, is correct beyond terms of order $\frac{1}{p}$.

The problem of making unstable elementary particles equivalent to resonances is investigated in a soluble model field theory. It is demonstrated that the irreducible part of the scattering amplitude develops a pair of complex conjugate poles on the second sheet of the energy plane, one of them corresponding to an $S$-wave resonance. The poles in the vertex and inverse propagator induced by these complex poles of the irreducible part of the scattering amplitude are such that the so-called Jin-MacDowell cancellation holds good. We show that under the conditions $Z_1=0\mathrm{}$ and $Z_3=0\mathrm{}$, the $S$-wave resonance completely replaces the unstable particle.

The dynamical group ${\tilde{G}}_5$ for relativistic quantum mechanics phenomenologically suggested by Aghassi, Roman, and Santilli is derived from the analysis of symmetry properties of Lagrangians and corresponding equations of motion for a free relativistic particle. All physical observables such as position, momentum, angular momentum, and mass squared are represented by well-defined operators which close the algebra of the dynamical group ${\tilde{G}}_5$. The unitary irreducible representations of this group, which are possible states of the physical system, are found. The particles accommodated in the single unitary irreducible representations have various spins starting from the lowest spin value and going up to infinity in integral steps. The mass-squared operator $P_{\mu }{P}^{\mu }$ lies in the enveloping algebra of ${\tilde{G}}_5$, and its eigenvalues are not necessarily quantized and can have any positive or negative values. It is pointed out that this group has several failures and thus it cannot be accepted as the reliable dynamical group for particles within relativistic quantum mechanics.

The algebraic realizations of chiral symmetry obtained by Weinberg are investigated for the case of $\mathrm{SU}\left(3\right)\mathrm{}$ symmetry. When only $p$-wave interactions are taken into account, the algebraic realization of the first superconvergence condition is given by the Lie algebra of the group $\mathrm{SU}\left(2\right)\mathrm{}\otimes \mathrm{SU}\left(6\right)\mathrm{}$. The mass matrix obtained from the second superconvergence relation corresponds to degenerate masses within each of the multiplets ($V=0,56\mathrm{}$) and ($V=2,56\mathrm{}$). The consequences of particle classification under the group $\mathrm{SU}\left(2\right)\mathrm{}\otimes \mathrm{SU}\left(6\right)\mathrm{}$ are enumerated.

Motivated by the observed scale invariance of high-energy inelastic electron-proton scattering, we study the constraints on a Compton amplitude that follow if it is invariant under the full group of conformal coordinate transformations. Although the conformal group contains operations that interchange spacelike and timelike coordinate intervals, a large class of manifestly causal amplitudes can be constructed. We find that the strict application of conformal symmetry reduces the number of independent covariants from four to two in the case of a spin-zero target. The two covariants of the conformal Compton amplitude remain independent when it is restricted to forward scattering. Hence, invariance under the full conformal group yields no constraints on the structure functions of inelastic electron scattering other than that already provided by simple dilation invariance.

The Veneziano-type formula for the scattering amplitude $F(t,s)\mathrm{}$ is derived in the form of an integral representation under the following assumptions: (1) $F(t,s)\mathrm{}$ has a fixed-$s$ unsubtracted dispersion relation, and (2) for $s$ fixed, $F(t,s)\mathrm{}$ has equidistant simple poles only. The crossing symmetry, $F(t,s)=F(u,s)\mathrm{}$, is seen to be nothing more than a relation specifying the characters of the weight function. It is demonstrated that the Veneziano-type integral representation is more convenient than the conventional Veneziano series. A Veneziano-type integral representation for the five-point function is derived under the following assumptions, which are natural extensions of the above two assumptions for the four-point function: (1′) The five-point function has a certain double-spectral representation, and (2′) it has equidistant simple poles only in the two relevant variables.

In this paper we present the neutrino luminosity due to a completely relativistic electron gas in the presence of an intense magnetic field. The neutrino emission rate is determined rigorously, under the assumption of the $V-A$ theory of universal weak Fermi interaction, by standard field-theoretical methods. It is shown that the neutrino radiation rate can be expressed as a function of temperature, density, and the field-strength parameter $\Theta =\frac{H}{H_c}$, where $H_c=\frac{m^2{c}^3}{e\hslash }=4.414\mathrm{}\times {10}^{13}$ G. Comparisons are made with previous approximate results due to others, and the reasons for discrepancies are discussed.

In this paper we have investigated some properties of the gravitational fields of magnetic type, using timelike congruences and the Bianchi identities. Null eigendirections of the conformal tensor are determined without any assumption regarding the nature of the sources of the gravitational field. Some relations between the kinematical quantities of the timelike congruences and the curvature of space-time have been derived. Our results reduce to those of empty space when the Ricci tensor is taken to be zero. These emptyspace equations lead us to a scalar potential which satisfies the field equation of Newton's theory of gravitation.

The classical self-energy problem for charged tachyons is more serious than that for charged bradyons. As a result, the theoretical basis for generally expected experimental properties of such objects is shaky.

An elementary model of the gravitational field in the presence of external sources is studied. This model has positivity requirements, invariance under transformations, and nonlinear equations of motion, all of which are analogs of similar properties in the full theory. The classical solutions of the "free" version of this model exhibit singularities, but these are removed in the quantum theory.

Weyl has found a particular class of static electromagnetic vacuum solutions for fields that depend on at most two spatial variables. The purpose of this paper is to report on another class of electromagnetic vacuum solutions which are not of Weyl's type. As with Weyl's case, these new solutions are generated from functions that formally satisfy the two-dimensional Newtonian Laplacian equation.