An investigation is made of the effects of an indefinite number of coupled channels on two-particle amplitudes. Central coupling, chain coupling, and combinations of them are considered. Some results are general, while others depend on a generalized potential model. The existence of strong resonance-producing effects is related to zeros of the $S$ matrix. Forms of Levinson's theorem are found for both the amplitude and $S$-matrix phases. It is shown that if the effective number of coupled channels increases linearly with the energy that (1) the partial-wave $S$ matrix can vanish asymptotically (the usual result is that $S_L\to 1$); and (2) Regge trajectories rise asymptotically as the square root of the energy. Special results of the MacDowell symmetry are noted for a finite number of coupled channels. With respect to a special (boundary condition) model it is found that Regge trajectories which cross the origin near $L=\frac{1}{2}$ should have much larger slopes than those crossing the origin near $L=1\mathrm{}$, in agreement with observation. Regge cuts result from the coupling of effective three-body channels.

The concept of scalar hyperplane-dependent fields is defined on the basis of the author's earlier treatments of the position operators for relativistic particles. Such fields are a special case of fields over homogeneous spaces of the Ponincaré group and are related to infinite families of integer spin particles with spin-dependent mass spectra. The intrinsic parity of the particles and degeneracy in the mass spectrum is examined. The Lagrangian formalism, including Noether's theorem, is developed and explored. It is shown that the existence of a Lagrangian formalism yields a normalization condition that is incompatible with the existence of space-like solutions. All is in preparation for the sequel to this paper which considers a specific model with Bose statistics and nondegenerate mass spectrum.

The formulation of the Dirac theory based on the group $O(4, 2)$, in which an internal additive quantum number $n$ appears, distinguishing particles and antiparticles, is discussed in detail. The substitution law then becomes an analytic continuation in $n$.

A general $n$-point amplitude based on tree diagrams which has Regge asymptotic behavior and poles in all channels, but which is not dual, is explicitly factorized, and the coupling of three or more Reggeons is computed. The degeneracy of levels of the factorization of the $n$-point function is $n^2$, the same as in the four-point function. Group-theoretical implications are briefly discussed.

A simplified version of the Bali-Chew-Pignotti amplitude for multi-Regge-pole particle production is given using the recently developed projection-operator technique to construct many-particle states. This amplitude is generalized to a multi-Toller-pole expansion to avoid singularities which otherwise occur when any momentum-transfer variable goes to zero. It is shown how to determine the $M$ value of a Toller pole by measuring the distribution as a function of a Treiman-Yang angle.

Interactions of composite (Reggeized) hadrons are investigated. We give a scheme of writing down trilinear vertices between hadrons that is consistent with Lorentz invariance. The vertices become local when the hadrons involved are put on their mass shells (physical points of the trajectories). A graphical method, similar to Feynman diagrams, is developed for writing down scattering amplitudes involving composite hadrons. The procedure for constructing scattering amplitudes is stated in terms of a few rules. The example of elastic scattering of spinless particles with the exchange of a single Lorentz pole is briefly discussed; it is shown that the diagram rules lead to the well-known expression for a scattering amplitude with rapidly decreasing residues.

We investigate the possibility of describing the interactions of composite ("Reggeized") hadrons in space-time. We construct a formalism giving a unified description of the spin states of composite hadrons. A Lorentz family of hadron Regge trajectories allows the introduction of local hadron amplitudes. Investigation of the covariance properties of the local hadron amplitudes with respect to proper complex Lorentz transformations, space and time inversion, charge conjugation, and gauge transformations establishes the principles according to which observed families of hadron states are described in terms of representations of the complex Lorentz group. We find that covariance under these transformations necessarily demands parity doubling ("MacDowell symmetry") for meson as well as fermion trajectories. In a world of parallel Regge trajectories (unbroken Lorentz symmetry), chirality is exactly conserved. We give a manifestly Lorentz-covariant description of the propagation of composite hadrons. Explicit expressions for the composite hadron propagators are given for chiral singlet mesons and chiral doublet fermions. Examples of physical assignments are briefly discussed. It is found in particular that particles of unit baryon number transform according to representations of the complex Lorentz group with Toller quantum number $M=\frac{1}{2}$.

An attempt to construct a theory that links the macroscopic and microscopic properties of matter is presented. This is done using a space of more than four dimensions. Equations for microscopic-particle fields are investigated in a fixed six-dimensional metric space. The topology of the space is solely responsible for the quantization of mass and charge. The metric contains terms which transform like the Yang-Mills $B$ field. These terms appear appropriately in all particle equations. Without the $B$ field, the symmetry group of the theory is P×SU(2)/Z(2). The presence of the B field lowers this symmetry. Particle mass spectra are presented for six-dimensional scalar, spinor, and vector fields, and a coupling-constant ratio is predicted. The later part of the paper deals with the cosmological implications of the microscopic model presented in the first part of the paper. It is shown that Einstein's equations for the metric are consistent if a massless cosmological vector field is introduced. A critique of previous higher-dimensional field theories along with a summary of the results of an eight-dimensional theory is given. Since all symmetries dealt with result as approximations to the equations of the model, the no-go theorems are not applicable. Nevertheless, the six- and eight-dimensional models contain the shadows of the $P\times \frac{\mathrm{SU}\left(2\right)\mathrm{}}{Z\left(2\right)\mathrm{}}$ and $P\times \frac{\mathrm{SU}\left(3\right)\mathrm{}}{Z\left(3\right)\mathrm{}}$ symmetries in all particle-field representations.

It has long been held that the divergences of field theory could be removed if the theory was formulated on a Hilbert space with an indefinite metric. The difficulty of this approach was that the theory then possessed channels with negative probability and therefore violated unitarity. In a recent paper, Lee and Wick have observed that this difficulty could be eliminated if negative-metric states had complex energy. In this paper, we produce a model which shows several of the difficulties associated with the Lee-Wick proposal. The existence of complex energies leads to interpretation difficulties and, in higher sectors, does not completely solve the unitarity problem. We use the model to show how the Lee-Wick conjecture can be extended to overcome the difficulties that are produced. We describe the physical meaning of "good" theories with an indefinite metric.

In this paper we study all elastic scattering amplitudes in scalar electrodynamics at high energies. The impact factors for the scalar meson and for the vector meson in this theory are explicitly calculated in the lowest order. We also show that the elastic amplitudes still satisfy an impact-factor representation after all vector-meson exchanges are summed, and an exponentiation, similar to that in quantum electrodynamics, is obtained.

Previous treatments on high-energy scattering in quantum electrodynamics are extended to diffractive inelastic processes. We explore the model in which the diffractive excitation proceeds through the exchange of vector mesons. Six inelastic impact factors are explicitly calculated in the lowest order: (1) scalar to scalar, (2) pseudoscalar to pseudoscalar, (3) vector to vector, (4) axial vector to axial vector, (5) scalar to axial vector, and (6) pseudoscalar to axial vector. For all six of these impact factors, the masses of the incoming and outgoing particles may be different; and for the last two impact factors, the spins and the parities of the incoming and outgoing particles may be different. The nonvanishing of the impact factors demonstrates that mass, spin, and parity can change during a diffractive process. We also conclude that $C$ (charge conjugation quantum number), $S$ (strangeness), $I$ (isotopic spin), $B$ (baryon number), and $G$ parity, etc., must remain the same during a diffractive process. Some general properties of the impact factor are also discussed.

Production amplitudes in quantum electrodynamics are studied in the high-energy limit. In particular, we calculate the impact factors for pair production and for bremsstrahlung explicitly. Numerous production amplitudes are then obtained in terms of these two impact factors. The method we use is the one of impact diagrams, which is applicable to all Feynman diagrams at high energies.

The discontinuity of the Regge trajectory obtained recently from a duality series by Kikkawa, Sakita, and Virasoro is investigated. The trajectory is shown to possess all the thresholds corresponding to two-particle intermediate states, i.e., Regge recurrences or daughters. A correction to a linear trajectory is also shown to arise by requiring the Born term (the Veneziano amplitude) to be modified to satisfy two-body partial-wave unitarity near the leading pole in the $J$ plane. This discontinuity is evaluated up to the third two-particle threshold and found to be identical to that obtained from the duality series. Above higher thresholds, such agreement is dependent on the function used to ensure factorization at internal vertices in each term of the duality series.

With the aid of a covariant spin formalism, the kinematic constraint equations for helicity amplitudes are studied in a systematic way for all mass configurations, including the case of zero-mass particles. The complete set of constraints at thresholds and pseudothresholds is given in a form convenient for calculation; that is, the coefficients of the helicity amplitudes are simple numerical ones (as opposed, for instance, to $D$ functions). For nonzero-mass particles, the reduction of the constraints in terms of total spin amplitudes is shown to follow. Amplitudes with different values of total spin are not related at thresholds or pseudo-thresholds.

In the usual method of regularization, the regulator axial-vector current is not conserved. Consequently, the regularized axial-vector current does not, in general, satisfy the ordinary divergence equation derived by using field equations. The anomalous axial-vector divergence equation of spinor electrodynamics is shown to be related to this difficulty of regularization. Using the chiral-invariant regularization and the generalized renormalization procedure developed previously, we show how to obtain an axial-vector vertex which obeys the normal Ward-Takahashi identity and is renormalizable to all orders of perturbation theory. Our general argument is illustrated by an explicit calculation of the fourth-order contribution to the axial-vector vertex from the anomalous triangle loop diagram. We also observe that the renormalized axial-vector vertex, which is well behaved from the point of view of perturbation theory, violates the requirement of generalized unitarity.

Constructing particles out of quarks (via infinite-component fields), we show that if an "elementary" particle has integral spin, baryon number $B\ne 0$, and a symmetric spin-isospin wave function, then its electromagnetic charge form factor vanishes.

Certain of the Reggeized Feynman diagrams suggested by Kikkawa, Klein, Sakita, and Virasoro that require integrals of infinite products of functions are examined. Methods are formulated for explicitly evaluating the infinite products that occur in simple examples of the class of one-loop nonplanar Reggeized Feynman diagrams whose duals are homeomorphic to a sphere with two windows. The infinite products are found to cause the integrals in these cases to diverge, necessitating a regularizing procedure. The integrals can be given meaning by a redefinition which produces an unwanted pole in the square of the sum of the momenta associated with either window, or with a certain type of smoothing function.

The relation between the algebra of current densities and the finite- or infinite-component wave equations is critically investigated. It is found that, at any arbitrary momentum, the charge-current density commutation relations can be satisfied by the solutions of the wave equation, but only in a trivial sense, viz., if $J_0\mathrm{}\left(0\right)\mathrm{}$ is taken to be unity, in which case the content of the current commutators is essentially 1. $J_{\mu }\mathrm{}\left(k\right)=J_{\mu }\mathrm{}\left(k\right)\mathrm{}$, i.e., empty. Furthermore, it is shown explicitly in an example that, starting from the covariant wave equation, this condition $J_0\mathrm{}\left(0\right)=1\mathrm{}$ can be satisfied only if it is made true by definition. The precise connection between the current algebra and infinite-component wave equations is discussed by the introduction of translation operators in momentum space.

The general theory of the propagation of pulse disturbances in an unstable, slightly inhomogeneous, and time-dependent medium is derived in complementary fashion to the eikonal theory of the propagation of stable waves. For a dispersion $D(\mathrm{k}, \omega ; \mathrm{a})$, where k is the wave number and $\omega$ is the (complex) frequency which is related to k for given values of the auxiliary parameters a (temperature, density, etc.) by the vanishing of $D$, the rate of change of propagation velocity of an element of the disturbance is found to be $\frac{\mathrm{dV}}{\mathrm{da}}=\mathrm{Re}(-\frac{D_{\mathrm{a}}}{k{D}_{\omega }})$ in one dimension. A similar equation holds in the case of a general $n$-dimensional system. This relation makes possible the construction of the complete pulse shape in fairly simple fashion for the case of a general slowly varying medium. The theory is illustrated by application to a particular unstable dispersion.

Coherence properties of the single-mode electromagnetic radiation emitted by a system of identical two-level atoms which is initially in the superradiant state are determined approximately. The approximation is valid only for short times when the state of the atomic system does not differ appreciably from the superradiant state. The electromagnetic field radiated spontaneously possesses full coherence in the sense of Glauber's definition, its density operator having the form of that for the ideal laser radiation. This coherence disappears when, because of external radiation, stimulated and spontaneous emission both take place.