The point-source limit is found for a previously proposed static model satisfying crossing symmetry and two-particle unitarity below three-particle thresholds. A numerical solution for this limit is obtained, and both renormalized and unrenormalized quantities are calculated. The renormalization constants are found to vanish; the relation of this to compositeness of the neutron in this model is discussed.

The behavior of commutator matrix elements at small distances is investigated in the framework of general quantum field theory. The equivalence between current density commutators and the commutators of a finite number of charge moments and a current density at equal times is proved by means of microcausality but without using the spectrum condition. If $N$ is the (finite) order of the commutator, then the equal-time limits of the commutators between all charge moments of degree higher than $2N+m-1$ ($m$ fixed between zero and $N$) and a current density vanish. The equal-time commutators between the first $2N+m-1$ charge moments and a current density exist if and only if the equal-time limit of the corresponding current densities exists and is a sum of $2N+m-1$ derivatives of the $\delta$ function in the space variables x. The coefficients of the individual $\delta$ functions are identical to the equal-time limits of the charge moments and one density. If the spectrum condition holds in addition, then $m$ is equal to zero.

Using the Koba-Nielsen formalism for the generalized beta function, a general form for factorization into two pieces is developed in an entirely Möbius-invariant way. Ward identities for the vertex in the general configuration are derived in a manner which uses only the Möbius invariance of that factor. Special cases (still Möbius-invariant) of this procedure include the multiperipheral, semimultiperipheral, and symmetric factorizations already known.

We show that scattering boundary conditions are incompatible with a monochromatic radiation field for the case of nonlinear Compton scattering. We demonstrate this by showing that (a) in the monochromatic limit, gauge invariance in a given order of the expansion of the $S$ matrix is destroyed, and (b) the physical (scattering) boundary condition, that of a pulse of radiation incident on a target electron, cannot be reconstructed from the monochromatic limit of the $S$ matrix. We then proceed to show by an example that the frequency profile of the scattered radiation is a function of both the intensity and line shape of the incident field. Another interesting feature of this calculation is that the profile of the photon scattered at a fixed angle is significantly broadened in comparison with the incident line shape. The worked-out example is a simple model, that of a neutral, scalar "electron" interacting with a bilinear scalar, massless external field, which contains all the important features of nonlinear Compton scattering. While from the point of view of gauge invariance it is sufficient to treat the external radiation field as a one-dimensional wave packet, for a complete description of the problem it is necessary to describe the incident radiation (quanta) in terms of normalizable states. An estimate of the breakdown of the plane-wave approximation is included.

The experimental meaning of coherent superposition of state vectors is analyzed. It is shown that the involvement of the measuring instrument is of basic importance and that it is erroneous to study superposition without it. From this analysis it is found that there are no such things as superselection rules. The meaning of the sign change of spinors under $2\pi$ rotation is considered, and it is found to have experimental consequences. The nonmeasurability of the relative parity of states with different $z$ components of angular momentum is shown. An appendix discusses the application of some of the considerations of the paper to decaying states and elementary particles.

The Dirac equation of an electron in a Coulomb field is shown to be equivalent to a certain subclass of Feynman diagrams taken within the eikonal approximation. The result is obtained via analysis of a two-body scattering amplitude, wherein the role of the heavier of the two particles involved is to generate the effective external potential in which the lighter particle propagates. The poles of the scattering amplitude are found to be the well-known relativistic binding energies.

A model of quantum-mechanical scattering is given in terms of the Abelian subalgebras of observables which are associated with the initial preparation and final measurement procedures. The scattering is described by an operator, the $S$ matrix, which defines an isomorphism between these two subalgebras. By using the concept of presymmetry for Galilean-invariant systems, these algebras are shown to be generated by the generalized momenta ′P which are related to the dynamical situation by $H{P}_{\mathrm{ac}}=\frac{{\left|{\mathrm{P}}^{\prime }\right|}^2}{2\mu }$. Here $H$ is the total Hamiltonian effective at the time of measurement, $P_{\mathrm{ac}}$ is the projection onto its absolutely continuous spectrum, and $\mu$ is the mass of the particle. The $S$ matrix which is defined in this way gives the same formal expressions for the differential cross sections as the usual one, and it is automatically unitary. Furthermore, since the differential cross section is defined as a relative probability in terms of the results of the two measurement procedures, it can be finite even when the classical cross section is infinite. Scattering can therefore be defined in this model for spherically symmetric potentials which decrease as slowly as $r^{-\epsilon }$, for any $\epsilon >0\mathrm{}$, provided that the potential is repulsive at large separations between the particles.

An integral representation for a set of infinitely many $N$-point amplitudes satisfying duality is presented. Two universal functions, which satisfy simple equations, enter into this representation. These equations have been studied previously and their solutions are well known. These generalized $N$-point amplitudes are not simply a series of the amplitudes already known. The latter form the set of special cases where only one of the two universal functions is involved. These amplitudes satisfy duality and invariance under cyclic permutation of the external momenta. Their resonance residues factorize and ancestors are absent. However, the possibility that the particle spectrum is richer than that of the ordinary amplitudes exists. It is also indicated how restrictions on these amplitudes may arise upon the implementation of unitarity.

We examine the single-loop charge renormalization of an $\mathrm{SU}\left(3\right)\mathrm{}\otimes \mathrm{SU}\left(3\right)\mathrm{}$ chiral-invariant $\sigma$ model. This requires that a systematic way of treating charge renormalization for models with the added complexity of $\mathrm{SU}\left(3\right)\mathrm{}$ be developed. The model we consider has two nonderivative four-point couplings and one nonderivative three-point coupling. We find that two of these three independent chiral-invariant couplings are not individually renormalizable, as their higher-order $\mathrm{SU}\left(3\right)\mathrm{}$ structure is not expressible in terms of the $\mathrm{SU}\left(3\right)\mathrm{}$ structure of their respective tree diagrams. Only when all three invariant couplings are considered simultaneously is the resultant $\mathrm{SU}\left(3\right)\mathrm{}$ structure given by the tree diagrams. Hence, each of the three charge renormalization constants is a function of all three coupling constants.

Methods are developed for constructing momentum-space amplitudes corresponding to nonpolynomial nonderivative interactions of a real scalar field. The methods give rise to a supergraph technique and rules for writing down matrix elements very similar to Feynman techniques. The methods are not established rigorously; at several points the argument requires certain analytic properties of Feynman integrands which, though plausible, can only be demonstrated rigorously for the zero-mass case. Asymptotic behavior, both in spacelike and timelike directions, is discussed. Rough arguments are given that indicate that the singularity structure of the amplitudes is likely to be consistent with unitarity.

It is asserted that the Sugawara theory of currents is equivalent to a canonical Lagranigan theory of massless scalar fields with a Yang-Mills type of interaction. The Sugawara currents are identified with the canonical currents associated with nonlinear transformations of the massless fields. These transformations arise naturally in the context of spontaneous symmetry breakdown. In models of spontaneously broken symmetry in which the symmetry-breaking scalar fields are elementary dynamical variables, that part of the stress-energy tensor containing the Goldstone bosons terms goes over into the Sugawara form when reexpressed in terms of the currents. However, additional terms relating to a massive field appear in the stress-energy tensor, and the massive field operator replaces the usual Sugawara constant. A Sugawara current theory is obtained by eliminating the massive field and retaining the Goldstone boson field only. The reverse is also true in that any Sugawara theory is necessarily equivalent to this canonical representation. Finally, it is shown that if the currents are coupled to gauge fields, a massive Yang-Mills field is obtained.

A theory of weak interactions is suggested which is free of the divergences associated with the fourfermion theory and the usual intermediate-vector-boson theory, without the use of cutoffs. The interaction Lagrangian is a universal, local coupling of the usual weak currents to a vector field operator which is a nonentire function of a minimal set of vector-meson fields of large mass (≳5 GeV). A summation procedure related to that of Efimov and Fradkin is developed for the analysis of such Lagrangians, and $S$-matrix elements are computed thereby. It is shown that the (two) parameters of the Lagrangian may be adjusted to produce finite fermion self-energies and finite second- and fourth-order graphs. A power-counting argument is given that this finiteness then persists to all orders without the use of cutoffs. It is further shown that the parameters may be chosen to produce agreement with the predictions of the Fermi theory at low-energies and momentum transfers. The high-energy deviations from Fermi theory predicted by this theory are analyzed. The problem of ambiguities associated with nonpolynomial Lagrangians is discussed. A prescription for the resolution of the ambiguity is given which is expected to ensure the unitarity of the theory in all orders.

The customary notion of superselection rule stipulates the vanishing of those matrix elements of the density matrix that connect states separated by the superselection rule, i.e., the relative phases of which are, according to the superselection rule, meaningless. It is shown that, if any two states for which an additive conserved quantity has different values are separated for all physical systems by a superselection rule at one time, the separation will persist for all times. It is concluded that the states with different electric (or baryonic) charges are so separated. The reason for the difference between electric charge and other additive conserved quantities, such as momentum, is brought out.

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.

A variation of the quasipotential approach of Logunov and Tavkhelidze is investigated. Two relativistic particles with or without spin are subjected to a mutual interaction that can be described (i) as generated by the exchange of field quanta or (ii) as a two-particle potential $V\left(\mathrm{r}\right)$ that depends on the relativistic three-dimensional coordinates introduced by Kadyshevsky. The framework includes the complete axiomatic structure of nonrelativistic quantum mechanics (except that the metric in Hilbert space is indefinite in the same way as in the static Klein-Gordon or Dirac theories) and is at the same time fully and explicitly covariant. Both the nonrelativistic and the classical limits exist and are susceptible to detailed interpretation. The case of two spinless particles with a "relativistic Coulomb" interaction is examined in detail. The wave equation, as well as the equation for the $T$ matrix, is in this case exactly soluble. Closed analytic expressions are given for transition form factors, including "photoproduction," the elastic scattering amplitude, and a production amplitude (bremsstrahlung).

In a classical system of particles interacting via neutral vector-meson exchange, at very high densities the pressure exceeds the relativistic energy density, and the speed of low-frequency sound waves exceeds the speed of light in vacuum. A quantum version of the same model, if it is stable against spontaneous pair production, can be neither ultrabaric nor superluminal, if, at high density, the correlation energy increases faster than the number of particles. Real matter, if it is stable at very high densities, is not expected to show noncausal sound propagation.

This paper extends the work of the preceding paper by including the gravitational (general relativistic) aspects of a magnetic field such as results from an initial uniform current which has been switched off. The solutions to the entire set of Einstein-Maxwell equations are expressed simply in terms of the "pure Maxwell solutions" obtained by neglecting gravitation. The latter solutions, in both graphical and analytical form, were found and discussed in the preceding paper; they are not simple functions of space and time. Appropriate initial conditions for the space outside the wire ($\rho >1\mathrm{}$) are established by setting time derivatives and electric field equal to zero in the Einstein-Maxwell equations, and solving. We then show that the time-dependent solutions are given by $B=\frac{B_M{h}_3}{h_0}$, and $E=\frac{E_M{h}_3}{h_0}$, where $B_M(\rho , \tau )$ and $E_M(\rho , \tau )$ are the time-dependent Maxwell fields given in the preceding paper, $h_0$, $h_1$, $h_2$, and $h_3$ are the gravitational-field-determining scale factors $h_{\alpha }\equiv {\left|g_{\alpha \alpha }\right|}^{\frac{1}{2}}$ associated with $\tau \equiv \frac{\mathrm{ct}}{a}$, $\rho \equiv \frac{r}{a}$, $\phi$, and $\zeta \equiv \frac{z}{a}$, respectively ($r$, $\phi$, and $z$ correspond to standard cylindrical coordinates); $a$ is the wire radius. The $h_{\alpha }$ are expressed in terms of the pure Maxwell field quantities $A_M$ and $V$ and a small universal constant $K\equiv {\left(\frac{4\pi G}{{\mu }_0{c}^4}\right)}^{\frac{1}{2}}\simeq 2.91\times {10}^{-19\mathrm{}}$ m/Wb by the relations $h_3=\frac{1}{cosh\left(K{A}_M\right)}, h_2=\frac{a_{\rho }}{h_3}, h_1=h_0=\exp \left(K^{2}V\right)h_3,$ $A_M(\rho , \tau )=-{\left(\mathrm{vector}\mathrm{potential}\right)}_M=A_M(\rho , 0)+\frac{a}{c}\int 0^{\tau }{E}_M(\rho , s)ds,$ $A_M(\rho , 0)=\frac{1}{2}b({\rho }^2-1) \left(\rho \le 1\right), A_M(\rho , 0)=b\ln \rho (\rho \ge 1), b\equiv a{B}_M(1, 0)=\frac{{\mu }_{0}I}{2\pi },$ $V(\rho , t)=\left(\frac{{\mu }_0}{\pi }\right) \left(\mathrm{field}\mathrm{energy}\mathrm{out}\mathrm{to} \rho \mathrm{per}\mathrm{unit} z\right)-\frac{1}{4}{b}^2=a^2\int 0^{\rho }\mathrm{xdx}\left[\frac{{E_M}^2(x, \tau )}{c^2}+{B_M}^2(x, \tau )\right]-\frac{1}{4}{b}^2.$

The methods and order of investigation presented in these two papers can be a practical modus operandi for finding realistic solutions to the Einstein-Maxwell time-dependent equations.

This paper gives the exact solutions, graphical and analytical, for an idealization of a phenomenon described in every textbook of physics: the decay of a magnetic field after its source current has ceased. The problem is solved for the cylindrically symmetric case where a uniform current in a long straight wire is suddenly brought to zero at an initial time $\tau =0\mathrm{}$. The probable reason why this apparently simple problem has not been solved before is that the mathematical details are complex. The solutions for the electric and magnetic fields which result from the decay of the initial magnetic field are classified according to the four regions of a space-time diagram in which they appear: Regions $A$ (inside wire) and $O$ (outside wire) are outside the light cone of a point on the boundary of the wire. In accord with causality, the magnetic and electric fields retain their initial values (the latter equal to zero) in region $O$; in region $A$ the magnetic field retains its initial value, but the electric field rises linearly with time, since its time derivative replaces the extinguished constant current density. Region $B$ is between the axis and the wave front from the boundary (which started at $\tau =0\mathrm{}$) after this wave front has passed through the axis and started outward again. Region $C$ is contained between the initially outgoing wave front which starts from the wire boundary and the initially ingoing wave front after it has passed through the axis and started outward again; it is therefore a region which the initial disturbance from the wire boundary has swept over only once instead of twice, in contrast to region $B$. In both regions $B$ and $C$ the analytical expressions for the electric and magnetic fields $\mathcal{E}$ and $\mathcal{B}$ are very complicated except along certain selected lines in space-time, e.g., at the wire axis ($\mathcal{E}={[\tau +{({\tau }^2-1)}^{\frac{1}{2}}]}^{-1\mathrm{}}$, $\mathcal{B}=0\mathrm{}$) or on the light cone of a wire boundary point ($\mathcal{E}=\tau [1-(\frac{2}{\pi }\left)invcos\right(\frac{1}{{\tau }^{\frac{1}{2}}}\left)\right]-\left(\frac{2}{\pi }\right){(\tau -1)}^{\frac{1}{2}}$, $\mathcal{B}=\frac{1}{2}(\tau -1)-\left[\frac{\tau }{\pi {(\tau -1)}^{\frac{1}{2}}}\right]\{1-[\frac{(2-\tau )}{{(\tau -1)}^{\frac{1}{2}}}\left]invsin{\left[\frac{(\tau -1)}{\tau }\right]}^{\frac{1}{2}}\right\}$). Quite apart from the specific nature of this problem, the investigations in this and the following papers illustrate a practical method for finding realistic solutions to the Einstein-Maxwell time-dependent equations. These solutions correspond to special cases—worked out explicitly, and in one case independently—of a general family of solutions of the time-dependent electrogravitational equations for cyclindrical symmetry. This general family of solutions has been discussed by Melvin and stachel.

The question of the maximum amount of energy which can be radiated by a collapsing spherical star is reexamined. On Newtonian theory, gravitational energy is negative and unbounded below, so that unlimited amounts of energy can be released. It is shown that, in the relativistic collapse of a star with non-negative energy density, self-closure always takes place before the star can release 100% of its initially positive mass energy. Moreover, under physically reasonable restrictions on the pressure, the 100% upper limit can be approached only if the star happens to pass through a very special and improbable momentarily static configuration first considered by Zel'dovich. It is concluded that for normal spherical collapse the efficiency of energy release must be low.

It is shown by example that causality violation can take place through the transmission of classical tachyons between three or more observers in such a way that each observer receives and emits only tachyons whose energies he measures to be positive. Such examples are not open to the ambiguities of interpretation proposed by Feinberg; they imply that a classical-particle description of tachyons is not physically viable.