The Riemannian geometries, which are derivable from a quadratic action principle, are generated by a new mathematical approach, based on the method of the Lagrangian multiplier. This changes the 10 differential equations of fourth order for the $g_{\mathrm{ik}}$ to 24 differential equations of second order for a new field quantity. This field quantity is a tensor of third order, antisymmetric in one pair of indices, in remarkable analogy to the fundamental $\Lambda$-tensor of Einstein's theory of distant parallelism. The field equations are tied together by 13 identities. The basic structure of the field equations for infinitesimal fields is investigated and their relations to Einstein's theory of gravity and Einstein's theory of distant parallelism discussed.

The decomposition of tensors into irreducible representations of the orthogonal groups is calculated for three and four dimensions. The connection is shown with the problem of the allowed values of ordinary and isotopic spin for a given symmetry of the spacial eigenfunction of a nuclear system.

Construction and discussion of phase equivalent potentials, i.e., of potentials which give equal phase shifts for $S$-waves. In particular, it is shown that the phase shifts do not determine the energy of bound states, and that there exist continuous families of phase equivalent potentials.

In this paper, a covariant field theory of the general type of the theory of relativity is brought into the canonical form and then quantized. Particles are assumed to be represented as singularities of the field. Primarily, we had to overcome two difficulties. First, the variational integral should be extended only over that space-time domain which is free of singularities. Since the location of the singular world lines cannot be known until after the integration of the field equations has been completed, we have introduced a second set of coordinates—called "parameters" in this paper—which will serve as variables of integration and in terms of which the motions of the particles can be arbitrarily prescribed. The second difficulty arises in that the expressions for the canonical momentum densities cannot be solved with respect to the partial time derivatives of the field variables; this circumstance precludes the construction of the Hamiltonian by the usual methods. Nevertheless, we have shown that a Hamiltonian exists, though it is not uniquely determined by the Lagrangian; our Hamiltonian contains an arbitrary linear combination of the eight algebraic relationships that exist between the canonical variables at each world point. The canonical field equations have the usual form. They are covariant if the choice of Hamiltonian is left open. The eight algebraic constraints on the canonical variables at each point are all integrals of the field equations. So are the Poisson brackets between the canonical variables (at the same time). When this system of equations and constraints is quantized, the property of general covariance can be used to carry out a proof of the covariance of the whole theory, including the commutation relations, that requires none of the computational effort usually required in theories that are merely Lorentz invariant. Once the system of equations has been completed, it turns out that the covariance goes much farther than was required originally. Because of the introduction of the parameters, the ordinary coordinates of space-time turn formally, at least, into dynamical variables, and the usual canonical transformations, with respect to which the theory is covariant, transform the coordinates, the original field variables, and the canonical conjugates of both together. The canonical conjugates of the coordinates are the expressions ordinarily interpreted as energy and momentum densities. The physical significance of these canonical transformations, which cause the world points to lose their identities, is not yet understood.

A set of postulates is formulated which leads to the relativistic wave equations of present quantum mechanics. The mathematical difference between the usual non-linear interaction terms and terms which introduce essential non-linearities in the equations is discussed. It is proved that every particle must possess an antiparticle unless at least one of the basic postulates is discarded. The connection between the rest mass of the particle, the minimal equation of the $\alpha$-matrices, and the general commutation relations of the $\alpha$-matrices is derived and discussed. It is proved that for a particle of spin ½ there is only one possible wave equation, while for a particle of spin 1 there are others besides the usual scalar and vector equations. One such example is given.

The formal method of regularization of mathematical expressions of sums of products of different types of $\delta$-functions is first applied to the example of vacuum polarization. It is emphasized that only a regularization of the whole expression without factorization leads to gauge invariant results. It is further shown, that for the regularization of the expression for the magnetic moment of the electron, a single auxiliary mass is sufficient, provided that different functions of the same particle (e.g., the photon functions $\overline{D}$ and $D^{\mathrm{}\left(1\right)\mathrm{}}$) are regularized in the same way and that the regularization of products of two electron functions is never factorized. The result is then the same as that of using Schwinger's method of introducing suitable parameters as new integration variables in the argument of $\delta$-functions, without using any auxiliary masses.

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A simple argument is given to explain Herglotz' theorem that a rigid body has three degrees of freedom only. A new example of rigid motion is given.

It is attempted to formulate the properties of localized states on the basis of natural invariance requirements. Chief of these is that a state, localized at a certain point, becomes, after a translation, orthogonal to all the undisplaced states localized at that point. It is found that the required properties uniquely define the set of localized states for elementary systems of non-zero mass and arbitrary spin. The localized functions belong to a continuous spectrum of an operator which it is natural to call the position operator. This operator has automatically the property of preserving the positive energy character of the wave function to which it is applied (and it should be applied only to such wave functions). It is believed that the development here presented may have applications in the theory of elementary particles and of the collision matrix.

For the purposes of atomic theory it is necessary to combine the restricted principle of relativity with the Hamiltonian formulation of dynamics. This combination leads to the appearance of ten fundamental quantities for each dynamical system, namely the total energy, the total momentum and the 6-vector which has three components equal to the total angular momentum. The usual form of dynamics expresses everything in terms of dynamical variables at one instant of time, which results in specially simple expressions for six or these ten, namely the components of momentum and of angular momentum. There are other forms for relativistic dynamics in which others of the ten are specially simple, corresponding to various sub-groups of the inhomogeneous Lorentz group. These forms are investigated and applied to a system of particles in interaction and to the electromagnetic field.

Exact solutions to the Dirac equations for an electron in an external field of an arbitrary plane wave are obtained by transforming plane wave solutions for a free electron by variable Lorentz matrices. These matrices are those which occur in the discussion of the classical orbits. An example shows that this method, which applies classically, may fail for the Dirac equations when the external field is a constant one.