Non-Linear Field Theories II. Canonical Equations and Quantization

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Abstract
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.

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