Search

Euclidean and Minkowski four-space uniqueness theorems are derived which yield a new perspective of classical four-vector fields. The Euclidean four-space uniqueness theorem is based on a Euclidean four-vector identity which is analogous to an identity used in Helmholtz’s theorem on the uniqueness of three-vector fields. A Minkowski space identity and uniqueness theorem can be formulated from first principles and the space components of this identity turn out to reduce to the three-vector Helmholtz’s identity in a static Newtonian limit. A further result is a uniqueness theorem for scalar fields based on an identity which is proved to be a static Newtonian limit of the zeroth or scalar component of the Minkowski space extension of the Helmholtz identity. Last, the three-vector Helmholtz identity and uniqueness theorem and their four-space extensions to Minkowski space are generalized to mass damped fields.

A study of radial conformal Killing fields (RCKF) in Minkowski space-time is carried out, which leads to their classification into three disjointed classes. Their integral curves are straight or hyperbolic lines admitting orthogonal surfaces of constant curvature, whose sign is related to the causal character of the field. Otherwise, the kinematic properties of the timelike RCKF are given and their applications in kinematic cosmology is discussed.

The classical Backlund transformation, which transforms surfaces of constant negative curvature, has been extended to surfaces in Minkowski three‐space. In this article the tangency requirement of the Backlund transformation is relaxed to obtain a transformation of surfaces of Minkowski three‐space with mean and Gaussian curvatures satisfying a linear relationship.

The decoupling theorem of quantum field theory is proved in Minkowski space. It states the vanishing property, in the distributional sense, of the renormalized Feynman amplitudes when any subset of the underlying masses are scaled to infinity. To prove the theorem we were able, in the process, to bound the corresponding integrals, in the e → +0 limit, by similar Euclidean integrals. All subtractions of renormalization are assumed to be carried out at the origin of momentum space with the degree of divergence of a subtraction coinciding with the dimensionality of the corresponding subdiagram.

By use of a finite volume, lattice approximation, we set up an approximation to the analytic continuation of a polynomial, self‐interacting boson quantum field theory from Minkowski space to Euclidean space. The infinite volume limit for various boundary conditions is shown to exist and to be asymptotic to the perturbation expansion in the coupling constant g at g=0. For g:φ4:d theory we prove mass renormalizability and show how, by use of Nelson’s reconstruction theorem, the corresponding Minkowski space quantum field theory can be obtained. We discuss, at least for d?4, how statistical mechanical techniques, used to analyze the Ising model in the critical region just above the critical temperature, can be used to compute the properties of quantum field theory.

The decoupling theorem of quantum field theory is studied in Minkowski space for theories which on experimental grounds may contain particles with vanishingly small masses. Rules are set up to prove the distributional vanishing property of the renormalized amplitudes when any subset of the underlying masses is scaled to infinity, and any subset of the remaining masses is scaled to zero. By careful estimates, the analysis in Minkowski space may be reduced to a similar one in Euclidean space. All subtractions of renormalization are carried out at the origin of momentum space with the degree of divergence of a subtraction coinciding with the dimensionality of the corresponding subdiagram.

The proof of the decoupling theorem of quantum field theory given earlier [E. B. Manoukian, J. Math. Phys. 26, 1065 (1985)] in Minkowski space, in the distributional sense, for theories involving particles with vanishingly small masses as well is extended under more general conditions, thus being applicable to a larger class of graphs. All subtractions of renormalization are carried out at the origin of momentum space with the degree of divergence of a subtraction coinciding with the dimensionality of the corresponding subdiagram.

Minkowski space–time is developed in terms of a set of undefined primitive elements called events, certain subsets of events called paths which correspond to the worldlines of free particles, and a temporal order relation on each path. Nine axioms describe the existence and uniqueness of paths, temporal order, connectedness, causality, collinearity, continuity, isotropy, and dimension.

The purpose of this paper is to prove one of Zeeman's conjectures that the group of homeomorphisms of the finest topology on Minkowski space that induces the 3‐dimensional Euclidean topology on spacelike hyperplanes is the one generated by the inhomogeneous Lorentz group and dilatations.

There is a discussion of (i) the transformation properties of hyperbolic hypersurfaces in Minkowski 4‐space, (ii) the association of hyperbolic world lines in Minkowski 2‐space with uniformly accelerated motion.

Utilizing the fact that the Schwarzschild and Kerr geometries have an intrinsically defined Minkowski space associated with them, we show that these Minkowski spaces (as the Kerr parameter varies) can be viewed as ``real slices'' in a complexified Minkowski space. The complex Weyl tensor of each member of the family can then be viewed as a single complex field on the complex Minkowski space. Further, the degenerate principle null vectors associated with each geometry can be considered as projections into the ``real slices'' of a complex null vector field in the complex Minkowski space. These results may be considered as clarifying earlier work on obtaining the Kerr metric from the Schwarzschild metric by a complex coordinate transformation.