Max Dresden

Physical algebras in four dimensions. I. The Clifford algebra in Minkowski spacetime

This paper provides a compact, unified framework for the description of physical fields in spacetime. We combine features of the traditional vector, matrix, tensor, spinor, quaternion, and dyadic methods into a simple easy-to-use scheme. The field description is both matrix free and coordinate free. This construction is achieved by using the differential forms of Minkowski spacetime to realize a Clifford algebra of dimension 16. We should note that this algebra is distinct from either the algebra of Dirac matrices, or the algebra of the Majorana matrices. A novel characteristic of the algebraic structure is that an inverse, and consequently division by a vector or an antisymmetric tensor field of any rank, is perfectly well defined. Products and inverses of all antisymmetric tensor fields in spacetime are worked out in terms of the usual vector notation. Among useful features of this scheme is the description of duals as products by a basis element. Moreover, this field description is intrinsically Lorentz covariant, and the algebraic product preserves the covariance. Many examples are given in order to illustrate the practical value of the formalism presented herein.

New perspectives on Kac ring models

It is proposed that the type of model first suggested by Kac in connection with problems of nonequilibrium statistical mechanics can be generalized and modified so that it can be directly applied to cellular automata. It is further noted that these same models can be used to illuminate some basic questions in the interpretation of quantum mechanics.

Reflections on “Fundamentality and Complexity”

It seems fitting that the impetus for the sketchy considerations presented here came from many luncheon conversations in Iowa City, where both Joseph Jauch and the author spent many pleasant and productive years.

Models in nonequilibrium quantum statistical mechanics

In this paper, quantum versions of statistical models are constructed. All aspects of the systems can be explicitly solved. It is possible to give magnetic realizations of these models. The most interesting conclusions are: (1) the state for time going to infinity is approached in an oscillatory manner in the quantum case; (2) in both classical and quantum cases, the exact description gives limiting states which remember the initial specifications; and (3) in these models, the time evolution generally cannot be described. even approximately, by a master equation.

Scalar field theories in curved space-time

A 3 field theory in a six-dimensional conformally flat space-time is studied at the two loop level. It is found that the state-dependent divergences do not cancel. As a result, the theory does not renormalize in the usual way. The question of why 3 is badly behaved in curved space-time and 4 is well behaved is discussed.

General Physical Principles and Non‐Linear Group Realizations

The basic question raised in this paper is the relationship between group theoretical and physical notions. In particular the physical significance of the mathematical entities occurring in group‐representations is examined in detail. For this reason a brief outline is presented of the mathematical definitions and status of non‐linear realizations of groups. Special non‐linear realizations of the Lorentz and Poincare groups are exhibited. The possible physical meaning of these realizations is discussed. It is shown that there is a fundamental interpretation question involved, which indicates that the identical formalism can describe a wide variety of physical phenomena. The classical (Wigner) theory of the representations of the Poincare group, allows the description of many‐particle systems in terms of elements and operators in tensor products of Hilbert space. An appropriate adaptation of this procedure suggests that the utilization of the non‐linear realization of the Poincare group, describes a relativistic many‐particle system in interaction. General requirements such as causality are shown to be compatible with the formalism of the non‐linear realizations.