Formal relativistic quantum theory of fourth-dimensional wave propagation (A)

Jurjen F. Koksma, W. Westra

The Feynman propagator used in the conventional in-out formalism in quantum field theory is not a causal propagator as wave packets are propagated virtually instantaneously outside the causal region of the initial state. We formulate a causal in-out formalism in quantum field theory by making use of the Wheeler propagator, the time ordered commutator propagator, which is manifestly causal. Only free scalar field theories and their first quantization are considered. We identify the real Klein Gordon field itself as the wave function of a neutral spinless relativistic particle. Furthermore, we derive a probability density for our relativistic wave packet using the inner product between states that live on a suitably defined Hilbert space of real quantum fields. We show that the time evolution of our probability density is governed by the Wheeler propagator, such that it behaves causally too.

Detlef Dürr, Dustin Lazarovici

Matthias Lienert

The present work deals with the idea of a multi-time wave function, i.e. a wave function with N space-time arguments for N particles. Firstly, a careful derivation of the necessity of multi-time wave functions in relativistic quantum mechanics is given and a general formalism is developed. Secondly, the physical meaning of multi-time wave functions is discussed in connection with realistic relativistic quantum theories, in particular the "Hypersurface Bohm-Dirac" model. Thirdly, a first interacting model for multi-time wave functions of two Dirac particles in 1+1 space-time dimensions is constructed. Interaction is achieved by means of boundary conditions on configuration space-time, a mechanism closely related to zero-range physics. This is remarkable, as a restrictive consistency condition rules out various types of interaction and consequently no rigorous interacting model was known before. Fourthly, the model is extended to more general types of interaction and to the N-particle case. Higher dimensions are also discussed. Finally, the "Two-Body Dirac equations" of constraint theory are placed within the context of the multi-time formalism. In particular, the question of probability conservation is critically discussed, leading to further implications both for fundamental and applied questions.

Peter Schwerdtfeger

C. A. Lopez, M. A. Pérez

A. O. Barut, S. Malin

The principles of quantum mechanics impose certain restrictions on acceptable position operators. The probability density as derived from the position operator should agree with the fourth component of the current vector. The various relativistic wave equations are reexamined from the point of view of this consistency test. These and the group theoretical considerations lead to the conclusion that only a Dirac particle can be considered to be elementary. The position operator of the infinite-dimensional theories is investigated and the advantages of theories using the unitary representations of the Lorentz group are discussed.