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arXiv:1012.3473

A Causal Alternative to Feynman's Propagator

Jurjen F. Koksma, W. Westra

Published Dec 15, 2010• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1012.3473

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Recently a photon position operator with commuting components and a particle density that is positive definite for both positive and negative frequency states has led to a first quantized theory of the photon. Here we extend this theory to a manifestly covariant form and show for the first time that, with invariant plane wave normalization, the position operator is Hermitian with real instantaneously localized eigenvectors. Reality of the field and wave function ensure causal propagation and zero net absorption of energy in the absence of charged matter. The position space wave function is the real part of the projection of the photon's state vector onto a basis of position eigenvectors. Manifest covariance and consistency with quantum field theory is maintained through use of the electromagnetic four-potential and the Gupta-Bleuler gauge condition.
An operational development of quantum theory is presented, from general correlations through single-particle theories to elementary quantum field theory, with a focus throughout on causal relations. The objective is to establish the extent to which quantum theories are consistent with the principle of relativistic causality (that no two events separated by a distance greater than their separation in time multiplied by the speed of light may have a causal influence on each other) and to examine the assumptions that this analysis requires. To do this, it is necessary to pay particular attention to the notions of spatial location and measurement. It is found that in a relativistically causal theory, any measurement made in a finite spatial region must have the capacity for particle creation. A number of derivations are presented, including some relativistic single-particle propagators and a Hamiltonian based on the square root of the Klein Gordon equation.
Drawing from the optimal transport theory adapted to the relativistic setting we formulate the principle of a causal flow of probability and apply it in the wave-packet formalism. We demonstrate that whereas the Dirac Hamiltonian impels a causal evolution of probabilities, even in the presence of interactions, the relativistic-Schr\"odinger model is acausal. We quantify the causality breakdown in the latter model and argue that, in contrast to the popular viewpoint, it is not related to the localization properties of the states.