BULLETIN DE L'ACADEMIE

POLONAISE DES SCIENCES

Serie des sciences physiques et

astron. Vol. XXVII, No2, 1979

by

Arkadiusz Z. Jadczyk

Summary: It is shown that to give a constitutive tensor for the vacuum is to give a conformal structure to space-time.

Keywords: light, electromagnetism, conformal structure, conformal, Hodge star, light-cone, vacuum, permeability, space-time, Maxwell equations, R. Haag, D. Kastler, I.M. Singer, Riemannian metric,bivector

Acknowledgments: This note is a solution to a problem raised in a discussion with Professors R. Haag, D. Kastler and I. M. Singer

History: In 1978, Daniel Kastler (CPT CNRS Marseille), and Rudolph Haag (University of Hamburg), organized a small private conference in Elmau, Austria. Elmau is a little village in Tirol where Daniel's wife, Lisa, was born. As I knew both of them, I was invited there too. Also present were Derek W. Robinson, Sergio Doplicher, and I. M. Singer.

While Haag, Kastler, Doplicher and Robinson are experts in algebraic quantum field theory, Singer is the world famous mathematician who contributed to different branches of differential geometry; (there is, for instance, the famous Atiyah-Singer theorem). I was working in both areas - algebraic methods and geometric methods.

During our meetings and excursions we discussed many topics, but I was mostly interested in the problems of fibre bundles, gauge theories, meaning of electromagnetic potentials, and also conformal invariance of electromagnetism. It is well known that Maxwell equations in vacuum are conformally invariant - which results in the fact that photons are massless. A geometrical expression of this fact is that space-time metric enters Maxwell equations only via Hodge star operator on two-forms. It was Singer who pointed out to us that in 2n dimensional space-time (2n=4 in our case), Hodge operator on n-forms is conformally invariant.

The natural question that arose was: can one read the conformal structure of space-time from Maxwell equations? And if so, is the light cone structure coded into Maxwell equations?

The question is important because light-cone structure determines nine of the ten components of the metric tensor. Thus, in other words, we were asking to what extent is gravity intertwined with electromagnetism?

To find the answer was not easy - as the several months I spent doing intense calculations proved. But, finally I was able to formulate a precise mathematical theorem and to provide a formal proof of it.

This paper gives the details. I demonstrate that in 4-dimensional space-time, every linear operator on 2-forms, which is self-adjoint (in a natural sense), and whose square is -1 (so that it behaves like the imaginary unit "i"), determines and is determined by a unique light-cone. This is a purely algebraic statement, but it can be applied, locally, point by point, to a differential manifold modeling of curved space-time.

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Contact: ajad@physik.uni-bielefeld.de

Note added: On August 30, 1999 I received the following email from Prof. F. W. Hehl:

Dear Professor Jadczyk,
Together with Yuri Obukhov, I have recently published a paper on the
relation of electrodynamics with the metric of spacetime (Phys.Lett. B 458
(1999) 466). From a colleague I heard that our paper may have something to
do with a paper which you published 20 years ago, namely 
> ELECTROMAGNETIC PERMEABILITY OF THE VACUUM AND LIGHT CONE STRUCTURE. By
> A.Z. Jadczyk (Wroclaw U.). 1979. Published in
> Bull.Acad.Polon.Sci.(Phys.Astron.) 27:91-94,1979 
Unfortunately, I have no access to the Bull. Acad. Polon.
Would you be kind enough to fax me a copy of your paper? I would be very
grateful to you. Thank you in advance.
Best regards,
Friedrich Hehl