What follows below is a faithful copy (with just a couple of spelling correction) of my letter to prof. Ph. Blanchard (University of Bielefeld) delineating the ideas behind the "Quantum Future Project". Some of these ideas have been already realized. Some other are still on the drawing table.
Florence, June 26, 1990
Dear Professor Blanchard,
Thanks for your letter of June 6. Included below is a more complete version of the Quantum Future Project. It is not written for the administration to get a support. It is written for you. With my best will I can not make it now more precise. I am working in this direction all the time and details change rather fast. It might be a lost of time to attempt more precision now. As you can see from the project below, there is no better place in the world to work on it than Bielefeld, and no more appropriate scientist to co-operate with than you. First, as you can see, the project goes slightly against the current fashion. Therefore, it needs open-mindness and some perspective in the vision. Second, it needs a good knowledge of several branches of mathematics which I am not an expert in, but you are. Third, I know from the few discussions that we have already had some time ago, that our perspectives on quantum theory are not far one from the other.
Quantum Future v. 0.7
Wigner, Karolazhyi, Diosi and others have analyzed incompatibilities between uncertainty relations ensuing from quantum theory and geometry of relativistic space-times. They all came to the conclusion that a contradiction is there. Einstein himself was well aware of this serious problem but was unable to find out an acceptable solution in his lifetime. One of the few ideas that he had thought of, but had no courage to study was that of a complex space-time. Today this concept is not that strange as a few decades ago. One of the lessons that we have learned from the work done on geometric quantization is that it works best on Kaehler manifolds. Thus "complex space-time" should really mean "relativistic phase-space", with a kind of the Born reciprocity principle (instead of locality and causality) as a basic guidance. Of particular interest are Kaehlerian symmetric domains of the type U(n,n)/(U(n) x U(n)), because of their relation to the conformal group and twistors. Their attractive feature is that they have a natural positive definitive Riemannian metric while a Lorentzian conformal structure arises automatically on their Shilov boundaries. Planck constant is naturally included in their geometry if they are to be interpreted as "phase spaces". Thus we propose to critically reconsider again the physical arguments of the above mentioned authors, with the aim to see if replacing space by such a complex manifold (and replacing the events by processes) will not remove the contradictions. A negative answer to this question should suggest a way out (through discretizing space-time, stochastic geometry or other). On the other hand, if the answer will turn out to be positive (i.e. a contradiction can be removed; and a preliminary analysis indicates that this is likely to be the case), then a standard scheme of geometric quantization, based on a linear quantization condition) have to be reconsidered, taking into account the fact that it is the measurement apparatus (with many microscopically random parameters) that determines a polarization. If a nonlinear quantization condition can be produced as a result of taking into account a feedback of the results of measurements on a state, then we may end up with a theory that describes "quantum jumps" and has a classical geometry (albeit of "process-space" rather than of space-time) in its roots. This program should be thought of as an alternative to the recently more and more fashionable research into "quantum geometry". The latter one is based on the idea that commutative geometry should be totally abandoned. Although mathematically attractive, it is, in our opinion, unacceptable from the physical point of view (as understood already by Niels Bohr). It should be however pointed out that certain mathematical constructions of the geometry of complex domains (like natural connections in holomorphic vector bundles), when interpreted in terms of their Shilov boundaries, lead to non-local geometrical objects that find their natural interpretation in terms of non-commutative geometry (cyclic cohomology developed by Bott, Connes, Kuntz, Kastler). Thus the program follows:
Quantum Future - Research project
For more than fifty years no satisfactory unification of quantum theory with relativity theory (including general relativistic theory of gravitation) has been constructed. Dynamical description of wave-packet reduction lies beyond the scope of present-day quantum theory. The theory does not predict the observed behavior of individual macroscopic bodies. On the other hand the Einstein’s relativistic theory of gravitation does not take into account the fundamental restriction on observability of space-time geometry that follows from quantum theory. One of the practical results of this type of discrepancy is that lot of money is being spent on high energy and nuclear physics experimental research- this because there is no theory powerful enough to tell where it is really worth while to look.
In answer to these difficulties several ways out of the dilemma has been already tried. These include
- experimental research testing the range of validity of quantum theory - based on Bell inequalities (Aspect...)
(i) Analyze critically the possible incompatibilities between quantum theory and point-manifold picture of relativistic space-time. Analyze the problem of measurability of space-time metric in the spirit of Bohr-Rosenfeld. Investigate the possibility that contradictions can be removed while remaining in a realm of commutative differential geometry, but at the cost of replacing the 4-dimensional space-time by another manifold, possibly space of processes rather than events; e.g. a complex conformal relativistic phase-space.
(ii) In case the contradictions are impossible to remove while remaining in the realm of commutative differential geometry, investigate possible ways out, e.g, stochastic geometry and its relation to non-commutative geometry.
(iii) If, on the other hand, obstructions can be removed and if a model of a relativistic semi-classical complex phase space is not excluded by the analysis, then adapt the scheme of geometrical quantization to the new physical picture arising; in particular investigate a possibility of describing measurement processes and "quantum jumps" by a nonlinear correction terms to the standard linear quantization condition (wave functions covariantly constant along a chosen polarization).
l) The name of the Hungarian physicist: Karo... is probably misspelled; I do not have the references with me now.
2) All the above have been written according to my best knowledge as of today. But, since I am working all the time in this direction, the details (but also the fundamentals) may change with any time. On the other hand, a few minutes of a discussion with you may lead to a much better understanding of what has to remain for a while a dream, and what has a chance to be realized with the time and resources available.
3) The statements about feed-back, nonlinearities, measurements may look for you somewhat fuzzy. They do so for me too. However, it is precisely a part of this project to work hard to remove this fuzziness. Even if they are fuzzy now, it is my deep belief that they point into important directions.
With best regards,