Critique of Geometro-Stochastic Theory

In his book on geometro-stochastic quantum mechanics, Prugovecki argues forcefully for a new approach to fundamental physics. He quotes extensively from the masters, and also from lesser people, and has a wonderful list of references. Along the way, he offers criticism of present theories and existing trends. To make the case for his new theory, he is obliged to say something about the other theories used by particle physicists: his theory is not is accordance with what Haag calls "Einstein causality", and Einstein is one of Prugovecki's heroes. I shall not explain, even if I could, what GS theory is; for that, buy his books! Rather, I shall try to answer his remarks about local quantum field theory and Haag's algebraic programme.

Prugovecki argues that local commutativity "has no physically truly meaningful relationship to the question of Einstein causality any more than... in classical relativistic theory. Indeed, in classical special relativistic theory, the commutativity of all observables is trivially satisfied". Here, Prugovecki has put his finger on the difference between the two concepts that Haag calls "Einstein causality" and "primitive causality". I believe that Prugovecki has confused the two. According to Haag, local commutativity is needed so that a measurement of of an observable localised in a space-time region O, and the consequent reduction of the state, causes no instantaneous influence in a region O' space-like to O. In the famous paper, R. Haag and D. Kastler, "An algebraic approach to quantum field theory", Jour. Mathematical Phys., 5, 848-861, 1964, the authors outline a proposal as to how a local measurement can be achieved mathematically. We measure A in the region O by applying a completely positive stochastic map, denoted T(A), to the algebra of all observables, with the properties that on the local algebra in O, the dual T*(A) of T(A) agrees with the conventional result of a measurement on the vector-states; that is, it gives the reduction of the wave-packet, into the mixed state consisting of eigenvectors of A, weighted with their transition probability from the initial state. On any observable B located space-like to the space-time region involved in measuring A, T(A) has no effect: T(A)B=B. It follows that the measurement of A has no effect on the states observed in the region where B is localised. Later work showed that local commutativity, namely the condition [A,B]=0, is a necessary ingredient for the general existence of such T(A). Notice that if the observer in O not only triggered the measuring apparatus for A, but also read the result of the measurement, then the reduced state on the algebra in O would become conditioned, and only the eigenprojection with the observed eigen value would remain in the state at O. This knowledge is not available to the observer at B, so the state on the algebra at O' is still unaffected by the measurement. If O communicated the result to O' by a signal, then this information would change the state at O', and it would collapse; this can't happen if they are space-like separated. To keep track of which state is assigned by whom, we must introduce information sets and treat the problem as in game theory; the algebras generated by the indicator functions of the information sets could be called "information algebras". This can be generalised to non-abelian algebras in the theory of quantum games. This is suggested in my book.

Why did Haag call this "Einstein causality"? I surmise that the term is short for "Einstein-Podolski-Rosen causality". In their EPR paper, Einstein, Rosen and Podolski suggest a thought experiment, which, they argued, showed that quantum mechanics is not "complete". Part of their argument assumes that any measurement of one observable, such as A above, will also reduce the quantum state of an observer looking at B. They were forced to this conclusion, because mixed states (that is, statistical mixtures represented by density matrices) were excluded, following Einstein's dictum, "God does not play dice". Thus arose a misconception, wherein the "influence" of a measurement was alleged to travel faster than light. Thus, in "The Emperor's New Mind", R. Penrose asserts that the experimental set-up used in O, namely, which observable is being measured, but not the result of the experiment, is instantaneously transmitted to O'. This is not what quantum theory says. The simple non-relativistic model of two particles of spin 1/2, uses a four dimensional Hilbert space H, the tensor product of two copies of spin-1/2 spaces; the EPR state is an entangled pure state on the algebra of all observables, but is a fully mixed state (2 × 2 density matrix, equal to 1/2 of the unit matrix) when viewed by the observer of A, and also when viewed by the observer of B; and the spin observables of one particle commute with those of the other. This commutativity allows the direct construction of the map T(A) for any spin observable A of the first particle, and having the above property of not changing the state of the second. The requirement that the effect of an observation in quantum mechanics does not travel faster than light can (arguably) be said to have originated in EPR, and so might deserve the name "Einstein causality".

The other concept, primitive causality, originated in the even earlier paper, that of R. Haag and B. Schroer ("The postulates of quantum field theory", Journ. Mathematical, Phys., 3, 248-256, 1962). It postulates that the observables localised in a region O generate the observables in the double cone causally defined by O. This is the quantum version of the concept of "domain of dependence" which holds for classical fields obeying hyperbolic partial differential equations; it requires (in classical physics) that non-linear terms be local, and that the order of the equation be finite. Then the characteristics of the equation define the "light cone" used to define the domain of dependence (and the range of influence). It is clear that Prugovecki would have preferred this idea, rather than local commutativity, to have been called "Einstein causality"; primitive causality does not involve any measurement; it is designed to imply that nothing can move outside the range of influence, and in particular, for the relativistic wave equations, nothing can move faster than light. Although Einstein can be said to have stimulated ideas related to this classical concept, he did not say anything about it in the quantum context, unlike his input into EPR.

Returning to GS, Prugovecki introduces a fundamental length (the Planck length), and abandons local commutativity. He says "Contemporary experimental high-energy technology...is still far from being able to probe energies and distances of "Planckian" orders of magnitude...Hence the choice between conventional models and their GS counterparts is not one that could be made...on the basis of experiment alone." This bold statement is a hostage to fortune; experimentallists are very clever people, and might be able to devise an experiment to detect the violation of "Einstein" causality, inherent in GS. One only needs to recall Salam's model, in which the proton was assumed to be unstable, with a life-time of 10³¹ years. Salam presented this model at the Rutherford Lab., on the occasion of Dr. Stafford's 25 th Christmas conference. Salam said in his talk that this hypothesis had no experimental consequences: no-one was going to be able to watch a proton for that long. Nevertheless, at Kamiokande, people watched 10³⁴ protons for 10 years, and none decayed; so the lifetime had to be revised upwards by four orders of magnitude, and it no longer served any purpose in Salam's model.

Closer to the question, is local commutativity exact, or is it violated at Planck lengths? is a very recent observation in astronomy. A supernova, which exploded maybe 4 milliard years ago, was observed; the photons of the various colours all arrived at the same time, within the time-span of the explosion. I would have expected this, since all photons travel at the same speed. However, the BBC report on the story said that "physicists" expected them to arrive at different times, because "they" believe that photons of different colour travel at different speeds. Who are these physicists? The story starts with the work of Amelino-Camilia ["Relativity in space-times with short-distance structure governed by an observer-independent (Planckian) length-scale", Intern. J. Modern Phys., D11, 35-, 2002; available at the archive, gr-qc/0012051]. This analysis suggested that Einstein's relation between the mass, energy and momentum of a particle (the dispersion relation) might be changed. The theory was called "double relativity", because not only is the speed c the same in all inertial frames, but also Planck's length is the same: it is the unique length that does not suffer from the Lorentz contraction. Various models were constructed, in one of which a massless particle might have a speed greater than c, depending ever so slightly on its colour; then c is the speed of photons in the limit of long wave-length. This was taken up by J. Magueijo, whose book, "Faster Than the Speed of Light: The Story of a Scientific Speculation" (Perseus, 2003; in the UK: Heinemann) was fortuitously published at the same time as both the BBC report, and an article in the New Scientist on double relativity. The author was confident that there would be no way to test the theory for several years, until a new facility in astronomy is finished; measuring the speed of light in the lab is not accurate enough. Nevertheless, the theory was refuted in quick time, the same week in which the book appeared, by the astronomical event mentioned.

Prugovecki makes a valid criticism of Haag's local quantum physics, which postulates that to any open set of space-time there is associated an algebra of observables; more, the operators assigned space-like separated regions are postulated to commute. Prugovecki points out that to probe small distances needs high energy, and that this cannot be made available by a small piece of apparatus lying in the region in question. It had been long understood that the operators assigned to an open set O of space-time were to represent the observables that could be measured by a piece of apparatus, switched on for a small interval of time, with the spatial extent of the apparatus lying in the space part of O during the time-interval needed for the measurement. This raison d'etre for Haag's theory has been advocated by Borchers. However, looking at a typical CERN measuring device one sees that to probe very small distances needs enormous bits of hardware. Thus, Prugovecki argues, Haag needs classical and non-relativistic concepts to give experimental meaning to localisation in small regions. In particular, the apparatus we use certainly interfers with the environment outside the causal domain of influence generated by the local observable being looked at. He concludes that local commutativity of space-like observables might not be valid for distances of the size of the Planck length.

In answer, we must admit that we cannot be sure that an observable assigned to a small region O can be observed from inside O. For example, who can say that we can measure the electric field, smeared with a test-function with support inside a region of space-time the size of a Planck volume, with a piece of apparatus smaller than the earth? We must provisionally weaken Haag's postulates. We could then understand that an observable A assigned to O can be observed by a large apparatus, but which still occupies a finite volume of space-time. Since the measurement cannot interfere with any other, B say, made at a large enough distance, we must then postulate that the commutator [A,B] vanishes for all B located far enough away. In GS, such commutators, albeit very small, are not zero. Prugovecki claims that no experiment in the forseeable future will be able to detect this violation of (Haag's) Einstein causality, or to show that it does not happen. Perhaps a clever astronomer will be able to show this to be too pessimistic.