I teach quantum physics to high school students with little prior exposure to any physics. They don't find the requirement of continuity "quite natural" until they have obtained an intuitive grasp of the fuzziness that fluffs out matter. We make this fuzziness our starting point - not an abstract mathematical principle but something that (i) makes sense before we ask mathematical questions and (ii) is essential for the existence of all things that "occupy" space. We then ask, how do we give mathematical expression to this fuzziness, and this takes us straight to the kinematical formalism of QM (quant-ph/0202149, section 2). Admittedly, I invoke Gleason's theorem and its recent generalizations, which these young students have to accept on faith. But they do get a clear idea where QM comes from. In a similar fashion we come to understand the dynamics (quant- ph/0202149, section 3) and special relativity (quant-ph/0202149, section 4). I believe that this is what both young students and the public should know and can understand about these fundamental theories: the contributions they make to the familiar world. Moreover, showing that the axioms of a fundamental theory are needed for some obvious feature of the familiar world is the only way in which such axioms can be explained, inasmuch as there is no underlying theory from which they could be derived.

In his contribution of 4 March Shelly Goldstein writes: "The radical implications of Bohmian mechanics have indeed been tested and confirmed, in the very experiments mentioned in the quotations."

But these "radical implications" are identical with the implications of standard QM?!

Concerning the issue of the consistency of QM with SR (raised in the same contribution): QM is about probabilities of possible events, including joint probabilities of possible events in spacelike separation. Probability assignments are based on actual events, but not on a unique set of actual events; the assignment basis can be any set of relevant actual events. Take an EPR-Bohm setting and the probability of a possible event e1 indicating that the spin of particle 1 is up with respect to a certain axis at the time t1. The assignment basis can be the fact that the composite system started out with zero spin, or it can be this *and* an event e2 indicating that the spin of particle 2 is up with respect to the same axis at the time t2. (No temporal order is implied.) The corresponding probabilities obviously differ, but this does not by any means imply that particle 1, or anything at the location of particle 1, changes in consequence of e. The switch from one assignment basis to another is not something that happens in time, for both assignments are timelessly true: It always has been and always will be true that the probability of e1 is p1 given the first assignment basis, and that it is p2 given the second assignment basis. Nor is the switch from p1 to p2 something that happens at any particular location: the probability for something to happen at a particular location is obviously not something that exists at any location.

So how could there be a conflict with SR? A conflict arises if one tries to fit the quantum world into the intrinsically and maximally differentiated spacetime of classical physics (or to build it bottom-up, on a classical spatiotemporal manifold) and asks for a causal explanation for the quantum- mechanical correlations. But the fuzziness of relative positions implies that the quantum world is not maximally differentiated spacewise or timewise (quant-ph/9903051, 0105097, 0109150, 0202148), and causality is "emergent" (see my contribution of 6 March). Trying to causally explain the correlations is putting the cart in front of the horse. The correlations are fundamental. There is no domain of underlying causal processes. It is the correlations that account for the limited usefulness of causal language; they explain why causal explanations work to the extent they do. They work in the macroscopic domain, where the correlations between property-indicating facts evince no statistical variations (dispersion).

To clarify the physical significance of SR in a fuzzy world, one should again look at what SR contributes to the familiar world. The relevant question is the physical origin of the metric. The physical basis of the temporal part of the metric is mass, which is essentially the rate at which the phase factor associated with a Feynman path cycles ("ticks") as a freely falling particle travels along it. (The sense in which a particle travels along its Feynman paths is explained in quant-ph/0202148.) What about the spatial part of the metric? Is there any other particle property, besides mass, on which it could be based? It doesn't seem so. But if the spatial metric too has its origin in the rates at which freely falling particles "tick" as they travel along their Feynman paths, a particle must also "tick" if it is at rest, and the action of a Feyman path must be the same for all inertial frames. This requires Lorentz invariance (quant- ph/0202149, section 4). So the quantum world, far from being inconsistent with SR, *requires* SR.

Ulrich Mohrhoff

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Published   2002.03.07