I am not going to go too hard on him, James S. Trefil, because he is such a fine author and I enjoy his work; but I must address an error that this physicist makes in one of his books, From Atoms to Quarks: An Introduction to the Strange World of Particle Physics (1980). (See my review of the book by clicking on this sentence.) I have chosen Trefil’s error for discussion, because he is a fine physicist, which makes for a good mark in proving a point, namely, that physics needs philosophy of physics to mind a number of problems that are not central to advancement of the science. These problems include the kind of conceptual one that will be mentioned —one that I hope other physicists do not err on— and conceptual problems in foundations, metaphysics, and so forth.
In the abovementioned book, Trefil gives a list of three things that the Heisenberg Uncertainty Principle (Δx ∙ Δp ≥ ћ) does not imply, and begins his list with: “It does not imply that the particle’s position cannot be measured exactly” (emphasis in the original) (1980, pg. 46). That’s fine, right? Nope. The left half of the inequality, Δx Δp, must be greater than or equal to the constant h-bar (ћ), which is planck’s constant divided by 2π —a non-zero number. The problem, then, is that, if Trefil’s thought is that the change in momentum can tend toward infinity (Δp –> ∞) as the change in position is zero (Δx = 0), then he is mistaken; and I do believe that this is was his thinking. It is untenable, because this is to propose: Δx ∙ Δp = ∞ ∙ 0 = 0 ≥ ћ. I hope that all non-mathematicians are math-literate enough to see the problem. In fact, I used the terms “tend toward infinity” for a reason. Infinity is a concept, not a number, so there is a philosophical sense in which what I wrote doesn’t quite make sense (and another reason why Trefil’s assertion doesn’t quite make sense); but I have treated “∞” as any other number, which, when multiplied by zero has a product of zero, and zero is not greater than h-bar. So you know that there has been an error, when Trefil says, “There is nothing in the uncertainty principle that says that Δx cannot be zero” (1980, 46). Trefil attempts to justify his position by saying that we are unable to get information about the momentum of a particle when the position is precisely known, so the lack of information makes it okay that the position is precisely known. Ignoring (what is, at least, ostensibly) circular reasoning, just ask: Regardless of what Δp is, isn’t it going to be some real value? If Δp is going to have some real value, the above holds: 0 ∙ Δp = 0.
That was more of a mathematical, though still philosophical, excursion, but I wanted to convince you that there is a serious and obvious problem with Trefil’s proposal. Now, we can dig a bit deeper. The first thing to think about is the fact that we are dealing with changes in position and momentum. Therefore, take a second and think the most obvious thoughts about what you must know to be the case if the change in position of an object is zero. Think about it. I hope you asked yourself, “Doesn’t some amount of time have to pass for me to know that there is no change in position (Δx = 0)?” The answer is “of course!” and this brings in a new problem: If you know that an object hasn’t gone anywhere (id est, Δx = 0) during some interval of time, then the change in velocity would be zero (in fact, if the particle is just sitting in one place, for any interval, the velocity would be zero, too), and thus the change in momentum would be zero. (Note that: m ∙ Δv = Δp) Now, if the reader has the intuition to do so, you might be able to imagine that the more confinement and localization that you try to impose upon a particle, the less favorable that would be for gauging changes in momentum, because you want to be able to stand back, so to speak (but, still, in some literal sense), so that you can survey the non-local behavior. Here, I mean “non-local” in the sense of relatively larger distances, that is (not in the traditional sense on quantum non-locality), as opposed to point-like instantaneous measurements in which one tries to pinpoint a particle’s location. As a matter of fact, what I have just tried to put into words is a property of mathematical spaces, not something special about the physical space of the universe, and one can derive the property from some fairly basic linear algebra, using the Schwarz Inequality and the Triangle Inequality. A long story short, Trefil’s proposal that a quantum particle’s position can be completely known is spurious, and his attempt to justify it by saying that complete certainty of position and complete uncertainty of the change in momentum fails to the mathematical reasoning discussed earlier.
To close with an anecdote, even when I was studying undergraduate physics, these philosophical issues jumped out at me. When I was taking electrodynamics under Hrvoje Petek, at the University of Pittsburgh, I became aware of laser trapping and laser cooling, so I discussed Heisenberg’s Uncertainty Principle with him. I expressed (basically) a variation of the above concerns, which put him in a prudent state of reflection on the issue, agreeing with my position. The central question I had was, “What will happen if you laser trap a neutron, and continually add energy to the system such that the position is precisely known?” He was reluctant to the point of cerebrating silence, unsure of what to say, and I proposed that the adding of energy would, at some critical total energy, would cause less control of the particle. That is, the kinetic energy of the particle would increase. He said that this was his tentative answer. I am not sure whether this calculation has been done (I imagine it has, by someone in the area of laser physics), but the point is that, even if you try to force the case, using a clever trick like laser trapping, you cannot get a particle to sit still. The discussion with Petek actually suggests that there may be an experimental critical infimum to Δx (and probably Δp).
Note related to this post: I will be blogging further on the matter of the uncertainty principle by way of discussing the quantization of length. In the next couple of months, expect a blog on Zeno’s Arrow Paradox and “at-at theory”, and, as soon as it is published, another blog with my comments on Dr. Amit Hagar’s (IU Bloomington) work on length quantization. Finally, as soon as a pre-print is available of the “Kantian Uncertainty Principle,” an article I am co-authoring with Dr. David Cale (West Virginia University), that will be posted, too.