A Dialogue on Position
| Scientific Paper | |
|---|---|
| Title | A Dialogue on Position |
| Read in full | Link to paper |
| Author(s) | Richard Oldani |
| Keywords | position |
| Published | 2002 |
| Journal | Galilean Electrodynamics |
| Volume | 12 |
| Number | 2 |
| No. of pages | 10 |
| Pages | 70-80 |
Read the full paper here
Abstract
A student who wants to "understand" quantum mechanics asks a physics professor how to determine the four-coordinate position of a particle. It soon becomes evident that Heisenberg's microscope experiment is totally inadequate as a model since the observer does not actually participate in the measurement process, and a procedure for measuring the time coordinate microscopically has never been defined. In fact, in a strict sense, quantum mechanics does not have a logically coherent method for determining position in even a single dimension. Their attempts to resolve these differences are an exercise in futility until the student finally realizes that before they can agree on anything they have to be able to communicate.
Overview
Oldani's paper, presented at the NPA/AAAS meeting in Santa Fe in April 1999 and published in Galilean Electrodynamics, is written as a Galilean dialogue rather than as a derivation. A Student asks a Professor a deliberately elementary question — how do you determine the position of a particle? — and refuses to be deflected from it for ten pages. There is almost no new mathematics; the only equations are standard ones that the Professor supplies, and the work of the paper is done by the Student's cross-examination of them.
The target is not the predictive success of quantum mechanics, which the dialogue nowhere disputes, but its operational coherence on a single concept. The Student's contention is that the theory uses one word, "position", for two incompatible things: a four-coordinate event in the laboratory frame, established after the fact by Compton-scattering detection, and a probability distribution referred to a particle-centred abstract space. On the mainstream account these are two aspects of the same observable, and the uncertainty relation is the bridge between them. Oldani's Student argues that no bridge has ever been built: the two are defined in different coordinate systems, the observer's own indeterminacy is omitted from Heisenberg's microscope, and the time coordinate has never been given a microscopic measurement procedure at all.
The argument
Past position and future position
The Professor opens with the standard answer. To find where an electron was, collide a high-energy photon with it and let detectors register both arrival times; the impact point and time follow. A footnote adds the important qualification that the accuracy of past position measurement is not limited by the uncertainty principle. Future position, by contrast, is governed by indeterminacy, illustrated by single-slit electron diffraction, where Δy · Δpy ≈ h.
The Student's first objection is that these two results are not of the same kind. One is four numbers in the laboratory frame; the other is a distribution over an ensemble. When the Professor answers that the flashes on the screen are realisations of the predicted positions, the Student presses the analogy with statistical mechanics: there, ensemble results can also be recovered from individual trajectories, so what makes the quantum case different? The Professor's reply — that quantum probabilities are "intrinsic to each particle" rather than obtained by averaging over microscopic coordinates — is the pivot of the whole dialogue, and the Student spends the rest of the paper testing it.
Which electron?
The test is indistinguishability. If a photon is fired at a multi-electron atom, does the uncertainty principle together with the shell distributions say which electron was localised? No — only the probability of striking one of them. Does a position measurement identify the electron as belonging to a particular hydrogen atom? No, because all hydrogen atoms are physically identical. The Student concludes that "future" in this usage refers to the experiment or the system, not to the particle, and that the distributions describe the relative density of the electron cloud — atomic structure — rather than anyone's future position.
The two spaces
The Professor falls back on Heisenberg's 1927 microscope thought experiment, Δx · Δp ≥ h, as a single-particle derivation. The Student's counter is geometrical. If localising one particle "defines the space in which the interaction occurs", and each measurement localises at most one particle, then that space contains exactly one object of known position. "Then quantum theory is formulated in the same manner as the Ptolemaic theory, except that an electron is designated to be the center of the universe instead of the earth."
He develops the point with a repeated analogy: latitude, longitude and altitude locate a point on the Earth's surface but cannot locate the Earth relative to the Sun; two coordinate systems are needed, and a fixed terrestrial point traces an irregular helix in the heliocentric one. He asks whether quantum theory makes any comparable distinction between atomic and laboratory coordinates. The Professor says that would require an independent coordinate system per particle and is "way too complicated"; quantum mechanics uses one space, and treats particles as singularities so that structure and motion can be described together. The Student replies that a genuine singularity could not carry spin, a magnetic moment or a diameter — and notes the price paid, since it is the singular electron that forces renormalisation in QED, a "sophisticated fudging process" the Professor concedes has not been resolved. The Student's proposal is that atomic and ordinary space have "physically distinct origins", the first describing matter's structure and the second our perception of it, so that results from one cannot simply be carried into the other.
Motion, and the missing clock
Turning to motion, the Professor says that quantum mechanics describes it as emission at one place and detection at another, with nothing said about the interval — "the particle occupies all possible paths" — while bound electrons are known to move because they have a well-defined angular momentum but have no trajectories. The Student objects that the theory uses "motion" when convenient and pleads abstraction when not, and points out that a non-relativistic Schrödinger solution assigns finite probability to a bound electron anywhere in the universe without its ever exceeding c.
The last section is the paper's most specific complaint and the one the abstract singles out. Bubble chambers and ideal microscopes measure time differences, not the time parameter against a continuous standard such as Greenwich Mean Time. Because a clock is periodic, using a clock of period τ to date an event introduces a minimum error Δt ≥ τ that no improvement in difference-measurement removes. The Student argues that measuring the half-life of a radioactive atom does not touch the internal processes producing the decay, so two distinct time parameters are needed — one structural, one for the event as recorded. The Professor's answer is that only time differences are physically meaningful, that two kinds of time would complicate the mathematics, and finally that the Student should take the question to the relativity specialist down the hall.
A footnote supplies the supporting calculation. From ΔE · Δt ≥ h with ΔE = hν, Δt ≥ h/hν = τ; from Δx · Δp ≥ h with Δp = h/λ, Δx ≥ λ. Hence, Oldani argues, a photon cannot be localised more precisely than one wavelength and one period — so the eye or detector that registers the microscope's photon carries its own irreducible position error, which Heisenberg's thought experiment simply omits. The Student closes by recalling that Einstein asked for a clarification of past and future position at the fifth Solvay conference in 1927 and was ignored; the Professor says goodbye.
Assessment
The dialogue form suits the subject, and the paper is at its best where it stays operational. Three of its observations are simply correct and are not always made clearly in textbooks: that retrodicted position is not limited by the uncertainty relation; that the ordinary derivations of the relation are ensemble statements whose extension to a single particle is an interpretive step rather than a measured result; and that Heisenberg's microscope treats the detection of the scattered photon as errorless while insisting that the scattering itself cannot be. The arithmetic in the footnote is right as far as it goes: with ΔE = hν one does get Δt ≥ 1/ν = τ, and with Δp = h/λ one does get Δx ≥ λ. The Bohr-radius figure quoted in passing, 0.5 Å, is also the right number.
The difficulties are of two kinds. The first is quantitative. The relations used throughout are Heisenberg's 1927 heuristic forms with a bare h; the rigorous Kennard-Robertson inequality is Δx · Δp ≥ ħ/2 = h/4π. Substituting that changes the footnote's conclusion by a factor of about 12.6: the bound becomes Δx ≥ λ/4π, not λ. Since the argument that the observer's indeterminacy is comparable to the electron's rests on the size of that bound, the factor matters, and the paper does not address it. The Student's related assertion that "real electrons have measurable diameters" is not supported by measurement: electron-positron scattering constrains the electron's size to below about 10−18 m with no positive result, and the classical electron radius of 2.8 × 10−15 m is a derived combination of constants, not an observed extent.
The second is dialectical. The Professor is not a strong opponent. He answers at three separate points with "everybody does it", "it would only complicate the mathematics" and "because no one understands quantum mechanics anyway", and he is sent packing at the end without having offered the standard replies actually available. The distinction between structural and laboratory time that the Student wants is close to the proper-time/coordinate-time distinction of relativity — which is why the Professor's referral down the hall is funnier than the paper seems to intend. The problem of one-particle-per-space is handled in ordinary quantum mechanics by configuration space and, in the many-particle case, by field operators defined at every point of a shared spacetime, which is precisely the "independent coordinate system for every particle" the Professor dismisses as too complicated. And renormalisation, presented here as an unresolved embarrassment, has since the 1970s been understood as a statement about the scale-dependence of couplings rather than as a device for hiding a singularity — the anomalous magnetic moment of the electron computed by that machinery agrees with measurement to better than one part in 1012, the most precise confirmation in physics.
None of this touches the paper's actual thesis, which is narrower than it first appears: that quantum mechanics has never supplied an operational procedure for measuring the time coordinate of a microscopic event against a continuous standard, and that its two senses of "position" have never been formally reconciled. That is a real gap in the presentation of the theory, and stating it plainly, without claiming to have closed it, is the paper's honest contribution.